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DEPARTM ENT O F PHYSICS

UNIVERSITY O F JYV �ASK YL�A

RESEARCH REPO RT No.1/2008

A M ETA G EO M ET R IC EN Q U IRY

C O N C ER N IN G T IM E,SPA C E,

A N D Q U A N T U M PH Y SIC S

B Y

D IEG O M ESC H IN I

Academ ic Dissertation

fortheDegree of

DoctorofPhilosophy

To bepresented,by perm ission ofthe

Faculty ofM athem aticsand NaturalSciencesofthe

University ofJyv�askyl�a,forpublicexam ination in

Auditorium FYS-1 oftheUniversity ofJyv�askyl�a on

9th February,2008,at12 o’clock noon

Jyv�askyl�a,Finland

February 2008

http://arxiv.org/abs/0804.3742v1

\Aristippus, the Socratic philosopher, shipwrecked on the coast of Rhodes and

having noticed sketchesofgeom etric�gures,itissaid m ay havespoken thusto his

com panions: ‘W e can hope for the best,for I see the signs ofm en.’ (Vitruvius,

De architectura,Book 6,Preface)" Frontispiece to (Euclid,1703). Translation by

H.J.Santecchia.Reproduced from (Heilbron,2000).

To M arkku, who showed m e the rare art

of ying keeping one’sfeeton the ground

Do space and tim e existalone and

Aspartand parcelofthe world,

Or are they m erely illusions

Ourm inds create and setunfurled?

The very products ofour brain

Like,say,the feeling ofdisdain?

Three hundred years ofbitter history

Have failed thisissue to resolve:

Cartesian vorticesand whirlpools,

Around which corpuscles revolve;

Contraptions builtoutofm aterial

Notonce upgraded from ethereal.

Today the ether hasnotleftus;

Itlivesin new,geom etric m odes:

Asloops and foam s,asfancy networks,

Asstringsand m em branes,linksand nodes;

For are these objects,say,these strings,

The seatofany realthings?

Are even spacetim e points ethereal?

An argum entabouta hole:

So wide to charm m ostvaried thinkers,

Butdeep enough to lose your soul;

Ofwhich philosophers are fond,

Butwhich it’stim e to step beyond.

Forgo the hole,the points,the ether.

Instead use things you can detect;

The sim plestthingsthan M an can think of,

W ith sources M an can clearly a�ect.

Find answers pastthe older helm

W ithin a non-geom etric realm .

A basic secretofthe cosm os

Itisour goalto better grasp.

A highly am bitiousundertaking;

M ighteven m ake the Reader gasp,

Or utterapprehensive shrieks:

\A task to dauntthe AncientGreeks!"

| Orutter criesofdisbelief

And hope this thesisto be brief.

A wish thatwe cannotconcede:

Ofm uch soul-searching there isneed.

Preface

Theprologueisthem ostem otivem om ent,wherethetension

accum ulated duringa yearofwork isreleased likea gunshot.

M iguelIndurain,Cycle SportM agazine

Thegoalofthisthesisistocontributeto theelucidation ofan elem ental,

age-old problem ,nam ely,thephysicalnatureofspaceand tim e.Thewords

of m y supervisor,Dr.M arkku Lehto, to whom I owe countless hours of

illum inating revelations, perhaps describe the spirit of this thesis better

than no other:

W eparticipatein oneofm ankind’sm ostchallenging intellectual

e�orts. The goal is to understand the biggest secrets of the

cosm os:thestructureofspaceand tim easwellastheconnection

ofthisstructure with thatofm atter. O ne m ay even hold these

secrets to be \greater than life," when one thinks that life can

be explained as an em ergent phenom enon,i.e.a phenom enon

which arises (probably very com plicatedly) via evolution from

theprim itivestructureofspaceand tim eand itsinteraction with

m atter.

The �rststep isto explain whetherspace and tim e can have in

generalan own structure.O rarespaceand tim ein theend only

hum an sensationslike,forexam ple,thefeeling \happy"?1

In thestudy ofthisproblem ,likein thatofallproblem senduring obsti-

nately through the ages,itsoon dawned on m e that the question to focus

on wasnottheim m ediately expected,\How do wego about�nding a solu-

tion?" butratherthe m ore carefuland m easured,\W hatis the problem ?"

M oreover,iteventually becam eapparentthatthesetwo questionswereone

and the sam e;thatan understanding ofthe essence ofa problem wasquite

aptto revealill-founded attem pts at solving it,as wellas pointim plicitly

to thepath ofa m ethodicalresolution.

Thisrealization,togetherwith m ounting wearinessofhackneyed,indul-

gentaccountsofthenatureand prospectsofquantum gravity| thediscipline

which currently occupiesitselfwith the issue ofspace and tim e| m ade an

1Private com m unication,2001.

vi P reface

unsym pathetic,bare-bones analysis ofthis �eld ofresearch as indispens-

able as it was enlightening. Itallowed m e to disentangle few facts from a

surfeit ofclich�ed beliefs (including even the occasionalextravaganza) that

populate the quantum -gravitationalworld| a referent-orphaned no-m an’s-

land oftheoreticalresearch. In particular,Ilearnt from this analysis that

the presentwork can beconsidered partofquantum gravity only in a very

shallow,conventionalsense,since itdoesnotshare with this�eld itsstan-

dard m otivations,m ethods ofenquiry,or desired destination. Ifquantum

gravity isto bereached by eitheroneofthethreegeom etricroadscurrently

envisioned,thiswork treadsnoneofthem .

Aftertwo and a halfm illennia ofbitterand fruitlessstruggle| the last

threeand ahalfcenturiesofwhich within thecontextofm odern science| the

problem ofthe physicalnature ofspace and tim e continues to be nothing

short ofan opinionated battle�eld,a vicious display ofparry and repost,

a twisted labyrinth with no exit anywhere to be seen. W e believe that

unravelling thisage-old problem requiresa return to a physicsbased on ob-

servations,togetherwith theconceptualoverthrow ofthegeom etricnotions

in term s ofwhich the philosophically and m athem atically m inded debate

continuesexclusively to rage,i.e.via a thorough departurefrom allform sof

geom etric thoughtand language,an idea we condense in the nam e \m eta-

geom etry."

Focused on m etageom etry, however,we notice with disquiet the om i-

nous fact that hum an beings carry with them an innate craving for geo-

m etric m odes ofthought and expression. This is not only revealed by an

overview ofphysicaltheories,oreven ofscienti�ctheories,but,withoutfear

ofoverstatem ent,onem ay even say ofallhum an endeavour.Hum an beings’

fascination forgeom etry easily growstothepointofenslavem ent;and som e-

tim eseven to theepitom icalextrem eofhaving G eom etry becom ea m odern

religion| physicists’incarnation ofG od.

W hile a solution to the ontologicalspacetim e problem cannotbe found

through philosophically m inded,labyrinthinediscussionsvoid offresh phys-

icalcontent,neithercan itbe found through the hit-and-m issinvention of

new spacetim estructureso� thetop ofone’shead;and especially notwhen

these proceduresare based solely,orm ostly,on purem athem aticalinsight.

In physics,m athem atics is a good servant but a bad m aster. Because it

is a physicalquestion that we have in our hands,its clari�cation requires

strong physicalintuition,the identi�cation ofrelevant physicalprinciples,

and a �rm anchorage in observationsand experim entsasan unconditional

starting point.

Itisin view ofallthisthatthiswork dealswith thequestion,\W hatdo

observations,quantum principles,and m etageom etric thought have to say

aboutspaceand tim e?" Itconsistsprim arily in thedevelopm entofnew,ele-

m entalphysicalideasand conceptsand,therefore,doesnotneed to abound

in com plicated technicalities.O bstinateproblem srequireconceptually fresh

starts.

3 3 3

vii

Literally speaking,this work would have been im possible without the

wisdom and extraordinary guidance ofm y supervisor,Dr.M arkku Lehto.

Notonly did hegently and unsel�shly show m etheway through them aze-

likepassagewaysofquantum -gravity research,unobtrusively fostering atall

tim estheindependentdevelopm entand expression ofm y own thoughts,but

also showed m e the di�erence between physics,m athem atics,and philoso-

phy.In so doing,hetaughtm eby exam ple(theonetruly e�ective teaching

m ethod)the m eaning ofgenuine physicalthinking| a surprisingly di�cult

activity rarely witnessed in frontiertheoreticalphysics.To him go m y in�-

nitegratitude and adm iration.

From a �nancialpointofview,them aking ofthisthesiswasassisted by

the supportofdi�erentorganizations. Ithank the Centre forInternational

M obility (CIM O )forsponsoringthebeginningofm y Ph.D.studies;and the

Ellen and ArtturiNyyss�onen Foundation,theFinnish CulturalFoundation,

and the Departm entofPhysics,University ofJyv�askyl�a,forallowing their

continuation and com pletion.

The research leading to this thesis was carried out during the period

Septem ber2001{July 2007 within thecontextofthelast-m entioned Depart-

m ent,whereIfound am plefreedom ofaction and betterworking conditions

than Icould have dream t of. Iparticularly want to thank its adm inistra-

tive sta� foritse�ciency and helpfulnessand m y room m ates,AriPeltola,

Johanna Piilonen,and AnttiTolvanen forpondering and assaying with m e,

som etim eslightheartedly,som etim esin allseriousness,the exactionsoflife

in thebrave new world ofacadem ia.

Lastbutnotleast,the actualm aterialcreation ofthisthesiswasm ade

notonlypossiblebut,m oreover,enjoyablebythetypesettingsystem LATEX|

an essentialtool,alm oston a parwith thedi�erentialcalculus,ofthem od-

ern m an ofscience;a toolwhich deservesm ore attention in ourtypeset-it-

yourselfworld ofam ateur desktop publishing than is usually paid to it in

thesciencesand hum anitiescurricula.

M an does not live by bread alone. I would also like to thank other

people who have indirectly contributed to the ful�llm ent ofthis project;

they are:m y parents,Stella M arisand O m ar,forbringingm eup within the

fram ework ofan excellent fam ily life and education| therein lie the seeds

ofallotherachievem ents;K aisuliina,forhercom panionship and warm th in

the joysand com m onplaces ofeveryday life alike;and Ulla and Teuvo,for

unconditionally furnishing the context ofa second fam ily in a foreign,yet

m arvellous,land.

I bring this prelude to a close with som e hope-ridden words2 to the

reader: ifyou should �nd this work sim ply readable,Ishallbe pleased;if

itshould give you a background on which to think new thoughts,Ishallbe

delighted.

Diego M eschini

Jyv�askyl�a,July 2007.

2Paraphrase ofFoss’s(2000,p.xiii).

A bstract

M eschini,Diego

A m etageom etric enquiry concerning tim e,space,and quantum physics

Jyv�askyl�a:University ofJyv�askyl�a,2008,xiv + 271 pp.

Departm entofPhysics,University ofJyv�askyl�a,Research Report1/2008.

ISBN 978-951-39-3054-7

ISSN 0075-465X

This thesis consists ofa enquiry into the physicalnature oftim e and

space and into the ontology ofquantum m echanics from a m etageom etric

perspective. Thisapproach results from the beliefthat geom etric thought

and language arepowerlessto fartherhum an understanding oftheseissues,

theirapplication continuing instead to restrictphysicalprogress.

The nature and assum ptionsofquantum gravity are analysed critically

throughout,including m isgivingsabouttherelevanceofthePlanck scaleto

itand itslack ofobservationalreferentin thenaturalworld.

The anthropic foundationsofgeom etry asa toolofthoughtare investi-

gated.Theexclusiveuseofgeom etricthoughtfrom antiquity topresent-day

physicsishighlighted and found to perm eateallnew attem ptstowardsbet-

ter theories,including quantum gravity and,within it,even pregeom etry.

Som eearly hintsattheneed to supersedegeom etry in physicsareexplored.

Theproblem oftheetherisstudied and found to haveperpetuated itself

up to the present day by transm uting its form from m echanical,through

m etric,to geom etric.Philosophically m inded discussionsofthisissue (Ein-

stein’shole argum ent)are found wanting from a physicalperspective.

A clari�cation ism adeofthephysical,m athem atical,and psychological

foundationsofrelativity and quantum theories. The form erisfounded ge-

om etrically on m easurem ent-based clock-reading separationsds.Thelatter

isfounded m etageom etrically on the experim ent-based conceptsofprem ea-

surem entand transition things,inspired in thephysically unexplored aspect

oftim easaconsciousness-related product.A conceptofm etageom etrictim e

is developed and coordinate tim e t is recovered from it. Discovery ofthe

connection between quantum -m echanicalm etageom etric tim eelem entsand

general-relativistic clock tim eelem entsds (classicalcorrelations)isdeem ed

necessary fora com bined understanding oftim ein theform ofm etageom et-

ric correlationsofquantum -m echanicalorigin.

Tim eisconjectured tobethem issinglink between generalrelativity and

quantum m echanics.

O P P O N EN T

Dr.Rosolino Buccheri

Institute for EducationalTechnology,NationalResearch Council,Palerm o

& Centro Interdipartim entalediTecnologie della Conoscenza,University of

Palerm o

SU P ERV ISO R

Dr.M arkku Lehto

Departm entofPhysics,University ofJyv�askyl�a

EX A M IN ER S

Dr.G eorge Jaroszkiewicz

SchoolofM athem aticalSciences,University ofNottingham

Dr.M etod Saniga

Astronom icalInstitute,Slovak Academ y ofSciences

R elated publications

I am probably both dissatis�ed with everything that I have

written and proud ofit.

Stanis law Lem ,M icroworlds

I.M eschini,D.,Lehto,M .,& Piilonen,J.(2005). G eom etry,pregeom e-

try and beyond.Studiesin History and Philosophy ofM odern Physics,

36,435{464.(arXiv:gr-qc/0411053v3)

II.M eschini,D.,& Lehto,M .(2006).Isem ptyspacetim eaphysicalthing?

Foundations ofPhysics,36,1193{1216.(arXiv:gr-qc/0506068v2)

III.M eschini,D.(2006).Planck-scalephysics:Factsandbeliefs.Toappear

in Foundations ofScience.(arXiv:gr-qc/0601097v1)

G uided by hissupervisor,Dr.M arkku Lehto,theauthorresearched and

wrote thevastm ajority ofArticle Iand the totality ofArticlesIIand III.

The contents ofthis thesis consist ofm aterialpartly drawn| with en-

hancem ents,abridgem ents,and am endm ents| from the above publications

(Chapters2,4,6,8,and 9),and partly previously unpublished (Chapters1,

3,5,7,and 10{13).In particular,thecontentsofChapters10and 12consist

ofenhanced versionsby theauthorofpubliclecturesgiven by Dr.Lehto at

the Departm ent ofPhysics,University ofJyv�askyl�a: Chapter 10 is based

on Relativity theory (9th January{26th April,2007);Chapter 12 is based

on Physical,m athem atical,and philosophicalfoundationsofquantum theory

(12th April{29th June,2005).

C ontents

P reface v

A bstract viii

R elated publications x

1 Q uantum gravity unobserved 1

2 T he m yth ofP lanck-scale physics 7

2.1 Facts . .. . . .. . .. . .. . . .. . .. . . .. . .. . . .. . 8

2.2 Beliefs .. . . .. . .. . .. . . .. . .. . . .. . .. . . .. . 13

2.3 Appraisal . . .. . .. . .. . . .. . .. . . .. . .. . . .. . 18

3 T he anthropic foundations ofgeom etry 22

3.1 Theriseofgeom etric thought . .. . .. . . .. . .. . . .. . 22

3.2 Theoriginsofform .. . .. . . .. . .. . . .. . .. . . .. . 23

3.3 Theoriginsofnum ber . .. . . .. . .. . . .. . .. . . .. . 24

3.4 Theoriginsofm agnitude . . . .. . .. . . .. . .. . . .. . 27

3.5 Theriseofabstractgeom etry . .. . .. . . .. . .. . . .. . 28

4 G eom etry in physics 32

4.1 G eom etry in relativity theory . .. . .. . . .. . .. . . .. . 36

4.2 G eom etry in quantum theory . .. . .. . . .. . .. . . .. . 38

5 N ew geom etries 42

5.1 Threeroadsto m ore geom etry .. . .. . . .. . .. . . .. . 44

6 P regeom etry 49

6.1 Discreteness . .. . .. . .. . . .. . .. . . .. . .. . . .. . 50

6.2 G raph-theoreticalpictures . . . .. . .. . . .. . .. . . .. . 52

6.2.1 Dadi�c and Pisk’sdiscrete-space structure .. . . .. . 52

6.2.2 Antonsen’srandom graphs . .. . . .. . .. . . .. . 53

6.2.3 Requardt’scellularnetworks .. . . .. . .. . . .. . 53

6.3 Lattice pictures . . .. . .. . . .. . .. . . .. . .. . . .. . 54

6.3.1 Sim plicialquantum gravity by Lehto etal... . . .. . 55

6.3.2 Causalsetsby Bom bellietal... . . .. . .. . . .. . 56

6.4 Num ber-theoreticalpictures . . .. . .. . . .. . .. . . .. . 57

xii C O N T EN T S

6.4.1 Rational-num berspacetim e by Horzela etal. .. . .. 58

6.4.2 Volovich’snum bertheory . .. . .. . .. . . .. . .. 58

6.5 Relationalorprocess-based pictures . . .. . .. . . .. . .. 59

6.5.1 Axiom atic pregeom etry by Perez Berglia�a etal. . .. 59

6.5.2 Cahilland K linger’sbootstrap universe . . . .. . .. 60

6.6 Q uantum -cosm ologicalpictures . . .. . .. . .. . . .. . .. 61

6.6.1 Eakinsand Jaroszkiewicz’squantum universe .. . .. 61

6.7 Theinexorability ofgeom etric understanding?.. . . .. . .. 62

7 T he geom etric anthropic principle and its consequences 64

7.1 Thegeom etric anthropic principle .. . .. . .. . . .. . .. 65

7.2 O ntologicaldi�culties .. . .. . . .. . .. . .. . . .. . .. 69

7.3 Psychologicaldi�culties . . .. . . .. . .. . .. . . .. . .. 70

7.4 Experim entaldi�culties . . .. . . .. . .. . .. . . .. . .. 72

8 B eyond geom etry 74

8.1 Cli�ord’selem entsoffeeling .. . . .. . .. . .. . . .. . .. 75

8.2 Eddington’snature ofthings . . . .. . .. . .. . . .. . .. 76

8.3 W heeler’spregeom etry .. . .. . . .. . .. . .. . . .. . .. 78

8.3.1 M ore bucketsofdust.. . . .. . .. . .. . . .. . .. 81

8.4 Einstein’sconceptualtoolsofthought . .. . .. . . .. . .. 83

8.5 W horf’slinguistic relativity .. . . .. . .. . .. . . .. . .. 85

8.6 Appraisal .. . .. . . .. . .. . . .. . .. . .. . . .. . .. 87

9 T he tragedy ofthe ether 89

9.1 Them echanicalether. .. . .. . . .. . .. . .. . . .. . .. 91

9.2 Them etric ethers. . . .. . .. . . .. . .. . .. . . .. . .. 96

9.3 Thegeom etric ether . .. . .. . . .. . .. . .. . . .. . .. 102

9.4 Beyond thegeom etric ether .. . . .. . .. . .. . . .. . .. 108

10 A geom etric analysis oftim e 119

10.1 Clock readingsasspatiotem poralbasis . .. . .. . . .. . .. 124

10.1.1 Eventsand particlesasgeom etric objects . . .. . .. 124

10.1.2 Theobserveraspsychologicalagent . .. . . .. . .. 127

10.1.3 Clocksand the lineelem ent .. . .. . .. . . .. . .. 128

10.1.4 Invariance ofthe lineelem ent . . .. . .. . . .. . .. 130

10.1.5 TheNewtonian line elem ent.. . .. . .. . . .. . .. 132

10.1.6 TheRiem ann-Einstein line elem ent. . .. . . .. . .. 133

10.1.7 From G aussian surfacesto Riem annian space .. . .. 134

10.2 Riem ann-Einstein physicalspacetim e . . .. . .. . . .. . .. 137

10.2.1 Towardslocalphysicalgeom etry .. . .. . . .. . .. 137

10.2.2 Localphysicalgeom etry from psychology . . .. . .. 139

10.2.3 Spacelike separation and orthogonality from clocks .. 142

10.3 G ravitation asspacetim e curvature .. . .. . .. . . .. . .. 148

10.3.1 Einstein’sgeodesic hypothesis. . .. . .. . . .. . .. 150

10.3.2 Theabsolute derivative . . .. . .. . .. . . .. . .. 154

10.3.3 Thecovariantderivative . . .. . .. . .. . . .. . .. 159

C O N T EN T S xiii

10.3.4 Thecurvaturetensor. . .. . .. . . .. . .. . . .. . 161

10.3.5 G eodesic deviation and therise ofrod length . . .. . 166

10.4 Thesourcesofgravitation . . . .. . .. . . .. . .. . . .. . 168

10.4.1 Thevacuum .. . .. . . .. . .. . . .. . .. . . .. . 168

10.4.2 Trem orsin the vacuum .. . .. . . .. . .. . . .. . 172

10.4.3 Thecosm ic vacuum . . .. . .. . . .. . .. . . .. . 175

10.4.4 Thedust-�lled cosm os . .. . .. . . .. . .. . . .. . 178

10.4.5 Theproblem ofcosm ology . . .. . . .. . .. . . .. . 180

11 Interlude: Shallw e quantize tim e? 184

12 T he m etageom etric nature of quantum -m echanical things

and the rise oftim e 188

12.1 Q uasigeom etry and m etageom etry . .. . . .. . .. . . .. . 190

12.2 Towardsm etageom etric quantum -m echanicalthings . . .. . 192

12.2.1 Experim entalfacetofpsychophysics .. . .. . . .. . 193

12.2.2 M etageom etric facetofpsychophysics . . .. . . .. . 198

12.2.3 Appraisal . .. . .. . . .. . .. . . .. . .. . . .. . 202

12.3 M etageom etric realm . . .. . . .. . .. . . .. . .. . . .. . 202

12.3.1 Prem easurem entthings .. . .. . . .. . .. . . .. . 202

12.3.2 Transition things .. . . .. . .. . . .. . .. . . .. . 208

12.4 Q uasigeom etric realm . .. . . .. . .. . . .. . .. . . .. . 212

12.4.1 Thetransition postulate and thetransition function . 212

12.4.2 A lead on probability . .. . .. . . .. . .. . . .. . 215

12.4.3 Thetransition-function postulate . . .. . .. . . .. . 216

12.5 G eom etric realm . .. . .. . . .. . .. . . .. . .. . . .. . 223

12.5.1 State vectorsfrom geom etric decom position . . . .. . 223

12.5.2 Derivative geom etric connotations . .. . .. . . .. . 225

12.5.3 G eom etric traces .. . . .. . .. . . .. . .. . . .. . 228

12.5.4 O bservables .. . .. . . .. . .. . . .. . .. . . .. . 229

12.6 Tim e . .. . . .. . .. . .. . . .. . .. . . .. . .. . . .. . 231

12.6.1 M etageom etric tim e . . .. . .. . . .. . .. . . .. . 232

12.6.2 Theriseofcoordinate tim e . .. . . .. . .. . . .. . 236

13 T im e,the last taboo 241

13.1 Theenigm a oftim e.. . .. . . .. . .. . . .. . .. . . .. . 241

13.2 Thetaboo oftim e .. . .. . . .. . .. . . .. . .. . . .. . 245

13.3 O verview and outlook . .. . . .. . .. . . .. . .. . . .. . 247

Epilogue 251

A ppendix A :To picture or to m odel? 257

A ppendix B :P reparation tim e 259

R eferences 263

C hapter 1

Q uantum gravity unobserved

Ifyou don’tknow where you are going,any road willtake

you there.

LewisCarroll,Alice’sadventures in W onderland

Thestudy ofthephysicalnatureofspaceand tim efallscurrently under

theconventionalsphereofquantum gravity| a �eld ofphysicsthathasbeen

abundantly described asan attem ptto com bine the clashing worldviewsof

generalrelativity and quantum m echanics,asthough piecesofafaultyjigsaw

puzzle.1 Itisnottheintention ofthisthesisto go into yetonem oreofthese

staledescriptions.W eareinstead interested in analysingcritically them ode

ofworking and assum ptionsofquantum gravity,thereby to understand why

itisnotyetpossibleto know even super�cially the answerto the question,

\W hatisquantum gravity?"

Paraphrasing Hardin (1969,p.vii)on population,the em erging history

ofquantum gravity is a story ofconfusion and denial| confusion experi-

enced,butconfusion psychologically denied. Q uantum gravity isa �eld of

physicswhosesubjectm atterhasnotbeen identi�ed atthetim e ofwriting

ofthis work. W hat is quantum gravity about? This question has no real

answerowing to the factthatno-one hasasyetidenti�ed any relevantob-

servations that could act to guide physics’next leap forward: either new,

unaccounted-for observations involving interactions of som e sort between

gravitation and quantum system sor well-known ones that,when analysed

from a new perspective, revealpreviously hidden features of the natural

world (cf.Einstein’s analysis oftim e m easurem ents as a source ofspecial

relativity). Because ofthisinde�niteness,there are,to varying degrees,al-

m ostasm any m eaningsattached to quantum gravity asthere are research

groupsofit.In otherwords,ignoranceaboutwhatquantum gravity isabout

leadsto uncertainty aboutwhatquantum gravity is.

In practice,however,this state ofutter disorientation is not perceived

to be as gloom y as allthat. In quantum gravity’s ivory tower,not only

has research happily becom e highly speculative and ruled by m atters of

1See e.g.(Butter�eld & Isham ,2000).

2 Q uantum gravity unobserved

internalnature,but,m uch m ore worryingly,hasto a large extentbegun to

display �ctionalfeatures.In theself-assured expositionsofquantum gravity,

speculation is not only apt to supplant m issing facts but is also capable

ofbecom ing fact itself: assum ptions originally proposed as possibilities or

beliefshave now,through concerted repetition,acquired the statusoffacts

irrespective ofthe question oftheir unattested physicaltruth. As a result

ofthisprocess,quantum gravity hasbecom e a clich�e-ridden �eld.

Them ostgeneralofthese clich�esholdsquantum gravity to bebasically

the joint consideration of \quantum " and \gravity." In other words, it

holds quantum gravity to be essentially a m echanicaltheory ofsom ething

physical(butforthem om entunknown)ofa certain typicalsizeand energy

evolving in tim ein typicalperiods(cf.thePlanck scale).Thisunwarranted

conclusion iseithersim ply read o� thelabel\quantum gravity," orisbased

on a stronger,m ostly unconscioushum an a�nity to m echanicalconceptsto

the detrim ent ofother ideas farther rem oved from the m echanicalsphere.

In truth,however,onecannottellatthisstage whatcom bination ofknown

physicsm ay be needed fora theory ofquantum gravity;even lesswhether

new,unforeseen physicalideasm ay benecessary instead.

Anotherclich�e involvesthe beliefthata theory ofquantum gravity will

(or worse,that it m ust2) solve certain,m ost,or allofwhat are presently

considered im portantopen problem sin frontierphysics.Forexam ple,that

quantum gravity will(orm ust)give a solution to the divergences ofquan-

tum �eld theory,unify thefourforcesofnature,explain the dim ensionality

of space, the value of the cosm ological constant, the nature of quantum

non-locality and ofsingularities,including black holesand theorigin ofthe

universe,recover generalrelativity as a low-energy lim it,possibly rewrite

the rulesofquantum m echanics,etc. Such an alm ighty characterization of

quantum gravity,however,seem stohavem orein com m on with expectations

and desiresthan with physicsrealistically achievable atone fellswoop.

Am ong these problem sawaiting solution,the various form s ofgravita-

tionalcollapsepredicted by generalrelativity and thecon�rm ed existenceof

quantum -m echanicalcorrelations�gurem ostnotably.3 To besure,M isner,

Thorne,and W heeler(1973)called gravitationalcollapse\thegreatestcrisis

in physics ofalltim e" (p.1196),while Stapp (1975) called Bell’s theorem

\the m ostprofound discovery ofscience" (p.271). Although one m ustnot

necessarily discard spacetim esingularitiesornon-localcorrelationsasm oti-

vationsforthe search forbettertheories,do we really need to increase our

knowledge oftheseissuesasa conditionalpartofa theory ofquantum grav-

ity? Thism ay orm ay notbethecase.ThecaseofNewtonian singularities,

for exam ple,was certainly not relevant to the invention ofrelativistic or

quantum theories,while,on theotherhand,theproblem ofelectrom agnetic

singularitieswasrelevantto thedevelopm entofquantum m echanics,asev-

idenced by Bohr’se�ortsto solve the problem ofthe collapse ofthe atom .

The sam e can be said ofquantum non-locality. Here again one cannottell

2See (Sm olin,2003,pp.10{11).

3See e.g.(M onk,1997,pp.3{9).

3

whether our knowledge ofthese correlations that defy the ideas ofspatial

locality should beincreased by atheory thatwillsupersedethecurrentones,

orwhetherthey sim ply are a trouble-free characteristic ofquantum theory

whose expounding isirrelevantto any future developm ents. W e sim ply do

notknow whatcurrentproblem squantum gravity willsolve,which it will

declare non-issues,and which itwillnoteven touch upon.W hatiscertain,

however,isthatany futuretheory thatm ay be called physicalwillexplain

theorigin and natureofthe observables itrefersto.

A related clich�eresultsfrom stretching thepreviousonebeyond allm ea-

sure. Itinvolves the beliefthata theory ofquantum gravity willnotonly

bea theory ofgravitation and quantum phenom ena,buta \theory ofevery-

thing" thatwillputan end to research in elem entalphysics. Thisdubious

idea,although nowadays not necessarily restricted to the particle-physics

perspective,ispresum ably inspired in the worldview o�ered by this�eld of

physics,accordingtowhich m icroscopicparticlesand theirparticle-m ediated

interactionsare the m ostbasic constituentsofnature.4 O n thisbasis,itis

believed that to account for all(known and presum ed) basic m icroscopic

constituents (particles,strings,or suchlike) and their m utualinteractions

is tantam ount to accounting in principle for \everything" in the physical

world.

To say theleast,thisconception isatoddswith anotherwell-established

�eld ofphysics,generalrelativity, from which we know that the m atter-

dependent geom etric properties ofspacetim e are as physicaland basic as

the particles found em bedded in its geom etric network. If,as it seem s to

be,the suspectidea ofa \theory ofeverything" was originally inspired by

the optim istic prospect ofa uni�cation ofthe four known interactions on

the basisofstring-theory considerations,5 then a cursory glance atgeneral

relativity leaves us wondering,And what about space and tim e? In the

lastanalysis,theparticle-physicsworldview originally inspiring a \theory of

4For approches from this direction,see e.g.(G reene,1999;W einberg,1993);for ap-

proaches from a generaldirection, see e.g.(Barrow, 1992; Breuer, 1997; Hartle, 2003;

Laughlin & Pines,2000;Tegm ark,1998;Tegm ark & W heeler,2001).5The origin of claim s regarding a \theory of everything" can be traced to a talk

by M ichael G reen advertised at O xford University in the m id-1980s. It is told that

Brian G reene \[w]alking hom e one winter day...saw a poster for a lecture explain-

ing a new-found ‘theory ofeverything.’" \It m ade it sound dram atic," G reene adds.

D ram atic, indeed, but to what purpose dram a in science? (Colum bia College To-

day,http://www.college.colum bia.edu/cct/sep99/12a.htm l. W ith variationsalso atEncy-

clopedia ofW orld Biography,http://www.notablebiographies.com /news/G e-La/G reene-

Brian.htm l)In truth,however,wecan tracetheorigin ofthephrase\theory ofeverything"

fartherback in tim e.In ashortstory �rstpublished in 1966,Stanis law Lem introduced the

concept| as could not be otherwise| as part ofharsh satire. His space-faring character

Ijon Tichy relatesthathisgrandfather

atthe age ofnine...decided to create a G eneralTheory ofEverything and

nothing could diverthim from thatgoal.Thetrem endousdi�culty heexpe-

rienced,from the earliestyearson,in form ulating concepts,only intensi�ed

after a disastrous street accident (a steam roller attened his head). (Lem ,

1985,p.258)

Thisisan origin thatdoesnotbode wellforthe idea in question.


everything" isbased on the presum ption thatto be m ore elem entalm eans

to be sm aller;butwhence should the elem entalconceptsofourm ostbasic

theories refer to sm allentities? This is also the presum ption behind the

Planck-scale considerationsunderlying quantum gravity.

Deeperreasonsto doubtany claim abouta \theory ofeverything" con-

cern epistem ologicaland ontologicalconsiderations. From an epistem olog-

icalperspective,physicaltheories are the productsofthe hum an intellect,

builtwith theconceptualtoolsofthe m ind.This,forone,entailsthatthe-

oriesare builtfrom lim ited experience outofthe conventional(overwhelm -

ingly geom etric),volatile conceptsofthe physicaltradition they belong to,

and alsothatonecan neverbecertain whetherallaspectsofnature| known

and unforeseeablealike| can even in principlebereduced tothoseofany ex-

isting theory.AsStanis law Lem (1984a),thatgreatdiscernerofthehum an

spirit,putit,\O ne hasto look through the history ofscience to reach the

m ostprobable conclusion: that the shape ofthings to com e isdeterm ined

by whatwe do notknow today and by whatisunforeseeable" (p.124).

But m uch m ore profoundly,the fact that our theories are nothing but

conceptualtools ofthought precludes the beliefthat a \theory ofevery-

thing" isa reasonable oreven m eaningfulhum an goalto pursue (or,worse

still,justaround the corner),because a toolofthought is a �lter that se-

lectsaccording to a built-in bias. Thism ustnecessarily be so,because the

jum ble ofim pressionsthatm ake up the innerand outerworld ofconscious

experiencecannotbem adesenseofasan indivisiblewhole.Heed theperti-

nentwordsfrom a di�erentcontextofanothergreatdiscernerofthehum an

spirit,biologist-ecologistG arrettHardin:

W e need to be acutely aware ofthe virtues and shortcom ings

ofthe tools ofthe m ind. In the light ofthat awareness,m any

publiccontroversiescan beresolved.

[...] O ur intellectualtools are �lters for reducing reality to a

m anageable sim plicity. To assertthat we have ever com pletely

captured reality in an equation or a string ofwords would be

extrem ely arrogant. Each �lter captures only part of reality.

M ost expertise is single-�lter expertise. W e com e closer to the

truth when we com pensate for the bias ofone �lter by using

others(which have di�erentbiases).(Hardin,1985,pp.20{21)

Any physicaltheory issuch a�lter,itsm esh currently builtoutofgeom etry;

indeed,a physicaltheory,asNorth (1990)putitwith a degree ofm odesty

currently in shortsupply,6 is\m erely an instrum entofwhatpassesforun-

derstanding" (p.407). \Everything" cannot be understood. Ultim ately,

the beliefin a \theory ofeverything" is based on what Lem (1984a) has

called \the m yth ofour [and our scienti�c language’s]cognitive universal-

ity" (pp.26{27)| in otherwords,on hum an folly.

6In a world wherescienti�ctheorieshavebecom em arketableconsum ergoods,m odesty

isa luxury we cannota�ord. Thisexplainswhy dram a isneeded in science. Forrelated

ideas,see (Lagendijk,2005).

5

In connection with his quotation above,to better grasp the reality of

a �nite overpopulated world,Hardin proposed the application ofa com bi-

nation ofseveral�lters: literacy,num eracy,and ecolacy. Hardin’s various

�lters,because ofbeing m any and di�erent,are �lters against folly| the

folly ofsingle-�lterexpertise.Thesingle-�lterexplain-allschem esproposed

by the advocates ofa \theory ofeverything," on the other hand,are then

ofnecessity �ltersof folly.

The only positive thing| yet far-fetched and, at any rate, of lim ited

scope| thatcan be said fora \theory ofeverything" com esfrom the onto-

logicalperspective;in other words,from a di�erentiation between hum an

knowledge and truth. Turning a blind eye to the built-in lim itations of

theoriesas�ltersforunderstanding,som e have argued fora \theory ofev-

erything" as follows. As m uch as physicaltheories m ay change,what is

trueofthem asexpressed in theirlaws neverchangesbeyond the degree of

accuracy with which these laws have been corroborated. For exam ple,as

m echanicaltheoriesweresuccessively im proved from G alilei,through K epler

and Newton,to Einstein and the fathers ofquantum m echanics,the laws

ofinertia,ofequivalence between restand gravitationalm ass,the inverse-

squarelaw ofgravitation,etc.,eitherrem ained untouched orwere m odi�ed

for new physicaldom ains but never banished. W hat is once known is,as

it were,known forever. Thus,a \theory ofeverything" could conceivably

contain lawsthatare absolutely and perfectly true| whetherwe know itor

not| ofthose aspectsofnature,both known and to be disclosed in the fu-

ture,thatitas a �lterm anages to capture| not,however,ofevery aspect

ofnature,which escapes any chosen m esh. This is the m ost a \theory of

everything" could be.

O ntologically speaking,then,weseethata\theory ofeverything"in this

severely restricted one-�ltersense isnota logicalim possibility,butonly an

utterly unreasonableexpectation judging from theconstantturm oilscience

has lived in during allofits history,including especially the (psychologi-

cally denied)utterdisorientation ofthe presentepoch. AsPeacock (2006)

enquires with regard to string theory,\Ifthe search for a unique and in-

evitable explanation ofnature hasproved illusory atevery step,isitreally

plausiblethatsuddenlystringtheory can m akeeverythingrightatthelast?"

(p.170).

Since when do scientists seriously concern them selves with the study

ofutter im plausibilities? (Perpetual-m otion m achines and astrology com e

m ostreadily to m ind.) Isthe enterprise ofnaturalscience aboutclaim ing

non-im possibilitiesand asking othersto dem onstrate a \no-go theorem " to

proveonetobelogically wrong? Thisis,atany rate,theturfofm athem atics

orprofessionalphilosophy,butnota gam ethatphysicsisready to play.O r

to borrow Hardin’s(1993,pp.40{43) words,in science (as in ecology;but

unlike in professionalphilosophy,as in econom ics) the \default status" is

\guilty untilproven innocent," which m eansthatthe\burden ofproof" lies

on the proponent of extravagant claim s (or \progressive developm ents"),

not on the receiver. Hardin asks,\Aren’t scientists supposed to be open-


W hat is quantum gravity?

Clich�e # 1: Q uantum gravity is

quantum \ gravity.

Answer# 1: Q uantum gravity is

known physics[ unforeseeablenew

physics.

Clich�e # 2: Q uantum gravity to

solve de�nite problem s ofcurrent

physics.

Answer# 2: Q uantum gravity to

explain origin and natureofitsob-

servables.

Clich�e # 3:Q uantum gravity isthe

\theory ofeverything."

Answer# 3: A \theory of every-

thing" issingle-�lterfolly.

Clich�e # 4: Q uantum gravity ap-

pliesatthePlanck scale.

Answer# 4: Q uantum gravity ap-

pliesatthescaleofitsobservables.

Table 1.1: Contrastbetween the typicaland ourown viewsregarding the nature

ofquantum gravity.Thesym bol\ representsthelogicaloperation AND,whilethe

sym bol[ representsthe logicaloperation O R.

m inded? Does notscience progressby exam ining every possibility?" And

forthrightly replies,\Theanswerto thesecond question isno.Asforbeing

open-m inded,how m uch open-m indednesscan we a�ord?" (p.40).

A fourth and �nalclich�e to be m entioned here consists in the general-

ized belief| regardlessofresearch approach orinvestigator| thatthePlanck

scale willberelevantto quantum gravity.Thisnotion iswithoutdoubtthe

m ostcherished,overarching them eofquantum gravity.Itgoesunchallenged

in virtually allresearches,and hasbecom esuch a central,yetworryingly lit-

tle exam ined,assum ption thatwe shallanalyse itin detailin the following

chapter.Asweseparatefactfrom beliefin connection with thePlanck scale,

we give a new look atsom eoftheotherissuesm entioned above.

Q uantum -gravity clich�es,sum m arized in Table 1.1,are �ctionalbeliefs

that have becom e facts through sheer repetition, in line with a reigning

politics ofpublication that,as Lawrence (2003) has rem arked,uncritically

favoursfashionableideas.Thissituation isfuelled by thefactthatthestan-

dard accounts ofquantum gravity are capable ofo�ering alluring vistasto

the hum an im agination,which are,in turn,invigorated by colourfulpop-

ularizations in m agazines,books,and visualm edia,for which the m aking

ofpro�tsetsthe m ain standard ofaction. Ultim ately,these beliefsrem ain

unchallenged dueto thelack ofa referentofquantum gravity in thenatural

world. O nly the recognition ofrelevantobservables,whethernew onesyet

unaccounted-for or old ones seen in a new light,willprovide the clue as

to whatquantum gravity isabout. And only knowledge ofwhatquantum

gravity isaboutwillspelloutwhatquantum gravity is.Forthisreason,the

question one should worry aboutisnotwhatquantum gravity is,butwhat

are the relevantobservables on which to build a quantum theory ofspace

and tim e.

C hapter 2

T he m yth ofPlanck-scale

physics

There was a tim e when educated m en supposed thatm yths

were som ething ofthepast| quaint,irrationalfairy talesbe-

lieved in by our credulous and superstitious ancestors, but

notby us,oh dear no!

G arrettHardin,Nature and M an’sfate

An overview of the current con�dent status of so-called Planck-scale

physicsisenough to perplex.Research on quantum gravity hasabundantly

becom e synonym ouswith it,with the Planck length lP ,tim e tP ,and m ass

m P being unquestionably hailed asthescalesofphysicalprocessesorthings

directly relevantto a theory ofquantum gravity.And yet,in the lastanal-

ysis,the only linking threads between Planck’s naturalunits,lP ,tP ,and

m P ,and quantum gravity ideasare(i)generally and predom inantly,dim en-

sionalanalysis| with allitsvicesand virtues| and,speci�ctosom eresearch

program m es,(ii)jointconsiderationsofgravitational(relativistic)and quan-

tum theories,such asad hoc considerationsaboutm icroscopic,Planck-sized

black holes or the quantization ofgravity| arenas in which the e�ects of

both quantum theory and generalrelativity areto berelevanttogether.

Dim ensionalanalysisisa surprisingly powerfulm ethod capable ofpro-

viding great insight into physicalsituations without needing to work out

or know the detailed principles underlying the problem in question. This

(apparently) suits research on quantum gravity extrem ely wellat present,

sincethephysicalm echanism sto beinvolved in such a theory areunknown.

However,dim ensionalanalysisisnotan all-powerfuldiscipline:unlessvery

judiciously used,theresultsitproducesarenotnecessarily m eaningfuland,

therefore,they should be interpreted with caution. So should,too,argu-

m ents concerning Planck-sized black holes or the quantization ofgravity,

considering,aswe shallsee,theirsuspectcharacter.

In thischapter,we putforward severalcautionary observationsagainst

the role uncritically bestowed currently on the Planck unitsas m eaningful

8 T he m yth ofP lanck-scale physics

scalesin a futurephysicaltheory ofspaceand tim e.O nerem ark consistsin

questioning the extentofthe capabilitiesofdim ensionalanalysis:although

itpossessesan alm ostm iraculousability toproducecorrectordersofm agni-

tudeforquantitiesfrom dim ensionally appropriatecom binationsofphysical

constantsand variables,itisnotby itselfa necessarily enlightening proce-

dure.A conceptually deeperrem ark hasto do with thefactthat,despiteits

nam e,a theory ofquantum gravity need notbethe outcom e of\quantum "

(h) and \gravity" (G ;c): it m ay be required that one or m ore presently

unknown naturalconstants be discovered and taken into account in order

to picture1 thenew physicscorrectly;oryetm oreunexpectedly,itm ay also

turn outthatquantum gravity hasnothing to do with one or m ore ofthe

above-m entioned constants.Them eaningoftheseobservationswillbeillus-

trated in the following with the help,where possible,ofphysicalexam ples,

through which we willcastdoubton the m ostly undisputed signi�cance of

Planck-scalephysics.W ewillarguethatquantum gravity scholars,eagerto

em bark on the details oftheir investigations,overlook the question ofthe

likelihood oftheir assum ptions regarding the Planck scale| thus creating

seem ingly indubitablefactsoutofm erely plausiblebeliefs.

Butbeforeundertakingthecriticalanalysisofthesebeliefs,letus�rstre-

view thewell-established factssurroundingdim ensionalanalysisand Planck’s

naturalunits.

2.1 Facts

The Planck units were proposed for the �rst tim e by M ax Planck (1899)

overa century ago.Hisoriginalintention in theinvention oftheseunitswas

to providea setofbasicphysicalunitsthatwould belessarbitrary,i.e.less

hum an-oriented,and m ore universally m eaningfulthan m etre,second,and

kilogram units.2

ThePlanck unitscan becalculated from dim ensionally appropriatecom -

binations ofthree universalnaturalconstants,nam ely,Newton’s gravita-

tionalconstantG ,Planck’sconstanth,and thespeed oflightc.Them ath-

em aticalprocedure through which Planck’s units can be obtained is not

guaranteed to succeed unless the reference set ofconstants fG ;h;cg is di-

m ensionally independent with respect to the dim ensions M ofm ass,L of

length,and T oftim e. Thisisto say thatthe M LT-dim ensionsofnone of

the above constants should be expressible in term s ofcom binations ofthe

othertwo.

The reference set fG ;h;cg willbe dim ensionally independent with re-

spectto M ,L,and T ifthesystem ofequations8<

:

[G ] = M �1 L3T�2

[h] = M 1L2T�1

[c] = M 0L1T�1

(2.1)

1D o scientistspicture the naturalworld,ordo they m odelit? Fora briefelucidation,

see Appendix A.2Foran accountofthe early history ofthe Planck units,see (G orelik,1992).

2.1 Facts 9

isinvertible.Following Bridgm an (1963,pp.31{34),we take the logarithm

ofthe equationsto �nd

8<

:

ln([G ]) = � 1ln(M )+ 3ln(L)� 2ln(T)

ln([h]) = 1ln(M )+ 2ln(L)� 1ln(T)

ln([c]) = 0ln(M )+ 1ln(L)� 1ln(T)

; (2.2)

which isnow a linearsystem ofequationsin thelogarithm softhevariables

ofinterest and can be solved applying Cram er’srule. For exam ple,forM

weget

ln(M )=1

�

��

ln([G ]) 3 � 2

ln([h]) 2 � 1

ln([c]) 1 � 1

��; (2.3)

where

�=

��

� 1 3 � 2

1 2 � 1

0 1 � 1

��= 2 (2.4)

isthe determ inantofthe coe�cientsofthe system .W e �nd

M = [G ]�1

� [h]1

� [c]1

� ; (2.5)

and sim ilarly for L and T. This m eans that the system ofequations has

a solution only if� isdi�erentfrom zero,which in factholdsin thiscase.

G ranted thepossibility to constructPlanck’sunits,we proceed to do so.3

ThePlanck length lP resultsasfollows.W e de�nelP by the equation

lP = K lG�h�c ; (2.6)

where K l is a dim ensionless constant. In order to �nd �,�, and , we

subsequently writeitsdim ensionalcounterpart

[lP ]= L = [G ]�[h]�[c] : (2.7)

In term sofM ,L,and T,we get

L = (M �1L3T�2 )�(M L

2T�1 )�(LT�1 ) : (2.8)

Nextwe solve for�,�,and by rearranging the right-hand side and com -

paring itwith theleft-hand side,to get� = � = 1=2,and = � 3=2.Thus,

thePlanck length is

lP = K l

rG h

c3= K l4:05� 10�35 m : (2.9)

TheothertwoPlanckunitscan becalculated bym eansofatotally analogous

procedure.They are

tP = K t

rG h

c5= K t1:35� 10�43 s (2.10)

3Although Cram er’s rule could be used to actually solve for lP ,tP ,and m P ,in the

following we shalluse a lessabstract,m ore intuitive m ethod.


and

m P = K m

rhc

G= K m 5:46� 10�8 kg: (2.11)

The crucialquestion now is: is there any physicalsigni�cance in these

naturalunitsbeyond Planck’soriginalintentionsofproviding a lesshum an-

oriented setofreferenceunitsforlength,tim e,and m ass? Q uantum -gravity

research largely takes for granted a positive answer to this question,and

confers on these units a com pletely di�erent,and altogether loftier,role:

Planck’s units are to represent the physicalscale ofthings relevant to a

theory ofquantum gravity,oratwhich processesrelevantto such a theory

occur.4

O ne m ay seek high and low forcogentjusti�cationsofthism om entous

claim ,which isto be found in innum erable scienti�c publications;m uch as

one m ight seek,however,one m ust always despairingly return to,by and

large,thesam eexplanation (when itisexplicitly m entioned atall):dim en-

sionalanalysis. For exam ple,Butter�eld and Isham (2000,p.37) account

for the appearance of the Planck units in passing| and quite accurately

indeed| as a \sim ple dim ensionalargum ent." Sm olin (1997),forhis part,

grantsthesuspectderivation ofthe Planck scale from dim ensionalanalysis

butgoeson to justify it:\Thism ay seem likea bitofa trick,butin physics

argum entslike this[dim ensionalanalytic]are usually reliable." He also as-

sures the reader that \severaldi�erent approaches to quantum gravity do

predict" (p.60) the relevance ofthis scale,but does not m ention that all

thesedi�erentapproachesusethesam e trick.Isdim ensionalanalysissuch a

trustworthy toolasto grantusde�niteinform ation aboutunknown physics

withoutneeding to look into nature’sinnerworkings,orhasourfaith in it

becom e exaggerated?

Two uncom m only cautious statem ents regarding the m eaning of the

Planck unitsare those ofY.J.Ng’s,and ofJ.C.Baez’s. Ng (2003)recog-

nized thatthe Planck unitsare so extrem ely large orsm allrelative to the

scaleswecan exploretoday that\ittakesa certain am ountoffoolhardiness

to even m ention Planck-scale physics" (p.1,onlineref.).Although thisisa

welcom e observation,itappearsto work only asa proviso,forNg im m edi-

ately m oved on to a study ofPlanck-scalephysicsratherthan to a criticism

ofit:\Butby extrapolating thewell-known successesofquantum m echanics

and generalrelativity in low energy,we believe one can stillm ake predic-

tionsaboutcertain phenom enainvolving Planck-scalephysics,and check for

consistency" (p.1).

Also Baez (2000) m ade welcom e critical observations against the hy-

potheticalrelevance ofthe Planck length in a theory ofquantum gravity.

Firstly,hem entioned thatthedim ensionlessfactor(heredenoted K l)m ight

in fact turn out to be very large or very sm all,which m eans that the or-

derofm agnitude ofthe Planck length asisnorm ally understood (i.e.with

K l = 1) need not be m eaningfulat all. A m om ent’s re ection shows that

thisproblem doesnotreally threaten theexpected orderofm agnitudeofthe

4See e.g.(Am elino-Cam elia,2003)forhypothetically possible rolesofPlanck’slength.

2.1 Facts 11

Planck units.An overview ofphysicalform ulassuggeststhat,in general,the

dim ensionlessconstantK appearing in them tendsto bevery large orvery

sm all(e.g.jK j� 102 orjK j� 10�2 )only when thephysicalconstantsenter

the equationsto high powers. Forexam ple,considera black body and ask

whatm ightbeitsradiation energy density �E .Sincethesystem in question

involvesa gas(kB �)ofphotons(c;h),oneproposes

�E = K �h�c�(kB �)

; (2.12)

where K � isa dim ensionlessconstantand � isthe absolute tem perature of

theblack body.Theapplication ofdim ensionalanalysisgivesthe result

�E = K �

(kB �)4

h3c3= K �4:64� 10�18 �4

J

K 4m 3; (2.13)

whereasthecorrectresultis

�E = 7:57� 10�16 �4J

K 4m 3; (2.14)

and thusK � = 163.ThevalueofK � doesnotresultfrom dim ensionalanal-

ysisbutfrom detailed physicalcalculations;thesevalueswillfrom now on be

given in parenthesesforthepurposeofassessingtheaccuracy ofdim ensional

analysis. The resultyielded by dim ensionalanalysis,then,istwo ordersof

m agnitudelowerthan the correctvalue of�E ,which representsa m eaning-

fuldi�erence for the physics ofa black body. This discrepancy,however,

could have been expected since itcan betraced back to the constantsh,c,

and kB entering theequation to high powers(3,3,and 4,respectively).5 In

thisrespect,thesituation appearsquitesafeforthePlanck units,excepting

perhapsthe Planck tim e,in whoseexpression centersto the power� 5=2.

M oreinterestingly,Baezalsorecognized that\atheory ofquantum grav-

ity m ight involve physical constants other than c, G , and ~" (p. 180).

This is indeed one im portant issue regarding the signi�cance of Planck-

scale physics| orlack thereof| which willbe analysed in the nextsection.

However,notwithstanding hiscautionary observationsto thee�ectthat\we

cannotprovethatthePlancklength issigni�cantforquantum gravity," Baez

also chose to try and \glean som e wisdom from pondering the constantsc,

G ,and ~" (p.180).

W hatthesecriticism sintend to pointoutisthecuriousfactthat,within

thewidespread uncriticalacceptanceoftherelevanceofPlanck-scalephysics,

even thosewho noticed thepossibleshortcom ingsofdim ensionalanalysisin

connection with quantum gravity decided tocontinuetopursuetheirstudies

in thisdirection| asifto seriously question thePlanck scalewastaboo.W e

enquire into the deeper reason behind this practice later on in Chapter 7.

Fornow,weconcentrateon theissueathand and ask:wheredo thecharm s

ofdim ensionalanalysislie,then?

\[D]im ensionalanalysis is a by-way ofphysics that seldom fails to fas-

cinate even the hardened practitioner," said Isaacson and Isaacson (1975,

5See (Bridgm an,1963,pp.88,95)forrelated views.


p.vii).Indeed,exam plescould bem ultiplied atwillfortheview thatdim en-

sionalanalysisisatrustworthy,alm ostm agical,tool.Consider,forexam ple,

a hydrogen atom and ask whatisthe relevantphysicalscale ofitsradiusa

and binding energy E .Dim ensionalanalysiscan readily provide an answer

apparently withoutm uch knowledge ofthe physicsofatom s,only ifone is

capable ofestim ating correctly which naturalconstantsare relevantto the

problem . The generalm annerofthisestim ation willbe considered in Sec-

tion 2.3;forthem om ent,weassum ethatone atleastknowswhatan atom

is,nam ely,a m icroscopic aggregate oflighternegative chargesand heavier

(neutraland)positivechargesofequalstrength,and whereonly thedetails

oftheir m utualinteraction rem ain unknown. W ithout this kind ofprevi-

ousknowledge,in e�ectassuringusthatwehaveaquantum -electrodynam ic

system in ourhands,theapplication ofdim ensionalanalysisbecom esblind.6

Arm ed with previousphysicalexperience,then,onenoticesthatonly quan-

tum m echanicsand theelectrodynam icsoftheelectron areinvolved,so that

theconstantsto considerm ustbeh,theelectron chargee,theelectron m ass

m e,and theperm ittivity ofvacuum �0.Equipped with theseconstants,one

proposes

a = K ah�e�m

e�

�0 (2.15)

and

E = K E h�e�m

�e�

�

0; (2.16)

where K a and K E are dim ensionlessconstants. A procedure analogous to

the oneabove forthe Planck length givesthe results

a = K a

�0h2

m ee2= K a1:66 �A = 0:529 �A (K a = 1=�) (2.17)

and

E = K E

m ee4

�20h2= K E 108 eV = � 13:60 eV (K E = � 1=8): (2.18)

Aswecan see,m eaningfulvaluesoftheorderofm agnitudeofthechar-

acteristic radius and binding energy of the hydrogen atom can thus be

found through the sheer power ofdim ensionalanalysis. W e believe that

itison a largely unstated argum entalong theselinesthatquantum -gravity

researchers’claim sregarding the relevance ofthePlanck unitsrest.

Regrettably,the physicalworld cannotbeprobed con�dently by m eans

ofthistoolalone,since ithasitsshortcom ings. Aswe see in the nextsec-

tion,these are that dim ensionalanalysis m ay altogether failto inform us

whatphysicalquantity theobtained orderofm agnituderefersto;and m ore

severely,m uch can go am issin the initialprocessofestim ating which nat-

uralconstantsm ustbeconsequentialand which not.Therefore,any claim s

venturing beyond thelim itsim posed by theseshortcom ings,although plau-

sible,m ustbebased on faith,hope,orbelief,and should notin consequence

beheld asstraightforward facts.

6See (Bridgm an,1963,pp.50{51).

2.2 B eliefs 13

2.2 B eliefs

O ur�rstobjection againsta securely established signi�cance ofthe Planck

unitsin quantum gravity liesin theexistence ofcasesin which dim ensional

analysisyieldsthe orderofm agnitude ofno clearphysicalthing orprocess

atall.

To illustrate this point, take the Com pton e�ect. A photon collides

with an electron,after which both particles are scattered. Assum e| as is

the case in quantum gravity| thatthe detailed physicsbehind the e�ectis

unknown,and use dim ensionalanalysis to predict the order ofm agnitude

ofthe length ofany thing orprocessinvolved in the e�ect. Since quantum

m echanicsand thedynam icsofan electron and a photon areinvolved here,

one assum es (quite correctly) that the naturalconstants to be considered

are h,c,and m e,where,like before,itishere essentialto be assured that

the physicalsystem under study is quantum -electrodynam ic. To �nd this

Com pton length lC ,we set

lC = K C h�c�m

e; (2.19)

whereK C isa dim ensionlessconstant,to �nd

lC = K C

h

m ec= K C 0:0243 �A = 0:0243 �A (K C = 1): (2.20)

W hatisnow lC thelength of? In thewordsofO hanian (1989),\in spiteof

itsnam e,thisisnotthe wavelength ofanything" (p.1039).

O hanian’s statem ent is controversial and requires a quali�cation. In

fact,one im m ediate m eaning ofthe Com pton length can be obtained from

the physicalequation in which it appears,nam ely, �� = lC [1 � cos(�)],

where� isthe wavelength ofthephoton and � isthe scattering angle.Itis

straightforward to seethatlC m ustbethechangeofwavelength ofa photon

scattered atrightangleswith thetargetelectron.In thism anner,am eaning

can befound to any constantappearing in a physicalequation by looking at

thespecialcasewhen thefunctionaldependenceissettounity.Perhapsthis

explainsO hanian’sdenialofany realm eaning to lC ,since via thism ethod

m eaning can beattached to any constantwhatsoever.7

In thissense,the m eaning ofconstantsappearing in physicalequations

can be learntfrom their very appearance in them . However,this takes all

charm away from dim ensionalanalysis itself,since such equations are not

provided by itbutbecom eavailableonly afterwell-understood physicalthe-

ories containing them are known;i.e.after the content ofthe equations is

related toactualobservations.In consequence,thediscovery ofthem eaning

7Baez(2000)explained anotherphysicalm eaningoftheCom pton length asthedistance

at which quantum �eld theory becom es necessary for understanding the behaviour ofa

particle of m ass m ,since \determ ining the position of a particle of m ass m to within

oneCom pton wavelength requiresenough energy to createanotherparticleofthatm ass"

(p.179). Here again, a new m eaning can be found for lC based on already existing,

detailed knowledge provided by quantum �eld theory.(Butsee observationson page 17.)


ofthePlanckunitsin atheoryofquantum gravity dependson such equations

being �rstdiscovered and interpreted againsta background ofactualobser-

vations.8 And yet,whatguaranteesthatthePlanck unitswillm eaningfully

appearin such a futuretheory? O nly thevery de�nition| supported,aswe

shallsee,by a hasty guess| ofquantum gravity as a theory involving the

constantsG ,h,and c.Thisbringsusto oursecond objection.

A m ore severe criticism ofthe widespread,unquestioned reign ofthe

Planck unitsin quantum -gravity research resultsfrom conceptualconsider-

ations aboutquantum gravity itself,including an exem plifying look atthe

history ofphysics.

Asm entioned above,itisa largely unchallenged assum ption thatquan-

tum gravity willinvolve precisely what its nam e m akes reference to| the

quantum (h)and gravity (G ;c)| and (i)nothing m ore or(ii)nothing less.

The �rstalternative presupposesthatin a future theory ofspacetim e,and

any observationsrelated to it,the com bination ofalready known physics|

and nothing else| willprove to be signi�cant. Ascorrect asitm ightturn

outto be,thisis too restrictive a conjecture foritexcludesthe possibility

ofthe need for truly new physics (cf.Baez’s observations). In particular,

isourcurrentphysicalknowledge so com plete and �nalasto disregard the

possibility oftheexistence ofrelevantnaturalconstantsyetundiscovered?

To m ake m atters worse,m ainstream quantum gravity only assum esG ,

h,and c as relevant to its quest,while paying little heed to other readily

available aspectsofthe Planck scale. Forexam ple,why notroutinely con-

siderBoltzm ann’s constant kB and the perm ittivity ofvacuum �0 (or how

aboutothersuitable constants?) to getPlanck’stem perature

�P = K �

s

hc5

G k2B

= K �3:55� 1032 K ; (2.21)

and Planck’scharge

qP = K q

phc�0 = K q1:33� 10�18 C (2.22)

asrelevantto quantum gravity aswell? Justbecauseonly length,tim e,and

m assappealto ourm ore prim itive,m echanicalintuition? O risitperhaps

becausethenotionsoftem peratureand chargearenotincluded in thelabel

\quantum gravity"?9

The second alternative above takes for granted that at least allthree

constantsG ,h,and cm ustplay a rolein quantum gravity.Although thisis

8Anotherpossibility isthatthe m eaning ofphysicalconstantscould be obtained from

sim pler,theory-independentm ethodsthrough which to m easure them directly;however,

thatwe have such a m ore directm ethod iscertainly nottrue today ofthe Planck units.

Neither is such an extrem e operationalistic stance required to give m eaning to them ;

it su�ces to have a (Planck-scale) theory ofquantum gravity m aking som e observable

predictions. But at least this m uch is necessary in order to m ake quantum gravity a

m eaningfulphysicaltheory,and thusdisentangle the claim s ofthiscurrently speculative

and volatile �eld ofresearch.9Forapproachesconsidering extended Planck scales,see e.g.(Cooperstock & Faraoni,

2003;M ajor& Setter,2001).

2.2 B eliefs 15

a seem ingly sensible expectation,itneed nothold true either,fora theory

ofquantum gravity m ay also be understood in lessconventionalways. For

exam ple,notasa quantum -m echanicaltheory of(general-relativistic)grav-

ity butasa quantum -m echanicaltheory ofem pty spacetim e,aswe explain

below.

In order to present the im port ofthe �rst view m ore vividly,we o�er

an illustration from the history ofphysics. Reconsider the hydrogen-atom

problem before Bohr’s solution was known and before Planck’s solution to

the black-body problem was ever given,i.e.before any knowledge ofthe

quantum ,and therefore ofh,was available.10 W ith no solid previous ex-

perience about atom ic physics,the atom would have been considered an

electrom agnetic-m echanical system , and dim ensional analysis could have

been (unjusti�ably)used to �nd theorderofm agnitudeofphysicalquanti-

tiessigni�canttotheproblem .Confronted with thissituation,anineteenth-

century quantum -gravity physicistm ay havedecidedly assum ed thatnonew

constantsneed play a partin yetunknown physicalphenom ena,proceeding

asfollows.

Since an electron and a (m uch m ore m assive) proton ofequalcharge

are concerned,the constants to be considered m ust be e,m e,�0,and the

perm eability of vacuum �0| and nothing else. This is of course wrong,

butonly to the m odern scientist who enjoys the bene�tofhindsight. The

inclusion of�0 isnotatallunreasonablesinceitcould havebeen suspected

thatnon-negligible m agnetic e�ectsplayed a part,too.

In orderto�nd theorderofm agnitudeoftheradiusaand bindingenergy

E ofthehydrogen atom ,one sets

a = K0ae

�m

�e�

0��0 (2.23)

and

E = K0E e

�m

�e�

�0�

�

0: (2.24)

Proceeding asbefore,one obtains

a = K0a

e2�0

m e

= K0a3:54� 10�4 �A (2.25)

and

E = K0E

m e

�0�0= K

0E 5:11� 105 eV: (2.26)

These expressions and estim ates for a and E are com pletely wrong (see

Eqs.(2.17)and (2.18)forthecorrectresultsand Table2.1foracom parison).

Thiscom esasno surpriseto today’sphysicist,who can tellata glancethat

10This dem and is,ofcourse,anachronistic since h was known to Bohr. This notwith-

standing,thehypotheticalsituation weproposeservesasthebasisfora com pletely plausi-

bleargum ent.Itisonly a historicalaccidentthattheproblem ofthecollapse oftheatom

was�rstconfronted with knowledgeofthequantum ,sincethelatterwasdiscovered while

studying black-body radiation,a problem independentoftheatom .Sincetheblack-body

problem doesnotlend itselfto theanalysiswehavein m ind,wepreferto taketheexam ple

ofthe hydrogen atom ,even ifwe m ustconsideritanachronistically.


Hydrogen atom Conscientious Haphazard

Radiusa K a�0h

2

m ee2 � 1:7 �A K 0

ae2�0m e

� 4� 10�4 �A

Binding energy E K Em ee

4

�20h2

� � 108 eV K 0E

m e

�0�0� � 5� 105 eV

Table 2.1: Contrasting resultsinvolving two characteristicfeaturesofthe hydro-

gen atom obtained via dim ensionalanalysis.A conscientiousapplication takesinto

accountthequantum -m echanicalnatureofthesystem ,whilea haphazard applica-

tion doesnotand assum esthehydrogen atom to bean electrom agnetic-m echanical

system .O nly the form ergivesreasonably approxim ateresults.

theconstantsassum ed tobem eaningfularewrong:�0 should notbethereat

all,and h hasnotbeen taken into account.How can then today’squantum -

gravity physicistsignore the possibility thatthey,too,m ay be m issing one

orm oreyetunknown constantsstem m ing from genuinely new physics?

Thesecond,lessconventionalview expressed above,nam ely,thatquan-

tum gravity need not be understood as a quantum -m echanicaltheory of

(general-relativistic) gravity, is supported by the following reasoning. In

view ofthe repeated di�cultiesand uncertaintiesencountered so farin at-

tem ptsto uncoverthequantum aspectsofgravity,onem ay wonderwhether

theissuem ightnotratherbewhetherspacetim ebeyond itsgeom etric struc-

ture m ay have quantum aspects. In particular and as we enquire further

on,m ay theconsideration ofquantum theory revealany physicalreality (in

theform ofobservables)thatem pty spacetim epossessesbutwhich classical

generalrelativity denies to them (cf.hole argum ent)? From this perspec-

tive,the possible quantum features ofgravity are not an issue and,ifone

standsbeyond thegeom etric structureofspacetim e| itselfthebearerofits

gravitationalfeatures(G )and causalstructure (c)| the constantsG and c

have no reason to arise in thetheory.

Asadvanced in the introduction,there existsom e approachesto quan-

tum gravity thatdo notrely on dim ensionalanalysisin orderto reproduce

Planck’slength and m ass.W estartby analysing onetypeofthese,nam ely,

thoseinvolving thenotion ofPlanck-scale black holes.Baez(2000,pp.179{

180), Thiem ann (2003, pp.8{9, online ref.), and to som e extent Saslow

(1998) have allindependently put forward the essence ofthe argum ent in

question from di�erentperspectives.In thelanguageofwavem echanics,the

gist ofthe idea is that in orderfor a wave packet ofm ass m to be spread

no m ore than one Com pton length lC = h=m c,its energy spectrum �E

m ust be spread no less than m c2,i.e.enough energy to create| according

to quantum �eld theory| a particle sim ilarto itself.O n the otherhand,in

orderforthesam ewave packetto createa non-negligiblegravitational�eld

(strong enough to interactwith itself),itm ustbespread no m orethan one

Schwarzschild radiuslS = 2G m =c2,i.e.beconcentrated enough tobecom e|

according to generalrelativity| a black hole. Both e�ects are deem ed to

becom e im portanttogetherwhen lC = lS,i.e.when the particle becom esa

2.2 B eliefs 17

black holeofm ass(hc=2G )1=2 = m P =p2 and radius(2G h=c3)1=2 =

p2lP |

a so-called Planck black hole,the paradigm atic denizen ofthe quantum -

gravitationalworld.11

Isthisa sound argum entorisityetanotherad hoc,ad lib arm chairex-

ercisedevised only to quench ourthirstfortangible\quantum -gravitational

things"? Regardless ofthe com plicated and physically controversialques-

tion whethersuch black holesactually exist,letusconcentratehereonly on

theconceptualaspectofthe issue.O urcriticism hasthreeparts.

Firstand irrespectiveofquantum -gravityideas,togiveoperationalm ean-

ing to the above statem ents about wave packets and their spreads, one

should talk abouta setofidentically prepared m icroscopicparticlesofm ass

m .Taking one m icroscopic system ata tim e and upon repeatedly m easur-

ing eitheritsposition oritsm om entum ,the statisticalresultsare found to

be spread according to the uncertainty relation 4h(� X )2ij ih(� P )2ij i �

jh[X ;P ]ij ij2 = ~

2.12 Butspeltoutthisway,theargum entlosesitsintuitive

appealand,with it,itsintended m eaning.W eseethatto preparea system

hasnothing to do with oursqueezing a particle into any sortoftightcon-

�nem ent,even lesswith ourneeding a certain m inim um am ountofenergy

to do so. Thus,neither the creation ofnew particles nor ofblack holes is

naturally connected with theuncertainty relationsatall.

Second and regardlessoftheabove,itisstraightforward to seethatthis

reasoning to the e�ect that Planck black holes m ust be paradigm atic of

quantum gravity assum esthe previously criticized view thatsuch a theory

m ustarise outofa com bination ofthe quantum and gravity.Itisreally no

surpriseto �nd that,on thissupposition,the Planck scale does resulteven

withoutthe intervention ofdim ensionalanalysis;now instead ofcom bining

universalconstantstogether,we com bine theories which contain them .

In fact, a variation of this way of proceeding is found, for exam ple,

in loop quantum gravity and in the canonicalquantization ofHam iltonian

black-holedynam ics.In them ,theappearanceofthePlanck length doesnot

resultfrom dim ensionalanalysis13 nor relying on the idea ofPlanck black

holes. The Planck length arises instead from a di�erent instance of the

said physically uncalled-forprocedureofnow com bining theories(instead of

sim ply constants) whose fundam entalconstants are G ,c,and h,i.e.from

quantizing generalrelativity in oneway oranother.W hy uncalled-for? Be-

cause no reasons are known as to why one should proceed thus| as ifone

were,say,to quantizethetra�con thestreetsorthesolarsystem | norare

11Planck’stim e isconnected with thisargum entaccording to extrapolationsofpredic-

tionsfrom quantum �eld theory in curved spacetim e,nam ely,thata Planck black hole is

unstablewith a lifetim eoftP � 10�43 s.However,thistheory,in which spacetim eistaken

to be classically curved but not \quantized," is com m only referred to as a sem iclassical

theory (ofquantum gravity)and paradoxically distrusted to hold atthePlanck scale.See

e.g.(Repo,2001,p.164).12In thiscontext,the substitutionsh(� P )

2ij i � (�E =c)

2(relativistic approxim ation)

and h(� X )2ij i = (h=m c)

2(Com pton spread)are used to get�E � m c

2=4� � m c

2.

13See (Rovelli, 2004, p.250) for an explicit m ention of this point in regard to loop

quantum gravity.


theconsequencesofquantization known tobeattested in thephysicalworld.

Allin all,whatthisanalysisshowsissom ething we already knew,nam ely,

thatthePlanck scalem ustberelevantto a theory which gluesthepiecesof

ourcurrentknowledge together.Butistherereason to do so?

Third,them annerin which theblack-holeargum entisconstrued appears

asnotoriously haphazard asthequantum -gravity application ofdim ensional

analysisalso forreasonsirrespective ofthe �rstcriticism .In e�ect,general

relativity is �rst assum ed nonchalantly in order to im plem ent the idea of

a Planck black hole,but this concept is subsequently taken to belong in

quantum gravity,which,according to theusualview,signalsthebreakdown

ofgeneralrelativity| thereby precluding the possibility ofblack holes!| at

the Planck scale. (O rcan there be black holes withoutgeneralrelativity?

Doesthe m ythical,unknown yettrue,theory ofquantum gravity allow for

them ?) Therefore,the reasoning heretoo seem sto beconceptually awed.

The welcom e view is som etim es expressed that the appropriateness of

thisnotion isuncertain.Forexam ple,Thiem ann shares(forhisown reasons)

our last point that the notion ofPlanck black holes m ust be conceptually

awed:

[A Planck black hole]is again [at]an energy regim e at which

quantum gravity m ust be im portant and these qualitative pic-

tures m ust be fundam entally wrong...(Thiem ann,2003, p.9,

online ref.)

Interestingly enough,Thiem ann takesthisnegative argum entoptim istically,

while wecan only take itpessim istically| asa bad om en.

G iven theconceptually uncertain natureand relevanceoftheseideas,we

concludethatoneshould take them with a pinch ofsalt,and notanywhere

m ore seriously than considerationsarising from dim ensionalanalysis.

2.3 A ppraisal

The state ofa�airs reviewed in the �rstpartofthe previoussection leads

to seriousdoubtsregarding the applicability ofdim ensionalanalysis. How

can weeverbesureto trustany resultsobtained by m eansofthism ethod?

O r as Bridgm an (1963) jocularly put it in his brilliant little book: \W e

are afraid that...we willgetthe incorrectanswer,and notknow ituntila

Q uebec bridge fallsdown" (p.8).Itistherefore worthwhile asking:where,

in the lastanalysis,did we go wrong?

In this respect, Bridgm an enlightened us by explaining that a trust-

worthy application ofdim ensionalanalysisrequiresthe bene�tofextended

experience with the physicalsituation we are confronted with as wellas

carefulthought:

W e shallthusultim ately be able to satisfy ourcritic ofthe cor-

rectness ofour procedure,butto do so requires a considerable

background ofphysicalexperience,and theexerciseofa discreet

2.3 A ppraisal 19

W ealth ofpreviousphysicalexperience

Positiveidenti�cation ofphysicalsystem

Carefulthoughtdeterm ining itskind

Selection ofconstantsand variables Q uantum gravity

Dim ensional-analyticpredictions

Figure 2.1: Stepsin the correctapplication ofdim ensionalanalysis. Thisisthe

only way to assureourselvesthatthepredictionsobtained by m eansofthism ethod

willlikely possess physicalrelevance. Dim ensionalanalysis in quantum gravity

becom eshaphazard by om itting the �rstthree stepsofthe procedure.

judgm ent. The untutored savage in the busheswould probably

notbeableto apply them ethodsofdim ensionalanalysisto this

problem and obtain resultswhich would satisfy us. (Bridgm an,

1963,p.5)

And further:

The problem [aboutwhatconstants and variables are relevant]

cannot be solved by the philosopher in his arm chair, but the

knowledgeinvolved wasgathered only by som eoneatsom etim e

soilinghishandswith directcontact.(Bridgm an,1963,pp.11{12)

In thislight,a conscientious application ofdim ensionalanalysis to the

a�airs of quantum gravity is hopeless. There does not exist so far any

realm ofphysicalexperience pertaining to interactions ofgravitation and

quantum m echanics,and therefore there isno previousphysicalknowledge

available and no basiswhatsoeveron which to base ourjudgem ent. Thus,

thequantum -gravity researcherproclaim ingtherelevanceofthePlanckscale

resem blesBridgm an’ssavage in the bushesorphilosopherin the arm chair,

notbecauseofbeinguntutored ornotwantingto soilhishands,butbecause

there are no observationalm eansavailable to obtain any substantialinfor-

m ation about the situation ofinterest. In consequence,quantum gravity

skipsessentialstepsin itsapplication ofdim ensionalanalysis,asshown in

Figure 2.1.

Bridgm an also dism issed any m eaning to be found in Planck’s units

considering the m annerin which they arise:

Theattem ptissom etim esm adetogofartherand seesom eabso-

lutesigni�cancein thesizeofthe[Planck]unitsthusdeterm ined,

looking on them as in som e way characteristic ofa m echanism

which isinvolved in theconstantsentering thede�nition.


The m ere fact that the dim ensionalform ulas ofthe three con-

stants used was such as to allow a determ ination of the new

units in the way proposed seem s to be the only fact ofsignif-

icance here,and this cannot be ofm uch signi�cance, because

the chancesare thatany com bination ofthree dim ensionalcon-

stantschosen atrandom would allow thesam eprocedure.Until

som eessentialconnection isdiscovered between them echanism s

which are accountable forthe gravitationalconstant,the veloc-

ity oflight,and thequantum ,itwould seem thatno signi�cance

whatever should be attached to the particular size ofthe units

de�ned in thisway,beyond thefactthatthesizeofsuch unitsis

determ ined by phenom ena ofuniversaloccurrence. (Bridgm an,

1963,p.101,italicsadded)

G orelik (1992),and von Borzeszkowskiand Treder(1988)have,in fact,crit-

icized theserem arksofBridgm an’son accountthat,indeed,theconnection

has now been found under the form ofquantum gravity. A m om ent’s re-

ection shows that this is a deceptive argum ent. O ne cannot legitim ately

contend that the new connection found is quantum gravity,and therefore

thatPlanck’sunitshavem eaningbecause,with nophenom enologicale�ects

to supportit,there is no physicalsubstance to so-called quantum gravity

besides(m ostly Planck-scalebased)theoreticalspeculations.Therefore,the

argum entisatbestunfounded and atworstcircular.

To sum up,the relevance ofPlanck-scale physics to quantum gravity

restson severaluncriticalassum ptions,which wehaveherecastdoubtupon.

M ostim portantly,thereisthequestion ofwhetherquantum gravity can be

straightforwardly decreed to bethesim plecom bination ofthequantum and

gravity. This excluding attitude does not appear wise after realizing,on

the onehand,thatblack-body physicsdid notturn outto bethe com bina-

tion oftherm odynam icsand relativity,noratom ic physicsthe com bination

ofelectrom agnetism and m echanics;and on the otherhand,thatquantum

gravity can also be interpreted in a lessliteralsense| forwhat’sin a nam e

afterall?| and regarded to include,forinstance,spacetim efeaturesbeyond

gravity.Dim ensional,quantization,and ad hoc analysesonly yield m eaning-

fulresultswhen the system studied and itsphysicsare wellknown to start

with. To presum e knowledge ofthe rightphysicsofan unknown system is

forquantum gravity to presum etoo m uch.

Further,we argued thatthephysicalm eaning ofthePlanck unitscould

only be known after the successfulequations ofthe theory which assum es

them | quantum gravity| were known.To achieve this,however,therecog-

nition and observation ofsom ephenom enologicale�ectsrelated to quantum

m echanics and spacetim e are essential. W ithout them , there can be no

trustworthy guide to oversee the construction of,and give m eaning to,the

\equationsofquantum gravity," and one m ustthen resortto wild guesses.

Ultim ately,herein liestheproblem ofdim ensional,quantization,and ad hoc

analysesasapplied to quantum gravity.Thefactneverthelessrem ainsthat

no such observableshave asyetbeen identi�ed.Aswe expressed earlier,it

2.3 A ppraisal 21

isonly through thediscovery and observation ofsuch phenom ena telling us

whatquantum gravity isabout thatwe willgradually unravelthe question

concerning whatquantum gravity is.

In view ofthisuncertainty surroundingPlanck’snaturalunits,webelieve

it would be m ore appropriate to the honesty and prudence that typically

characterizethescienti�centerprise,nottoabstain from theirstudyifnotso

wished,butsim plytoexpresstheirrelevancetoquantum gravity asahum ble

belief,and notas an established fact,as isregrettably today’s widespread

practice.

C hapter 3

T he anthropic foundations

ofgeom etry

W e can hope for the best,for Isee the signsofm en.

M arcusVitruviusPollio,De architectura,Book 6.

Leaving fornow quantum gravity behind,we begin thestudy ofgeom e-

try| a discipline which is not only ubiquitous in quantum gravity but,in

fact,is basic to allofphysics and even to a wealth ofeveryday cognitive

tasks.

W hat are the nature and origins ofgeom etry? How ubiquitous is it,

really? Is it a toolofthought or a G od-given feature ofthe world? Is its

use bene�cialto furtherhum an understanding? To whatextent? How has

geom etry fared in physics so far in helping our understanding? And how

does geom etry fare as quantum gravity attem pts to attain by its m eans a

soundercom prehension ofthephysicalm eaningofspace,tim e,and quantum

theory? Thesequestionswilloccupy usin the com ing chapters.

3.1 T he rise ofgeom etric thought

Them eaning oftheword \geom etry" isfarwiderthan itsetym ologicalroot

suggests.Thatistosay,them eaningattributed to\geom etry"ism uch m ore

com prehensiveand basicthan therelatively advanced notionsincluded in a

\m easurem entoftheearth."

To be sure,in order to im agine a triangle or a circle,no quantitative

conceptoflength isneeded,forsuch �guresm ay wellbethoughtofwithout

any need to establish whatthe lengthsofthe sidesofthe triangle orwhat

the circle radiusm ightbe. And yet,no-one would contend thata triangle

ora circle isnota geom etric object.

W e could therefore lay down theconceptualfoundationsofgeom etry in

the following way. G eom etry is builtout oftwo basic elem ents. The �rst

and m oreprim itiveofthem isthatofgeom etricobjectssuch as,forexam ple,

3.2 T he origins ofform 23

points distan e arrow︸︷︷

︸length areasurfa e volumespatial figureFigure 3.1: Som egeom etricobjectsand associated geom etricm agnitudes.

point,line,arrow,plane,square,circle,sphere,cube,etc.G eom etricobjects

stand fortheidealized shape orform ofthe objectsofthe naturalworld.

G eom etricobjects,in turn,display geom etricproperties.Thesearequal-

itative attributesabstracted from geom etric objectsand possibly shared by

severalofthem .Exam plesofgeom etric propertiesare atness,curvedness,

straightness,roundness,concavity,convexity,closedness,openness,connect-

edness,disconnectedness,orientability,orthogonality,parallelism ,etc.G eo-

m etricobjectsand som eoftheirgeom etricproperties(thoseinvariantunder

continuousdeform ations)can befound attherootofthestudy oftopology.

The second,m uch m ore advanced ingredient that geom etry consists of

isthatofgeom etric m agnitudes.Asitsnam esuggests,thesem agnitudesin-

volvetheexpression ofaquantity,forwhich theconceptsofnum ber and unit

m easure areprerequisite.G eom etric m agnitudesariseasresultofa process

by m eansofwhich geom etric objectscan be attributed di�erentnotionsof

size or m easure. The m ost basic and paradigm atic exam ple ofa geom et-

ric m agnitude islength;thisappliesto a geom etric object(e.g.line,curve)

ora partofit(e.g.edge ofa polygon),and its originalm eaning is thatof

countinghow m any tim esaunitm easurecan bejuxtaposed alongtheobject

in question. O thergeom etric m agnitudes,shown with theircorresponding

geom etric objectsin Figure 3.1,are distance,area,and volum e.

3.2 T he origins ofform

According to Sm ith (1951,pp.1{4),the existence ofgeom etric form can

be traced back,as it were,to the m ists oftim e,ifone considers form s to

bepresentin naturein them selves,and notassom ebody’sinterpretation of

nature.In thisview,thegalacticspirals,theellipsesoftheplanetary orbits,

the sphere ofthe m oon,the hexagonsofsnow akes,etc.,are allgeom etric

displaysofnatureand antecede any presence oflife on earth.

Isnature,then,intrinsically geom etric? Itisalltoo tem pting to regard

nature with hum an eyes,even when talking abouta tim e when no sentient

beingsexisted. Isitnotrathera hum an prerogative to interpretthe world

geom etrically? O r, paraphrasing Eddington (1920, p.198), to �lter out

geom etry from ajum bleofqualitiesin theexternalworld? Donotgeom etric

24 T he anthropic foundations ofgeom etry

objects actually arise through idealization of sensory perceptions by the

m ind to becom e geom etric objectsofhum an thought? W e believe thatthe

history ofgeom etric thoughtbeginswith the �rstconsciousand purposeful

appreciations ofshape orform . G eom etry isto be equated with geom etric

thoughtratherthan with geom etricexistence,and itsoriginsm ustbelocated

only afterthe adventoflife. Geom etry is a toolofthought,nota property

oftheworld.

Can we now consider,forexam ple,the geom etric patternsdisplayed on

thesurfaceoftreeleavesorthoseofasea-shellorspiderweb asprim itivedis-

playsofgeom etricthoughtby living beings? Hardly.Itistoo far-fetched to

say thata plantshowsgeom etricpatternsbecauseitthinksgeom etrically|

plantshaveno brain.To a lesserdegree,thesam eappliesto spiders,which,

although endowed with brains,appearas no great candidates for abstract

thought either. As Sm ith (1951) put it,\W ho can say when or where or

underwhatin uencesthe Epeira,agesago,�rstlearned to trace the loga-

rithm icspiralin theweaving ofherweb?" (p.5).W ho can say,indeed,that

the Epeira everlearntto do purposefully,insightfully,such a thing atall.

W e reserve the capacity to think geom etrically to hum ans,disregarding

in so doing also m ankind’s closest cousins,the apes (or the m ore distant

dolphins),as no other anim albut M an engages in geom etric thought in a

deliberate,conscious,and purposefulway;no otheranim alisdriven to geo-

m etric thoughtby theneed to conceptualize theworld around,norpsycho-

logically rewarded in actually achieving the sought-forunderstanding after

a geom etric m anner. Anim als can certainly tellapart di�erent m an-m ade

(once again,for the purpose ofunderstanding anim als) geom etric �gures,

but this does not qualify as an anim alinstance of geom etric endeavour.

Although m ost living creatures share with hum ans sim ilar sense organs|

especially,butnotexclusively,thosethatallow vision| itistheworkingsof

a particularly m oulded brain interpreting stim uliand striving to conceptu-

alize the world thatresultin geom etric understanding. The deeperreason

why hum an brainscan| or even m ust?| interpretthe world geom etrically

is butan open question. As we proceed through the com ing chapters,we

shallanalyse geom etry asa hum an conceptualtoolofthoughtfurther.

3.3 T he origins ofnum ber

The �rstappreciationsofform and sym m etry m ustbe asold ashum anity

itself. W hat about the �rst appreciations ofnum ber? In a rudim entary

sense close to butnot properly called counting,these m ay be taken to be

even older than hum anity. According to Sm ith (1951,p.5),som e form of

pseudo-countingexistsam ongm agpies,chim panzees,and even som einsects.

M orerecently,alsoHauser(2000)hasdescribed instancesofapes,birds,and

rats capable ofsom e form ofcounting \up to" a sm allnum ber. Allthese

cases,however,only involve subitization,i.e.telling ata glance thenum ber

ofobjectspresentin a sm allgroup. The factthatanim alscannot\count"

beyond aboutfour| thelim itforaccuratesubitization| m ay in factbeseen

3.3 T he origins ofnum ber 25

as evidence that they are not actually counting. However,Hauser relates

that,under hum an-induced training,som e anim als can learn to count in

exchange forfood.

In this light, the view according to which the ability to count exists

in anim als in a way that precedes hum anity is analogous to the previous

one| thatdisplaysofgeom etric form are found in inertnature,plants,and

anim als| and sim ilarly m istaken.Theinnateanim alability torecognizedif-

ferencesin thenum berofobjectsin a sm allgroup isfarfrom whatwem ean

by counting:an abstractprocessinvolving a deliberate actofenum eration.

Theability to count,aided today by an advanced developm entoflanguage

and radix system s,com es second nature to us,and we are thussom etim es

m isled into seeing ourown re ection in otheranim als.

The�rsthum an needsforcounting m ighthave arisen outofa desire to

know thesizeofone’shousehold,aswellasthatofhunted anim als,orthatof

them em bersoftheenem y.O riginally,however,them em bersoflargegroups

were notreally counted,butsim ply considered as \m any" and designated

by som e collective noun such as\heap ofstones," \schoolof�sh," and the

like. Hauser has proposed that the developm ent ofa m ore advanced form

ofcounting isto befound in theem ergence ofhum an trade.

Them ostfam ous| although notnecessarily theoldestorm ostm eaning-

ful| archaeologicalrecord ofthehum an useofcounting istheIshangobone,

found nearLake Edward,on the frontierofUganda and Zaire,and dating

back to about18000 b.C.in the UpperPaleolithic.The10-centim etre-long

bone,shown in Figure 3.2,displaysa seriesofm arkscarved in groupsand

placed in three rows. Naively, these m arkings have been interpreted as

follows.Row (i)showsgroupsof21 (= 20+ 1),19 (= 20� 1),11 (= 10+ 1),

and 9 (= 10� 1)m arks,which hintsata decim alnum bersystem ;row (ii)

containsgroupsof11,13,17 and 19 notches,which are the prim e num bers

between 10 and 20;row (iii)showsa m ethod ofduplication. Furtherm ore,

the m arkingson rows(i)and (ii)each add up to 60,and those on row (iii)

to 48,possible evidence ofknowledge ofa sexagesim alnum bersystem . A

speculative,yetpainstakingly thorough and ratherconvincing study carried

outby M arshack (1972),however,revealed thatitism oreplausibleforthis

bone| together with num erous other engraved specim ens from the Upper

Paleolithic,such astheBlanchard bone,shown in Figure3.3| to bea lunar-

phasecounterdisplaying a record of�ve and a halflunarm onths.

Theoriginally intended m eaning ofthese carved objectsrem ainsnever-

thelessuncertain.In thisrespect,Flavin wrote:

Ancientstraightlines,whetherparietalorportable,defy forensic

interpretation becausewecan’tpenetratetheirconstituentphe-

nom enologicalsim plicity. M arshack attem pts to turn m arques

de chasse (\hunting m arks") into lunar calendars,eschewing a

tally ofprey for a tally ofdays (or nights). As allofthe arti-

facts presented by M arshack as evidence for Upper Paleolithic

lunar calendars are individual,seem ing to share no consistent

notationalpattern,every artifactrem ainsuniquely problem atic.


Figure 3.2: Three views ofthe 10-centim etre-long Ishango bone dating back to

about18000 b.C.Itdisplaysnotchesin threerows,naively interpreted asevidence

ofUpper-Paleolithicknowledgeofprim enum bers,a decim aland sexagesim alnum -

ber system ,and duplication. According to A.M arshack,the bone consists ofa

lunar-phase counter displaying a record of�ve and a halflunar m onths. O ther

interpretations,such astalliesofprey orm eredecoration,arealso possible.Repro-

duced from (M arshack,1972).

M uch like every generation receivesa di�erentinterpretation of

Stonehenge,perhaps years from now the engraved item s from

the Upper Paleolithic willbe envisioned as som ething beyond

ourcurrentim agining.(Flavin,2004)

In any case, as far as evidence ofan ability to count is concerned,both

m arkings representing a sim ple tally ofprey and m arkings representing a

m ore com plex tally ofdays are equally signs ofit;not so,however,ifthe

m arkingsin question were sim ply perfunctory decoration orthelike.1

Returning to the developm ent ofcounting and following Sm ith (1951,

pp.7{14),a prim itive notion ofcounting was available m uch earlier than

wordsexisted forthenum bersthem selves.Prim itively,countingconsisted in

explicitly associatingtheobjectstobecounted with,forexam ple,the�ngers

ofthe handsor m arkson the ground oron bone. Traces ofsuch counting

m ethodscan stillbefound in thehabitsand languagesofindigenoustribes

livingtoday.Forexam ple,theAndam ans,an O ceanictribe,only havewords

for\one" and \two" butm anagetokeep counting up toten by tappingtheir

nosewith their�ngersin succession and repeating\and thisone." TheZulu

tribesexpressthenum bersix by saying\taking thethum b"and thenum ber

1W hen it com es to interpreting isolated pieces of archaeological evidence, a cau-

tious attitude such as Flavin’s has m uch to be recom m ended for. In M arch 2004,

BBC News reported the �nding, in the K ozarnika cave in northwest Bulgaria, of

an anim al bone over a m illion years old that displays several m arkings in the form

of seem ingly purposeful straight lines. Furtherm ore, \another anim al bone found at

the site is incised with 27 m arks along its edge" (BBC News, http://news.bbc.co.uk/

2/hi/science/nature/3512470.stm ). O ught one to conclude that Hom o erectus already

had the capacity to countand keep a tally ofhis�ndings,including recordsofthephases

ofthe m oon?

3.4 T he origins ofm agnitude 27

Figure 3.3: Partialview ofthe11-centim etre-long Blanchard bone,found in Abri

Blanchard,in the Dordogne region of France, and dating back to about 25000

b.C.in theUpperPaleolithic.Itdisplayscircularnotcheslaid outin a m eandering

pattern,interpretedbyA.M arshackasarecordofatwo-and-a-quarterlunarm onth.

Reproduced from (M arshack,1972).

seven by saying \he pointed." Finally,in the M alay and Aztec tongues,in

that ofthe Niu�es ofthe Southern Paci�c,and in that ofthe Javans,the

num ber nam es m ean literally \one-stone," \two-stone," etc.; \one-fruit,"

\two-fruit," etc.;and \one-grain," \two-grain," etc.,respectively.

Progressively,theability to nam ethenum bersarose.Sincecounting up

to a considerably greatnum berwould haveneeded theinvention ofasm any

di�erentwords(and latersym bols),som esystem wasneeded to handleand

organize these num bers. Universally,this solution was found in the use of

a radix system ,i.e.using a certain am ountofnum bersand then repeating

them overand again.

3.4 T he origins ofm agnitude

Like form and num ber,theidea ofsize m ustalso beasold asM an him self,

and asan anim al-like rudim entary conception even older.A predatory ani-

m al,forexam ple,m usttakeinto accountthesizeofitsprey beforedeciding

whether it is sensible to attack it alone,or whether it should do so with

the help ofthe pack. However,this is an instance ofthe qualitative (big,

sm all)oreven com parative (bigger,sm aller)appreciation ofsize,butnotof

quantitative size(threearm slong),asisnecessary forthebirth ofgeom etric

m agnitudesand thequantitative conception ofgeom etry.

Thus,itwasonly afterM an wasarm ed with both hisprim itive,innate

powerto recognizeform and sym m etry in natureand a later,m oreinvolved

capacity to count,that he had the necessary tools to develop the notion

ofgeom etric m agnitude and m easurem ent. The com bination ofthese two

skillsproceeded through the realization thatone need notonly be satis�ed

with knowing whethera naturalobjectwasbiggerorsm allerthan another


(the enem y is taller than m yself),but that their size could be quanti�ed

and stated with a num ber. For the m agnitude of length| the paradigm

ofa geom etric m agnitude| the process leading to it can be described as

follows. Ifa naturalobject is geom etrically idealized and thought ofas a

geom etric object(the enem y asa straightline),one could then counthow

m any tim esacom paratively sm allerunitm easurecould bejuxtaposed along

the object in question. In addition to the naturalobject to be m easured,

also the successfulim plem entation of a unit m easure (a m easuring rod)

from a naturalobject(a treebranch)required thegeom etric idealization of

the latter (a tree branch as a straight edge). The quantitative size ofthe

geom etrically objecti�ed naturalobjectcould then begiven by m eansofthis

unit-m easure-related num ber.Now| and only now| isgeom etry developed

enough to live up to itsetym ologicalduty of\m easuring theearth."

3.5 T he rise ofabstract geom etry

Theriseofm oresophisticated geom etric ideasbeyond form and m agnitude

is recorded to have had its origin in the Neolithic Age in Babylonia and

Egypt. It appears fair to say that the oldest geom etric construction that

rates higher in abstraction than other com m on uses ofgeom etric thought

is so-called Pythagoras’theorem . Evidence ofknowledge ofPythagorean

triples2 and ofthe Pythagorean theorem can be found in the Babylonian

cuneiform textPlim pton 322 and in the Yale tabletYBC 7289,which date

from between 1900 and 1600 b.C.in theLateNeolithicorEarly BronzeAge.

In the preserved part ofPlim pton 322,shown in Figure 3.4,there are

�vecolum nsofnum bersin thesexagesim alsystem .Counting from theleft,

the �fth one shows the num bers 1 to 15,the fourth one repeats the word

\num ber"(ki),whilethesecond and third show thewidth yand thediagonal

z ofa rectangle.Ifwe calculate z2 � y2 = x2,x turnsoutto be(aftersom e

corrections) a whole num ber divisible by 2,3 or 5,and a m issing colum n

for x can be reconstructed with con�dence. This helps explain the �rst

(partly m issing)colum n showing eitherv = (y=x)2 orw = (z=x)2 = 1+ v2.

It has been inferred from this that the Babylonians m ust have calculated

Pythagorean triplesoftheform (1;v;w)satisfying the equation

w2 � v

2 = 1 (3.1)

and then obtained triples(x;y;z)m ultiplying by x.Finally,ithasalso been

noticed thatw + v isin allcasesa regularsexagesim alnum ber,which m eans

thattheBabyloniansm ay have written (w + v)(w � v)= 1 and then found

w � v by inverting the regularnum berw + v.3

The Yale tabletYBC 7289,shown in Figure 3.5,showsa square whose

side ism arked o� as30 unitslong,and one ofitsdiagonals attributed the

2A Pythagorean triple consists ofan ordered triple (x;y;z),where x,y,and z satisfy

the relation x2+ y

2= z

2.

3See (van derW aerden,1983,pp.2{5).

3.5 T he rise ofabstract geom etry 29

Figure 3.4: Plim pton 322 tablet at Colum bia University. This Babylonian

cuneiform tablet shows evidence ofknowledge ofPythagorean triples as early as

1900{1600b.C.Verticalstrokes\_",aloneorgrouped,representthenum bersfrom

1 to 9;horizontalstrokes\< ",alone orgrouped,represent10,20,30,etc. Repro-

duced from http://www.m ath.ubc.ca/� cass/courses/m 446-03/pl322/pl322.htm l

num bers 1,24,51,10 and 42,25,35 (in base 60). Ifwe �x the oating point

and assum e that the �rst num ber is 1;24,51,10,then its equivalent in the

decim alsystem is

1+ 24=60+ 51=602 + 10=603 = 1:414212963: (3.2)

This is a rem arkably good approxim ation ofp2 = 1:414213562:::. The

length ofthe diagonal42;25,35 (42.42638889) can then beobtained as

42;25;35 = 30� 1;24;51;10 (30�p2): (3.3)

Thistabletconstitutes archaeologicalproofbeyond the shadow ofa doubt

that the Babylonians knew Pythagoras’ theorem , not just Pythagorean

triples. In particular,they knew thatp2 isthe length ofthe diagonalofa

squareofside1,since they used thisknowledge to solve forthe diagonalof

a square ofside30.

There exist records showing knowledge ofPythagorean triples by the

Egyptians,too. The papyrus Berlin 6619 dating from around 1850 b.C.

contains a m athem aticalproblem in which the sides oftwo squares are to

befound knowing thatonesideisthree-quartersoftheotherand thattheir

totalareasadd up to 100 squared units;i.e.y = 3x=4 and x2+ y2 = 100.In

the solution,itisrecognized thatthe totalarea isalso thatofa square of

side10units,and thereforethetriple(8,6,10)isrecognized aswell.However,


Figure 3.5:YaleYBC 7289tabletattheYaleBabylonianCollection.ThisBabylo-

nian cuneiform tabletshowsincontestableevidenceofknowledgeofPythagoras’the-

orem asearly as1900{1600b.C.,overa thousand yearsbeforePythagoras.Repro-

duced from http://www.m ath.ubc.ca/people/faculty/cass/Euclid/ybc/ybc.htm l

(BillCasselm an,photographer).

no triangles are m entioned here. van derW aerden (1983,p.24)suggested

that the Egyptians m ay have learnt about Pythagorean triples from the

Babylonians. Also in support of this view, Boyer and M erzbach (1991)

wrote: \It often is said that the ancient Egyptians were fam iliar with the

Pythagorean theorem ,butthere is no hintofthis in the papyrithat have

com e down to us" (p.17).

All this being said, what purpose did the discovery and use of the

Pythagorean theorem and itsassociated triangles and triplesserve ancient

M an? Boyerand M erzbach (1991)suggested that,although pre-Hellenistic

m athem atics was m ostly ofa practicalsort and people \were preoccupied

with im m ediateproblem sofsurvival,even underthishandicap therewerein

Egyptand Babyloniaproblem sthathavetheearm arksofrecreationalm ath-

em atics" (p.42).Asim ov (1972,pp.5{6),along sim ilarlines,placed therise

of(theartsand of)pureknowledgein hum an curiosity or,asheeloquently

putit,\theagony ofboredom ,"stem m ingfrom thefactthatany su�ciently

advanced brain needsto bekeptengaged in som eactivity.O n thepractical

side,itis also possible thatthe Pythagorean geom etric construction could

haveoriginally represented som ething m orepracticalto theancients,either

in connection with theirreligiousbeliefs,orasa calculationaltoolfortheir

astronom icalproblem s,orboth.

To sum up,whattheanalysisofthischapterrevealsisthatnatureisnot

geom etricby essence,butthatwhatseem tobenature’suniversalgeom etric

features are,in fact,m an-m ade,a direct consequence ofM an’s innate ca-

pacity to recognizeform ,num ber,and m easure.Thisability liesattheroot

3.5 T he rise ofabstract geom etry 31

ofhum ans’pastand presentscienti�c achievem ents: these stem essentially

from therelentlessgeom etrization thathum anshave m ade and continue to

m ake ofthenaturalworld,aswe seein thenextchapter.

C hapter 4

G eom etry in physics

He deals the cards to �nd the answer,

The sacred geom etry ofchance,

The hidden law ofa probable outcom e;

The num berslead a dance.

G ordon Sum ner,Ten sum m oners’tales

W epresented a view ofgeom etry thatisfounded on thehum an concep-

tionsofgeom etricobjectsand geom etricm agnitudesattached toorbetween

them .W eexem pli�ed thisview ofgeom etry by m eansofspatially extended

geom etricobjects,and theattribution ofquantitativesizetothem by m eans

ofrod m easurem entsoflength and itsderivatives. In physics,however,we

dealwith various kinds of m agnitude and their m easurem ents, of which

length and itsm easurem entare butone case. W hataboutthese then? Is

geom etry entwined with allphysicalm easurem ents?

G iving a clear-cut answer to this question is di�cult,because with it

we enter a realm where the presence ofgeom etry is tied not just to the

way a m easurem entism ade butalso to how itsresultsare interpreted the-

oretically. To begin with,rod m easurem ents oflength are an intrinsically

geom etry activity because,as we said,a m easuring rod m ust necessarily

be a straightedge. Thisfactm akeslength and distance1 paradigm atically

geom etric m agnitudes. W hat about radar-based m easurem ents oflength?

Are these geom etric? W e m ay now say that,as an activity,they are not,

because there isnothing essentially geom etric aboutthe instrum ent(e.g.a

clock)needed to m ake the m easurem ent.Asa conception,however,length

continuesto bea geom etric m agnitude.How so?

In thiscase,geom etry com esin asan im m ediateconcom itantinterpreta-

tion.Thehum an m ind visualizeslength asspatialextension,and them ental

pictureofageom etricobject(straightline)whose property isthelength just

1D istance can be regarded as equivalent to length because it is in essence the sam e

conception but applied to geom etric objects di�erently. Length refers to the unidirec-

tionalsize ofa geom etric object(e.g.a line,an edge),while distance refersto the size of

the unidirectionalspace between the endpoints ofthe sam e object or between points as

independentgeom etric objects(e.g.pointsofspace).

33

m easured im m ediately arises.In a m oreabstractbutequally vivid way,the

sam eisdonefortheresultsofotherm easurem ents,now spatially unrelated,

which asactivitieshavingnothingtodowith geom etry.Forexam ple,theob-

servationsoftim evia clock readings,probability via sim plecounting,speed

via speedom eterreadings,spin via Stern-G erlach device readings,etc.,are

again geom etry-unrelated observations or procedures. The hum an m ind,

however,with itsstrong penchantforvisualization in term sofextended ob-

jectsin som eform ofspace,takesthephysicalquantitiesresultingfrom these

m easuring activitiesand givesthem a geom etric theoreticalbackground.In

thisway,observationsofvariousnaturesarelikened to theonewhich weex-

perienceasm ostfam iliar,nam ely,observationsoflength and itsderivatives.

Letussee som e exam ples.

Although tim e intervals are m easured with clocks and space intervals

(custom arily) with rods,physics likens tim e to space depicting it as a co-

ordinate param eter along which m aterial objects do not m ove but ow.

Thus,m easuring tim eintervalsbecom estheoretically equivalentto m easur-

ingstretchesofspace,and hencethephrase\alength oftim e"in thesenseof

a fourth spatialdim ension. Thisgeom etric interpretation oftim e wasonly

deepened by relativity theory,which,super�cially atleast(seeChapter10),

fusedtim eand spaceintoonegeom etricwhole.Anotherexam pleisfurnished

by probability.W hilethisconceptonly hastodowith countingpositiveand

negative outcom es,itisunderstood in physicaltheoriesin oneoftwo ways:

as Isham (1995,p.14{17) has explained,in classicalphysics,probability

islikened to a ratio ofvolum es(the volum e ofsuccessfuloutcom esdivided

by that oftotaloutcom es),so that each outcom e is represented by a unit

volum e;2 in quantum physics,on theotherhand,probability islikened toan

area resulting from the squared overlap oftwo arrow state vectors.A third

exam ple is provided by velocity, which is depicted as an extended arrow

tangentto thetrajectory curveofa m aterialparticle,and whosem agnitude

isthen interpreted asthe length ofthisarrow. A fourth exam ple is found

in quantum -m echanicalspin,which,likevelocity,ism entally pictured asan

arrow and itsm agnitudeinterpreted to beitslength.

A case farther detached from physics also provides an enlightening ex-

am ple. In the theory ofcoding,one is concerned with sending m essages

overa noisy inform ation channeland with being ableto recovertheoriginal

inputfrom a possibly distorted output. Evidently,no intrinsically geom et-

ric objects or m agnitudesare involved in this activity. However,these are

readily constructed to ease understanding ofthe situation. A word w is

thoughtofasan arrow vector ~w = (w0;w1;:::;wn�1 )and a distance func-

tion,called Ham m ing m etric,is de�ned for any pair ofwords w and x as

m axfijwi6= xi;0 � i� ng,the m axim um num berofdi�ering com ponents

2In fact,one practicalm ethod to calculate � through the m easurem entofprobability

consists in drawing a circle ofunit-length radius on paperand subsequently enclosing it

within fourwalls form ing a square whose sidesare two unitsoflength. By random ly re-

leasing a sm allobjectinto thebox,countinghow m any tim esitlandswithin theboundary

ofthecircle,dividing by thetotalam ountoftrials,and m ultiplying by four(squared units

oflength),an approxim ate value of� can be obtained.

34 G eom etry in physics

physicalm agnitude geom etricm agnitude

physicalthing geom etricobject

shape

geom etricextension

Figure 4.1: Two processes ofgeom etrization in physics. A physicalm agnitude

resulting from a m easurem entbecom es a geom etric m agnitude when shape (geo-

m etric object) is attached to it;or a physicalthing becom es a geom etric object

when geom etricextension (length & co.) isattached to it.

between ~w and ~x.Aftercertain assum ptionsaboutthe nature ofthe chan-

nel,a received word w iscorrected searching forthe prede�ned codeword c

thatisclosestto itin the above sense,i.e.di�ering from a codeword in the

leastnum berofelem ents.

Forsom e physicalm agnitudes,however,thisunderlying processofgeo-

m etric objecti�cation isnearly nonexistent. M assand charge,forexam ple,

appear as exceptions ofphysicalm agnitudes that our m inds do not seem

to �nd useful(at least not yet) to theoretically represent as the extension

ofany speci�c geom etric object. This notwithstanding,one m ay (and in

practice usually does)stillview these m agnitudessim ply asthe extension,

or m easure,of segm ents of an underlying m etric line, in analogy with a

tim eline orgraduated tape.From thisperspective,any physicalm agnitude

m ay besaid to bea geom etric m agnitude.

In physicalpractice,geom etrization can alsotakeplacein adi�erentway.

Instead ofstartingwith aphysicalm agnitudethatbecom esageom etricm ag-

nitude via the attachm entofan underlying shape,itissom etim esthe case

thattheoreticalportrayalstartswith an abstractobject| possibly inspired

directly in experience,in which casewehavea physicalthing| withoutany

shapetospeak of,butwhich isturned intoageom etricobjectby attributing

to itgeom etric extension in the form oflength and itskin (distance,area,

volum e,overlap). W e shallsee exam plesofthisshortly in connection with

quantum theory,and lateron in connection with pregeom etry. Both types

ofprocessareschem atized in Figure4.1.

As a result of these processes of geom etrization, we witness today a

language ofphysicswhere geom etric conceptsare ubiquitous,asthey have

been since the birth ofscience,when G alilei(1957)expressed thatwithout

geom etric concepts \it is hum anly im possible to understand a single word

of[the book ofnature]";thatwithoutthem \one wandersaboutin a dark

labyrinth" (p.238). Today as then,ifone tried to �nd a physicaltheory

thatdoesnotincludegeom etricconceptions,onewould bebound forcertain

failure.

For exam ple, on the qualitative side, we �nd geom etric objects such

as spacetim e points,straight and circular trajectories,arrow state vectors,

planetary surfaces,crystallattices,and so forth. O n the quantitative side,

we�nd geom etricm agnitudessuch asspacetim eintervals,lengths ofvectors

35

spa etime points &spa etime intervalsP

Q

dsPQ

R

SdsRS

T U

dsTU

M

m︸

︷︷︸

‖~F‖parti le traje tory& for e strength︸

︷︷

︸

l rystal latti e &latti e unit lengthFigure 4.2: Som egeom etricobjectsand associated geom etricm agnitudesused in

physics.

(e.g.strength ofa force),areas ofsurfaces(e.g.probability ofa quantum -

m echanicaloutcom e),volum esofspaces(e.g.probability ofa classicalout-

com e),and so on (Figure4.2).Although allthegeom etricconceptsofphys-

icalscience appear im m ediately fam iliar to us on account ofour bend of

m ind,theirlevelofdetachm entfrom the thingswe can perceive can range

greatly. Som e are developed from straightforward observations, whereas

others bear a high levelofabstraction. The farther detached the concept

isfrom whatwe can directly observe and experim entwith,the m ore trou-

blesom e the ascertainm ent ofits physicalm eaning becom es (cf.spacetim e

pointsand state vectors).This,in turn,can lead to philosophicalproblem s

abouttheirontology| in otherwords,to considerable confusion.

Beforem oving on to thestudy ofthegeom etricfoundationsofstandard

relativity and quantum theories,we note the existence ofa geom etric con-

ceptthatisparticularly noteworthy foritsrole in physics. W e referto the

inner product. In physicalcontexts,the innerproductplays a m uch m ore

essentialrole than the prim itive idea oflength or distance. This is partly

because ofa m athem aticalreason and partly because ofa physicalreason.

M athem atically,theinnerproductprovidesa foundation fortheem ergence

ofthree key quantitative geom etric notions;in bracketnotation,these are:

theoverlap ofjAiand jB i,hAjB i(with parallelism and orthogonality arising

asspecialcases);the length ofjAi,hAjAi1=2;and the distance between jAi

and jB i,hA � B jA � B i1=2 (Figure4.3).Physically,being itselfan invariant

scalar,the innerproductprovidesa directconnection with m easurem ents,

theresultsofwhich are invariantsaswell.In physics,the representation of

invariantscalarsasinnerproductsisalwaysasuresign oftheintroduction or

existence ofa geom etric outlook upon the objectofstudy.The advantages

and pitfallsofthisway ofproceeding willcom e to lightin laterchapters.

Nextweanalysethegeom etriccharacterofrelativity and quantum theo-

riesasconventionally understood.W eshallseethatboth theoriesaredeeply

geom etric at their m ost elem entallevel,and even that the m otivation be-

hind theirgeom etric character issim ilar. Thisview,however,seem sto be

atoddswith com m on belief,asAshtekar’sstatem entreveals:


α

|A〉

|B〉〈A|A〉1/2 cos(α)|B〉

︸

︷︷

︸

〈A|A〉1/2

︸︷︷︸

〈A|A〉1/2 cos(α)︸︷︷︸

〈A|B〉

︸

︷︷

︸

〈A − B|A − B〉1/2

Figure 4.3: Four quantitative geom etric notions: length of arrow vector jAi,

projection ofjAi on arrow vector jB i,distance between jAi and jB i,and inner

productbetween jAiand jB i.

[W ]e can happily m aintain a schizophrenic attitude and usethe

precise,geom etric picture ofreality o�ered by generalrelativity

while dealing with cosm ologicaland astrophysicalphenom ena,

and the quantum -m echanicalworld ofchance and intrinsic un-

certainties while dealing with atom ic and subatom ic particles.

(Ashtekar,2005,p.2,italicsadded)

In the next two sections,we attem pt to m aintain an expository attitude

towardsthefoundationsofrelativity and quantum theoriesasconventionally

understood,asourpresentpurposeisonlytoexposetheirgeom etricnatures.

In laterchapters,weshallreturn to thesetopicsand dealwith them from a

m ore criticalpointofview.

4.1 G eom etry in relativity theory

Historicalconsiderations aside,we can view relativity theory asessentially

built upon severalelem entalgeom etric concepts. First and forem ost,the

theory deals with physicalevents,which are idealized as points belonging

to a four-dim ensionalspacetim econtinuum em bracing thepast,thepresent,

and thefuture.A point-eventin thefour-dim ensionalspacetim econtinuum

ischaracterized analytically bym eansofasetoffournum bers(x1;x2;x3;x4)

called thepoint-eventcoordinates.

The theory also dealswith m assive particlesand photons,which italso

idealizesaspoints;aparticleis,m oreover,am ovingpointthatby itsm otion

in thespacetim econtinuum determ inesa sequenceofeventscalled a world-

line,x� = x�(u) (� = 1;2;3;4),where u is a one-dim ensionalparam eter.

Theidea oftheworldlineofa m assiveparticlealso includestheidealization

oftheobserverasa m assive pointthatm oveswhilem aking observations.

O n thesefoundations,relativity theory buildsitsconceptualconnection

with physicalm easurem ents.Thisisdoneby m eansofthecentralconceptof

thedi�erentialspacetim e intervalds.Thedi�erentialintervaldsA B givesa

4.1 G eom etry in relativity theory 37

quantitativem easureoftheseparation in spacetim ebetween two neighbour-

ingeventsA and B .Itistraditionally understood thatds,beingaspacetim e

interval,ism easured with rodsand clocks. Because tim e isthoughtasab-

sorbed asa fourth coordinatex4 on a parwith thethreespatialcoordinates

x1;x2;x3 (with thespeed oflightcasa transform ation factor),theinterval

isconventionally regarded geom etrically,asa generalized distance between

neighbouringevents.Thedi�erentialintervalisalso called theline elem ent;

thisisadenom ination inspired in thegeom etricpictureoftwogeneralevents

joined by a worldline com posed ofstraight piecesdx,whoselengthsds add

up to the �niteseparation �s.

In relativity theory,the assum ption is m ade that dsA B depends ana-

lytically on the coordinates x� of point-event A and hom ogeneously (of

�rst degree) on the displacem ent vector,or arrow,dx� form A to B ;i.e.

ds = f(x;dx),where f(x;dx)is som e function hom ogeneous in dx,and x

denotesthesetoffourcoordinates.Besidesthedem and ofhom ogeneity,the

form off(x;dx)isnotentirely freebecausethescalards,beingam easurable

quantity,m ustbe an invariant. Aswe m entioned previously,thissituation

suggeststheuseofan innerproduct.Thenew assum ption isnow m adethat

the relation between ds,on the one hand,and x and dx,on the other,is

an inde�nite quadratic form ,nam ely,ds =p�g�� (x)dx

�dx�,3 i.e.the line

elem entisan invariantinnerproductbetween thedisplacem entvectorsand

som e function g�� (x) ofx. This quadratic form is in fact the foundation

on which the restofrelativity theory isbuilt. (Butto whatkind oftheory

would the sim plerlinearchoice ds = g�(x)dx� lead us? Thisisa question

weshallponderlateron.)

A study ofRiem annian space,in conjunction with itsgeneralization to

spacetim e,allows one to determ ine that the function g�� (x) is a second-

order covariant tensor with a clear geom etric m eaning: it is the m etric of

the spacetim e continuum , and as such the form al(m athem atical) source

ofthe spacetim e-continuum quantitative geom etry. The m etric is apt to

inform ushow m uch two vectors(in particular,two contravariantdisplace-

m ent vectors) at the sam e event A(x) overlap;on this basis,we can �nd

the generalized length,or norm ,ofa vector at A(x),the generalized dis-

tance between two vectors atA(x),and,via integration,the separation or

generalized distance between any two events.

TheNewtonian law ofinertia isgeneralized in relativity theory via Ein-

stein’s geodesic hypothesis. According to it,a free (m assive or m assless)

particlefollowsageodesic in spacetim e,i.e.astraightline in spacetim e,such

thattheseparation �sbetween two eventsjoined by thislineism axim ized.

O n the otherhand,the intrinsic shape ofa spacetim e geodesic followed by

a free particle isgiven by an absolute property ofspacetim e,itscurvature.

Curvature is set in a one-to-one correspondence with gravitation as a

phenom enon asfollows.Absolute spacetim e curvatureischaracterized,not

sim ply by som e tensor(e.g.g�� (x)),butby som e suitable tensor equation

3Here � = � 1 ensures that ds � 0 always,since the quadratic form is not positive

de�nite.


(e.g.R ��

(x)= 0) to be speci�ed later on. W hen this equation vanishes

identically (R ��

(x)� 0),i.e.when thetensorisnulleverywhere,spacetim e

is at and there are no gravitationale�ects;when this equation does not

vanish identically (R ��

(x)6� 0),i.e.when thetensorisnotnulleverywhere,

spacetim e iscurved and there aregravitationale�ects.

In view ofthe pervasiveness and essentialness ofgeom etric notions in

relativity theory,itseem sunderstandablethatthistheory hascom etobere-

garded asepitom ically geom etric,and \m oregeom etric" than itsNewtonian

predecessor. Butisthisreally so? Hasgeneralrelativity truly geom etrized

gravitation? Isphysicsm oregeom etric now aftergeneralrelativity? Notat

all.W ith generalrelativity,gravitation only acquiresa new geom etricchar-

acter.New because the Newtonian description ofgravitation asa vectorial

force wasalso a geom etric concept,nam ely,a directed arrow in Euclidean

space with a notion ofsize (strength) attached,acting between point-like

m assive bodiesa certain Euclidean distance apart.

In the currentclim ate ofm isunderstanding regarding the essentiallack

ofnovelty introduced by generalrelativity as a geom etric theory,it is not

withouta m easure ofsurprisethatwe �nd thata correctassay ofthe uno-

riginalgeom etric statusofrelativity wasm ade by itsown creatoroverhalf

a century ago. In a letter to Lincoln Barnett dated from 1948,Einstein

considered asfollows:

Ido notagreewith theidea thatthegeneraltheory ofrelativity

isgeom etrizing Physicsorthe gravitational�eld. The concepts

ofPhysicshave always been geom etricalconceptsand Icannot

see why the gik �eld should be called m ore geom etrical than

f.[or]i.[nstance]theelectrom agnetic�eld orthedistancebetween

bodies in Newtonian M echanics. The notion com es probably

from thefactthatthem athem aticalorigin ofthegik �eld isthe

G auss-Riem ann theory ofthe m etricalcontinuum which we are

wontto look atasa partofgeom etry.Iam convinced,however,

thatthedistinction between geom etricaland otherkindsof�elds

isnotlogically founded.4

Having disclosed the geom etric essence| butnotgeom etric novelty| of

relativity theory,buthaving leftitsphysicalfoundationsasyetunanalysed,

we now turn ourgaze to quantum theory and its geom etric essence.

4.2 G eom etry in quantum theory

Q uantum theory studiesthe m easurable propertiesofm icroscopic physical

system s,such as m icroscopic m aterialparticles. Unlike relativity theory,

quantum theory does not �nd it essentialto geom etrically conceptualize

particles aspoints,since the classicaltrajectory,orworldline,ofa particle

playsno role in it. G eom etric conceptshave in quantum theory a di�erent

4Q uoted from (Earm an,G lym our,& Stachel,1977,p.ix).

4.2 G eom etry in quantum theory 39

physicalsource| yetthesam eform alexpression m otivated by needssim ilar

to those ofrelativity theory.

In order to ascertain the properties ofa m icroscopic system ,one m ust

setup m easurem ents,which soon revealthatthese system sdisplay a non-

classicalfeatureofsuperposition.W estudy asan exam plethem easurem ent

ofthepolarization ofphotons.FollowingDirac(1958,pp.4{7),setup aplate

which only transm itslightpolarized perpendicularly to itsopticalaxis,and

directatita low-intensity beam oflight,polarized atan angle � with the

above axis, such that photons arrive one at a tim e. O ne �nds that the

intensity oftransm itted light, and therefore oftransm itted photons,is a

fraction sin2(�) ofthe totalintensity,suggesting that polarization cannot

be a classicalproperty in the following sense: it cannot correspond to a

tiny m aterialstick attached to photonsthatisblocked by the plate unless

aligned perpendicularly to its opticalaxis,for then no photons could pass

through.Indeed,againstclassicalexpectations,som eintensity isperceived,

signalling the unim peded passage ofa fraction ofthe originalphotons: on

arrivalattheplate,each photon reorganizesitselfpolarization-wiseaseither

com patible orincom patible with it.5 M oreover,because the interaction at

thepolarization plateacts,to allintentsand purposes,likea black box,one

cannotpredictwhatthefateofeach individualphoton willbewith certainty,

butonly with a certain probability.Thisprobability isan invariantscalar.

Thetask ofquantum theory isto providea link between thisprobabilis-

tic experim entalresult and som e theoreticalfeature attributable to each

m icroscopicphoton.O n thebasisofthesaid non-classicalbehaviour,quan-

tum theory pictureseach photon to bein a polarization statejP i,such that

itisa com plex-num ber(see below)linearsuperposition

jP i= ctjPti+ cajPai (4.1)

oftwo polarization states,jPtiand jPai,associated with transm ission and

absorption with respectto a given plate. The superposition feature ofjP i

leadsdirectly to thesuggestion thatquantum -m echanicalstatesarevectors,

since it holds physically that the linear com bination oftwo states is also

a state, but this is not to say (yet) that state vector jP i is a geom etric

object.Thisinterpretation,however,providesa ready-m ade foundation for

theintroduction ofgeom etry into quantum theory,which takesplacein two

steps.

Forthe construction ofjP iasa linearsuperposition to m ake sense and

be useful,the com plex coe�cients c t and ca should indicate (qualitatively

atleast)to whatdegree,on arrivalata given plate,theoriginalstatejP iis

com patible with the one associated with transm ission jPtiand to whatde-

greewith theoneassociated with absorption jPai,respectively.A geom etric

5A display of the non-classical nature of polarization can be produced in a purely

phenom enological| and thereforem uch m orestriking| way.Placetwo plateswith optical

axes orthogonalto each other,so that no light willbe perceived through both plates.

Insertthen a third plate obliquely between the previoustwo and observe now som e light

passing through thethreeplates.The addition ofan extra �lter allowsm ore lightto pass

through| a phenom enon thatgoesagainstclassicalintuition.


〈Pt|P 〉|Pt〉

〈Pa|P 〉|Pa〉

|P 〉

θa

θt

|〈Pt|P 〉|2

= cos2(θt)

|〈Pa|P 〉|2

= cos2(θa)

〈P |P 〉 = 1

Figure 4.4: Riseofquantum -m echanicalprobabilitiesfrom Pythagoras’theorem .

pictureisim m ediately created in which statevectorsareim agined asm etric

vectors or extended arrows,and their m utualcom patibilities expressed in

term s oftheir overlaps or,in other words,the inner product. The linear

superposition ofEq.(4.1)isthusrecastgeom etrically as

jP i= hPtjP ijPti+ hPajP ijPai; (4.2)

with jPtiand jPaiim agined asm utually orthonorm aland with unitarrow

vector jP iim agined as being projected or collapsed onto them . The inner

producte�ectively turnsin principle shapeless state vectors into extended

arrows,because the idea ofoverlap inducesa need forgeom etric objectsin

orderforusto projectonto oneanother(butwhy should weoverlap arrows

and not surfaces or spatial�gures?). The inner product being a com plex

num ber,however,wecannotyettalk ofgeom etric m agnitudesin a physical

sense.

The second step in the geom etrization ofquantum theory com es from

the way it is linked to the experim entally obtained probability. O n the

one hand,probability isa realscalarthatshould arise from som e invariant

function f(jP i;jPti;jPai)ofstate vectors;on the otherhand,state vectors

so faronly com bineto producecom plex num bers6 hPijP i(i= t;a).Further

6Asjusti�cation forthe signi�cantbutotherwise arbitrary requirem entthatthe coef-

�cientshP ijP ibecom plex num bers,D irac(1958,p.18)o�ered thefollowing explanation.

W ithin the contextofthe polarization experim entabove,he argued that,given thatthe

length ofstate vectors is irrelevant (norm alization condition),in a linear superposition

c1j i+ c2j�i oftwo states only the ratio c3 = c2=c1 carries a m eaning,and so c3 m ust

be com plex (two realparam eters),i.e.c1 and c2 m ust be com plex,since two indepen-

dent realnum bers are necessary to specify a generalstate of plane polarization. But

why should two com plex num bers(fourrealcoe�cients)be needed in a superposition of

three linearly independentstate vectors,etc.? Anotherproposalforthe need ofcom plex

num bers was given by Penrose (1998, pp.62{63). He stressed the feature of com plex

4.2 G eom etry in quantum theory 41

geom etric analysis succeeds in �nding com m on ground. O n the basis of

Pythagoras’theorem ,onecan seethatthethreestatevectorsjP i;jPti;and

jPaiareconnected by a real-num berrelation,

X

i

jhPijP ij2 = kjP ik2; (4.3)

solongaswethinkthem asarrows(butnotsurfacesorspatial�gures).Now,

because the length ofjP i is a geom etric invariant and equalto unity,this

relation becom essuitablefora probabilisticinterpretation.Theprobability

ofeach outcom ejPiiisgiven bythesquared length,orarea,oftheprojection

ofjP ionto jPii,and the probability ofobtaining any outcom e addsup to

unity (i.e.a unitarea),asshown in Figure 4.4.

It is in this m anner that probability, an invariant7 scalar that only

has to do with counting positive and negative outcom es, is geom etrized

on the basis of the inner product and Pythagoras’ theorem (in general,

in n-dim ensions) in order to provide the hum an m ind with geom etric un-

derstanding of,in principle,geom etry-unrelated observations. Pythagoras’

theorem com esthusfrom itshum blerNeolithic usesto serve hum an under-

standing in yetanotherfacetofM an’senquiriesinto nature| to instate,in

thiscase,Sum ner’s\sacred geom etry ofchance."

conjugation between a com plex num ber� and itsconjugate ��in connection with quan-

tum inform ation and itsrelation to classicalinform ation.Heargued thattheexistence of

quantum inform ation,which doesnotrespectcausality,isonly possible via the existence

ofcom plex am plitudesand theirconjugates,which can be interpreted as going forwards

and backwards in tim e: hPijP i from the present state jP i to the future state jPii and

hPijP i� = hP jPiifrom thefuturestatejPiito thepresentstatejP i.O netheotherhand,

classicalinform ation| in otherwords,probability| resultsfrom m ultiplying the com plex

am plitude � by itscom plex conjugate ��,obtaining thusa realnum ber. In Chapter12,

we give a deeper justi�cation ofwhy com plex num bers are needed to describe quantum

phenom ena.7By quantum -m echanicalprobability being an invariant,wem ean thatitisan absolute

property ofa large ensem ble ofidentically prepared m icroscopic system s(relative to the

testsm easuring apparatusessubm itthem to).Thedenom ination \invariant" isused here

in order to highlight the independence of m easurable quantum -m echanicalprobability

from the unobservable state vectors from which ittraditionally arises. The nam e is also

used todraw attention tothesim ilarconceptualway in which both quantum and relativity

theoriesintroduce the invariantresultsofm easurem entsvia the innerproductofentities

deem ed m ore elem ental from a logico-axiom atic| but not physical| perspective. The

denom ination,however,isnotintended to hintata relativistic connection. Form ore on

the relativistic interpretation ofa single quantum -m echanicalm easurem ent,see (Eakins

& Jaroszkiewicz,2004,pp.4,9).

C hapter 5

N ew geom etries

Itwould be fascinating if,ultim ately,we build allthe funda-

m entalthingsin the world outofgeom etry.

Interviewer to AbdusSalam ,Superstrings: A theory ofev-

erything?

In the light of the previous chapter, it is not surprising to �nd that

physics is not only built out ofgeom etry,but also that it seeks progress

through the invention ofnew geom etries. Ptolem y’s geocentric system of

sphereswas im proved upon by Copernicus’heliocentric system ofspheres,

in turn replaced by K epler’s ellipses. Newton’s Euclidean space and tim e

structure were im proved upon by M inkowski’s at spacetim e structure,in

turn replaced by Riem ann and Einstein’scurved one.In them aiden �eld of

quantum m echanics,headway wasm adeastheearly collection ofscattered

resultswassetin a progressively m oreintuitive geom etric fram ework:from

Heisenberg-Born-Jordan m atrix m echanics,passing through Schr�odinger’s

wave m echanics,and ending with Dirac’sbracketalgebra,whose geom etric

featureswe exam ined in the previouschapter.

In early attem pts at uni�cation of gravitation and electrom agnetism ,

W eyl, Eddington,K aluza,K lein,and Einstein and collaborators (M ayer,

Bergm ann,Bargm ann,Strauss)attem pted to im prove on generalrelativity

through theintroduction ofnew geom etries:generalized Riem annian geom -

etry,a�ne geom etry,�ve-dim ensionalspacetim e,a non-sym m etric m etric

tensor,teleparallelism ,�ve-vectorson four-dim ensionalspacetim e,bivector

�elds,a com plex-valued m etrictensor,etc.1 Thisfruitlessquestfornew ge-

om etriesaim ingtooverthrow thereigningorderin spacetim ephysicscontin-

uestoday unim pared in quantum -gravity research program m es,asfollows.

According to unanim ous agreem ent by decree in the physics com m u-

nity,2 there are three roadsto quantum gravity. The �rstroad isthe road

ofstring theory, which seeks progress through the geom etric concepts of

strings,m em branes,deca- and endecadim ensionalm etric spacetim es,etc.

1See (G oenner,2004;K ostro,2000,pp.98{101,115{148).

2See e.g.(Chalm ers,2003;Sm olin,2001,pp.10{11).

43

The second road isthe road ofloop quantum gravity,which seeksprogress

through thegeom etricconceptsofloopsofspace,spin networks,spin foam s,

etc. And the third road to quantum gravity is the politely acknowledged

road ofeverythingelse| a wastedum p,asitwere| thatdoesnot�tthe�rst

two characterizationsand that,beyond politeness,onegetstheim pression,3

can besafely ignored.Pursuing thislessglittering third road,we can m en-

tion pregeom etricians,who,inspired in W heeler,alsocontinuetofurtherthe

sam egeom etrictrend (only now unwittingly)in thesearch forlesscom m on

keysto a quantum theory ofspace and tim e.

Itisalso custom ary in any ofthese�eldsforyetnewer geom etriesto be

readily invoked to explain any di�culties that their tentative new geom e-

tries,attem pting to explain well-established older ones,m ight have given

riseto.W oit(2002)exem pli�esthisstate ofa�airswhen hewrites,\String

theoristsask m athem aticiansto believe in the existence ofsom e wonderful

new sort ofgeom etry that willeventually provide an explanation for M -

theory" (p.110).Thatisto say,in thefaceofdi�cultieswith thetentative

new geom etriesof�vedistinctstringtheories,researchersnaturally resortto

ayetnewer,m oreconvoluted geom etry on which tobaseso-called M -theory,

and with the help ofwhich problem atic issuesm ightstillbe resolved. The

essence ofthe present situation is captured rather accurately by Brassard

(1998),when (forhisown reasons)herem arksthat\Them ostdi�cultstep

in thecreation ofa scienceis...the�ndingofan appropriategeom etry.Itis

thedi�culty thatisfacing thephysicistswho aretrying to m ergequantum

physicswith generalrelativity..." (x 3.4).

M ay thisprocedure of�tting nature into evernewergeom etric m oulds,

essentially physically unenlightening since the establishm entofgeneralrel-

ativity,4 then reach a naturalend? Thisquestion m ustbe answered in the

negative so long asthehum an m ind continuesto beexclusively gripped,as

itis,by geom etric m odesofthought,rem aining obliviousto otherm anners

oftheoreticalre ection.Butwhatwould bethesym ptom sofa condition in

which thephysicalproblem sthatconfrontus| whetherpertaining to quan-

tum theory, spacetim e theories, or both| were im m une to our relentless

geom etric attacks? Trapped by our own conceptualtools ofthought,we

would witness(i)inexhaustible philosophicalargum entoverold topicscast

in thesam erecurringgeom etriclanguage,and (ii)increasingly com plicated,

m athem atically orphilosophically inspired fram eworksbuiltoutofgeom et-

rically \elegant," \self-consistent" concepts yet without physicalreferents.

Such sym ptom sare,in fact,already visible;the �rstcategory ispopulated

by self-perpetuating philosophicaldiscussions on Einstein’s hole argum ent

and spacetim e ontology (substantivalism vs.relationalism ,the m eaning of

spacetim epoints,etc.),aswellasinterpretationaldebateon them eaning of

quantum -theoreticalgeom etric concepts (the state vector and its collapse,

3See e.g.(Chalm ers,2003)and (Rovelli,2003,p.37).

4The geom etric form ulation ofquantum m echanics,which reached its fullform with

D irac’sbracketform ulation,isnothere considered precisely speaking enlightening,since

the presentcontroversies ofquantum theory stem from a lack ofphysicalm eaning in its

geom etric concepts,aswe shallsee lateron.

44 N ew geom etries

probability,etc.); and the second category is inhabited by m uch ofwhat

presently goesby the nam eofquantum gravity.

Isitreasonableto continue along a tangled geom etric path when itgets

m oretwisted thefartheritistrodden? Areourdeveloped geom etricinstincts

leading usin thesam eom inousdirection thatthegeom etricinstinctsofthe

Aristotelian tradition led thePtolem aicgeocentricsystem ofperfectspheres?

Forovera m illennium since Ptolem y,an arm y ofspheresinterlocking with

ever-m ounting com plexity wasa pricegladly paid to preservean astronom y

based on the philosophicalopinion thatspheresare perfect(and the earth

all-im portant) yet \fraught with cosm ological absurdities" (Saliba, 2002,

p.364). Confronted by so m uch absurd com plexity,Ptolem y excused itas

natural,considering thatnothing shortofunderstanding the m ind ofG od

wasin question here:

Now let no-one,considering the com plicated nature ofour de-

vices,judgesuch hypothesesto beover-elaborated.Foritisnot

appropriate to com pare hum an [constructions]with divine,nor

to form one’sbeliefsaboutsuch greatthingson thebasisofvery

dissim ilaranalogies.5

Thelaym an’spointofview,when hedaresto voice itunintim idated by the

pro�ciency ofthe expert,is usually closer to the truth. In the thirteenth

century,K ing Alfonso X ofSpain wasm uch lessim pressed by Ptolem aicas-

tronom y than Ptolem y him self.\IftheLord Alm ighty had consulted m ebe-

foreem barking upon Creation," rem arked theking (perhapsapocryphally),

\Ishould have recom m ended som ething sim pler."

The story ofAristotelian-Ptolem aic astronom y strikesa resonantchord

with quantum gravity. Saving the distances, it warns against blindness

towardsourgeom etricbiases,whetherinnateorinherited from aphilosophy

built upon opinion,as wellas against the rationalization ofcom plexity in

thenam eofthefundam entality ofthetask athand and theinexorability of

geom etry in dealing with it.

How farisquantum gravity ready to go,and how m uch to sacri�ce,to

save itsgeom etric prejudices?

5.1 T hree roads to m ore geom etry

String theory doesnot�nd faultwith itsthoroughly geom etric character|

and, in the context of a paradigm atically geom etric science, why should

it?| m ostelem entally given by thevery conceptofa string and them ultidi-

m ensionalm etric spacetim esto which they belong.W hatism ore,itsm ore

vehem entproponentspredicate the correctness ofthe theory on a sense of

\elegance" and \beauty" based on featuresofitsgeom etric portrayalthat,

by theirvery geom etric nature,tend to bepsychologically appealing.In his

revealingarticle,G opakum ar(2001)pondersthe\alm ostm ystical"(p.1568)

5Q uoted from (Saliba,2002,p.366).

5.1 T hree roads to m ore geom etry 45

connection between geom etry and thelawsofnatureand,em bracing it,cor-

rectly describesprogressin string theory astheevolution and growth ofits

geom etric edi�ce,together with the geom etric insights to be found in its

newly builtstoreys. He also foretells| and we are likely to agree with him

in view ofcurrenttrends| m ore\innovationsin geom etry" (p.1572)in the

progressivereform ulation ofstringtheory.Thedesirability offurtheringthis

geom etrictrend in string theory (and in physicsasa whole)isperhapsbest

epitom ized in the alm ost m ysticalwordsofan anonym ousreporterduring

an interview to AbdusSalam :

Clearly stringtheory isdeeply rooted in geom etry.Isupposeone

could say thatsciencestarted with geom etry (ifonegoesback to

ancientG reece).Itwould be fascinating if,ultim ately,we build

allthe fundam entalthingsin the world outofgeom etry.6

These strong views notwithstanding,and as in other quantum -gravity

endeavours(seebelow),even in expositionsofan adm ittedly geom etricthe-

orysuch asthisone,onepuzzlingly witnessesisolated,casualallusionstothe

possible lim itations ofgeom etric theorizing. G opakum ar asks,\Are there

geom etricalstructuresthan can replace ourconventionalnotions ofspace-

tim e? O r perhaps geom etry em erges only as an approxim ative notion at

distanceslarge com pared to thePlanck length?" (p.1568).Lacking appro-

priate context or further explanation,one can only attribute a vague and

am biguousm eaningto\geom etry" in thistypeofrem arks.Ifby \geom etry"

ism eantany form ofgeom etric concept,then quantum gravity answersthe

form erquestion with a resounding yes,and thelatterwith a resounding no;

butifby \geom etry" ism eantsim ply a continuousm anifold,then quantum

gravity answerstheform erquestion m ostprobably with a no,and thelatter

m ostprobably with a yes.

In loop quantum gravity,on the other hand,the standpoint regarding

geom etry ism uch m oreconfusing.Itsproponentsnotrarely butoften claim

that\the very notion ofspace-tim e geom etry is m ostlikely notde�ned in

thedeep quantum regim e" (Perez,2006,p.17),orthata \quantum gravity

theory would...be free ofa background space-tim e geom etry" (Ashtekar,

2005,p.11),butwe then repeatedly witness expositions ofloop quantum

gravity as a straightforward quantum theory ofgeom etry.7 Despite their

lack ofclarity,whatthese phrasesintend to m ean issim ply that,afterthe

lessonsofgeneralrelativity,oneshould notassum ean absolutely �xed back-

ground spacetim e in quantum gravity,butnotthatthere isno background

geom etry in theloop-quantum -gravity quantization ofgeneralrelativity;else

witnessthethoroughly geom etricform ulation ofloop quantum gravity in ev-

ery exposition ofit: from elem entalarrow-vector triads8 to the celebrated

\quantum geom etry" ofspin networks in the form oftheir quantized area

6In (D avies& Brown,1988,p.178).

7Seee.g.(Ashtekar,2005,p.13),(Sm olin,2001,pp.75{191)and (Rovelli,2004,p.262).8Because ofitsinviability forquantization,talk ofthe m etric �eld isat�rstbanished

and instead supplanted by talk ofa (m athem atically disem bodied) \gravitational�eld"

asan elem entalphysicalconcept.Butbecause loop quantum gravity desperately needsa

46 N ew geom etries

and volum e.Like string theory,then,loop quantum gravity also progresses

via the invention ofnew geom etries.

At this point,it is worthwhile noting that som e loop-quantum -gravity

scholarsdonotonlydenytheevidentgeom etricfoundationsoftheir�eld but,

m ore seriously,also those ofgeneralrelativity (and by im plication,should

oneunderstand,ofloop quantum gravity?).Sm olin (1997)explicitly claim s

that \while the analogy between spacetim e,as Einstein’s theory describes

it,and geom etry is both beautifuland useful,it is only an analogy" with

\little physicalcontent" (p.333)and that

Them etaphorin which spaceand tim etogetherhaveageom etry,

called the spacetim e geom etry, is not actually very helpfulin

understanding the physicalm eaning ofgeneralrelativity. The

m etaphorisbased on a m athem aticalcoincidencethatishelpful

only to those who know enough m athem atics to m ake use of

it...(Sm olin,2001,p.59)

Sm olin’swordsare inspired in the general-relativistic feature ofdi�eom or-

phism invariance,according to which the spacetim e m etric g�� (x) has no

absolute m eaning,as its values change from one coordinate system to an-

other. G eom etrically speaking,g�� (x) has no absolute m eaning because

spacetim epointsP (cf.x)arenotphysically realand failto individuatethe

m etric �eld. And this iscorrect,butto inferthat generalrelativity is not

essentially a geom etrictheory becausethem etric�eld hasnoabsolutephys-

icalm eaning isutterly m istaken| a confusion caused by theconnotation of

words.W hile tensorsare relative to a coordinate system ,tensorequations,

however, are not. The reason g�� (x) plays no physicalrole in relativity

theory is that it does not as such satisfy any tensor equation. Riem ann’s

curvature tensor R �� (x),on the other hand,does satisfy a tensor equa-

tion,Einstein’s �eld equation,and thus has absolute (i.e.independent of

coordinate system )m eaning asa geom etric property ofspacetim e,nam ely,

itscurvature.

W eshowed thegeom etricfoundationsofrelativity in thepreviouschap-

ter,and weshallstillsay m oreaboutthisissuelateron,butforthem om ent

letusstressthattheroleofgeom etry in generalrelativity isneitheran anal-

ogy,nora m etaphor,nora m athem aticalcoincidence,nordevoid ofphysical

content. Even atitsm ore basic level,generalrelativity isessentially about

a geom etric network ofinvariant intervals dsA B (norm ofthe di�erential

displacem entarrows dxA B )between point-eventsA(x)and B (x+ dx),and,

as forits physicalcontent,we could notm ake any theoreticalsense ofthe

physicalm easurem entsds withoutthisunderlying geom etric structure.

The helpfulness ofthe geom etric foundations ofthe theory for under-

standing itsphysicalm eaning istherefore enorm ous;in thisregard,we can

m athem atical�eld in any case,a new m athem aticalconcept| rechristened \gravitational

�eld"| appears at the starting point ofthe theory,nam ely,triads ofarrow vectors EaI

(a;I = 1;2;3)whosetraceTr(EaIE

bI)= det(g)g

ab(an innerproducton theinternalindices

I)m athem atically recreatesthe previously banished m etric �eld.

5.1 T hree roads to m ore geom etry 47

perhaps do no better than heed the words ofits creator,who,during an

addressto thePrussian Academ y ofSciencesin 1921,openly acknowledged:

Iattach specialim portancetotheview ofgeom etry which Ihave

justsetforth,because withoutitIshould have been unable to

form ulate the theory ofrelativity. W ithout it the following re-

ection would have been im possible:| In a system ofreference

rotating relatively to an inert system , the laws of disposition

ofrigid bodies do not correspond to the rules ofEuclidean ge-

om etry on account ofthe Lorentz contraction; thus ifwe ad-

m it non-inert system s we m ust abandon Euclidean geom etry.

(Einstein,1983b,p.33)

And duringhisK yoto lecturea yearlaterin 1922,Einstein (1982)expressed

wordstothesam ee�ect:\Ifound thatthatthefoundationsofgeom etry had

deep physicalm eaning in thisproblem " (p.47).M oreover,neitherisittrue

thatthegeom etricform ulation ofgeneralrelativity is\helpfulonly to those

who know enough m athem atics" since,as has been m ade plain tim e and

again,Einstein was led to this geom etric form ulation despite not knowing

enough m athem aticsto expresshisphysicalinsightsin m athem aticalterm s.

Finally,not even Sm olin’s (1997) claim that he has him self\described

generalrelativity while m aking no m ention ofgeom etry" (p.333) is true,

ashisown description ofthistheory isin term sof\the geom etry ofspace

and tim e" (p.234)and \the m etric ofspacetim e" (pp.235,239). Sm olin’s

rem arks leave one feeling uneasy with a bitter aftertaste: ifcorrect,why

introducesuch revolutionaryideasonlyasashortendnoteorin passingin his

books? Can onereally doaway with 90yearsofphysicalunderstandingsince

Einstein,and with nearly 200 years ofm athem aticaldevelopm entrelevant

to relativity theory sinceG aussand Riem ann,in a footnote? Thiscan only

bethe bene�tofspeaking from an authoritative position.

Finally and m ostunexpectedly,also pregeom etricians attem ptto m ake

progressvia theinvention ofnew geom etries.By design,pregeom etry isthe

attem pttoexplain theorigin ofgeom etry in physicaltheoriesfrom aconcep-

tually di�erentbasis,likeam aterialcontinuum �ndsexplanation in term sof

atom s.Theuseofgeom etricconceptsisthusproscribed ifpregeom etry isto

liveup to itsintended goal.Unwittingly,however,becausephysicsknowsof

no otherm ode ofoperation than geom etry,allideasforpregeom etries(ex-

cepting a few ofW heeler’s)turned outto befraughtwith geom etricobjects

and m agnitudes: graphs,networks,lattices,and causalsets with m etrics

determ iningthedistances oftheedgesbetween theirpoint-likeconstituents.

In theerroneousbeliefthatm ereengagem entin pregeom etry would turn its

productspregeom etric,the geom etric nature ofthese building blocks went

unnoticed, and the task of pregeom etry was tacitly reinterpreted as the

search forthe origin ofthe geom etric spacetim e continuum from a discrete

geom etric foundation. The case ofpregeom etry is so exem plifying that it

deservesa chapterofitsown,to which we now turn.

W e close thischapterwith an illustration (Figure 5.1)ofthe geom etric

road physicshasfollowed so far,and suggestan unexplored alternative.

48

New

geometries M E T A G E O M E T R Y

G E O M E T R YPto

lem

aicm

echa

nics

Galile

anm

echa

nics

New

toni

anm

echa

nics

The

rmod

ynam

ics

Qua

ntum

theo

ry

Copernican

mechanics

Keplerian

mechanics

Electrom

agnetism

Relativity

theory

Quantum

gravity

String

theo

ry

Loop quantum gravity

Pregeometry

Figure 5.1: The language ofphysicsisthe languageofgeom etry,and progressin physicshasalwaysbeen sought,whethersuccessfully ornot,

via the invention ofnew geom etries.Thisisstillthe casenowadayswith the latestendeavoursin the �eld ofquantum gravity.W e subm itthere

existsa di�erent,unexplored road,the treading ofwhich can help usunderstand the physicalworld in waysdeeperthan any am ountoffurther

branching ofthe geom etricroad iscapableof.Thisisthe road ofm etageom etry.

C hapter 6

Pregeom etry

Curious things, habits.People them selves never knew they

had them .

Agatha Christie,W itnessfor the prosecution

Afterhaving dweltin them eaning ofgeom etry and witnessed itsubiqui-

tousrolein physicsand attem ptsatnew physics,weanalysethosetheoret-

icalpicturesthathave explicitly proposed to forgo geom etric ideasin their

depiction ofspace and tim e,and which go by the nam e ofpregeom etry| a

very aptnam eindeed,fortheword isevocativeofpositioning oneselfbefore

geom etry.In pregeom etry,thegeom etricspacetim econtinuum isconsidered

a awed physicalpicture,and itisattem pted to replaceitwithoutassum ing

any othertype ofm anifold forspacetim e (in particular,withoutquantizing

generalrelativity asa presupposed starting point);instead,spacetim e isto

bedepicted vianon-geom etricconstructsontologically priortothegeom etric

onesofitspresentdescription.

Indeed,if,likeus,oneisinterested in going beyond geom etry in physics,

whatbetterplace than in pregeom etry to search forthe realization ofthis

goal? Has not pregeom etry, after all, already argued in earnest for the

overthrow ofgeom etric explanations,and even ful�lled the very goals that

we have setourselvesin thiswork? Have wesuddenly realized,on reaching

Chapter6,thatthe work we have em barked on isatbestsuper uousand

atworstobsolete?

Hardly. A criticalstudy soon reveals thatthe above prelim inary char-

acterization ofwhatiscalled pregeom etry1 isforthem ostpartincongruous

with itsactualarchetypes.Thatisto say,despiteitsclaim sto thecontrary,

pregeom etry surreptitiously and unavoidably fellprey to the very m ode of

description itendeavoured to evade,asevidenced by itsall-pervading geo-

m etric pictures. Indeed,the distinctive feature ofpregeom etric approaches

isthat,asintended,they drop the assum ption ofspacetim e being correctly

described by a m etric continuum ,butonly to replace itwith som e type or

other ofgeom etric-theoreticalconception.

1See e.g.(M onk,1997,pp.11{18).

50 P regeom etry

Nothing supportsthisstatem entm oredram atically and vividly than the

wordsofG ibbs’(1996),who explainsthe m ode ofworking ofpregeom etry

thus:\O nceithasbeen decided which propertiesofspace-tim earetobedis-

carded and which areto rem ain fundam ental,thenextstep isto choosethe

appropriategeom etric structuresfrom which the pregeom etry isto be built"

(p.18,italics added). In thisuse ofthe term ,pregeom etry issynonym ous

with pre-continuum physics,although notsynonym ouswith pregeom etry in

the propersem antic and historic sense ofthe word| the sense itsinventor,

J.A.W heeler,endowed itwith and which itscurrentpractitionersattem pt

to ful�ll;in term s ofW heeler’s (1980) two m ost striking pronouncem ents:

\a conceptofpregeom etry thatbreaksloose atthe startfrom allm ention

ofgeom etry and distance" and \...to adm itdistanceatallisto give up on

the search forpregeom etry" (p.4).2

O n the basis of the incongruity between G ibbs’and W heeler’s state-

m ents,it is striking to com e across rem arks such as Cahilland K linger’s

(1996),who tellusthat\The need to constructa non-geom etric theory to

explain thetim eand spacephenom enahasbeen stronglyargued byW heeler,

under the nam e ofpregeom etry. G ibbs has recently com piled a literature

survey ofsuch attem pts" (p.314).Atthe reading ofthese words,one isat

�rstquitepleased and subsequently nothing butdisheartened.For,whereas

itistruethatW heelervowed fora non-geom etric theory ofspace and tim e

and nam ed it pregeom etry,it could not be farther from the truth to say

thatthe pregeom etric theoriesdevised so far(forexam ple,those in G ibbs’

or M onk’s survey) refrain from the use ofgeom etric concepts;to the con-

trary,they usethem extensively,asisourpurposeto show.

Atthe risk ofdispleasing som e authors,in the show-piece analysisthat

follows,attem ptsto transcend thespacetim e continuum willbeunderstood

as pregeom etric in tune with what is called pregeom etry,even when this

designation is not used explicitly in every program m e. In such cases,this

way of proceeding is som ewhat unfair, since the following works willbe

criticized chie y as failed expressionsofa genuine pregeom etry (butm any

also for lack ofunderlying physicalprinciples). W here they exist,explicit

m entions ofpregeom etric intentions willbe highlighted; lack ofany such

m ention willbeacknowledged aswell.

6.1 D iscreteness

O ne category could have em braced m ost of| ifnot all| the works in this

analysis.W ithoutdoubt,thebest-favoured typeofprogram m eto dealwith

new ways of looking at spacetim e is, in generalterm s, of a discrete na-

ture. The exact m eaning of\discrete" we willnotattem ptto specify;the

2The criticism to pregeom etry to be putforward here isnotbased m erely on a verbal

objection. It consists ofan objection to the activities ofpregeom etricians proper,given

thatthey explicitly setoutto avoid geom etry| and fail. Since thisgoalisdepicted very

wellindeed by thenam egiven tothe�eld,thecriticism appliesto theword \pregeom etry"

justaswell.Itisonly in thissense thatitisalso a verbalobjection.

6.1 D iscreteness 51

sim ple intuitive connotation ofthe word as som ething consisting ofsepa-

rate,individually distinct entities3 willdo. A choice ofthis sort appears

to be appealing to researchers;�rstly,because having castdoubtupon the

assum ption ofspacetim e as a continuum ,its reverse,discreteness ofsom e

form ,isnaturally envisaged;secondly,becausequantum -theoreticalconsid-

erationstend to suggest,atleaston an intuitive level,thatalso spacetim e

m ustbebuiltby discrete standards.

Lastbutnotleast,m any �nd som e ofthe specialpropertiesofdiscrete

structuresquiteprom ising them selves.Am ong otherthings,such structures

seem tobequitewell-suited toreproduceacontinuum in som elim it,because

they naturally supportgeom etric relationsasa built-in property.Thiswas

originally Riem ann’srealization.Asto the possibleneed forsuch a type of

structure,hewrote in an often-quoted passage:

Now it seem s thatthe em piricalnotions on which the m etrical

determ inationsofspace are founded,the notion ofa solid body

and ofaray oflight,ceasetobevalid forthein�nitely sm all.W e

arethereforequiteatliberty tosupposethatthem etricrelations

ofspacein thein�nitely sm alldo notconform to thehypothesis

ofgeom etry;and weoughtin facttosupposeit,ifwecan thereby

obtain a sim plerexplanation ofphenom ena.

The question ofthe validity ofthe hypotheses ofgeom etry in

thein�nitely sm allisbound up with thequestion oftheground

ofthe m etric relationsofspace. In thislastquestion,which we

m ay stillregard asbelonging to the doctrine ofspace,isfound

the application of the rem ark m ade above; that in a discrete

m anifoldness,the ground ofits m etric relations is given in the

notion ofit,whilein acontinuousm anifoldness,thisground m ust

com e from outside. Eithertherefore the reality which underlies

space m ust form a discrete m anifoldness,or we m ust seek the

ground ofits m etric relations outside it,in the binding forces

which actupon it.(Riem ann,1873,p.11,online ref.)

Riem ann can therefore be rightly considered the father ofalldiscrete

approachestothestudyofspacetim estructure.Heshould notbeconsidered,

however,them entorofan authenticpregeom etry since,ashehim selfstated,

quantitatively geom etric relationsare given in the very notion ofa discrete

m anifold,which,through theserelations,becom esageom etricobjectitself.4

3M erriam -W ebsterO nline D ictionary,http://www.m -w.com4In Article I,the sign ofgeom etry in pregeom etry was taken to be the appearance

of the geom etric m agnitude length. Here, attention is also drawn to the presence of

geom etric objects. Although these appearsom etim esin self-evidentways,atothertim es

theyappearat�rstasabstractobjects,which becom e geom etricobjectsviatheattachm ent

of geom etric m agnitudes to them . For exam ple, a \set of objects with neighbouring

relations" becom es a graph, i.e.a set of points with joining lines, when a distance is

prescribed on the abstract relations. In quantum m echanics,we saw that state vectors

likewisebecom egeom etricobjects(arrows)afterthespeci�cation oftheircom plex-num ber

overlaps(innerproduct)and theirsquared realm oduli(areas).

52 P regeom etry

Theabsoluteveracity ofhisstatem entcan bewitnessed in every attem ptat

pregeom etry,notasoriginally designed by W heeler,butasitofitselfcam e

to be.(W heeler’spregeom etry willbeexam ined in Chapter8.)

6.2 G raph-theoreticalpictures

G raphs are a m uch-favoured choice in attem pts at pregeom etric schem es.

Following W ilson (1985,pp.8{10,78),a graph ofa quitegeneraltypecon-

sists ofa pair (V;E ),where V is a possibly in�nite set ofelem ents called

verticesand E isa possibly in�nitefam ily ofordered pairs,called edges,of

notnecessarily distinctelem entsofV .In�nitesetsand fam iliesofelem ents

allow foran in�nitegraph,which can be,however,locally-�nite ifeach ver-

tex has a �nite num ber ofedges incident on it. Fam ilies (instead ofsets)

ofedgesperm itthe appearance ofthe sam e pairsm ore than once,thusal-

lowing m ultipleedgesbetween thesam epairofvertices.An edgeconsisting

ofa link ofa single vertex to itselfrepresentsa loop.Finally,ordered pairs

allow forthe existence ofdirected edgesin thegraph.

A particularreason forthepopularity ofgraphsin pregeom etry isthat,

along the lines of Riem ann’s thoughts, they naturally support a m etric.

W hen a m etric is introduced on a graph,alledges are taken to be ofthe

sam e length (usually a unit)irrespective oftheiractualshapesand lengths

asdrawn on paperorasconjured up by theim agination.Thisdistinguishes

graphsfrom certain typesoflattices,which have a m ore literalinterpreta-

tion,asexplained below.

6.2.1 D adi�c and Pisk’s discrete-space structure

Dadi�cand Pisk(1979)wentbeyond them anifold assum ption byrepresenting

\space asa setofabstractobjectswith certain relationsofneighbourhood

am ongthem "(p.346),i.e.agraph;in it,theverticesappeartocorrespond to

space points. Thisapproach assum esa m etric as naturally inherited from

the graph,and a de�nition ofdim ension| based on a m odi�cation ofthe

Hausdor� dim ension| thatisscale-dependent.The graph jG irepresenting

spacetim e is required to be unlabelled in its points and lines, and m ust

be characterizable by its topological structure only. O perators by and b

are de�ned forthe creation and annihilation oflines,such thatjG ican be

constructed from the vacuum -state graph j0iby repeated application ofby.

This is a Fock-space fram ework,which includes a second m etric (on state

vectors)aspartofitsbuilt-in structure.

W hatled Dadi�c and Pisk to thischoice ofgraph-theoreticalapproach?

They argued that in analysing what is essentialin the intuitive notion of

space,they found thisto be the existence ofsom e objectsand the relation

ofneighbourhood between them . Based on this,they proceeded to con-

structtheirdiscrete-spacestructure.However,thism otivationalbasisisnot

grounded on any furtherphysicalprinciples. Q uantum -theoreticalideas|

in particular,a Fock-space m ethod| are used in this approach only as a

6.2 G raph-theoreticalpictures 53

form alism to dealwith graphs,theirstates,and theirevolution.

G eom etric conceptsare here clearly assum ed. They are typi�ed by the

naturalm etricde�ned on thegraph on theonehand,thatofFock spaceson

theother,and notleastby agraph asacom positegeom etricobject(vertices

and edges)itself.Thisfactalone would m ake thisapproach a questionable

caseofan authenticpregeom etry,although weacknowledgethatitsauthors

did notso argue.

6.2.2 A ntonsen’s random graphs

Antonsen (1992) devised a graph-theoreticalapproach thatisstatisticalin

nature.Space isidenti�ed with a dynam icalrandom graph,whose vertices

appearto beassociated with spacepoints,and whoselinkshaveunitlength;

tim e is given by the param eterization ofthese graphs (spaces) in a m eta-

space. Prim ary point-creation and point-annihilation operators,ay and a,

are de�ned,as wellas secondary corresponding notions for links,by and

b;probabilities for the occurrence ofany such case are either given or in

principlecalculable.According to Antonsen,thisfram ework doesnotreally

assum e traditionalquantum m echanics although it m ight look like it. He

claim ed,echoing perhapstheway ofproceeding ofDadi�cand Pisk,thatthe

operatorsand the Hilbertspace introduced by him are only form alnotions

withoutdirectphysicalinterpretationsbesidesthatofbeingabletogenerate

a geom etric structure.

This approach is,according to its author,\m ore directly in tune with

W heeler’soriginalideas"(p.8)and,in particular,sim ilartothatofW heeler’s

\law withoutlaw" (p.9)in thatthelawsofnatureareviewed asa statisti-

calconsequence ofthe truly random behaviourofa vastnum berofentities

workingatadeeperlevel.O nem usthavereservationsaboutthisview,since

Antonsen isalready working in a geom etric context(a graph with a notion

ofdistance),which isnotwhatW heelerhad in m ind.M oreover,theredoes

notappear to be any principle leading Antonsen to introduce this kind of

graph structure in the �rstplace. He som etim es (pp.9,84) m entions the

need to assum e as little as possible,perhapsoverlooking the fact thatnot

only the quantity butalso the nature ofthe assum ptionsm ade| asfew as

they m ightbe| bearsasm uch im portance.

Antonsen is therefore the �rst claim ant to pregeom etry who calls for

the use of geom etric objects (a graph) and m agnitudes (distance) in its

construction;forthisreason,hisschem eisrathersuspiciousasan authentic

expression ofpregeom etry.

6.2.3 R equardt’s cellular netw orks

Requardtalso wentbeyond the m anifold assum ption by m eansofa graph-

theoreticalapproach. At a deeper level,space is associated with a graph

whose nodes ni are to be found in a certain state si (only di�erences of

\charge" si� sk are m eaningful),and its bondsbj in a state Jik that can

be equalto 0,1,or � 1 (vanishing or directed bonds). In contrast to the

54 P regeom etry

preceding schem e,here spacetim e pointsare notassociated with the graph

verticesbutwith \densely entangled subclustersofnodes and bonds ofthe

underlying network or graph" (Requardt & Roy, 2001, p.3040). A law

for the evolution in the \clock-tim e" ofsuch a graph is sim ply introduced

(pp.3042{3043) on the grounds that it provides,by trialand error,with

som e desired consequences. Requardt(2000,p.1) then assum ed that this

graph evolved from a chaotic initialphaseor\big bang" in thedistantpast

in such a way asto have reached a generally stable phase,associated with

ordinary spacetim e.

A peculiarity ofthis schem e is that it seeks to understand the current

structure ofspacetim e as em ergent in tim e,5 i.e.as a consequence ofthe

prim ordialeventsthatarebelieved to havegiven riseto itin the�rstplace.

In thisrespect,itcould have also been classi�ed asa cosm ologicalschem e,

although notasaquantum -cosm ologicalonesincequantum m echanicsisnot

assum ed:Requardt’s(1996)goalisto \identify both gravity and quantum

theory as the two dom inant but derived aspects ofan underlying discrete

and m oreprim ordialtheory..." (pp.1{2).

A lack ofguiding principlesm ustagain becriticized.Requardt’s(1995)

choiceofapproach seem storeston \acertain suspicion in thescienti�ccom -

m unity thatnaturem ay be‘discrete’on thePlanck scale" (p.2).Therefore,

heargued,cellularnetworksm akea good choicebecausethey arenaturally

discrete and can,m oreover,generate com plex behaviourand self-organize.

However,itm ustbe said thatbasing a physicalchoice on a generally held

suspicion whoseprecise m eaning isnotclearm ay notbevery cautious.

Although Requardt(1995) intended to produce a pregeom etric schem e

(\W hatweareprepared toadm itissom ekind of‘pregeom etry’"(p.6))and

strovefor\asubstratum thatdoesnotsupportfrom theoutset...geom etrical

structures" (p.2),he soon showed| in totalopposition to the earlier re-

m arks| thathisgraphs\havea naturalm etricstructure" (p.15).Heintro-

duced thusa very explicitnotion ofdistance.Requardtwasconcerned with

avoiding theassum ption ofacontinuum ,and in thishesucceeded.However,

geom etricnotionsarenotm erely continuum notions,asargued earlier.This

attem ptatpregeom etry su�ersthereforefrom thesam egeom etric a�iction

asthepreviousone.

6.3 Lattice pictures

Atan intuitivelevel,(i)alatticeconsistsofaregulargeom etricarrangem ent

ofpointsorobjectsover an area orin space.6 From a m athem aticalpoint

5As is typicalof argum ents considering the dynam ic em ergence of tim e (also as in

spacetim e),itisunclearin whatdynam ic tim e the em ergenttim e issupposed to appear.

This hypothetical,deeper dynam ic tim e,when considered at all(discrete ticks ofclock-

tim e forRequardt),isneverthelessalwaysthe sam e kind ofsim ple and intuitive external

param eterasthenotion oftim eitattem ptsto giveriseto.Theend resultisthen to trade

one little-understood external-param etertim e foranother.6M erriam -W ebsterO nline D ictionary,http://www.m -w.com

6.3 Lattice pictures 55

ofview,(ii)a latticeconsistsofa partially ordered sethL;� iin which there

existsan in�m um and a suprem um forevery pairofitselem ents.

The�rstnotion oflatticegoeshand in hand with thesam eideaasused in

Reggecalculus.In it,an irregularlattice isused in thesenseofan irregular

m esh whoseedgescan havedi�erentlengths.Forthisreason,such a lattice

cannotbeagraph,wherealllinksareequivalenttooneanother.Thesecond

notion oflattice is,to a degree,connected with the approach to spacetim e

structurecalled causalsets.Although,strictly speaking,a causalsetisonly

a locally �nite,partially ordered setwithoutloops,when italso holdsthat

there existsa unique in�m um and suprem um forevery pairofitselem ents

(existenceofa com m on pastand a com m on future),thecausalsetbecom es

a lattice in thesecond senseoftheterm .Despitethefactthatthey arenot

rigorously always a lattice,causalsetshave been included here in orderto

also exem plify,in as close a fashion as possible,the use ofa lattice ofthe

second type in theconstruction ofan alternative structureforspacetim e.7

6.3.1 Sim plicialquantum gravity by Lehto et al.

Lehto (1988),and Lehto,Nielsen,and Ninom iya (1986a,1986b)attem pted

theconstruction ofa pregeom etric schem ein which itwasconjectured that

spacetim epossessesa deeper,pregeom etricstructuredescribed by threedy-

nam icalvariables,nam ely,an abstractsim plicialcom plex ASC,itsnum ber

ofvertices n,and a real-valued �eld ‘ associated with every one-sim plex

(pair ofvertices) Y . It is signi�cant that,up to this point,the approach

hasa chance ofbeing truly pregeom etric since the �elds‘ are not(yet) to

beunderstood aslengthsofany sort,and theverticescould also bethought

as(orcalled)elem entsorabstractobjects(cf.abstractsim plicialcom plex).

However,the introduction ofgeom etric conceptsfollowsim m ediately.

Next,an abstractsim plicialcom plex issetinto a uniquecorrespondence

with ageom etricsim plicialcom plex G SC,aschem eofverticesand geom etric

sim plices.By piecing togetherthese geom etric sim plices,a piecewise-linear

space can be builtso that,by construction,itadm itsa triangulation. The

crucialstep is now taken to interpret �eld ‘ as the link distance of the

piecewise-linearm anifold. The conditionsare thussetforthe introduction

ofa Regge-calculuslattice with m etric gRC�� given by thepreviously de�ned

link lengths.

Traditionally,theprim aryideaofReggecalculusistoprovideapiecewise-

linear approxim ation to the spacetim e m anifolds of generalrelativity by

m eansofthe gluing together offour-dim ensionalsim plices,with curvature

concentrated only in theirtwo-dim ensionalfaces.In thisapproach,however,

theapproxim ation goesin theopposite direction,sincesim plicialgravity is

now seen asm orefundam ental,having thesm ooth m anifoldsofgeneralrel-

ativity as a large-scale lim it. M oreover,whereas the basic concept ofthe

7Causalsetscan also be classi�ed asgraph-theoretical,considering thatthey can also

be thought ofas consisting ofa locally �nite,loopless,directed graph without circuits,

in the sense thatitis not possible to com e back to a starting vertex while following the

allowed directionsin it.

56 P regeom etry

Regge calculusconsistsin the link lengths,which once given are enough to

determ inethegeom etry ofthelatticestructure,Lehto etal.also introduced

the num berofverticesn asa dynam icalvariable.

A quantization ofthe Regge-calculus lattice issubsequently introduced

by m eans ofthe Euclidean path-integralform alism . A pregeom etric par-

tition function Z sum m ing over ASC,�elds ‘,and vertices n is �rst pro-

posed asa form alabstraction to begeom etrically realized by a correspond-

ing Regge-calculus partition function Z RC sum m ing over piecewise-linear

m anifoldsand link lengths‘.Indeed,quantization can be viewed asone of

the reasonswhy the notion oflength m ustbe introduced in orderto m ove

forwards.

Two reasons can be o�ered for the choice ofthe starting point ofthis

fram ework.O n theonehand,beingconcerned with theproduction ofagen-

uinepregeom etry,an abstractsim plicialcom plex waschosen on thegrounds

that,assuch,itisabstractenough tobefreefrom geom etricnotionsintrinsic

to it.O n theotherhand,an abstractsim plicialcom plex with a variableset

ofelem entswasseen �tto providean appropriatesetting in which to study

furtherand from a di�erentpointofview the character ofdi�eom orphism

invariance| a usefulconcept to help us rethink the ontology ofspacetim e

points (see Chapter 9). The requirem ent ofdi�eom orphism sym m etry of

the functionalintegralm easure with respect to a displacem ent ofvertices

resulted in a free-gas behaviour of the latter as elem ents of a geom etric

lattice.Any pairofverticescould then with high probability have any m u-

tualdistance,helping to avoid the rise oflong-range correlationstypicalof

�xed-vertex lattices.

In one sense,thiswork isno exception am ong allthose analysed in this

chapter.G eom etricconceptssuch aslink length haveto beassum ed to real-

ize the abstractsim plicialcom plex asa geom etric object,a Regge-calculus

lattice,bringing itthuscloserto a m orefam iliargeom etric setting and ren-

dering it�tforquantization.However,despitetheassum ption ofgeom etry,

a distinctive feature ofthis work is that it recognizes clearly pregeom etric

and geom etric realm s and,even though itcannotdo withoutthe latter,it

keeps them clearly distinguished from one another ratherthan,asis m ore

com m on,soon losing sightofalldi�erencesbetween them .

6.3.2 C ausalsets by B om belliet al.

Bom belli,Lee,M eyer,and Sorkin (1987)proposed thatatthesm allestscales

spacetim e is a causal set: a locally �nite set of elem ents endowed with

a partialorder corresponding to the transitive and irre exive m acroscopic

relation thatrelatespastand future.

In thisfram ework,causalorderisviewed aspriorto m etricrelationsand

nottheotherwayaround:thedi�erentialstructureand theconform alm etric

ofa m anifold are derived from causalorder. Because only the conform al

m etric(ithasno associated m easureoflength)can beobtained in thisway,

the transition ism ade to a non-continuous spacetim e consisting ofa �nite

6.4 N um ber-theoreticalpictures 57

butlargenum berofordered elem ents,a causalset.In such a space,sizecan

bem easured by counting.

W hat considerations have led to the choice of a causalset? In this

respect,Sorkin explains:

The insightunderlying these proposalsis that,in passing from

the continuous to the discrete, one actually gains certain in-

form ation,because \volum e" can now be assessed (asRiem ann

said)by counting;and with both orderand volum e inform ation

present,wehaveenough torecovergeom etry.(Sorkin,2005,p.5,

online ref.)

Thisisa valid recognition resting on the factthat,given thatthe general-

relativisticm etricis�xed bycausalstructureand conform alfactor,adiscrete

causalset willbe able to reproduce these features and also provide a way

for the continuous m anifold to em erge. But is Sorkin here following any

physicalprinciplethatleadshim totheconclusion thatacausal-setstructure

underliesspacetim e,oristhe proposalonly m athem atically inspired?

As regards geom etry, volum e is already a geom etric concept, so that

\to recover geom etry" above m ust m ean \to recover a continuous,m etric

spacetim e," not\geom etry" in thestrictersenseoftheword.Thecausal-set

pictureusesothergeom etricconceptsaswell.Thecorrespondenceprinciple

between a m anifold and a causalset from which it is said to em erge,for

exam ple,reads:

The m anifold (M ;g) \em erges from " the causalset C [ifand

only if]C \could have com e from sprinkling points into M at

unitdensity,and endowing the sprinkled pointswith the order

they inheritfrom the light-cone structure ofg." (Sorkin,1991,

p.156)

The m entioned unitdensity m eansthatthere existsa fundam entalunitof

volum e,expected by Sorkin to be the Planck volum e,10�105 m 3. Further-

m ore,healso considersthedistance between two causal-setelem entsx and

y asthenum berofelem entsin thelongestchain joining them (p.17),turn-

ing the ordering relations into lines and x and y into points. Thus,ifa

causalsetis to be seen asa pregeom etric fram ework,due to its geom etric

assum ptions,itm akesagain a questionable case ofit. However,we do not

know itsproponentsto have staked any such claim .

6.4 N um ber-theoreticalpictures

Som einvestigatorshaveidenti�ed thekey toan advancem entin theproblem

ofspacetim e structure to lie in the num ber �elds used in current theories

orto be used in future ones. Butter�eld and Isham (2000,pp.84{85),for

exam ple,arrived attheconclusion thattheuseofrealand com plex num bers

in quantum m echanicspresupposesthatspaceisa continuum .According to

58 P regeom etry

thisview,standard quantum m echanicscould notbeused in theconstruction

ofany theory thatattem ptsto go beyond such a characterization ofspace.8

6.4.1 R ational-num ber spacetim e by H orzela et al.

Horzela,K apu�scik,K em pczy�nski,and Uzes(1992)criticized those discrete

representationsofspacetim e thatassum e an elem entary length,and which

m ay furtherm oreviolaterelativisticinvariance,on thegroundsthatthefor-

m erisexperim entally notobserved and thelatteris,in fact,experim entally

veri�ed.9 They proposed to starttheiranalysisby studying the actualex-

perim entalm ethod used to attribute coordinates to spacetim e points,the

radio-location m ethod.

Their starting point is that the m easured values of the coordinates

t= (t1 + t2)=2 and x = c(t2 � t1)=2 of an event in any system of refer-

ence are alwaysrational(ratherthan real)num bers,since both the tim e of

em ission t1 and reception t2 ofthe radio signalare crude,straightforward

m easurem ents m ade with a clock.10 Subsequently,they showed that the

property ofbeing rationalispreserved when calculating the coordinatesof

an eventin anothersystem ,aswellasin the calculation ofthe relative ve-

locity ofthis system . Therefore,they m aintained that Lorentz invariance

holds in a spacetim e pictured in term s ofrational-num ber coordinates. In

addition,since the rationalnum bersare dense in the realnum bers,such a

spacetim e freesitselffrom any notion ofelem entary distance. Ifspacetim e

m ust be described in a discrete m anner,the rational-num ber �eld is then

seen asthe key to a betterunderstanding ofit.

W hile Horzela etal.suggested thatthisline ofinvestigation should be

furthered by m eansofalgebraic m athem aticalm ethods,they adm itted not

knowing atthetim eoftheirwriting ofany guiding physicalprincipleto do

this. Asa consequence,since they do notpropose to go asfarasbuilding

a m orecom plete spacetim e picture based on theirprelim inary �ndings,the

usualcriticism sbeing m ade in thischapterdo notapply. Furtherphysical

guiding principles are lacking, but then this has been acknowledged and

action taken in accord with it;geom etric ideas are ofcourse present,but

then no realattem pt to transcend the fram ework ofrelativity was m ade

afterall.

6.4.2 Volovich’s num ber theory

Volovich (1987) argued that, since at the Planck scale the usualnotion

ofspacetim e is suspected to lose its m eaning,the building blocks ofthe

8Butdo notrealnum berssim ply arise asthe probability ofquantum -m echanicalout-

com esasthelim itingrelativefrequency between successesand totaltrials? Thesearequite

unproblem atic observationsm ade in the realworld,so why should we fabricate problem s

worrying aboutwhatm ightgo wrong in the\quantum gravity regim e" where \space and

tim e m ightbreak down" (Isham & Butter�eld,2000,p.6,online ref.)?9See (Hill,1955)fora precursorofthisprogram m e.

10For this idea to work,it is also essentialthat the value c ofthe speed oflight be

understood asthe resultofa \crude" m easurem ent.

6.5 R elationalor process-based pictures 59

universecannotbeparticles,�elds,orstrings,fortheseare de�ned on such

abackground;in theirplace,heproposed thatthenum bersbeconsidered as

thebasicentities.Atthesam etim e,Volovich suggested thatforlengthsless

than thePlanck length,theArchim edean axiom 11 m aynothold.Puttingthe

two ideastogether,he asked why should one constructa non-Archim edean

geom etry overthe �eld ofrealnum bersand notoversom e other�eld?

To m ake progress, Volovich proposed as a general principle that the

num ber�eld on which atheory isdeveloped undergoesquantum uctuations

(p.14).Hem aintained thatthism eansthattheusual�eld ofrealnum bers

is apparently only realized with som e probability but that, in principle,

therealso existsa probability fortheoccurrenceofany other�eld.Volovich

furtherm orespeculated thatfundam entalphysicallawsshould beinvariant

undera change ofthe num ber�eld,in analogy with Einstein’sprinciple of

generalinvariance (p.14).

It is di�cult to see what principle led Volovich to propose this pro-

gram m e;it appears that only analogies have guided his choices. It is also

di�cultto com etoterm swith hisopinion that\num bertheory and thecor-

respondingbranchesofalgebraicgeom etryarenothingelsethan theultim ate

and uni�ed physicaltheory" (p.15). Finally,the role ofgeom etry in this

fram ework is crystal-clear: instead ofattem pting to reduce \allphysics"

to Archim edean geom etry over the realnum bers,it is proposed that one

should ratherattem ptto reduce\allphysics"to non-Archim edean geom etry

overa uctuating num ber�eld.Forthisreason,although Volovich did not

so claim ,his schem e could,again,hardly be considered a true instance of

pregeom etry.

6.5 R elationalor process-based pictures

Having found inspiration in Leibniz’s relationalconception ofspace,som e

researchershaveproposed thatspacetim eisnotathingitselfbutaresultant

ofa com plex ofrelationsam ong basicthings.O thers,on a m orephilosoph-

icalstrand,have found inspiration in Heraclitus’principlethat\allis ux"

and have put forward pregeom etric pictures in which, roughly speaking,

processesare considered to bem ore fundam entalthan things.Thisview is

som etim eslinked with thecurrently m uch-dissem inated idea ofinform ation,

especially in itsquantum -m echanicalform .

6.5.1 A xiom atic pregeom etry by Perez B erglia�a et al.

PerezBerglia�a,Rom ero,and Vucetich (1998)presented an axiom aticfram e-

work that, via rules ofcorrespondence, is to serve as a \pregeom etry of

space-tim e." Thisfram ework assum estheobjectiveexistenceofbasicphys-

icalentities called things and sees spacetim e not as a thing itselfbutas a

resultantofrelationsam ong those entities.

11TheArchim edean axiom statesthatany given segm entofa straightlinecan beeven-

tually surpassed by adding arbitrarily sm allsegm entsofthe sam e line.

60 P regeom etry

In orderto derive from thism ore basic substratum the topologicaland

m etric properties of M inkowskian spacetim e, a long list of concepts and

axiom sispresented.Butforthesaid generalrequirem entsofobjectivity and

relationality,these\ontologicalpresuppositions" (p.2283)appeararbitrary

and inspired in purely m athem aticalideasratherthan stem m ing from som e

physicalprincipleconcerning spacetim e.

Am ongtheseconcepts,wecentreourattention on som ethingcalled ontic

spaceE o.Although itsexactm eaningcannotbeconveyed withoutreviewing

a greatdealofthecontentsofthearticle in question,itcould beintuitively

taken to be a form of space connected with the basic entities, which is

prior(in axiom atic developm ent)to the m ore fam iliargeom etric space E G

ofphysics| M inkowskian space,in thiscase. Perez Berglia�a etal.proved

(pp.2290{2291) thatthe ontic space E o ism etrizable and gave an explicit

m etricd(x;y)forit,adistance between things,turning,again,relationsinto

linesand thingsinto points. Finally,by m eansofan isom orphism between

E o and a subspace dense in a com plete space,the geom etric space E G is

obtained asthiscom pletespaceitself,whileatthesam etim eitinheritsthe

m etric ofE o.

O nenoticesonceagain thestealthy introduction ofam etricforpregeom -

etry,with theensuing geom etricobjecti�cation ofearlierabstractconcepts,

in orderto derive from it furthergeom etric notions. Thisprocedure m ust

be criticized in view thatthisapproach explicitly claim s to be an instance

ofpregeom etry.

6.5.2 C ahilland K linger’s bootstrap universe

Cahilland K linger(1997)proposed to give an accountofwhatthey rather

questionably called \the ‘ultim ate’m odelling ofreality" (p.2). This,they

claim ed,could only bedoneby m eansofpregeom etricconceptsthatdo not

assum e any notions ofthings but only those ofprocesses since,according

to them ,the form ernotions(butnotthe latter)su�erfrom the problem of

their always being capable offurther explanation in term s ofnew things.

O ne m ust wonder for what m ysterious reason certain processes cannot be

explained in term sofothers.

As an em bodim entofthe concepts above,these authors used a \boot-

strapped self-referentialsystem " (p.3)characterized by a certain iterative

m ap thatrelieson thenotion ofrelationalinform ation B ij between m onads

iand j.In orderto avoid thealready rejected idea ofthings,thesem onads

are said to have no independentm eaning butforthe one they acquire via

the relationalinform ation B ij. O nce again,it is evident that this m athe-

m atically inspired starting pointdoesnotreston any physicalfoundation.

Dueto reasonsnotto bereviewed here,theiterative m ap willallow the

persistenceoflarge B ij.These,itisargued,willgive riseto a tree-graph|

with m onads as nodes and B ij as links| as the m ost probable structure

and as seen from the perspective ofm onad i. Subsequently,so as to ob-

tain (by m eansofprobabilisticm athem aticaltools)a persistentbackground

structureto beassociated with som eform ofthree-dim ensionalspace,a dis-

6.6 Q uantum -cosm ologicalpictures 61

tance between any two m onads is de�ned as the sm allest num ber oflinks

connecting them in thegraph.

\Thisem ergent3-space," Cahilland K lingerargued,\...doesnotarise

within any a priori geom etricalbackground structure" (p.6). Such a con-

tention is clearly m istaken for a graph with its de�ned idea ofdistance is

certainly a geom etric background forthe reasonswe have pointed outear-

lier. Thus,yet another m athem atically inspired pregeom etric schem e falls

prey to geom etry.

6.6 Q uantum -cosm ologicalpictures

Q uantum -cosm ologicalapproaches tackle the problem ofspacetim e struc-

ture from the perspective ofthe universe as a whole and seen as a neces-

sarily closed quantum system .Forthem ,theproblem oftheconstitution of

spacetim e isbound up with the problem ofthe constitution ofthe cosm os,

custom arily dealing aswellwith the problem ofhow they interdependently

originated.

6.6.1 Eakins and Jaroszkiew icz’s quantum universe

Jaroszkiewicz (2001) and Eakinsand Jaroszkiewicz (2002,2003)presented

a theoreticalpictureforthequantum structureand runningoftheuniverse.

Its�rstclassofbasicelem entsareeventstates;thesem ay orm ay notbefac-

tored outasyetm orefundam entaleventstates,depending on whetherthey

are directproducts(classicity) ofyetm ore elem entary event states j i,or

whetherthey areentangled (non-separability)eventstatesji.Thesecond

classofbasicelem entsarethetestsactingon theeventstates.Thesetests�,

represented by Herm itian operators�,providethetopologicalrelationships

between states,and endow the structure ofstates with an evolution (irre-

versibleacquisition ofinform ation)via \statecollapse" and with a quantum

arrow oftim e. A third com ponent is a certain inform ation content ofthe

universe| som etim es,although notnecessarily,represented by thesem iclas-

sicalobserver| at each given step in its evolution (stage),which,together

with the present state ofthe universe,determ ines the tests which are to

follow.Thus,the universe isa self-testing m achine,a quantum autom aton,

in which the traditionalquantum -m echanicalobserverisaccounted forbut

totally dispensed with.

Eakinsand Jaroszkiewicz(2002,p.5),likeRequardtbutaftertheirown

fashion,also attem ptto understand thepresentstructureofspacetim e asa

consequence ofprim ordialevents,which in thiscase can be traced back to

whattheycallthe\quantum bigbang." In thiscase,however,theem ergence

ofspacetim eism oresophisticated than in Requardt’s,sincehereithappens

in thenon-param eterintrinsic tim e given by state-vectorcollapse.

The subject of pregeom etry as pioneered by W heeler is also touched

upon (Eakins& Jaroszkiewicz,2003,p.2).Theview thatthecharacteristic

featureofpregeom etricapproachesconsistsin \avoidingany assum ption ofa

62 P regeom etry

A uthor G eom etric G eom etric

objects m agnitudes

Dadi�c & Pisk vertices,edges overlap,length,distance

Antonsen� idem length,distance

Requardt� idem idem

Lehto etal.� idem idem

Bom bellietal. idem length,distance,volum e

Horzela etal. asin sp.rel. asin sp.rel.

Volovich asin m od.phys. asin m od.phys.

Perez Berglia�a etal.� points,lines length,distance

Cahill& K linger� idem idem

Eakins& Jaroszkiewicz� asin q.m ech. asin q.m ech.

Nagels� points,lines length,distance

Stuckey & Silberstein� idem idem

Table 6.1: Sum m ary ofthe use ofgeom etric concepts in the pregeom etric ap-

proachesanalysed in this chapter,plus two m ore discussed laterin Section 8.3.1.

Approachesthatm ention pregeom etry explicitly are m arked with an asterisk (� ).

Abbreviations:Specialrelativity (sp.rel.),quantum m echanics(q.m ech.),m odern

physics(m od.phys.).

pre-existingm anifold"(p.2)isalsostated there,although itisnowherem en-

tioned thatthenotion ofdistance,according to W heeler,should beavoided

as well. In any case,these authors explain that their task is to reconcile

pregeom etric,bottom -up approaches to quantum gravity with other holis-

tic,top-bottom onesasin quantum cosm ology.Finally,sincethetraditional

m achinery ofquantum m echanicsisassum ed in theapproach,soistherefore

geom etry. Again,this fact underm ines the status ofthis program m e as a

genuine representative ofpregeom etry.

O fallthe above approaches,we �nd this one to be the m ost appeal-

ing. Itgivesclearexplanationsaboutthe nature ofitsbuilding blocksand

puts forward judicious physicalprinciples and m echanism s| as contrasted

to purely m athem atically inspired ones| by m eansofwhich the traditional

notionsofspacetim e and quantum non-locality m ay em erge.

In Table 6.1, we sum m arize the use of geom etric concepts in allthe

pregeom etricpicturesanalysed in thischapter,plustwom orediscussed later

on in Chapter8.

6.7 T heinexorability ofgeom etricunderstanding?

By way ofappraisal,wetakenoteofcertain viewsexpressed by Anandan,as

they constitutetheabsoluteepitom eofthem odeofworking ofpregeom etry

witnessed so far.Anandan suggests

aphilosophicalprinciplewhich m aybeschem atically expressed as

O ntology = G eom etry = Physics.

6.7 T he inexorability ofgeom etric understanding? 63

Thelastequality hasnotbeen achieved yetby physicistsbecause

we do nothave a quantum gravity.Butitishere proposed asa

philosophicalprinciplewhich should ultim ately besatis�ed by a

physicaltheory.(Anandan,1997,p.51)

Anandan appears to have taken the geom etric essence ofpresent physical

theoriesasa ground to argueahead that,correspondingly,any successfulfu-

turephysicaltheory m ustofnecessity begeom etric.(Som uch forW heeler’s

pregeom etry asa road to quantum gravity!) Such ishisconviction thathe

raisesthisidea to the statusofa principle thatany physicaltheory should

\ultim ately" follow. But does it ensue from the fact that physics has so

fardescribed nature by m eansofgeom etry thatitwillcontinue to do so in

the future? Physicsm ust describe nature geom etrically com e whatm ay|

whence such an inevitability? M oreover,in whatway has the �rstpartof

thequoted equality,asim plicitly stated,already been achieved? Sim ply be-

causeitisa factthatreality itselfisgeom etric? W earerem inded ofSm ith’s

view,according to which form isan inherentpartofnature.W hy doesthis

idea keep com ing back to hauntus?

Thus,we witnesshere an overstated version ofthe spiritim plicitly un-

derlying the cases analysed through this section. Ifin the latter the use

ofgeom etry was,so to speak,accidentalor non-intentional,in Anandan’s

statem entwe�nd theexplicitclaim m aterialized thatgeom etry m ustbethe

m odeofdescription ofphysicalscience,in particular,concerning a solution

to the problem ofthe quantum structure ofspacetim e. No stronger grip

ofgeom etry on physics| and on the hum an m ind| could possibly be con-

ceived. Butis this conception justi�ed? And ifnot,whatis the source of

thiswidespread m isconception?

The connection underlying Chapters3{6 now com es to the fore. From

the m ists of antiquity to the forefront of twenty-�rst century theoretical

physics,geom etry rem ains (for better or worse,successfully or vainly) the

enduring,irreplaceable toolby m eansofwhich hum ansportray the world.

Ironically enough,even in a �eld thatsetsitselfthe task ofexplaining the

origin ofgeom etry in physics and nam es itselfpregeom etry,one is to �nd

them ostexplicitdisplaysofgeom etric understanding.In thenextchapter,

westartto considerthereasonsbehind thispeculiarstate ofa�airs.

C hapter 7

T he geom etric anthropic

principle and its

consequences

The m ore I study religions the m ore I am convinced that

m an never worshipped anything buthim self.

SirRichard FrancisBurton,The book ofthe thousand nights

and a night,Vol.10

J.Ellis(2005)described the state ofa�airsdocum ented in thetwo pre-

vious chapters rather insightfully when he said,\Following Einstein,m ost

theoreticalphysicistsassign acentralroletogeom etricalideas"(p.57).Ellis’

recognition isquite valid,butafterpondering geom etry to the extentthat

wehavedone,weareready to m akea few speci�cations.Although itistrue

thatphysicistscurrently draw theirgeom etricinspiration toacertain extent

from the success ofEinstein’s work| for it doeslook m ore geom etric than

thephysicaltheoriespreceding it| wehaveseen thatthegeom etricdrivein

physics is rooted m uch deeperand m uch fartherback in tim e than in any

twentieth-century intellectualproduct.M oreover,one m ay even push Ellis’

claim further and,without fear oferror,say that,with the exceptions of

the nextchapter,itisessentially allphysiciststhatassign a centralrole to

geom etric ideas.How centralisthisrole?

Pedersen (1980), for exam ple,tells us that \G eom etry is the connec-

tion between the realworld and the m athem aticswe use to solve problem s

aboutthatrealworld," and reportson m athem atician Ren�eThom ’snotion

that\G eom etry isthem ostfundam entalabstractrepresentation ofthereal

world" (Hilton & Pedersen,2004). Pedersen’sstatem entisan accurate de-

scription ofthecontingentm annerin which geom etry hassofarhappened to

beused by hum ans,butitshould notbeunderstood asa description ofthe

necessity ofgeom etric theorizing. Thom ’srem ark,on the otherhand,errs

in itscon ation ofcontingency with necessity,nam ely,thatthelanguage of

geom etry isnotsim ply a parochialpicturing m eansofhum ans,buttheuni-

7.1 T he geom etric anthropic principle 65

versaltoolby m eansofwhich any theorist(hum an orotherwise)atany tim e

(now and forever)m ustrepresentany aspect(known and unforeseeable)of

the physicalworld. W hat is the origin and reason for this exaltation of

geom etry?

7.1 T he geom etric anthropic principle

The astounding e�ectiveness ofgeom etry in the portrayalofthe natural

world hassinceantiquity been interpreted by m any to signify thatthenat-

uralworld isgeom etric in itself. The roots ofthispersuasion are found in

ancientG reece and are represented by Plato’s thought;in particular,they

are best epitom ized by a notable phrase attributed to him ,nam ely,\G od

eternally geom etrizes."

Instead ofPlato,Foss(2000,p.36)attributestheorigin ofthisbeliefto

Pythagoras,and calls it the \Pythagorean Intuition." However,Pythago-

ras’view wasthat\reality isnum ber" orthat\atitsdeepestlevel,reality

ism athem aticalin nature,"1 which isnotequivalentto itsbeing essentially

geom etric. Jam m er (1969), on the other hand,does capture the distinc-

tion,togetherwith the Platonic originsofthegeom etric physicaltradition,

when he writes,\W ith Plato physics becom es geom etry,just as with the

Pythagoreansitbecam e arithm etic" (p.14).2

Two m illennia later,hardly anything had changed. The sam e opinion

asPlato’sbutwithoutitsreligiousovertoneswasarticulated by G alilei,for

whom the book ofnature

iswritten in thelanguageofm athem atics,and itscharactersare

triangles,circles,and othergeom etric�gureswithoutwhich itis

hum anly im possible to understand a single word ofit;without

these,one wanders about in a dark labyrinth. (G alilei, 1957,

p.238)

W hence so m uch em phasison theessentialnessofgeom etry?

These statem ents propounding the fundam entality ofgeom etry can be

understood asoriginating from a geom etric form ofthenotoriousanthropic

principle. O rdinarily,this principle asserts that \the universe m ust be as

itisbecause,otherwise,we would notbe here to observe it." In a sim ilar

vein,Plato’s words epitom ize a strong geom etric version ofthe anthropic

principle,which could bephrased thus:\G od m usthavem adetheuniverse

geom etrically because,otherwise,wewould notbeable to describeitaswe

do." G alilei’s wordsem body a weaker version ofthe sam e principle: \The

universe m ust be geom etric because,otherwise,we would not be able to

1Encyclopaedia Britannica,http://www.britannica.com

2The con ation ofm athem aticswith geom etry isan errorcom m only incurred in.For

exam ple,in connection with the \Pythagorean Intuition," Foss (2000) writes: \why has

geom etry becom e the m odeling toolofscience? Is it because Pythagoras was right in

thinking that reality itselfis som ehow deeply m athem aticalby nature?" (p.87). For a

sim ilarerror,see (Stewart,2002,p.5).

66 T he geom etric anthropic principle and its consequences

describe it as we do." As com m on sense is apt to reveal, the reasoning

expressed by anthropic principles is nevertheless inadequate, considering

thatthefactthatM an can dosom ething| in theform ercase,sim plyexist;in

thelatter,devisesuccessfulgeom etrictheories| isacontingentconsequence,

nota cause,ofthe featuresofthe world.

Despite its ine�ectualness and basically awed nature, the anthropic

principlecontinuestobewidely upheld today in both itsgeom etricvarieties.

Itsweakerversion �ndsexpression in statem entssuch asBrassard’s(1998)

\A science has to be based on a geom etry" (x 3.4), whereby physics is

posited asabranch ofgeom etry| a view thatputsthecartbeforethehorse.

It also �nds expression in Anandan’s (1997) previously analysed principle

\Reality = G eom etry = Physics" (p.57) that any physicaltheory should

ultim ately follow. Its strong version is in turn exem pli�ed by straightout

religious assertions such as \G od is a geom eter."3 The strong version of

thegeom etric anthropicprinciplehaseven been speltoutin itsfullexplicit

form | and shockingly,too| by Stewart(2006):\O nly a G eom eterG od can

create a m ind that has the capacity to delude itselfthat a G eom eter G od

exists" (p.203).

Has G eom etry perchance becom e a m odern religion| physicists’latest

incarnation ofG od? Unexpectedly,this rhetoricalquestion �ndsa willing

reply from Sm olin (1997), who answers forthrightly in the positive: \the

beliefthatatitsdeepestlevelreality m ay be captured by an equation ora

geom etricalconstruction isthe private religion ofthe theoreticalphysicist"

(p.178);heeven goeson tosaythat\an education in physicsorm athem atics

isa little like an induction into a m ysticalorder" (p.179). Butifthisisa

statem ent offact(and itm ay wellbe),then itisofa sad fact,and notof

one we should besym pathetic oform ention lightly.

Not even twenty-�rst century physics| m uch though it m ay secretly

presum e| hasridden itselfofthereligiousm indsetoriginally partofnatural

philosophy.Peacock (2006),forexam ple,rightly criticized physicistsin gen-

eralforsubscribing to thereligiousview that\G od wrotetheequations" of

nature revealed (notm ade)by ourtheories,and Susskind in particularfor

transferring \the quasi-religious awe [from intelligent design]to string the-

ory,whose m athem aticalresults he repeatedly describes as ‘m iraculous’"

(p.170). And Horgan (1996) hit the nailnearly on the head when he at-

tributed physicists’calling G od a geom eter to their being \intoxicated by

thepoweroftheirm athem atical[geom etric]theories"(p.77;seealsop.136).

Farfrom G od ornature,we place the origin ofgeom etry in hum an na-

ture;m orespeci�cally,in thecharacterofthehum an brain.Itisnotbecause

theworld isgeom etric itself(a crypticnotion,atany rate),orbecauseG od

m ade it so,that our theories ofit speak the language ofgeom etry; it is

becauseevolutionary vicissitudeshavem oulded thehum an brain such that,

for better or worse,successfully or vainly,we write geom etric theories of

the world. Indeed,itappearsrather m ore evidentthatG alilei’s geom etric

3See e.g.(Stewart & G olubitsky,1993) and (Stewart,2002,p.5);see also (Pollard,

1984,p.881).

7.1 T he geom etric anthropic principle 67

book ofnature is not pre-existent butwritten by M an| who eternally ge-

om etrizesbut,ingrained ashiswaysofthinking are,isunable to recognize

thehandwriting ashisown.

W ithouthesitation,weplacethesourceofhum an geom etricabilities| or

even,com pulsions| in thecharacterofthebrain and notelsewhere,such as

in thatofthesenseorgans,especially eyesight.Itisquitepossiblethatifwe

were stillendowed with the fam iliar �ve senses butourbrainswere essen-

tially di�erent,we would notdevelop any geom etric concepts and insights

into nature.An acquaintance with the self-related life experiencesofHelen

K eller,the fam ousdeaf-blind wom an who overcam e hersensory handicaps

to becom e a fullhum an being,givesobservationalsupportto thisview.

K eller (1933) tells us about her m ind’s naturalability to grasp allthe

established,conventionalthoughtsand conceptsofseeingand hearingpeople

despite her own,practically life-long,visualand hearing disabilities. She

wrote:\Can itbe thatthe brain isso constituted thatitwillcontinue the

activity which anim atesthesightand thehearing,aftertheeyeand theear

have been destroyed?" (p.63).And furtherre ected:

Theblind child| thedeaf-blind child| hasinherited them ind of

seeing-and-hearing ancestors| a m ind m easured to �ve senses.

Thereforehem ustbein uenced,even ifitisunknown tohim self,

by the light,colour,song which have been transm itted through

thelanguageheistaught,forthecham bersofthem ind areready

to receive thatlanguage.(K eller,1933,p.91)

K ellerdiscoversin herpersonalexperience thatherm ind naturally bridges

the gap to that ofher seeing-and-hearing peers by producing appropriate

conceptualcorrespondences,such asbetween a \lightning ash," which she

cannotexperience,and a \ ash ofthoughtand itsswiftness" (p.92),which

shecan;and shenotesthatno m atterhow farshewillexercisethism ethod,

it willnot break down. K eller tells us that she conceives ofthe world in

essentially thesam em annerasdohernorm alpeers,and thatsheknowsthis

from the factthatshehasno di�culty in com prehending otherpeople;she

distinctly knowsthatherconsciousness| prim arily a productofherbrain|

isin essencethesam easthatofsensorially unim paired people.\Blindness,"

shem etaphorically decreesfrom hervantageposition,\hasnolim itinge�ect

upon m entalvision"(p.95).K eller’sawe-inspiringwords,then,bearoutthe

pointin question in �rst-hand m anner,nam ely,thatitisprim arilythebrain,

notthesenses,thatleadsustoourparticulardescription and understanding

ofthe world. Change a m an’ssenses,orto a certain extentdeprive him of

them ,and his worldview willnot change; but change a m an’s m ind and

witnesshisvision,hearing,and touch acquire unsuspected new m eanings.

G eom etricthoughtisno exception to thisrule.Shapeand extension are

notpropertiesweabstractany m orefrom thesightofobjectsthan from the

sounds they m ake or the way they feel. As one can infer from the above

account,the nature ofthe m ind’sm eansofcontactwith the outerworld is

to a large extentirrelevantto the form ation ofgeom etric conceptions| the

task ofcreating these restsessentially upon the brain.


It is som etim es told that a �sherm an whose �shing net is loosely knit

willonly catch big �sh.Butwhileweintellectually understand them eaning

ofthis statem ent,we failto feelits m eaning in our bones. After all,we

can im agine a m ore tightly knit net that would catch the sm aller �sh in

the ocean;we can easily see on the other side ofthis arti�cially im posed

lim itation;wecan easily seehow m istaken itisto inferan ocean populated

exclusively by big �sh from the character of the net| and so the m oral

eludes us. The extent to which iteludes usis directly proportionalto the

di�culty we experience in trying to com e to term swith a statem entofthe

type,\Snow akes are not inherently hexagonal," because snow akes look

invariably hexagonalto us,and we�nd ithard| and pointless| to conceive

ofthem otherwise.Thisisnottosuggestthatthehexagonality ofsnow akes

isan illusion,butthatitisone possible true description ofthem ,justlike

bigness is one true description of�sh in the ocean| the net does not lie,

only possibly m isleads.

W hen the geom etric worldview,instead ofstem m ing from an outside

�lter such as a �shing net,stem s from the innerm ost recesses ofour own

selves,itislittlewonderthatitwillgounrecognized astheconceptualtoolit

is,and beinstead con ated with G od,nature,and reality them selves.W hat

is m ore,it willcontinue to pass unrecognized as a non-essentialtooleven

after we have caught a glim pse ofits possible shortcom ings. Foss (2000)

m akes a striking case ofthissituation in hisbook on consciousness. After

insightfullyrecognizingtheexclusively geom etricm odeofworkingofscience,

he goes on to note that qualia (i.e. the quality of conscious experience)

have escaped scienti�c understanding because science dealswith the world

only geom etrically. But such is the grip ofthe geom etric worldview,and

such is the strength ofthe taboo on any thought that would cast doubt

upon it,thatFoss’svaluablerealization soon regressesand becom esavictim

ofitself. Instead ofconcluding that scienti�c descriptions need not then

always be geom etric,he writes: \The crucialinsight is that even though

science m ust m odelthings geom etrically, this does not entailthat it can

only m odelgeom etric things,geom etric properties,orgeom etric aspectsof

reality" (p.70).(W e return to Foss’sideaslateron in Chapter12.)

Butwhence m ust science picture thingsgeom etrically,beyond the con-

tingency that it presently happens to do so? Is it not m ore naturalto

conclude that som e things (e.g.qualia)| which are neither geom etric nor

non-geom etricin them selves| defy geom etricrepresentation sim ply because

theirphysicalnatureissuch thatitdoesnotallow foritscastingintogeom et-

ricm oulds? Itisgripped by an indeliblegeom etricim perativeasthey probe

the physicalworld that,no m atter how speculative and non-conservative

today’s scientists m ight be, they continue to pound away with the only

artillery they can conceive| geom etry,m ore geom etries,new geom etries.

Returning once again to quantum gravity,we ask: would it be fair to

say that, after three-quarters of a century of disorientation and frustra-

tion in quantum gravity,the disclosure ofa deeper layer ofthe nature of

thingsthrough new geom etrieshasreached a dead-end? That,afternearly

7.2 O ntologicaldi� culties 69

a century ofphilosophicalpuzzlesconcerning the ontology ofstate vectors

and spacetim e points,m ore and new geom etric deliberations on quantum

and spacetim e ontology can take usno further? Thatwith m ore and ever-

renewed geom etric m odesofthoughtwecan only continueto im prison our-

selvesin a labyrinth ofourown m aking?

In the rest ofthis chapter,we expound som e ofthe consequences for

theoreticalphysicsofbeliefin thegeom etric anthropicprinciple.

7.2 O ntologicaldi� culties: State vectors and

spacetim e constituents

Attitudestowardsthe state vector and itscollapse are asvaried ashum an

im agination.Theontologicaldependenceofquantum theory on states,com -

bined with thehum an yearningtogain insightexclusively through geom etric

visualization,hasled to a frantic search forthe physicalm eaning ofthear-

row statevector.Isham (1995,p.83)subm itssom eofthem any possibilities:

does the state vector refer to an individualsystem ,a collection ofidenti-

calsystem s,our knowledge ofthe system (s),the result ofa m easurem ent

on a system ,the resultsofrepeated m easurem entson a collection ofthem ,

the de�niteness with which a system possesses a value,etc.? Further,to

m ake sense ofthe collapse ofthe geom etric state vector,potentially physi-

calm echanism shave been proposed,such asgravity orconsciousness,that

m ightcauseit;othershave in denialengineered a m etaphysicalescape to a

m yriad ofparalleluniverses;etc.

Notonly doesoneuniversesu�cetogiveanswerstotheontologicalques-

tionsofquantum theory but,wesubm it,noteven physicalin uencesbeyond

quantum m echanicsneed be invoked forthe task,so long aswe are willing

to reexam inethegeom etricfoundationsoftheportrayalofphenom ena and,

in line with a long physicaltradition,are ready to start the developm ent

of theoreticalconcepts from physicalthings we are fam iliar with (obser-

vations,experim ents) instead of,as is today m ore com m on,m athem atical

contrivanceswithoutphysicalreferents.

Attitudestowardstheontology ofem pty spacetim e arein no shortsup-

ply either. These arise prim arily as a sequelofEinstein’s hole argum ent,

which,looking atwhatisleftbeyond them etricstructureofspacetim e(i.e.

its quantitatively geom etric sca�old) concluded that what rem ains,space-

tim e points(i.e.itsqualitatively geom etric constituents),have no physical

m eaning.Asa falloutofthisargum ent,professionalphilosophersand philo-

sophically m inded physicists continue to devise,as far as logic willallow,

ingenious stratagem s to furnish the geom etric objecti�cation ofspacetim e

with new m eanings(substantivalism );orthen to reassure usthatEinstein

wascorrectin the�rstplace,thatspacetim epointshavein e�ectnophysical

m eaning,and thatrelationsbetween particlesand �eldsin spacetim e isall

there is to it (relationalism ).4 Butis a solution to be sought by m eans of

4Fora review,see e.g.(Earm an,1989,esp.pp.175{208).


an increase in ourknowledge within the very geom etric tradition thatgave

rise to the problem ? Ifthese philosophicalbattles have no end in sight,it

isbecause,in allprobability,the concepts used to seek solutionslie atthe

rootofthe di�culties.

Addressing theproblem ofspacetim e ontology from a di�erentperspec-

tive| yetwith equallydoubtfulphysicalrelevance| we�ndquantum -gravity

endeavours.Here,dueto conceptualdiscontentwith thegeneral-relativistic

spacetim e picture,com plicated geom etric edi�ces representing new space-

tim e structuresare industriously builtin an alm ostexclusively m athem at-

icalspirit;only subsequently is physicalm eaning sought for these psycho-

logically com pelling geom etric creations,butwith starting pointsbased on

observationally disem bodied concepts,physicalm eaning islaterhard to�nd

regardlessofhow \beautiful" or\self-consistent" these geom etric creations

m ay be. As we said in Section 2.2,not even the quantization ofgeneral

relativity,when m athem atically viable,isa reliable starting point,so long

astheconsequencesofquantization failto beattested in thephysicalworld.

7.3 Psychologicaldi� culties: T im e and space

Besides�ndinginspiration in thephysicalworld ratherthan in purely m ath-

em aticalrealm s,quantum gravity m ay also bene�tfrom reconsidering the

possibilities contained in the elem entalquestion that drives it,\W hat are

spaceand tim e?" Thisquestion isalwaystacitly assum ed tobesynonym ous

with \W hatisthe geom etry ofspace and tim e?" butthisreading only cir-

cum scribesresearch to a narrow context.Isitthoroughly unthinkablethat

there m ay be m ore aspects to the physicalworld than those that can be

captured in geom etric language? W e see in thislightthat\W hatarespace

and tim e?" isneither,asweknow,asim plequestion toanswerbutnoteven,

asSm olin (2001) suggests,\the sim plestquestion to ask" (p.1). Research

stilluniquely focuses on the geom etry ofspace and tim e,while rem aining

obliviousto any alternatives.

As for the geom etric answers o�ered,it is noteworthy that the treat-

m ents given to space and tim e are in generaldisunited,and that,m ore-

over,neither treatm ent is as radically new,unconventional,or di�erent as

thecharacterizationsm adeby quantum -gravity researcherswould have one

believe. Although quantum gravity endeavours for an elucidation of the

quantum -m echanicalstructure ofspace and tim e,the classicalconcept of

tim easan externalone-param eterisso deep-rooted in physics,and so m uch

m oresothan theclassicalconceptofspaceasathree-dim ensionalcontinuum

ofpoints,thatitiscom m on forquantum -gravity venturesto inventnotions

ofdiscreteorquantized spacethatarenon-classicalto varying degreesbut,

when it com es to \dynam ics," som e form ofexternaltim e m arking para-

m etric evolution issurreptitiously added to the picture. The psychological

di�culty to think ofspaceotherthan geom etrically (asa m etricnetwork of

points),and oftim ein waysotherthan geom etrically (asa param etricline)

and classically (asan externalparam eter)revealsitself,in di�erentways,in

7.3 P sychologicaldi� culties 71

allgeom etric roadsto quantum gravity.

In string theory and m ost pregeom etries,space is depicted less classi-

cally than tim e,yetclassically enough.W hetherdescribed asa wrapped-up

m ultidim ensionalm etric m anifold,orasa graph,a network,a lattice,ora

causalset,the di�erence with the classicalnotion ofspace is only one of

adding dim ensions,changing the topology and m etric,orm aking a contin-

uum discrete in literal-m inded ways.Asfortim e,dynam icevolution either

stilltakesplacealongacontinuousparam eteror,when any new treatm entis

given to tim eatall,thisisattheoutsidepostulated to bediscrete(in steps

ofPlanck’s tim e, naturally),but stillan externalparam eter nonetheless.

Thediscretization ofboth space and tim e norm ally com esfrom forcing the

classicalnotionsto com ply with com m on preconceptionsin sim ple-m inded

m anners,ratherthan astheresultofa process(e.g.quantization)or,better

still,som e observationally inspired idea thatwould naturally lead to it.

In loop quantum gravity,on the other hand,space is represented by a

quantum -m echanicalstatejsi,called a spin network,which isnotputin by

hand butarisesfrom a quantization ofgeneralrelativity.5 In thissenseand

unlikein theinstancesm entioned above,spaceasperloop quantum gravity

m ay be called a genuinely non-classicalspace.6 It is here,then,that the

contrastwith thetreatm entoftim e standsoutin sharpestrelief.Although

Rovelli(2004)insistson presenting a \tim eless" loop quantum gravity,this

isfarfrom true.In hisbook Quantum gravity,a m agnum opus born ahead

oftim e,loop quantum gravity7 is presented in term s ofa m athem atical,

unobservableparam eter� which actstoorderspin networksand alongwhich

thesestatesofspaceevolvedynam ically.8 O nem ay callparam eter� tim e,or

onem ay callitwhatonewill;Rovelli,determ ined to do withouttim ecom e

what m ay,calls it \an artifact ofthis technique" (p.112) referring to the

Ham iltonian form alism .Tim easan unobservableparam eteralso appears|

now withoutanyproviso| asacoordinatevariabletwhen dynam icevolution

isrepresented by m eansofthespin-foam (i.e.sum -over-histories)form alism ;

spin networksaresaid to beem bedded in a four-dim ensionalspacetim eand

to m ove \upward along a ‘tim e’coordinate" (p.325). Finally, tim e also

appears as a partialobservable q (also t);ofallofthe above,this is the

only type oftim e param eterthatcan bem easured directly by m acroscopic

system scalled clocksbut,beingexternal,itisuselessasan intrinsicm easure

oftim e forany \quantum -gravitationalsystem " purportedly atthe Planck

scale.(In the m acroscopic theory ofrelativity,clock m easurem entsdo play

5The question rem ains,however,to what degree the state vector itselfis \put in by

hand" (orbetter:by m ind)in a quantum theory.W e give an answerin Chapter12.6Exam plesoflessliteral-m inded pregeom etriesin thissenseareprovided by D adi�cand

Pisk (1979)and Antonsen (1992),who view graphsthem selvesasquantum statesjG i;and

by Jaroszkiewicz(2001),whoconsidersa topologicalnetwork whoseelem entsarequantum

statesj iand testoperators �.7Presum ably the quantum theory ofgravity? So m uch forthethreeroadsto quantum

gravity| oreven two!(Cf.Rovelli’squotation on page 73.)8M ostnotably,theprobability am plitudeW (sjs

0)ofatransition from js

0itojsiisgiven

in term sof� ashsjR

exp(i�H )d�js0i,with suitable integration lim its in �,and where H

isthe Ham iltonian.


an essentialphysicalrole,aswe see in Chapter10.)

Despite the psychologicaldi�culties involved in dealing with the con-

cept oftim e other than classically,it is com m on for researchers to ignore

the m atter(whetherpurposefully orunwittingly),presenting theirtheories

asifin them spaceand tim ewereon equally non-classicalfootings.W itness

the following com m entsgoing back and forth withoutryhm e orreason be-

tween \space" and \space and tim e," or\space" and \spacetim e": Sm olin

(2001) writes,\Is there an atom ic structure to the geom etry ofspace and

tim e...? [...] Both loop quantum gravity and string theory assert that

there isan atom ic structure to space" (pp.102{103,italics added;see also

p.170);in the sam e vein,Rovelli(1998) says,\A knotstate isan elem en-

tary quantum ofspace. In thism anner,loop quantum gravity tiesthe new

notion ofspace and tim e introduced by generalrelativity with quantum m e-

chanics" (p.19,italics added);last butnot least,W einberg (1999) writes,

\Thenatureofspaceand tim e m ustbedealtwith in auni�ed theory.Atthe

shortestdistancescales,space m aybereplaced byacontinually reconnecting

structureofstringsand m em branes| orby som ething strangerstill" (p.36,

italics added). And tim e? Is tim e,in the end,just a classicalparam eter

even in concept-sweeping quantum gravity? O r is there any non-classical,

non-geom etric conceptoftim e intrinsic to quantum -m echanicalsystem s?

A step awayfrom thisconceptualcon�nem entwastaken byJaroszkiewicz

(2001).Hepictured thepassageoftim ein anon-param eterand intrinsically

quantum -m echanicalway,asquantum ticks determ ined by the irreversible

acquisition ofinform ation atevery quantum -m echanicalm easurem ent. Al-

though attractive,this idea is potentially treacherous as long as it is con-

nected with thegeom etricform ulation ofquantum theory.Forifa quantum

tick corresponds to the collapse of the state vector, one could| enslaved

to geom etric thinking and param eter tim e| im m ediately ask: how,where,

and when doesj (t)icollapsewhen a m easurem entoccurs? Ultim ately,the

idea ofa quantum tick and the geom etric state-vectorform alism are m utu-

ally uncongenial.In Chapter12,we leave geom etry behind in orderto �nd

a non-param eter concept oftim e that is intrinsic to quantum -m echanical

system sand from which the origin oftim e tcan beunderstood.

To posethequestion,\W hatarespaceand tim e?" and,withheld by the

shacklesofthe geom etric anthropic principle,attem pta thousand and one

geom etric answers to it rem inds one ofthe irony in Shakespeare’s Ham let

words: we could be bounded in a theoreticalnutshelland count ourselves

kingsofan in�nitespace oftheoreticalpossibilities.

7.4 Experim entaldi� culties: T he Planck scale

Anotherinstance ofthe grip ofgeom etric thoughton quantum -gravity the-

orizing isrevealed by the notoriousissueofthe Planck scale.Asm any and

varied asthegeom etric approachesto quantum gravity m ay be,a detached

observerisgreatly surprised to �nd thatalltheoreticalpicturesnevertheless

agree on a seem ingly elem entalpoint: the relevance ofthe Planck scale.

7.4 Experim entaldi� culties 73

Theobserver’ssurprisem ountsashehearsofthepettinessofthisscale,and

wonders why allthese endeavours into the unknown should im pose such

stringentlim itson theirarea ofrelevance,awkwardly cornering theirexper-

im entalhopes to the probing ofthe secluded,far-o� depths ofspace and

tim e(and heightsofenergy).Doesthisagreem entstem from a shred ofde-

pendableshared knowledge obtained via observation ofthephysicalsystem s

ofinterest? O rdoesitratherstem | in the absenceofany guiding observa-

tionsoreven positiveidenti�cation ofthephysicalsystem understudy| from

shared ignorance com bined with thedesireto yetm akea statem ent? Aswe

learntin Chapter2,therelevanceofthePlanck scaleto quantum gravity is

notjusti�ed by the use ofdim ensional,quantization,and ad hoc analyses:

theirapplication ism eaningfulonly after thesystem studied iswellknown,

butratherhopelessin theabsence ofpreviousknowledge ofthese system s.

In view ofits shaky foundations,why is yet so m uch thrust put into

the essentialnessofthisscale? A deeperanswerm ay behere considered by

takingintoaccountthepsychologicalm ake-up ofthehum an researcher.The

hum an m ind,asitseeksto reach outbeyond thelim itsofknowledge,craves

nonetheless the fam iliarity ofm echanicalconcepts and intuition,the ease

ofgeom etric tools,and the com fort and safety ofgeom etric containm ent.

In thisparticularinstance,thisled to itstrapping itselfin the m arginality

ofa 10�105 -cubic-m etre cage| and intellectualsanctuary,asitwere| from

which itisunwilling to peek out,m uch lessescape.In thislight,thePlanck

scalestandstoday astheleitm otiv ofhum an beings’psychologicalneedsas

they attem ptnew physics.Itisthescalethatcam em osthandy to do a job

already in direneed ofbeing�lled| ifthegravitationalconstantG ,Planck’s

constanth,and thespeed oflightchad notcom bined to produceit,another

scale would have been ordained to take thevacantrole.

But,alas! So long as we continue to be ushered to \get in lane now"

by following either one of the \two obvious routes to quantum gravity"

(Chalm ers,2003)hand in hand with thePlanck scaleand itspresum ed phe-

nom enology,and so long aswe rejoice in notbeing \lostin a m ultitude of

alternative theories" because we have \two welldeveloped,tentative theo-

ries ofquantum gravity" (Rovelli,2003,p.37);in other words,so long as

reiteration and endorsem entofconventionalideasby largecom m unitiescon-

tinuesto be taken asa certi�cate ofinnovativenessand worthiness,and so

long asself-assured exposition ofphysically uncertain ideascontinuesto be

m istaken forrelevanceand correctness,weshallrem ain farfrom thegenuine

revolution in physicalthoughtwhoseabsolutenecessity isyettheinclination

ofm ain-road Planck-scale-bound researchersto insistently proclaim .

But while the value ofthis attitude to physics is doubtful,its extra-

physicalvalues are clearer,for,as Hardin (1993) haspointedly putit,\In

thelearned world,asin others,repetition and self-advertizem entcontribute

m ightily to the am assing ofa reputation" (p.223). After all,as another

critic ofscience said,\in science,...despite appearances,a discovery needs

credentialslouderthan itsown m erit" (Lem ,1984a,p.50).

C hapter 8

B eyond geom etry

Describing the physicallaws without reference to geom etry

issim ilar to describing our thoughtwithoutwords.

AlbertEinstein,\How Icreated thetheory ofrelativity"

Throughouttheprecedingchapters,wewitnessed how essentialandubiq-

uitous geom etry is in hum an theorizing. This notwithstanding,it is even

this cherished realm that,as early as over a century ago,som e proposed

m ay need to be transcended in the search for new physics| whether it be

a bettertheory ofspacetim e structureor,m oregenerally,theuncovering of

a deeperlayer ofthe nature ofthings. To this rare attem pt to go beyond

geom etry,stated in each case with varying degrees ofexplicitness,belong

the viewsespoused by Cli�ord in 1875,by Eddington in 1920,by W heeler

in thedecadesbetween 1960 and 1980,and by Einstein in thelatterpartof

hislife.

O fthese fourscholars,only W heeler m akes an explicitcase forthe su-

perseding ofgeom etry in physics with his well-known proposals for a pre-

geom etry. Cli�ord and Eddington,on the otherhand,do notm ake a very

explicitcase fortranscending geom etric thought,although one can see the

seedsofthisidea in their works,too. Einstein,forhispart,doesnottake

issuewith geom etry in particular,butworriesin generalaboutthenefarious

e�ectsofbeing unconsciously ruled by ourwell-established conceptualtools

ofthought,whateverthese m ay be.

Einstein’sre ectionsbring up the question ofthe in uence oflanguage

on thought,a notorious issue that has been m ost fam ously dealt with by

linguistBenjam in L.W horf. Hisviews,when conservatively interpreted in

the light ofthe unchallenged reign ofgeom etry in physics,provide a new

angleto understand theorthodox behaviourofphysicistscurrently engaged

in thecreation ofnew physicsand,in particular,to uncoveranotherreason

behind thefailureofpregeom etry to follow thedirectionsm arked outby its

own road m ap.

8.1 C li� ord’s elem ents offeeling 75

8.1 C li� ord’s elem ents offeeling

Viewsthatseem to advocate the provisionalcharacter ofgeom etric expla-

nationsin physicaltheoriesdate atleastasfarback as1875.Forthen,not

only did Cli�ord express his better-known beliefthat m atter and its m o-

tion are in factnothing butthe curvature ofspace| and thusreducible to

geom etry| but also lesser-known ideas about m atter and its m otion| and

so perhaps,indirectly,geom etry| being,in turn,only a partialaspect of

\the com plex thing we callfeeling."

W ithin the contextofa generalinvestigation in term sofvortex-atom s,

the m echanicalether,and other speculative ideas ofthe day| ideas which

the m odern physicistfeelsarcane butwhich,atthe sam e tim e,give rise in

him to the om inousthoughtwhetherthisisthe way m ostoftoday’sspec-

ulation willlook a century from now| we rescue an idea thatwe still�nd

m eaningfultoday. Cli�ord (1886b,pp.172{173) asked aboutthe existence

ofsom ething thatisnotpartofthe\m aterialorphenom enalworld" such as

m atterand itsm otion,butthatisits\non-phenom enalcounterpart," som e-

thing to which one can ascribe existence independently ofourperceptions.

Cli�ord wrote:

Theanswerto thisquestion isto befound in the theory ofsen-

sation;which tellsusnotm erely thatthereisa non-phenom enal

counterpart ofthe m aterialor phenom enalworld,but also in

som e m easure what it is m ade of. Nam ely, the reality corre-

spondingtoourperception ofthem otion ofm atterisan elem ent

ofthecom plex thing wecallfeeling.W hatwem ightperceiveas

a plexusofnerve-disturbancesisreally in itselfa feeling;and the

succession offeelingswhich constitutesa m an’sconsciousnessis

the reality which produces in our m inds the perception ofthe

m otions ofhis brain. These elem ents offeeling have relations

ofnextnessorcontiguity in space,which are exem pli�ed by the

sight-perceptions ofcontiguous points;and relations ofsucces-

sion in tim e which are exem pli�ed by allperceptions. O ut of

thesetwo relationsthefuturetheoristhasto build up theworld

asbesthe m ay.(Cli�ord,1886b,p.173)

In an essay published threeyearslater,Cli�ord (1886a)spellsoutthem ean-

ingheattributestothephrase\elem entary feeling" and thelogicalinference

leading him to seethem astheprim ordialconstituentsoftheworld.Hear-

guesthatasthe com plex feelings thatconstitute ourconsciousnessalways

occurin parallelwith m aterialnerve-disturbancesin thebrain,and because

the evolution oflife form sa continuum from inorganic m atterto the high-

estlife form s,a non-arbitrary line cannotbe drawn only pastwhich one is

justi�ed in inferring theexistenceofconsciousfeelingsin otherorganism s|

i.e.ejects,facts ejected from one’s consciousness and attributed to others.

Cli�ord concludesthat\every m otion ofm atterissim ultaneouswith som e

ejective factoreventwhich m ightbepartofa consciousness" (p.284),but

76 B eyond geom etry

that in its m ost elem entary form is not and does not need a conscious-

nessto supportit. He arrives thusat the idea ofan elem entary feeling as

som ething non-phenom enal,som ething \whose existence is not relative to

anything else" (p.284).

Cli�ord pursued afterthisfashion theconviction thatm atterand m otion

do not lie at the bottom of things, but that, on the contrary, they are

them selves an aspect ofdi�erent entities| elem entary feelings| and their

m utualrelations, contiguity and succession, of which the contiguity and

succession ofm atter are an im age. Ifone so chooses,both the said basic

constituents and their two relations m ay be understood to be geom etry-

unrelated, as neither form nor size is attributed to the form er or to the

latter. In fact,this we shalldo when,in Chapter 12,we take Cli�ord’s

dictum in the lastsentence quoted above in earnest,and try \to build the

quantum -m echanicalworld asbestwe m ay" outofm etageom etric physical

things,bearing a resem blanceto consciousfeelings,and thetwo relationsof

nextnessand succession reinterpreted in a strictly m etageom etric sense;in

particular,withoutany reference to space or tim e. It is in these elem ents

offeeling,then,in these thingsbeyond m atterand possibly geom etry,that

Cli�ord believed onecan catch aglim pseofadeeperlayeroftheconstitution

oftheworld.

W hen in addition weconsiderCli�ord’s(1886b)furtherwordsindicating

a possibleway to apply theidea of\contiguity offeelingsin space," nam ely,

\There are lines ofm athem aticalthought which indicate that distance or

quantity m ay com eto beexpressed in term sofposition in thewidesenseof

analysis situ" (p.173),the allusion to dispensing with typically geom etric

concepts such as distance becom esclearer still. O ne m ay then say thatin

Cli�ord we�nd the(perhapsunwitting)germ oftheideatom oveaway from

geom etric conceptsin physics| indeed,a sortofforefatherofthe lateridea

ofpregeom etry.

8.2 Eddington’s nature ofthings

Forty-�ve years later,Eddington (1920,pp.180{201) expressed viewsin a

sim ilarspiritasabove regarding whathecalled thenatureofthings,while,

atthesam etim e,m oreexplicitly casting doubton theextentofthepowers

ofgeom etric theorizing. In analysing generalrelativity, he identi�ed the

point-eventsasitsbasicelem ents,and theintervalasan elem entary relation

between them .Asto the natureofthelatter,Eddington wrote:

Its [the interval’s]geom etricalproperties...can only represent

one aspectoftherelation.Itm ay have otheraspectsassociated

with features ofthe world outside the scope ofphysics. Butin

physicsweareconcerned notwith thenatureoftherelation but

with the num berassigned to expressitsintensity;and thissug-

gestsa graphicalrepresentation,leading to a geom etricaltheory

ofthe world ofphysics.(Eddington,1920,p.187)

8.2 Eddington’s nature ofthings 77

Along these lines(in agreem ent with the idea thatsize inducesshape),he

suggested thattheindividualintervalsbetween point-eventsprobablyescape

today’sscalesand clocks,thesebeing too rudim entary to capturethem .As

a consequence,in generalrelativity one only dealswith m acroscopic values

com posed of m any individualintervals. \Perhaps," Eddington ventured

furtherin an allusion to transcending geom etric m agnitudes,

even the prim itive intervalisnotquantitative,butsim ply 1 for

certain pairsofpoint-eventsand 0 forothers.Theform ula given

[ds2 = g��dx�dx�]isjustan averagesum m ary which su�cesfor

ourcoarse m ethodsofinvestigation,and holdstrue only statis-

tically.(Eddington,1920,p.188)

O ne should notbe confused by the association ofthe num bers1 and 0 to

theprim itiveintervalsand think thatsuch num bersrepresenttheirlengths.

In thepassage,thenon-quantitativeness oftheprim itiveintervalsisclearly

stated.W hatthe1 and 0 representism erely whethertheintervalsexistor

not,any other designation being equally satisfactory forthe purpose. Ed-

dington’sinsightinto the need to transcend geom etric notionsisreinforced

by hisfurtherrem arks:\[W ]ecan scarcely hopeto build up a theory ofthe

natureofthingsifwe take a scale and a clock asthe sim plestunanalysable

concepts" (p.191).Itisnothing shortofrem arkableto �nd expressed such

views,on theonehand,inspired in thegeneraltheory ofrelativity while,on

the other,in a sense contrary to its spirit,only �ve years after the publi-

cation ofthe latter.Eddington’savowal,however,only seem sto beforthe

overthrow ofgeom etric m agnitudes but not ofgeom etric objects,since he

doesnotseem to �nd faultwith the conceptsofpoint-events and intervals

aslines.

Eddington also stressed therole ofthehum an m ind in the construction

ofphysicaltheories. He argued that,out ofthe above prim itive intervals

(taken aselem entsofreality and notofa theory ofit)between point-events,

a vastnum berofm ore com plicated qualitiescan arise;asa m atteroffact,

however, only certain qualities of allthe possible ones do arise. W hich

qualitiesareto becom eapparent(in one’stheories)and which notdepends,

according to Eddington,on which aspectsofthese elem entary constituents

oftheworld (and notofa theory ofit)them ind singlesoutforrecognition.

\M ind �ltersoutm atterfrom them eaninglessjum bleofqualities," hesaid,

as the prism �lters out the colours of the rainbow from the

chaotic pulsations ofwhite light. M ind exalts the perm anent

and ignoresthe transitory... Isittoo m uch to say thatm ind’s

search forperm anencehascreated theworld ofphysics? So that

the world we perceive around uscould scarcely have been other

than itis? (Eddington,1920,p.198)1

1Eddington talks about the m ind creating the world ofphysics,but one �ndssuch a

thing a daring exploit,forhow could the m ind create the physicalworld? O rto putitin

the wordsofD evittand Sterelny (1987),\how could we,literally,have m ade the stars?"


In thesewords,we�nd theechoofaprovokingstatem entwem adeearlier

on,nam ely,that \snow akes are not inherently hexagonal," butthat it is

ourm indswhich,to usenow Eddington’sphrasing,�lterouttheirgeom etry

from what are to us a rem aining jum ble ofindiscernible qualities. M ind

exaltsgeom etry and ignoresthenon-geom etric| so m uch so thatno word is

aptto describethatwhich isignored otherthan by clum sy negation.M ind

exaltsand �ltersoutgeom etry because itiswhatitknowshow to do,just

as a prism naturally decom poses white light according to the frequency of

itscom ponentparts.W hitelight,however,m ay by theuseofothersuitable

instrum ents be understood not in term s ofcom ponent frequencies,but in

term s ofits intensity or its polarization. Since we only have one kind of

intellect at our disposal, in order to �nd out what m ay be the possible

counterpartsofintensity and polarization in thiscom parison,wem ustlearn

to apply the m ind to the physicalworld in new ways.

Asfortheway thingsstand today,paraphrasingEddington’slast-quoted

question,we ask: isittoo m uch to say thatm ind’s geom etric instinct has

created allthe currenttheoriesofthe physicalworld? So thatourtheories

ofthe world could scarcely have been otherthan they are?

8.3 W heeler’s pregeom etry

W heeler(1964,1980),M isneretal.(1973),and Patton and W heeler(1975)

expressed pioneering ideas on what they called pregeom etry already four

decades ago. Since we understand W heeler to be by farthe m ain contrib-

utorto thisidea,the kind ofpregeom etry analysed in thissection iscalled

W heeler’spregeom etry.In ordertoavoid confusion,werem arkyetagain| in

addition to the early com m entsofChapter6| thatW heeler’spregeom etry

is not really whatlater on cam e to be known by the sam e nam e,i.e.pre-

continuum physics,butthe m ore radicalstance ofgoing beyond geom etry

in a propersense.

His basic dem and am ounts to the rejection of geom etric concepts in

order to explain geom etric structure. Indeed,W heeler (1980) advocated

\a conceptofpregeom etry thatbreaksloose atthe startfrom allm ention

ofgeom etry and distance," he was wary ofschem es in which \too m uch

geom etricstructureispresupposedtolead toabelievabletheoryofgeom etric

structure," and he clearly recognized that \to adm it distance at allis to

give up on the search for pregeom etry" (pp.3{4). This m eans that one

surprisingly �ndsthevery inventorofpregeom etry generally invalidating all

theschem esanalysed in Chapter6,asfarasthey areto bean expression of

hisoriginalpregeom etry.

W hat are the grounds for these strong pronouncem ents ofW heeler’s?

Firstly,heenvisaged,2 am ongotherthings,spacetim ecollapsein theform of

thepresum ed bigbangand bigcrunch,black holes,and asupposed foam -like

(p.200).Rather,whatthem ind createsistheoriesof theworld and into theseinventions

itputs,naturally,allitspartialitiesand prejudices.2See (Patton & W heeler,1975,p.547)and (M isneretal.,1973,p.1201).

8.3 W heeler’s pregeom etry 79

structure ofspacetim e on sm allscales,asindicatorsthatspacetim e cannot

beacontinuousm anifold,sinceithastheability tobecom esingular.Feeling

forthesereasonsthatthespacetim econtinuum needed rethinking,heposed

thequestion:

Ifthe elastic m edium is built out ofelectrons and nucleiand

nothing m ore,ifcloth isbuiltoutofthread and nothing m ore,

we are led to ask out ofwhat \pregeom etry" the geom etry of

space and spacetim e are built.(W heeler,1980,p.1)

A readingofW heelersuggeststhatthereason why heseeksa\pregeom etric

building m aterial" (M isner et al., 1973, p.1203) in order to account for

thespacetim econtinuum instead ofsim ply a \discretebuilding m aterial" is

that he believes that genuine explanations about the nature ofsom ething

do not com e about by explicating a concept in term s ofsim ilar ones,but

by reducing it to a di�erent,m ore basic kind ofobject. Thisis evidenced

in the passage: \And m ust not this som ething,this ‘pregeom etry,’be as

far rem oved from geom etry as the quantum m echanics ofelectrons is far

rem oved from elasticity?" (p.1207).

W hatwereW heeler’sproposalsto im plem entthisprogram m eto go be-

yond geom etry? His attem pts include pregeom etry as binary-choice logic,

pregeom etry asa self-referentialuniverse,and pregeom etry asa Borelsetor

bucketofdust.3

An early attem ptatpregeom etry based on binary choice wasW heeler’s

\sewingm achine," into which ringsorloopsofspacewerefed and connected

orleftunconnected by them achineaccordingtoencoded binary inform ation

(yesorno)foreach possiblepairofrings(pp.1209{1210).W heelerconsid-

ered thecoded instructionstobegiven from withouteitherdeterm inistically

orprobabilistically,and then asked whetherfabricsofdi�erentdim ension-

ality would arise,and whetherthere would be a higherweightforany one

dim ensionality to appear. As a design for pregeom etry,this idea succeeds

in staying clear ofgeom etric m agnitudesbutfails,atleastsuper�cially,to

avoid geom etric objects: W heeler’sloopsofspace have no size and yetare

rings.W e say thisfailureissuper�cialbecausetheseringsorloops,so long

asthey have no size,could as wellbe called abstract elem ents. Ifnothing

else,the m ention ofrings is witness to the fact that geom etric m odes of

thoughtarealwayslurking in the background.

A subsequent idea stillbased on binary-choice logic was pregeom etry

asthe calculusofpropositions(pp.1209,1211{1212). Thisconception was

m uch less picturesque and rather m ore abstract. W heeler exploited the

isom orphism between the truth-values ofa proposition and the state ofa

switching circuitto toy with theidea that\ ‘physics’autom atically em erges

from the statistics ofvery long propositions,and very m any propositions"

(Patton & W heeler,1975,p.598) in therm odynam ic analogy. In thatref-

erence and in (W heeler,1980),however,the expectation that pure m ath-

3See (D em aret,Heller,& Lam bert,1997,pp.157{161) for an independent review of

W heeler’spregeom etry.


em aticallogic alone could have anything whatsoever to do with providing

a foundation for physics was sensibly acknowledged as m isguided. In this

case,perhaps due to its abstractness and totaldivorcem ent with physics,

thisconception ofpregeom etry istruly freeofallgeom etry in a strictsense.

For allthis schem e was worth,it is at least rewarding to �nd it has this

feature,instead ofquickly proceeding to negate its intended pregeom etric

nature.

Later on,inspired in self-referentialpropositions,W heeler conceived of

the idea ofpregeom etry asa self-referentialuniverse.Aswith the previous

approaches,W heeler(1980)adm itted tohavingnom orethan avision ofhow

this understanding ofgeom etry in term s of(his) pregeom etry m ight com e

about.Twobasicingredientsin thisvision appeartobe(i)eventsconsisting

of a prim itive form of quantum principle: investigate nature and create

reality in sodoing(seebelow)and,m oreintelligibly,(ii)stochasticprocesses

am ong these events producing form and dependability outofrandom ness.

Hisvision readsthus:

(1)Law withoutlaw with no before beforethe big bang and no

afteraftercollapse.Theuniverseand thelawsthatguideitcould

nothaveexisted from everlasting to everlasting.Law m usthave

com einto being...M oreover,therecould havebeen no m essage

engraved in advance on a tablet ofstone to tellthem how to

com e into being. They had to com e into being in a higgledy-

piggledy way,astheorderofgenera and speciescam einto being

by theblind accidentsofbillionsupon billionsofm utations,and

asthe second law oftherm odynam icswith allitsdependability

and precision com esinto being outoftheblind accidentsofm o-

tion ofm olecules who would have laughed at the second law if

they had everheard ofit.(2)Individualevents.Eventsbeyond

law. Events so num erous and so uncoordinated that aunting

their freedom from form ula,they yet fabricate �rm form . (3)

These events,notofsom e new kind,butthe elem entary actof

question to natureand a probability guided answergiven by na-

ture,thefam iliareveryday elem entary quantum actofobserver-

participancy. (4)Billions upon billionsofsuch acts giving rise,

via an overpowering statistics,to theregularitiesofphysicallaw

and to theappearanceofcontinuousspacetim e.(W heeler,1980,

pp.5{6)

Earlier,Patton and W heeler(1975,pp.556{562)had elaborated furtheron

this.They considered two alternativesto theproblem ofspacetim ecollapse,

nam ely:(a)taking into accountquantum m echanics,theeventofuniversal

collapsecan beviewed asa \probabilisticscattering in superspace." In this

way,the universe isreprocessed ateach cycle,butthe spacetim e m anifold

retainsitsfundam entalm odeofdescription in physics.Thisalternativewas

notfavoured by theseauthors,butinstead (b)consideringasan overarching

guiding principle that the universe m ust have a way to com e into being

8.3 W heeler’s pregeom etry 81

(cosm ogony), they suggested that such a requirem ent can be ful�lled by

a \quantum principle" of \observer-participator."4 According to it, the

universe goes through one cycle only,but only ifin it a \com m unicating

com m unity" arises thatcan give m eaning to it. Thisuniverse is thusself-

referential,which wasthestandpointfavoured by these authors.5 Asfaras

pregeom etric ideasare concerned,thisvision ofa self-referentialcosm ology

succeedsin eschewing allkindsofgeom etric conceptsagain dueto itsbeing

conceived in such broad and generalterm s. Regardless ofits worth and

in the generalcontext ofparadoxicalgeom etric pregeom etry,let this be a

welcom e feature ofW heeler’se�ort.

RegardlessofW heeler’sactualim plem entationsofhispregeom etry and

any criticism sthereof,the im perative dem and thatthe foundationsofsuch

a theory should becom pletely freeofgeom etricconceptsrem ainsunaltered.

The factthatso m any (including,to the sm allestextent,W heelerhim self)

have m isconstrued his idea ofa program m e for pregeom etry is a tell-tale

sign ofsom ething beyond sim ple carelessness in the reading ofhis views.

Indeed,thelarge-scalem isinterpretation witnessed in Chapter6 m ay justas

wellstem from the lack ofallbackground forthoughtthatone encounters

as soon as one attem pts to dispense with geom etry. Physics knows ofno

other m ode of working than geom etry, and thus physicists reinterpreted

W heeler’s new idea in the only way that m ade sense to them | setting out

to �nd geom etry in pregeom etry.

8.3.1 M ore buckets ofdust

Astwo�nalclaim antstopregeom etry,weturn totheworksofNagelsand of

Stuckey and Silberstein in the lightoftheirclose connection with another,

partly new approach to pregeom etry by W heeler. In the works ofthese

authors,two independent attem pts are m ade to build again \space as a

bucketofdust" afterW heeler’s(1964,pp.495{499)earliere�ort.Thisear-

liere�ort,clearly connected with theabove-m entioned \sewing m achine" in

characterand purpose,consisted in starting with a Borelsetofpointswith-

outany m utualrelationswhatsoeverand assem bling them into structuresof

di�erentdim ensionality on thebasisofdi�erentquantum -m echanicalprob-

ability am plitudesattributed to the relationsofnearestneighbourbetween

the points. W heelerdism issed hisown trialbecause,am ong otherreasons,

thesam e quantum -m echanicalprinciplesused to de�neadjacency rendered

the idea untenable,since points that had once been neighbours would re-

m ain correlated afterdeparting from each other(p.498). W e observe once

m ore thatthisidea,in itsuse ofthe conceptofpoints(cf.earlierrings),is

notfreefrom geom etricobjects,butthatthis aw ism inim ized by W heeler’s

4This principle is tantam ount to the �rst ingredient (i) above,and perhaps helps to

clarify it.5Thislatterproposalseem stoconfusenaturewith ourtheoriesofit.Itisindeed correct

to say thatonly a com m unicating com m unity ofbeingscan give m eaning to the displays

ofnature,but it is rather som ething else to contend that a com m unicating com m unity

createstheworld in so doing;once again:\how could we,literally,havem adethestars?"


A uthor B asic objects B asic relations

Cli�ord feelings nextness,succession

Eddington points,intervals statisticalinteraction

W heeler

Sewing m achine rings nextness(bin.choice)

Borelset points nextness(prob.am p.)

Calculusofpropositions propositions statisticalinteraction

Self-referentialuniverse actofobservation idem

Table 8.1: Sum m ary oftheuseofbasicobjectsand relationsin thenon-geom etric

orpartly non-geom etricapproachesanalysed.G eom etricconceptsappearin italics.

Abbreviations:Binary (bin.),probability am plitude (prob.am p.).

avoidanceoftheintroduction ofany quantitativegeom etry.Asforthem en-

tioned quantum am plitudes,theseneed nothavegeom etricm eaning aslong

asthey do notarisefrom an innerproductofstatevectors.In Table8.1,we

sum m arize the use ofbasic objectsand relationsin the non-geom etric and

partly non-geom etric approacheswe have discussed.

Nagels’(1985) attem pt started o� with a schem e sim ilar to W heeler’s,

where\theonly structureim posed on individualpointsisa uniform proba-

bility ofadjacency between two arbitrarily chosen points" (p.545).Heonly

assum ed thattheseprobabilitiesarevery sm alland thatthetotalnum berof

pointsisvery large(and possibly in�nite).Hethusbelieved hisproposalto

satisfy thealwaysdesired requirem entof\a barem inim um ofassum ptions"

and very naturalonesatthat.However,m uch too soon hefellprey to quan-

titative geom etry.G iven thisisa pregeom etric fram ework directly inspired

in W heeler,itisstartling to com e acrossthefollowing early rem arks:

W ithouta background geom etry,the sim plestway to introduce

distance isto specify whetherornottwo given pointsare\adja-

cent." [...] W e m ay then say thattwo \adjacent" pointsare a

distance of1 unitoflength apart. (Nagels,1985,p.546,italics

added)

Forhispart,Stuckey introduced in hisattem pt

a pregeom etry thatprovidesa m etricand dim ensionality overa

Borelset(W heeler’s \bucketofdust")withoutassum ing prob-

ability am plitudes for adjacency. Rather,a non-trivialm etric

isproduced overa BorelsetX pera uniform ity base generated

via the discrete topologicalgroup structuresover X . (Stuckey,

2001,p.1,italicsadded)

Thus,both authorsm anaged to createpregeom etricm etrics,i.e.m etrics

thatwillgive a notion ofdistance for pregeom etry and by m eansofwhich

they hoped to obtain the usualspacetim e m etric. This assertion is sup-

ported by Stuckey and Silberstein’s(2000)words:\ourpregeom etricnotion

ofdistance" (p.9)and \theprocessby which thism etricyieldsa spacetim e

8.4 Einstein’s conceptualtools ofthought 83

m etric with Lorentz signature m ust be obtained" (p.13). Stuckey (2001)

m oreoverasserted thatthepregeom etriesofNagels,Antonsen,and Requardt

have,arguably,overcom e the di�cultiesrepresented by the presupposition

of\too m uch geom etric structure" as previously brought up by W heeler,

since their assum ptions are m inim aland \the notion oflength per graph

theory is virtually innate" (p.2). Arguably,indeed,because the assum p-

tion oflength is totally contrary to the spiritofpregeom etry according to

itsvery inventor,and itsintroduction actsto geom etrically objectify what

could have otherwisebeen pregeom etric schem es.

Alas,whathasbecom e ofW heeler’s reasonable dictum : \to adm itdis-

tanceatallisto giveup on thesearch forpregeom etry," orhisdem andsfor

\break[ing]loose atthe startfrom allm ention ofgeom etry and distance,"

and forpregeom etry to be \asfarrem oved from geom etry asthe quantum

m echanics of electrons is far rem oved from elasticity?" O r yet to quote

W heelerin two otherearly illum inating passages:

O ne m ight also wish to accept to begin with the idea ofa dis-

tance,oredgelength,associated with apairofthesepoints,even

though thisidea isalready a very greatleap,and one thatone

can conceive oflatersupplying with a naturalfoundation ofits

own.

[...][T]heuseoftheconceptofdistancebetween pairsofpoints

seem sunreasonable...[L]ength isanyway notanaturalideawith

which to start.Thesubjectofanalysishereis\pregeom etry," so

the conceptoflength should be derived,notassum ed ab initio.

(W heeler,1964,pp.497,499)

Hasallthissim ply gone into oblivion?

The question,indeed,rem ains| why the need ofa m etric forpregeom -

etry? Should not pregeom etry produce the traditionalspacetim e m etric

by non-geom etric m eans alone? Should it not go even \beyond W heeler"

and do without any sort of geom etric objects too? This state of a�airs

com esto show to whatextentthecraving ofthehum an m ind forgeom etric

explanations goes;at the sam e tim e,one catches a glim pse ofthe psycho-

logicaldi�cultiesthatm ightbeencountered in any attem ptthatgenuinely

attem ptsto go beyond such explanations,so deeply rooted in the m ind.

8.4 Einstein’s conceptualtools ofthought

Einstein,alongside his contributions to physics,also entertained thoughts

about the validity ofthe \conceptualtools ofthought" we use to portray

theworld.Hebeginstheforeword to M ax Jam m er’sConceptsofspace with

a crisp and clearre ection,which we herequote atlength:

In the attem pt to achieve a conceptualform ulation ofthe con-

fusingly im m ensebody ofobservationaldata,thescientistm akes

use ofa whole arsenalofconceptswhich he im bibed practically


with his m other’s m ilk; and seldom ifever is he aware ofthe

eternally problem atic character of his concepts. He uses this

conceptualm aterial, or, speaking m ore exactly, these concep-

tualtoolsofthought,assom ething obviously,im m utably given;

som ething having an objective value of truth which is hardly

ever,and in any case not seriously,to be doubted. How could

he do otherwise? [...] And yet in the interests ofscience it is

necessary overand overagain to engage in the critique ofthese

fundam entalconcepts,in orderthat we m ay notunconsciously

be ruled by them .Thisbecom esevidentespecially in those sit-

uations involving developm ent ofideas in which the consistent

useofthetraditionalfundam entalconceptsleadsustoparadoxes

di�cultto resolve.6

W esym pathizedeeply with thesewords,which eloquently bringforth oneof

theunderlying them esofthiswork when weidentify theconceptualtoolsof

thoughtreferred to with ouringrained and ubiquitousgeom etric concepts.

Einstein indirectly questioned certain geom etric conceptswhen,asPais

(1982,p.347) recounts,in the early 1940s he considered whether partial

di�erentialequationsm ay proveunsuitablefora m oreelem entaldescription

ofnature. Thisidea appearsto have been inspired in the face ofquantum

m echanics. Einstein (1935,pp.148,156) notes that quantum theory does

away with the possibility ofdescribing the world fundam entally in term s

ofspatiotem poral�elds subject to causallaws given by di�erentialequa-

tions,since quantum -m echanicalsystem s are described by state functions

thatare not�eldsin spacetim e and whose behaviourisnotstrictly causal.

Thisview isrelevantin thisconnection when weconsiderthattranscending

the �eld concepttogetherwith di�erentialequationsm eansleaving behind

the qualitative geom etry inherentin spatiotem poralpointsP (locality and

causality;cf.�eld as f(P )) and the quantitative geom etry inherentin the

values (physicalquantities) ofthe �eld.7 Although J.Ellis (2005), refer-

ring to Einstein’sidea,correctly assertsthat\M odern theoristscan hardly

be accused ofexcessive conservatism ,but even they have not revived this

startling speculation!"(p.56),itiseven thisstartling notion ofleaving �elds

and di�erentialequationsbehind thatshould bepartofaplan todowithout

geom etry entirely.

Finally,in this context ofreservation towards the lim itations im posed

by ourtoolsofthought,itisdi�cultto know how to interpreta deeperpro-

nouncem ent ofEinstein’s (1982) (which opens this chapter) now in direct

connection with geom etric conceptsin general.Explaining how hehad cre-

ated thetheory ofrelativity,hesaid:\Describing thephysicallawswithout

reference to geom etry is sim ilar to describing our thoughtwithout words"

(p.47).IsEinstein heresuggesting that,despitebeing possibly ruled by it,

doingphysicswithouttheassistanceofageom etriclanguageisashopelessly

inviable asdescribing ourthoughtswithoutwords,ordoeshe m ean to say

6In (Jam m er,1969,pp.xi{xii).

7See also Einstein’squotation on page 38.

8.5 W horf’s linguistic relativity 85

thatitwould be ashard,yetnotnecessarily fruitless,to do so? Perhapsit

iswisestnotto seek too m uch m eaning in theseotherwiserem arkablewords

and sim ply take them atface value,asa correctindication ofthe factthat

geom etry isindeed the only physicallanguage so faravailable| dispensing

with it,therefore,leavesuswordless.

However,thiscircum stanceneed notbeasinescapableasitseem s.Ein-

stein’scom parison only holdstrue untilthe non-geom etric wordsofa non-

geom etriclanguageareinvented and becom eavailable,allowingourphysical

thoughts to be expressed anew in term s ofthem . Further on in this the-

sis,weshallseethatthetask ofgoing beyond geom etry,although m entally

strenuous,isnotonly feasible butm oreoveressentialto bettercom prehend

som easpectsofthephysicalworld.

8.5 W horf’s linguistic relativity

LinguistBenjam in L.W horfisbestknown forhisnow notorious principle

oflinguistic relativity. A judiciousinterpretation ofthisprinciple provides

uswith quitea straightforward fram ework to understand thesourceofEin-

stein’sapprehensionsregardingourconceptualtoolsofthoughtand ourown

rem arksthatthoughtbeyond geom etry can be buttressed by a supporting

language: language,whetheritbe scienti�c orotherwise,isaptto greatly

in uencethoughtboth positively and negatively.W horfwrote:

Itwasfound thatthebackground linguisticsystem ...ofeach lan-

guage is notm erely a reproducing instrum entfor voicing ideas

butratherisitselftheshaperofideas...W edissectnaturealong

lines laid down by our native languages. The categories and

types that we isolate from the world ofphenom ena we do not

�nd there because they stare every observerin the face;on the

contrary,theworld ispresented in akaleidoscopic ux ofim pres-

sionswhich hasto be organized by ourm inds| and thism eans

largely by thelinguistic system sin ourm inds.

Thisfactisvery signi�cantform odern science,foritm eansthat

noindividualisfreetodescribenaturewith absoluteim partiality

butisconstrained to certain m odesofinterpretation even while

he thinkshim selfm ostfree...W e are thusintroduced to a new

principleofrelativity,which holdsthatallobserversare notled

bythesam ephysicalevidencetothesam epictureoftheuniverse,

unless their linguistic backgrounds are sim ilar,or can in som e

way becalibrated.(W horf,1956,pp.212{214)

O ne ought to be carefulas to what to m ake ofW horf’s rem arks,as their

signi�cance could go from (i)an im plication ofan im m utable constraint of

language upon thought with the consequent fabrication ofan unavoidable

world picturetowhich oneisled by thelanguageused,to(ii)m ilderinsinua-

tionsaboutlanguagebeing\only" ashaperofideasratherthan atyrannical

m aster.How faristhe in uenceoflanguage on thoughtto betaken to go?


In this,wesharetheviewsofDevittand Sterelny’s.In theirbook,they

explained thecircular| although notviciously so| processin which thought

and language interact;they wrote:

W e feela pressing need to understand our environm ent in or-

der to m anipulate and controlit. This drive led our early an-

cestors,in tim e,to express a prim itive thought or two. They

grunted orgestured,m eaning som ething by such actions.There

wasspeakerm eaning withoutconventionalm eaning.O vertim e

the grunts and gestures caught on: linguistic conventions were

born. Asa resultofthistrailblazing itism uch easier foroth-

ersto have those prim itive thoughts,forthey can learn to have

them from the conventionalways ofexpressing them . Further,

they haveavailablean easy way ofrepresenting theworld,a way

based on those conventionalgesturesand grunts. They borrow

theircapacity to think aboutthingsfrom thosewho created the

conventions.W ith prim itivethoughtm adeeasy,thedrivetoun-

derstand leadsto m ore com plicated thoughts,hence m ore com -

plicated speakerm eanings,hencem orecom plicated conventions.

(Devitt& Sterelny,1987,p.127)

In the�rstplace,thism eansthat,asonecould havealreadyguessed,thought

m ustprecede any form oflanguage,orelse how could the latterhave com e

intobeing? Secondly,itshowsthatthought,asasourceoflinguisticconven-

tions,isnotin any way restricted by language,although thecharacteristics

ofthelatterdo,in fact,facilitate certain form softhoughtby m aking them

readily available,in the sense thatthe thoughtprocessesofm any can now

bene�tfrom existing conceptsalready m ade by a few others. Thisisespe-

cially thecasein science,which aboundsin instancesofthiskind.Forexam -

ple,thoughtaboutcom plex num bersisfacilitated by the already invented

im aginary-num ber language convention \i =p� 1," just like the thought

ofa particle being atdi�erentplacesatthe sam e tim e isfacilitated by the

already invented state-vector-related conventionsofquantum m echanics.

At the sam e tim e, an existing language can also discourage certain

thoughts by m aking them abstruse and recondite to express (Devitt &

Sterelny, 1987, p.174). This does not m ean that certain things cannot

be thought| they eventually can be| but only that their expression does

notcom eeasily asitisnotstraightforwardly supported by theexisting lan-

guage.Asan exam ple ofthisinstance,we can m ention the reverse casesof

theaboveexam ples.Thatistosay,theevidentdi�culty ofconjuringup any

thoughtsaboutcom plex num bersand delocalized particlesbefore theform al

conceptsabovesupportingthesethoughtswereintroduced intothelanguage

ofscienceby som especi�cindividuals.Thefact,however,thatsuch linguis-

ticconventionswerecreated eventually showsthatthoughtbeyond linguistic

conventionsispossible| essential,m oreover,forthe evolution ofscience.

O neisthusled to theview thatthepicturethatphysicsm akesofnature

is not only guided by the physicist’s im agination, but also by the physi-

cist’s language| with its particular vices and virtues. This language,due

8.6 A ppraisal 87

to itsconstitution attained aftera naturaldevelopm ent,favoursgeom etric

m odes ofthought,while it appears to dissuade any sort ofnon-geom etric

contem plation. Pregeom etric schem es have shown thisfact clearly: im agi-

nation was able to produce varied creative fram eworks,butthey allspoke

the sam e geom etric language despite the attem ptto avoid it.O nly against

thislanguage-favoured background did physicists’m indsroam e�ortlessly.

The reason why hum ans have naturally developed prim itive geom etric

thoughts,leadingtogeom etricm odesofexpression,in turn reinforcingm ore

geom etric-likethinking,and so forth| in conclusion,developing a geom etric

understanding| can only be guessed at. It is reasonable to suppose that

geom etricinsightshould beconnected with thenaturalprovision ofan early

evolutionary advantage,ratherthan with a specially designed toolto devise

physicaltheories.Butaslittleasgeom etricinsightm ay havebeen designed

for high form s ofabstract conceptualization,it is the only insight we can

e�ortlessly availourselves of,and so the prospect ofdispensing with geo-

m etricthoughtin physicalscience isnotencouraging,forwhen geom etry is

lost,m uch islostwith it.W e close thissection,then,with theencouraging

wordsofotherswho also believed that,m entally strenuousasitm ightbe,

thisendeavourisworth pursuing:

And isnotthesourceofanydism ay theapparentlossofguidance

that one experiences in giving up geom etrodynam ics| and not

only geom etrodynam icsbutgeom etry itself| asa crutch to lean

on as one hobbles forward? Yet there is so m uch chance that

thisview ofnature isrightthatone m usttake itseriously and

explore itsconsequences.Neverm ore than today doesone have

theincentiveto explore[W heeler’s]pregeom etry.(M isneretal.,

1973,p.1208)

8.6 A ppraisal

A long way hasso farbeen travelled.Starting o� in therealm ofgeom etry,

we laid down its foundations in the form ofgeom etric objects and m agni-

tudes,and concluded thatphysicaltheoriesare woven outofthese concep-

tionsbecauseofa basichum an predilection to conceptualizetheworld after

a geom etric fashion. Indeed,we witnessed the essentially geom etric nature

ofallphysicaltheorizing,from Babylonian clay diagram sthrough Ptolem aic

astronom y to m odern science,and continuing with the attem pted im prove-

m ents oftwenty-�rst-century frontier physics despite the lack ofphysical

counterpartsforitslatestgeom etricentities,and despitethem ounting com -

plexity ofthegeom etric edi�cesbuiltupon them .

Furnishing a particularly conspicuous instance ofquantum -gravity re-

search,wevisited theabodeofpregeom etry.Although judging by itsnam e

and goals,pregeom etry wasthelastplacein which onewould haveexpected

to�ndgeom etry,theform erneverthelessfellpreytothelatter.Pregeom etry,

in itsneed foritsown geom etric m eansto explain theorigin ofgeom etry in


physics,cannotbutraisethedisquieting question:isthehum an m ind so de-

pendentupon geom etricm eansofdescription thatthey cannotbeavoided?

O rhaspregeom etry nottried hard enough? In any case,weconcluded that

pregeom etry hasfailed to live up to the sem antic connotationsofitsnam e

and to the originalintentions ofits creator,as wellas to the intentions of

itspresent-day practitioners,only to becom e a considerable incongruity.

Subsequently,wesurveyed thevistasofa land fartherbeyond.Di�erent

older suggestions for the need to overcom e geom etry in physicswere anal-

ysed,including those viewspertaining to thefatherofpregeom etry proper.

In general,thisproposalstem m ed from thedesireto reach into yetanother

layerofthenatureofthings,in conjunction with therealization thatevery-

day and scienti�c norm allanguage| i.e.geom etric language| were likely to

stand in ourway.

After this criticalgeneralsurvey ofthe geom etric physicaltradition|

from them istsofprehistory,ancientBabylonia and G reece,through therise

and golden age ofclassicalphysics,to contem porary m odern physicsup to

itspushingfrontiers| itisnow tim eto takea deeperlook atspaceand tim e

according to ourbestavailableknowledgeofthem .In otherwords,itisnow

tim etotakeastep,asitwere,backwardsand leavethespeculativedoctrines

ofquantum gravity behind to study whatclassicaland m odern physicshave

to say aboutthe physicalnature ofspace and tim e. Thisinexorably leads

usto theproblem oftheether.

C hapter 9

T he tragedy ofthe ether

The essence ofdram atic tragedy is notunhappiness. Itre-

sidesin the solem nity ofthe rem orseless working ofthings.

Alfred N.W hitehead,Science and the m odern world

Uponhearingtheword \ether,"them odernphysicist�ndshislipsalm ost

involuntarily form ing a knowing sm ile. The vision ofnineteenth-century

physicists with their convoluted theoretical pictures of this all-pervading

lum iniferousm edium and theirexperim entally vain attem ptsto observethe

m otion ofthe earth through it com es to his m ind,and he cannot butfeel

sorryforhisprecursorsem barkingon such awild-goosechase.Fortheywere,

afterall,only chasing a ghost:whatisthem aterialm edium in which light,

gravity,heat,and the electric and m agnetic forces propagate? Today we

know better.Thelum iniferousethersim ply doesnotexist.Electrom agnetic

and gravitationalinteractionsneed no m edium to propagate,and so wecan

putthisnotoriousidea,greatly troubled sinceitsvery inception,to rest.O r

can we?

Notreally.Contrary to com m on belief,theetherstillhauntsphysicsat

the turn ofa new century. Notin the form ofa m echanicalm edium | that

ideaisdead and buried| butin adi�erent,transm uted guise.Thatthisisso

isnotreadily acknowledged,becausethebeliefthatonly ourancestorswere

capableoffalling prey to theirown conventionalconceptionsoverpowersall

capability to look atourown position with anything resem bling objectivity.

O ur knowing sm iles, however, slowly give way to chagrin as we start to

recognize ourselves in the re ection ofthe past. W hat kind ofether still

hauntsphysics? A quick prelim inary survey ofthehistory ofthisconceptis

aptto naturally lead usto an outline ofthepresentpredicam ent.

Theneed foran etherasa m aterialm edium with m echanicalproperties

�rstbecam eapparenttoDescartesin the�rsthalfoftheseventeenth century

in an attem ptto avoid any actionsthatwould propagate,through nothing,

from a distance.During itshistory ofroughly threecenturiesin itsoriginal

conception,the idea ofthe ether m anaged to m aterialize in endless form s

via the works ofcountless investigators. Its m ain purpose ofproviding a

90 T he tragedy ofthe ether

m edium through which interactions could propagate rem ained untouched,

buttheactualpropertieswith which itwasendowed in orderto accountfor

and unifyan ever-increasingrangeofphenom enawerem utually incongruous

and dissim ilar.Neveryielding to observation and constantly confronted by

gruelling di�culties,them echanicaletherhad to reinventitselfcontinually,

butitsvery notion staggered nota bit.

Afteralm ost300yearsofbitterstruggle,them echanicalethereventually

gavein.The�rststep ofthischangetook placein thehandsofLorentz,for

whom the etherwasa sortofsubstantialm edium thata�ected bodiesnot

m echanically butonly dynam ically,i.e.due to the fact that bodiesm oved

through it. Drude and Larm or further declared that the ether need not

actually besubstantialatallbutsim ply spacewith physicalproperties.The

second and decisive step in thisdem echanization ofthe ether wasbrought

aboutbyPoincar�eand Einstein through ideasthateventually took theshape

ofthe specialtheory ofrelativity. In Einstein’s (1983a) own words,this

change\consisted in taking away from theetheritslastm echanicalquality,

nam ely,itsim m obility" (p.11).

Farfrom beingdead,however,theetherhad only transm uted itscharac-

ter| from a m echanicalsubstanceto an inertialspacetim e characterized by

an absolutem etricstructure.Theneed forregardinginertialspacetim easan

ethercam eafternoticing thatem pty spacetim e,despitebeing unobservable

and unalterable,displayed physicalproperties:itprovided an absolute ref-

erencenotforvelocity butforacceleration,and itdeterm ined thebehaviour

ofm easuring rodsand clocks,sim ilarto thefunction Newton’setherealab-

solute space had had before.

The natureofthisnew,special-relativistic etherunderwentyetanother

changewith thegeneraltheoryofrelativity,astheetherbecam ethedynam ic

and intrinsic m etric structure ofspacetim e (Einstein,1961,p.176). This

was so signi�cant a change that it m odi�ed the very ideas of ether and

em pty spacetim e.By m aking them etricstructurealterable,itactually put

an end to its status as a genuine ether. And by m aking the m etric �eld

a content ofspacetim e, it did away with em pty spacetim e, since now to

vacate spacetim e m eantto be leftwith no m etric structure and,therefore,

with nothing physicalatall:only an am orphous,unobservable substratum

ofspacetim e points,whose physicalreality was denied by Einstein’s hole

argum ent.From thisstandpoint,Einstein concluded thatem pty spacetim e

cannotpossessany physicalproperties,i.e.thatem pty spacetim e doesnot

exist.

W hatdegreeofcertainty can weattach tothisconclusion? Arespacetim e

points truly super uous entities,such that they neither act nor are acted

upon? Farfrom it. Spacetim e pointsP actby perform ing the localization

of�eldsf(P ),i.e.by giving thelocation P ofphysicalpropertiesf;in other

words,spacetim epointsarepartand parcelofthe�eld conception in physics,

on which theelem entalphysicalcategoriesoflocality and causality rest.And

yet spacetim e points cannot be acted upon such that their displacem ent

would produce a m easurable e�ect. So did the banishm ent ofthe second

9.1 T he m echanicalether 91

ether| the m etric ether| give way to em pty spacetim e and the birth ofa

third ether| the geom etric ether.

Einstein’s conclusion as to the physicalunreality ofem pty spacetim e

m ustbeviewed with caution also asa resultofm oregeneralconsiderations.

Thisconclusion restssolely on theclassical,geom etricfram ework ofthegen-

eraltheory ofrelativity,and thisisallvery well,butifweseek betterphysics

weshould notrestrictourselvesunduly.M ay quantum theory haveanything

new to add to theconceptofspacetim e pointsand em pty spacetim e? M ay,

m oreover,the adoption ofa m etageom etric outlook reveala deeperlayerof

thisontologicalproblem ,currently hiding beneath itsgeom etric facevalue?

Bethisasitm ay,when heeding thehistory oftheether,onething isatany

rate certain: forem pty spacetim e to lay any claim on physicalreality,the

recognition ofobservablesisan unconditionalm ust.

9.1 T he m echanicalether

W estartby tracing therich history oftheether,starting herefrom itsolder

conception as a m aterialsubstance,passing through its virtualdisappear-

ance afterthe progressive rem ovalofitsm echanicalattributes,and ending

with itsnew form ofan im m aterialm etricsubstratum ,analysed in thenext

section. For the historicalreview ofthis section,the very com prehensive

work ofW hittaker(1951)willbefollowed asa guideline.1

Ren�eDescartes(1596{1650)wasthe�rstto introducetheconception of

an etherasa m echanicalm edium . G iven the beliefthataction could only

be transm itted by m eans ofpressure and im pact,he considered that the

e�ects at a distance between bodies could only be explained by assum ing

the existence ofa m edium �lling up space| an ether. He gave thusa new

m eaning to this nam e,which in its originalG reek (�;

��) had m eant the

blue sky orthe upperair;however,the elem entalqualities the G reekshad

endowed this concept with,nam ely,universalpervasiveness (above the lu-

narsphere)and indestructibility (cf.una�ectability),2 rem ained untouched

throughoutitsm odern history. Descartes’ether wasunobservable and yet

itwasneeded to accountforhism echanistic view ofthe universe,given his

assum ptions.3 Atthe sam e tim e,the notion ofan etherwasrightfrom its

inception entwined with considerationsaboutthetheory oflight.Descartes

him selfexplained thepropagation oflightasatransm ission ofpressurefrom

a �rsttype ofm atterto befound in vorticesaround starsto a second type

ofm atter,thatofwhich hebelieved the etherto beconstituted (pp.5{9).

1Page num bers in parentheses in this section are to Vol.1 of this reference unless

otherwise stated.2K ostro (2000)writes:\Theether�lled and m adeup thesupra-lunaruniverse.Earth,

water,air,and �re were com posed ofprim ary m atter and form ,thus being capable of

transform ationsfrom oneinto theothers,whereastheetherwasso perfectthatitwasnot

capable ofbeing transform ed,and wasthusindestructible" (p.3).3W ith the invention ofthe coordinate system ,D escartes was,at the sam e tim e,the

unwitting precursorofthe laterconception ofthe etherasa form ofspace.


Thehistory oftheethercontinued tied tothetheory oflightwith Robert

Hooke’s (1635{1703) work. In an im provem ent with respect to Descartes’

view,heconceived oflightasa wavem otion,an exceedingly sm allvibration

oflum inousbodiesthatpropagated through atransparentand hom ogeneous

etherin a sphericalm anner.Hookealso introduced thusthefruitfulidea of

a wave front(pp.14{15).

Isaac Newton (1642{1727) rejected Hooke’swave theory oflighton the

groundsthatitcould notexplain the rectilinearpropagation oflightorits

polarization (seebelow).In itsplace,Newton proposed thatlightconsisted

ofraysthatinteracted with the etherto produce the phenom ena ofre ec-

tion, refraction, and di�raction, but that did not depend on it for their

propagation. He gave severaloptions as to what the true nature oflight

m ightbe,one ofwhich wasthatitconsisted ofparticles| a view thatlater

on would beassociated with hisnam e;nevertheless,astothenatureoflight,

he \letevery m an here take hisfancy." Newton also considered itpossible

fortheetherto consistofdi�erent\etherealspirits," each separately suited

forthe propagation ofa di�erentinteraction (pp.18{20).

Regarding gravitation in the context ofthe universallaw ofattraction,

Newton did notwantto pronouncehim selfasto itsnature.Henonetheless

conjectured that it would be absurd to suppose that gravitationale�ects

could propagate without the m ediation of an ether. However, Newton’s

eighteenth-century followers gave a twist to his views; antagonizing with

Cartesians due to their rejection ofNewton’s gravitationallaw,they went

asfarasdenying theexistenceoftheether| originally Descartes’concept|

and attem pted toaccountforallcontactinteractionsasactionsatadistance

between particles(pp.30{31).

Christiaan Huygens(1629{1695)wasalsoasupporterofthewavetheory

oflightafterobserving thatlightraysthatcrosseach otherdo notinteract

with one anotheraswould beexpected ofthem ifthey were particles.Like

Hooke, he also believed that light consisted of waves propagating in an

ether that penetrated vacuum and m atter alike. He m anaged to explain

re ection and refraction by m eans ofthe principle that carries his nam e,

which introduced the concept ofa wave front as the em itter ofsecondary

waves. As to gravitation, Huygens’idea ofa Cartesian ether led him to

accountforitasa vortex around the earth (pp.23{28).

An actualobservation that would later have a bearing on the notions

ofthe nature oflight and ofthe ether was thatofHuygens’regarding the

polarization oflight. He observed that light refracted once through a so-

called Icelandiccrystal,when refracted through a second such crystal,could

orcould notbeseen dependingon theorientation ofthelatter.Newton cor-

rectly understood thisresultasthe�rstlightray beingpolarized,i.e.having

propertiesdependenton thedirectionsperpendiculartoitsdirection ofprop-

agation. He then concluded that this was incom patible with light being a

(longitudinal)wave,which could notcarry such properties(pp.27{28).

Another thoroughly Cartesian account of the ether was presented by

John BernoulliJr.(1710{1790)in an attem ptto providea m echanicalbasis


forhisfather’sideason therefraction oflight.Bernoulli’setherconsisted of

tiny whirlpoolsand wasinterspersed with solid corpusclesthatwerepushed

aboutbythewhirlpoolsbutcould notastrayfarfrom theiraveragelocations.

A source oflight would tem porarily condense the whirlpoolsnearestto it,

dim inishing thustheircentrifugale�ectsand displacing thesaid corpuscles;

in thism anner,a longitudinalwave would bestarted (pp.95{96).

In the m idstofa generalacceptance ofthe corpusculartheory oflight

in the eighteenth century,also Leonhard Euler (1707{1783) supported the

view ofan etherin connection with a wavetheory oflightafternoticing that

light could not consist ofthe em ission ofparticles from a source since no

dim inution ofm asswasobserved.M ostrem arkably,Eulersuggested that,in

fact,thesam eetherserved asam edium forboth electricity and light,hinting

forthe �rsttim e ata uni�cation ofthese two phenom ena. Finally,he also

attem pted to explain gravitation in term s ofthe ether,which he assum ed

to have m ore pressure the fartherfrom the earth,so thatthe resulting net

balanceofetherpressureon a body would push ittowardsthecentreofthe

earth (pp.97{99).

At the turn ofthe century,the wave theory oflight received new sup-

portin thehandsofThom asYoung(1773{1829).W ithin thistheory,Young

explained re ection and refraction in a m ore naturalm annerthan the cor-

pusculartheory and,m ore im portantly,also accounted successfully forthe

phenom ena ofNewton’srings(and hinted atthecauseofdi�raction)by in-

troducing an interference principleforlightwaves.ItwasAugustin Fresnel

(1788{1827) who,in 1816 and am idst an atm osphere ofhostility towards

the wave theory,m anaged to explain di�raction in term s ofHuygens’and

Young’searlier�ndings(pp.100{108).

Young and Fresnelalso provided an alternative explanation ofstellar

aberration,which had been �rstobserved by Jam esBradley (1692{1762)in

1728 while searching to m easurestellarparallax and which had so farbeen

explained in term softhe corpusculartheory oflight. Young �rstproposed

thatsuch e�ectcould beexplained assum ingthattheearth did notdragthe

etherwith it,so thattheearth’sm otion with respectto itwasthecauseof

aberration. Subsequently,Fresnelprovided a fullerexplanation thatcould

alsoaccountforaberration beingthesam ewhen observed through refractive

m edia. Following Young,Fresnelsuggested that m aterialm edia partially

dragged the ether with them in such a way that the latter would pick a

fraction 1� 1=n2 (wheren isthem edium ’srefractiveindex)ofthem edium ’s

velocity. So farthe etherwasviewed asa som ewhatnon-viscous uid that

could be dragged along in the inside ofrefractive m edia in proportion to

their refractive index, and whose longitudinalexcitations described light

(pp.108{113).

Considerationsaboutthepolarization oflightwould bring along funda-

m entalchanges to the conception ofthe ether. AsNewton had previously

observed,the properties ofpolarized light did not favour a longitudinal-

wave theory oflight. Inspired by the results ofan experim ent perform ed

by Fran�cois Arago (1786{1853) and Fresnel,Young hit on the solution to


the problem ofpolarization by proposing thatlightwas a transverse wave

propagating in a m edium .Fresnelfurtherhypothesized thattheetherm ust

then be akin to a solid and display rigidity so as to sustain such waves

(pp.114{117).

The factthatonly a rigid ethercould supporttransverse wavesrobbed

theidea ofan im m obile,undragged etherofm uch ofitsplausibility,sinceit

washard toim agineasolid m edium ofsom esortthatwould notbe,atleast,

partially dragged by bodiesm oving through it.G eorge Stokes(1819{1903)

roseup tothischallengebyprovidingapictureoftheetherasam edium that

behaved likea solid forhigh-frequency wavesand asa uid forslow-m oving

bodies.Asa uid,Stokes’etherwasdragged by m aterialbodiessuch that,

in particular,itwasatrestrelative to theearth surface(pp.128,386{387).

M ichaelFaraday (1791{1867) gave a new dim ension to the ether con-

ception by introducing the notion of�eld,which in hindsightwasthe m ost

im portant concept to be invented in this connection.4 In studies of the

induction ofcurrents,oftherelation ofelectricity and chem istry,and ofpo-

larization in insulators,heputforward theconceptsofm agneticand electric

linesofforceperm eating space.Heintroduced thustheconceptofa �eld as

a stressin theetherand presentwhereitse�ectstook place.Faraday went

on tosuggest| prophesyingtherelativisticdevelopm ent| thatan etherm ay

notbe needed ifone were to think ofthese linesofforce,which he under-

stood as extensions ofm aterialbodies,as the actualcarriers oftransverse

vibrations,including light and radiantheataswell. O rthen that,ifthere

wasan lum iniferousether,itm ightalso carry m agnetic forcesand \should

have other uses than sim ply the conveyance ofradiations." By including

also them agnetic�eld asbeing carried by theether,Faraday added now to

Euler’searlierprophecy,hinting forthe�rsttim eattheconception oflight

asan electrom agnetic wave (pp.170{197).

Anotherunifyingassociation ofthistypewasm adeby W illiam Thom son

(1824{1907),whoin 1846com pared heatandelectricity in thattheisotherm s

oftheform ercorresponded to theequipotentialsofthelatter.Hesuggested

furtherm ore that electric and m agnetic forces m ight propagate as elastic

displacem ents in a solid. Jam es Clerk M axwell(1831{1879), inspired by

Faraday’sand W .Thom son’sideas,strove to m ake a m echanicalpicture of

theelectrom agnetic�eld byidentifyingstatic�eldswith displacem entsofthe

ether(forhim equivalentto displacem entsofm aterialm edia)and currents

with their variations. At the sam e tim e,M axwell,like G ustav K irchho�

(1824{1887) before him , was im pressed by the equality of the m easured

velocity oflightand thatofthe disturbancesofthe electrom agnetic theory

and suggested that light and electrom agnetic waves m ust be waves ofthe

sam e m edium (pp.242{254).

So far,thetheoriesofM axwelland Heinrich Hertz(1857{1894)had not

4And thisnotwithout a sense ofirony. At�rst,itwas on the �eld,a physicalentity

existing on its own and needing no m edium to propagate, that the overthrow of the

m echanicalether would rest;however,later on Einstein would reinstate the ether as a

(m etric)�eld itself.


m adeany distinction between etherand m atter,with theform erconsidered

astotally carried along by the latter. These theorieswere stillin disagree-

m entwith Fresnel’ssuccessfulexplanation ofaberration in m ovingrefractive

m edia,which postulated a partialetherdrag by such bodies.However,ex-

perim entsto detectany m otion oftheearth with respectto theether,such

asthoseby AlbertM ichelson (1852{1931)and Edward M orley (1838{1923),

had been negative and lent support to Stokes’theory ofan ether totally

dragged atthe surfaceoftheearth (pp.386{392).

Notcontentwith Stokes’picture,in 1892 Hendrik Lorentz (1853{1928)

proposed an alternativeexplanation with a theory ofelectrons,which recon-

ciled electrom agnetic theory with Fresnel’slaw.However,Lorentz’spicture

ofthe ether was that ofan electron-populated m edium whose parts were

m utually at rest; Fresnel’s partial drag was therefore not allowed by it.

Lorentz’s theory denied the ether m echanicalproperties and considered it

onlyspacewith dynam icproperties(i.e.a�ectingbodiesbecausetheym oved

through it),although stillendowed itwith a degree ofsubstantiality.5 The

negativeresultsoftheM ichelson-M orley experim entwerethen explained by

Lorentz by m eans ofthe existence ofFitzG erald’s contraction,which con-

sisted in a shorteningofm aterialbodiesby a fraction v2=2c2 oftheirlengths

in thedirection ofm otion relativeto theether.Thus,theetherwould cease

being a m echanicalm edium to becom e a sortofsubstantialdynam ic space

(pp.392{405).

Neartheend ofthenineteenth century,theconception oftheetherwould

com pletetheturn initiated by Lorentzwith theviewsofPaulDrude(1863{

1906) and Joseph Larm or(1857{1942),which entirely took away from the

etheritssubstantiality.Drude(1894,p.9)declared:

Justasonecan attributeto a speci�cm edium ,which �llsspace

everywhere,the role ofinterm ediary ofthe action offorces,one

could do withoutit and attribute to space itselfthose physical

characteristicswhich are now attributed to theether.6

Also Larm or claim ed thatthe ether should be conceived as an im m aterial

m edium ,nota m echanicalone;thatone should notattem ptto explain the

dynam icrelationsso farfound in term sof

m echanicalconsequencesofconcealed structurein thatm edium ;

we should ratherrestsatis�ed with having attained to theirex-

act dynam icalcorrelation, just as geom etry explores or corre-

lates,withoutexplaining,the descriptive and m etric properties

ofspace.7

Larm or’sstatem entisso rem arkablethatitwillreceivem oreattention later

on in Section 9.4.

5See (K ostro,2000,p.18)and (K ox,1989,pp.201,207).

6Q uoted from (K ostro,2000,p.20).

7Q uoted from (W hittaker,1951,Vol.1,p.303).


Despite the seem ing super uousnessofthe ether even taken as a �xed

dynam icspace,Lorentzheld fasttotheetheruntilhisdeath,hopingperhaps

that m otion relative to it could still som ehow be detected (K ox, 1989).

O thers,likePoincar�eand Einstein,understood therepeated failed attem pts

to m easurevelocitieswith respectto theetherasa clearsign thattheether,

in fact,did not exist. HenriPoincar�e (1854{1912) was the �rst to reach

such conclusion;in 1899 he asserted that absolute m otion with respect to

the ether was undetectable by any m eans, and that optical experim ents

depended onlyon therelativem otionsofbodies;in 1900heopenlydistrusted

theexistenceoftheetherwith thewords,\O urether,doesitreally exist?";

and in 1904 heproposed a principleofrelativity (Vol.2,pp.30{31).

ItwasAlbertEinstein (1879{1955)whoin 1905 provided a theory where

hereinstated theseearlierclaim sbutwith anew,lucid interpretationalbasis.

In particular,Einstein considered

The introduction of a \lum iniferous ether"...to be super u-

ousinasm uch asthe view here to be developed willnotrequire

an \absolutely stationary space" provided with specialproper-

ties...(Einstein,1952b,p.38)

Thus,in thehandsofPoincar�e and Einstein,the ether had died.

9.2 T he m etric ethers

Beliefin thenonexistenceoftheetherwould notlastvery long,foritwould

soon rise from itsashes| transm uted. A rebirth ofthe etherasan inertial

m edium wasadvocated byEinstein aboutadecadeafter1905on thegrounds

that,withoutit,em pty spacecould nothave any physicalproperties;yetit

displayed them through thee�ectsofabsoluteacceleration,aswellase�ects

on m easuring rodsand clocks.

Itisa well-known factthatallm otion sim ply cannotbereduced to the

sym m etric relationship between any two reference system s. Newton (1962,

pp.10{12),through therotating-bucketand revolving-globes(thought)ex-

perim ents,realized that the e�ect ofacceleration is not relative,and that

rotation in em pty spaceism eaningful.Asaresult,heheld on totheneed for

a space absolutely at restwith respectto which this absolute acceleration

could be properly de�ned.8 Einstein (1952a,pp.112{113) was im pressed

by the sam e fact,which he brought up by m eans ofthe rotating-spheres

thought experim ent. The conclusion forces itself upon us that som e ref-

erence fram es are privileged. In Newtonian space and special-relativistic

spacetim e,these are the inertialfram es;in generalrelativity,these are the

freely-falling fram es.

However,regarding the classicaland special-relativistic cases,Einstein

(1952a,p.113) com plains that no \epistem ologically satisfactory" reason

8In fact,only a fam ily ofinertialspaceslinked by G alilean transform ationswould have

su�ced.

9.2 T he m etric ethers 97

that is \an observable fact ofexperience" can be given as to why or how

space singles out these fram es and m akes them special. Speaking for a

laym an whose naturalintelligence has not been corrupted by a study of

geom etry and m echanics,hesaysin bluntreply to the m echanist:

You m ay indeed be incom parably welleducated. But justas I

could neverbe m ade to believe in ghosts,so Icannotbelieve in

the gigantic thing ofwhich you speak to m e and which you call

space. Ican neithersee such a thing norim agine it. (Einstein,

1996b,p.312)

InertialNewtonian space and inertialspecial-relativistic spacetim e are

m etric substrata,each characterized by a m etric structure determ ined by

an invariant quadratic form 9 (not just a m etric). The Newtonian m etric

substratum isdeterm ined by

dl2 = E abdxadxb (a;b= 1;2;3); (9.1)

whereE ab = diag(1;1;1),and thespecial-relativistic m etric substratum by

ds2 = �� dx�dx� (�;� = 1;2;3;4); (9.2)

where �� = diag(1;1;1;� 1) and � = � 1 ensures that ds2 � 0. In both

cases,the m etrics E ab and �� are constant,in the sense that coordinates

can always be found such that their values do not depend on the space

or spacetim e coordinates x. Do Newtonian space and special-relativistic

spacetim e have anything in com m on with the lum iniferousether?

Justlikethelum iniferousetherneverdisplayed any property intrinsicto

itsm aterialnature| m ostnotably,the etherwind always had nullrelative

velocity| so do these two m etric substrata failto display any non-nullor

non-trivialproperties intrinsic to their own m etric nature| m ost notably,

they are atand,therefore,structurally (m etrically) sterile. For exam ple,

gently release two m assive particles and observe their m utualseparation

rem ain constantasaresultofinertialm ediahavingnoactive e�ecton them ;

inertialm edia only actdynam ically,by having bodiesm ove through them ,

butnotstatically,asaresultofonly beingplaced in them .Further,justlike

thelum iniferousetheracted asam edium thatm adethepropagation oflight

and otherinteractions possible,yethad an enigm atic origin and could not

bealtered in any way,so do thesetwo m etricsubstrata actasthesourceof

physicale�ectsby determ ining inertialfram es;however,because they have

no sources,they cannot be acted upon in any way,and so their reason of

beingcannotbedeterm ined in aphysicalsense.TheNewtonian and special-

relativistic m etricsubstrata arethusin precisecorrelation with theoriginal

characterization oftheolderm echanicalether,and wearecom pelled to hold

them asa new,m etric em bodim entofthe ether.

Not content with the absolute nature of special-relativistic spacetim e

insofar as it could not be a�ected,Einstein looked for sources via which

9Form oreon thephysicaland m athem aticalm eaningofthisconcept,seeSection 10.1.4.


thephysicalpropertiesdisplayed by spacetim ecould bein uenced and thus

no longer�xed and beyond reach. W hile dealing with the rotating-spheres

thoughtexperim ent,hewrote:

W hatisthe reason forthisdi�erence [sphericaland ellipsoidal]

in the two bodies? No answercan beadm itted asepistem ologi-

cally satisfactory,unlessthereason given isan observable factof

experience...[T]he privileged space R 1 [inertial]ofG alileo...is

m erelyafactitiouscause,and notathingthatcan beobserved...

Thecausem ustthereforelieoutside thissystem ...[T]hedistant

m asses and their m otions relative to S1 and S2 [the spheres]

m ustthen beregarded asthe seatofthecauses(which m ustbe

susceptible to observation)ofthe di�erentbehaviourofourtwo

bodiesS1 and S2.They takeovertheroleofthefactitiouscause

R 1.(Einstein,1952a,pp.112{113)

In the m etric contextofrelativity theory,thism eantsearching fora space-

tim ewhosem etricstructurecould bechanged by thedistribution ofm atter

and which had gravitation asitsactive m etrice�ect.O n accom plishingthis,

Einstein wasthe �rstphysicistsince the conception ofthe etherto bring it

to fullphysicalaccountability,i.e.notonly to observe itsactive e�ectsbut

also to controlits structure. In other words,after two conceptualrevolu-

tions,Einstein wasthe �rstphysicistto observe and controlthe ether.But

thisissurely a contradiction in term s,and so theetherdied again.Einstein

had uprooted itforthe second tim e,butthistim e with the righttool| not

denial,observation.

Einstein neverthelesscontinued tocallthegeneral-relativisticm etricsub-

stratum an ether,because he seem ingly m eant the word prim arily in its

prim itivesenseofa ubiquitoussubstratum with physicalpropertiesregard-

less ofwhether its e�ects had now turned from passive to active and its

naturechanged from absolute to alterable.10

W hen we say that the m etric ether becam e fully accounted for as a

consequence ofthe second relativistic revolution,we generally m ean that

the m etric ether (i) becam e observable asa resultofits non-trivialm etric

structure,displaying now an active geom etric e�ectin the form ofgravita-

tion;and (ii) lost its absolute character when,through the �eld equation,

its intim ate linkage to m atter as its source was established. But did the

dynam ic,alterablenatureofthegeneral-relativisticm etricsubstratum ,now

determ ined by the invariantquadraticform

ds2 = �g�� (x)dx�dx�; (9.3)

putthem etric�eld g�� (x)on an equalfooting with otherphysical�elds? Is

g�� (x),in fact,nothing else than the new gravitational�eld? The answers

to thesetwo questionsarenotasstraightforward aswem ightwish,because

m athem aticalconcepts,otherwise ofaid,also blurourphysicalvision.

10See(K ostro,2000)fora usefulsourceofhistoricalm aterialon Einstein and theether.


O n thesurface,them etric�eld g�� (x)isakin tootherphysical�elds,like

the electric �eld E a(x)orm agnetic �eld B a(x)| or,better,to the electro-

m agnetictensor�eld F�� (x).In e�ect,whereasm atteractsasthesourceof

theform er,chargeand currentactasthesourcesofthelatter.Butthiscom -

parison ism isleading.How could them etrictensorbethegravitational�eld

ifitisnoteven nullin the special-relativistic case,i.e.when g�� (x)= �� ?

Should we perhaps concede that one should not take the correspondence

so literally but continue to claim that it holds,for exam ple,by claim ing

som ething like\the‘gravitational�eld’isnullwhen the‘gravitational�eld’

g�� (x) is constant"? Even ifwe accepted this contradiction in term s,it

would not be workable. The values ofg�� (x) change from one coordinate

system to another,and even when there exists a coordinate system in at

spacetim e with respect to which g�� (x)= constant holds,there are m any

othercoordinatesystem sin which itdoesnothold.In otherwords,thereis

no tensorequation,m eaningfulin allcoordinatesystem s,linking them etric

tensorassuch tothestress-energytensorT�� ,which characterizesthem atter

distribution in spacetim e;there is no straightforward invariantconnection

between g�� (x)and gravitation.Thesam eanalysisappliesto theview that,

ifnotthem etric �eld,then theChristo�elsym bolf�� g m ustbethegravi-

tational�eld;itcan be actually null(notjustconstant)in gravitation-free

spacetim e,but this value too changes from one coordinate system to an-

other. In consequence,equations g�� (x)6= constant and f�� g 6= 0 tellus

nothing aboutthe existence ornonexistence ofa gravitational�eld.

By looking attheactual�eld equation,

R �� (x)�1

2g�� (x)R(x)= �

8�G

c4T�� (x); (9.4)

where R �� is the Riccitensor and R the Ricciscalar,we notice that,ifa

correspondence with a \gravitational�eld" m ust be m ade,it is in general

the curvature tensor R ��

(ofwhich the Riccitensorand curvature scalar

are contractions)thatshould be identi�ed with it. Itisonly the curvature

tensor,acom plicated com bination ofsecond derivativesand productsof�rst

derivativesofthem etrictensor,11 that(i)can besetin directcorrespondence

with theexistenceofgravitationale�ectsin allcoordinatesystem s,and (ii)

has m atter as its source by m eans ofwhich it can be altered. Com pare

with the case ofelectrom agnetism ,where the electrom agnetic tensor �eld

F�� (x) (i) can be set in a one-to-one correspondence with the existence

of electrom agnetic e�ects in allinertial coordinate system s, and (ii) has

the four-currentJ�(x) (charge and current)as its source according to the

inhom ogeneousM axwellequations

@F �� (x)

@x�= �0J

�(x); (9.5)

where�0 isthe perm eability ofvacuum .

11Thephysicaldim ensions[x

�]�2

ofcurvaturearisefrom thedim ensionlessm etrictensor

in thisway.


The confusion of the m etric �eld g�� (x) with the \new gravitational

�eld" isrooted in thefactthat,in generalrelativity,g�� (x)isthem ostfun-

dam ental�eld in m athem aticaldevelopm ent. Forgetting thatm athem atics

isin physicsonly a usefultoolto aid and guideourthought,wefallprey to

thebeliefthatwhatism athem atically fundam entalm ustalso bephysically

fundam ental. This is not so. The gravitationale�ects that we experience

and observearenotin directcorrespondencewith them etric�eld butwith

thecurvatureofspacetim e.In thenextchapter,weshallseethatthem etric

�eld is,in fact,closely connected with (butnotequivalentto)an observable

factofexperience,butnotin connection with curvature| in connection with

clocks.Finally,a supplem entary cause ofm isdirection,whosepsychological

in uence should notperhapsbe underestim ated,isthatthe m etric �eld is

denoted with the letter \g" for \geom etry," suggesting at the sam e tim e

\gravitation."

Can now the analogy ofthe m etric and the lum iniferous ether be fur-

thered in such a way that,by considering cases that are not actually the

casein thephysicalworld| casesinterm ediatebetween the�xed, atspecial-

relativistic m etric substratum and the alterable,curved general-relativistic

one| we m ay gain betterunderstanding ofthe etherconcept? W hatprop-

erties are characteristic of an ether, and what m eaning do we attach to

them ?

Ifinstead ofhaving a �xed, atspacetim e,we had a �xed spacetim e of

everywhere-constantcurvature such as,forexam ple,a sphericalspacetim e,

would weconsiderthisa m etricether,and whatwould beitsanalogousm e-

chanicalcounterpart? A curved spacetim e possessesa property thata at

spacetim e lacks,nam ely,itdisplaysan active physicalproperty on account

ofitsnon-trivialm etric structure,i.e.ofitscurvature,which we experience

as gravitationale�ects: bodies are a�ected not dynam ically,because they

m ovethrough thissubstratum ,butstatically,becausethey areplaced in this

substratum . This would be analogous to having had a lum iniferous ether

with an active m echanicalproperty,nam ely,an etherwind constantevery-

where. Since thisisprecisely whatwassoughtasevidence ofthe existence

ofthe old ether,we m ay as wellconclude that such a m etric substratum

could notbe an ether at all(in the negative sense ofthe word). And yet,

are we not nonetheless troubled by the fact that gravitation as the result

ofa �xed spacetim e curvature hasa physically unexplained origin? W ould

notnineteenth-century physicistshavebeen likewisedisturbed athaving at-

tained to them easurem entofan im m utableproperty (etherwind)ofphysi-

cally unaccountable origin? In both cases,afterm om entary exultation,the

need toknow theorigin oftheseotherwiseG od-given m ediawould takeover.

(Butsee page 178 fora possibleway to view thisissueasno puzzle.)

W hat would happen if,generalizing the above,we were confronted by

a spacetim e ofvariable curvature,butthatis�xed once given,such as,for

exam ple,a sphericalspacetim e glued sm oothly to a hyperbolic one? Here

gravitation would vary in di�erentregionsofspace;bodiesreleased atrest

would gradually converge towardseach otheratsom e placesand gradually


divergefrom each otheratotherplaces.Finding ourselvesunableto m odify

this variable active e�ect of the m etric spacetim e substratum ,we would

in this case be even m ore puzzled than before,and the need to account

for its physicalsource (whether it had any or not) would be even m ore

com pelling. Analogously, ifthe lum iniferous ether had displayed a wind

velocity that varied in speed and direction at di�erent locations ofspace,

theneed to accountfortheorigin ofthese\upperwindsoftheG ods" would

have becom e even m ore pressing.

Now,do these these two kindsofobservable yetabsolute m edia qualify

as ethers or not? W e see that they are m idway between the idea of a

genuineether,which isunobservable,hom ogeneous,and absolute,and anon-

ether,which isobservable,inhom ogeneous,and m alleable.The�rstofthese

�ctitious constructions is,in fact,observable,hom ogeneous,and absolute,

whilethesecond isobservable,inhom ogeneous,and absolute.Becausethey

share partofthe fullcharacterization ofether,we callthem partialethers

ordem iethers.12

W e m ove forwards in the analysis of the ether conception by asking

next:did the m etric etherdie a clean death in the handsofobservation as

gravitation and a�ectability via itstwo-way interaction with m atter,ordid

itleave behind a di�cultlegacy in the em ptinesslying beneath it?

G eneralrelativity changed not only the notion ofether,but also that

ofem pty spacetim e. Another feature ofg�� (x) is that it depends on the

spacetim e coordinates x (geom etrically speaking,on its points),so that it

doesnotgiveriseto an im m utablem etricbackground thatcan beassociated

with em pty spacetim e,as�� did in specialrelativity.O n thecontrary,the

m etric�eld g�� (x)appearsnow asan intrinsic content13 ofspacetim e.The

rem ovalof this special�eld m eans that the m etric structure is gone, so

thatnotan em pty spacetim e with physicalpropertiesbut,rather,nothing

physicalrem ainswithoutit(Einstein,1961,p.176).

Upon rem ovalofthe m etric content ofspacetim e,what rem ains,what

we can stillcallem pty spacetim e,are the spacetim e pointsP (cf.x).Now,

are we certain that the am orphoussubstratum form ed by these geom etric

objects displays no physicalproperties at all? Ifthis were so,the ether

problem in physics would be de�nitely solved,perhaps not forever but at

leastso faraswecan see,i.e.within thereigning physicalworldview.

However,luck doesnotseem tobeon ourside| faithfultoitshistory,the

ether rises from its second ashes transm uted yet once m ore,as ifenacting

itstragic destiny ofim m ortality.W hy tragic?

In theclassicessay,\Thetragedy ofthecom m ons,"Hardin (1968)brings

ourattention to them eaning of\tragedy" assom ething beyond sim pleun-

happiness.Hebasestheuseoftheword \tragedy" in W hitehead’sobserva-

12D em i,from Latin dim idius,dis-\apart" + m edius \m iddle":one thatpartly belongs

to a speci�ed typeorclass.(M erriam -W ebsterO nline D ictionary,http://www.m -w.com )13Einstein (1961) distinguished space from its contents with the words: \[S]pace as

opposed to ‘what �lls space,’which is dependent on the coordinates..." (p.176,italics

added).Thisdistinction isbased on theidea thatanythingthatdependson thespacetim e

coordinates(orpoints)m ustbe in spacetim e.


tion that

The essence ofdram atic tragedy is notunhappiness. Itresides

in thesolem nity ofthe rem orselessworking ofthings...Thisin-

evitablenessofdestiny can only beillustrated in term sofhum an

lifeby incidentswhich in factinvolveunhappiness.Foritisonly

by them thatthe futility ofescape can be m ade evidentin the

dram a.(W hitehead,1948,p.17)

Like Hardin’s grazing com m ons butin reverse,so does the ether rem orse-

lessly follow an inevitable destiny| endlesslife.

W hat passive physicalproperties do spacetim e points display that we

m ust stillcallthem an ether? And,ifendowed with physicalproperties,

why can spacetim e points not be observed? W hat is this third rise ofthe

etherallabout?

9.3 T he geom etric ether

W ecallthesubstratum ofspacetim epointsthegeom etricether.Butbefore

we can m ove on,the readerasks:are notthe m etric ethersalso geom etric?

They are indeed,butin di�erent ways. Revealing the ways in which they

aresim ilarand thewaysin which they di�ercan help usseewhatthethird

rise ofthe ethershareswith itsancestorsand in whatitisdi�erent.

W e have characterized geom etry ascom prising two essentialparts,geo-

m etricobjectsand geom etricm agnitudesattached to them .W hatwecalled

m etric ethers each possessed an inherent quantitative geom etric structure

given by itsinvariantquadraticform .Thism etricstructuredeterm ined spa-

tialorspatiotem poralgeom etricextension,dl2 ords2,in spaceorspacetim e.

A m etric ether is for this reason endowed with a quantitatively geom etric

structure.

Spacetim e points,on the other hand,are geom etric objects but,after

the rem ovalofg�� (x),no quantitatively geom etric structure is any longer

placed on them .Spacetim e pointsconsistonly ofa qualitatively geom etric

m edium whoseelem entsbearnom utualquantitativerelations.W eacknowl-

edge,then,thatthem etricethersarealso geom etricethers,and so thatthe

classi�cation ofspacetim e pointsas\geom etric ether" is,strictly speaking,

too inclusive.However,them oreaccurate denom inations\qualitatively ge-

om etric ether" or\geom etric-object ether" do nothave a good ring about

them ,and considering that\m etric ether" bringsoutthe essentialcharac-

teristic ofthe ethers ofthe previous section,we shallvalue naturalness of

expression overtaxonom ic precision.

Spacetim e pointsare actors notacted upon. W hatis the physicalrole

they play? To �nd it,weshould notlook forpassiveoractivem etrice�ects,

even lessform echanicalones,becausethesegeom etric objectsform neither

a m etricnora m echanicalsubstratum .Thephysicalpropertiesdisplayed by

spacetim epointsareasfarrem oved from theonesattributed to theinertial

ethers as these were from those attributed to the m echanical ether. To

9.3 T he geom etric ether 103

becom eaware ofthem ,wem ustlearn to seethisnew geom etric etherin its

properlight.

Spacetim e points are inherent in the physicalconcept of�eld f(P ) by

perform ing theirlocalization| or,m oreaccurately,by perform ing thelocal-

ization ofphysicalquantities.Regarding theintrinsicconnection thatholds

between spacetim e pointsand the �eld,Auyang writes:

Thespatiotem poralstructureisan integralaspectofthe�eld...

W ecan theoretically abstractitand thinkaboutitwhileignoring

thedynam icalaspectofthe�eld,butourthinking doesnotcre-

ate thingsofindependentexistence;we are notG od. (Auyang,

2001,p.214)

ShefurthercriticizesEarm an’s(1989)rem arksthat\W hen relativity theory

banished theether,thespace-tim e m anifold M began to function asa kind

ofdem aterialized etherneeded to support�elds" (p.155).

W eagreewith Auyang’sview that\spacetim e" (spacetim epoints)isnot

m erely a substratum on which to m athem atically de�ne �elds. Spacetim e

points are inherent in �elds inasm uch as they perform the physical task

oflocalizing physicalquantities. This is what,after W eyl(1949),Auyang

(2001,p.209)called the\this" or\here-now" aspectofa �eld,additionalto

its\thus"orqualitativeaspect.W ealsoconcurwith Auyangin thatisolated

points are the illusive creations ofour geom etric thinking,but should we

o�hand renouncethepossibilitythatem ptyspacetim e| beyond itsgeom etric

description| be realon itsown,and notsim ply an illusion created by the

m ind? M ightwenotstilldiscoverthatithasactiveobservablee�ects,ifwe

learntto probeitwith di�erentconceptualtools?

In perform ingthelocalization of�elds,spacetim epointsgivefoundation

to theconceptoflocality (here-now)ofphysicalquantities.Thisis,further-

m ore,the foundation on which the causality (from here-now sm oothly to

there-then) ofallphysicallaws is grounded,and which also goes hand in

hand with the notion oftim e as a one-param eter along which things ow

causally.W ithoutthe�eld asalocalconcept,notheoreticalexplanation can

begiven fortheprim ary physicalproperties oflocality and causality,whose

e�ects we passively observe in the workings ofthe world. In currenttheo-

ries,we m ake sense ofthese perceived properties by attributing to em pty

spacetim e a point-like constitution,and then holding locality and causality

to beconsequencesofit.

In question hereisnotjustan \etherofm echanics"oran \etherofspace-

tim e theories" but an ether of�eld-based physics,ofalllocaland causal

physics.Thereis,then,no partofcurrentphysicsthatthe geom etric ether

doesnottouch,with theexception| notsurprisingly| ofthatpartofquan-

tum m echanics that cannot be understood within the �eld conception. In

quantum physics,the�eld conceptsurvivespartly (although notasa space-

tim e �eld)in the state vector j (x;t)iand itscausalSchr�odingerequation

ofm otion,butitcannotbe applied atthe global,non-causalcircum stance

ofitspreparation orm easurem ent,when itsaid to \inexplicably collapse."


And conversely,ifwefeltthatlocality and causality arenotgood m ouldsto

describethenaturalworld,wecould notm ovepastthem withoutgiving up

on the local�eld.

O n the other hand,however,active spacetim e points cannot be acted

upon such thattheirdisplacem entwould causeany observablee�ects.This

conclusion forced itselfupon Einstein (1996a)asaconsequenceofan analysis

he called the hole argum ent.Thisanalysiswasbroughtback into the spot-

lightby professionalphilosophersofscienceafteraboutsix decadesofincon-

spicuousness,and istoday the objectofluxuriantphilosophicaldebate| so

luxuriant,in fact,thatthe argum enthasram i�ed beyond recognition,and

one wonderswhetherm ore harm than good hasbeen done by resurrecting

them atter.O riginally,Einstein o�ered thisargum entin 1914 asproofthat

the equations ofgeneralrelativity could not be generally covariant,14 be-

cause accepting the alternative consequence ofthe hole argum ent was at

�rstunthinkable.However,thispath led to di�culties,and abouttwo years

later Einstein (1952a) re-em braced general covariance while at the sam e

tim e m aking his peace with the hole argum ent by accepting the unthink-

able,nam ely,thatgeneralcovariance \takesaway from space and tim e the

lastrem nantofphysicalobjectivity" (p.117). In otherwords,thatem pty

spacetim e hasno physicalm eaning.15

W e review the train ofthoughtleading Einstein to thisconclusion.Be-

cause the �eld equation is expressed in generally covariant form ,ifg�� (x)

is a solution to it corresponding to the m atter source T�� (x),then so is

g0��(x0)with corresponding m attersource T0

��(x0)forany continuouscoor-

dinate transform ation x0= x0(x). Here the sam e m etric and m atter �elds

are being described in two di�erentcoordinate system s,both ofwhich are

viableon accountofthe�eld equation beingexpressed in generally covariant

form .Droppingtheprim ed coordinatesystem S0com pletely,how toexpress

g0��(x0)and T0

��(x0)in the unprim ed coordinate system S? Thisisdoneby

replacing x0by x,so thatg0��(x)and T0

��(x)describenow with respectto S

the �eldsaswere earlierdescribed with respectto S 0.Thistransform ation

isthe active equivalentofthe originalcoordinate transform ation. Itworks

by replacing x0 by x,asif,within S,pointx0 wasdisplaced to pointx via

the transform ation x(x0) = x (here x = x0�1 ),i.e.as ifthe backdrop of

spacetim e pointswasm oved asa whole(Figure 9.1).

For an arbitrary transform ation x(x0) = x as here considered,a new

valid solution g0��(x) can be obtained in this active way only because,in

generalrelativity,g�� (x)isnotabsolutebutvarieswith T�� (x)(seebelow).

The resultis thatboth g�� (x)with source T�� (x)and g0��(x)with source

T0��(x) are solutions to the �eld equation in the sam e coordinate system .

Herearetwo di�erentm etric�eldsatthesam epointx,which isacceptable

so long aseach isattached to a di�erentm atter source and,therefore,are

14Ultim ately,the equations ofallspacetim e theories can be expressed in generally co-

variant form ,butgeneralrelativity alone adm its only a generally covariant form ulation.

See (Friedm an,1983,pp.46{61).15Forahistorically com pletereview oftheholeargum ent,seee.g.(Stachel,1989,pp.71{

81),where relevantquotationsand a fulllistofearly referencescan be found.


S

O

S ′

O

S

O

S ′

O

gαβ(x) g′αβ(x′)

gαβ(x)

g′αβ(x)

g′αβ(x′)

x′ → x

Figure 9.1: Sam e m etric �eld seen from S and S0 (top). S0 is dropped;points

x0 are m oved to x to express the m etric �eld in S as previously expressed in S0

(bottom ).

notsim ultaneously valid atx.

Now H is a so-called hole in spacetim e in the sense that within this

region no m atterispresent,i.e.T�� (x)= 0;outside H ,on the otherhand,

T�� (x)isnon-null.Furtherm ore,g�� (x)isthespacetim em etric,necessarily

non-nullboth inside and outside the hole,and � is an active coordinate

transform ation (akin to x before)with the property thatitisequalto the

identity outside H , di�erent from the identity inside H , and continuous

at the boundary. The unchanged g�� (x),with unchanged source T�� (x),

is now a solution outside H , but inside H , where no m atter is present,

both g�� (x) and g0��(x) are two di�erent and coexisting solutions at the

sam e point x (Figure 9.2). Einstein reasoned that this im plies the �eld

equation isnotcausal,sinceboth coexisting m etric�eldsin H areproduced

by the sam e source outside H . To avoid this,Einstein postulated that,

their m athem atical di�erences notwithstanding, g�� (x) and g0��(x) m ust

be physically the sam e. In otherwords,Einstein postulated thatthe point

transform ation x(x0)= x leading from g0��(x0)to g0

��(x)isphysically em pty,

i.e.there is no physicalm eaning to be attached to (the displacem ent of)

spacetim e points.

W e add parenthetically thatonly the hole argum entapplied to general

relativity leads to this problem . In specialrelativity,for exam ple,where

the m etric �eld �� isabsolute and doesnotresultasa solution to a �eld

equation,a new �eld �0��(x) can be obtained from �0

��(x0) by the active

m ethod x(x0) = x of m oving the spacetim e points only if x0 and x are

106 T he tragedy ofthe etherSpa etime{gαβ(x), Tαβ(x)}

HoleTαβ(x) = 0

{gαβ(x), 0}

{g′αβ(x), 0}

Figure 9.2:Schem aticrepresentation oftheholeargum entin coordinatelanguage.

otherwiseconnected by specialtransform ations xL thatleave �� invariant:

thesearetheLorentztransform ations.Asaconsequence,thequadraticform

rem ainsinvariant:

ds2 = �� (x)dx�dx� = ��

0�� (x)dx

�dx� = ds02: (9.6)

In the general-relativistic case,where no specialtransform ationsexistthat

leave g�� (x)invariant,

ds2 = �g�� (x)dx�dx� 6= �g

0�� (x)dx

�dx� = ds02; (9.7)

but then again this is no violation ofthe invariance ofds2,because the

source ofgeom etric structurehaschanged too,from T�� (x)to T0��(x).16

Einstein was therefore right to declare that em pty spacetim e does not

havephysicalproperties| butonlyifby\physicalproperties"weunderstand

observable (active) e�ects arising from their displacem ent. Locality and

causalevolution in param etertim e,on theotherhand,arenothing butthe

passive physicale�ects that we attribute to the point-like constitution of

em pty spacetim eand to thebehaviourofthelocal�eldsthepointsarepart

of.Doesnotallthisring a bell? Theetherrepeatsitshistory| tragically|

once again.

Despite the oddsagainstthem ,Friedm an (1983,pp.216{263) nonethe-

less argued for the physicalreality ofbare and isolated spacetim e points.

According to him ,thecoreofan explanation ofnaturalphenom ena isto be

abletoreduceawidevarietyofthem toasinglefram ework(Friedm an,1974),

so that whatone is required to believe isnota vastrange ofisolated rep-

resentationalstructures,buta single,all-encom passing construct.However,

in order to provide scienti�c understanding,theoreticalentities with su�-

cientunifying power m ustbetaken to beofa literalkind.Thus,Friedm an

argued thatdenying theexistenceofspacetim epoints| them selvesessential

forgeodesics to exist| can only lead to a loss ofunifying and explanatory

powerin spacetim e theories.

Aslogically im peccableasthistypically philosophicalargum entm ay be,

itisatoddswith history itselfand with theinterestsofphysicalscience.If

16Form ore details,see (Friedm an,1983,pp.46{61)and (Stachel,1989,p.78).


anythinghasbeen learntfrom thenarration ofthehistory ofthem echanical

ether,itshould bethis:no otherconception gripped the m indsofso m any

illustriousm en ofscience forlongerand m ore strongly than thatofthe lu-

m iniferousether.Itwasalwaysheld to bephysically realand uni�ed a wide

rangeofphenom ena(light,heat,gravitation,electricity,and m agnetism )de-

spiteitsrelentlessly unobservableexistence.And yet,attheopening ofthe

twentieth century,a scienti�crevolution rendered itsuper uousand useless,

and so,with no furtherado and hardly enough m ourning to do justice to

itsnearly three hundred yearsofage,itwasdeclared nonexistentand duly

buried.Science quickly forgetsitswaste productsand m oveson.

Evidently, no spell of tim e during which a conception proves to be

successfulis long enough to declare it realbecause ofits utility or unify-

ing power. W hat is to guarantee that today’s vastly successfulspacetim e

points| thegeom etric ether| willnotrun,in theirown duetim e,thesam e

fate as their m echanicaland inertialancestors? Ifany concept that is of

aid in physics is to be held real,nothing m ore and nothing less should be

dem anded ofitthatitbeobservable,i.e.thatithaveactive physicale�ects

and beopen to hum an m anipulation and control.

W hile professionalphilosophy does not try to overcom e the geom etric

ether,neitherdoesquantum gravity.A study ofthisdiscipline revealsthat

itdoesnottry to determ inefrom whereobservablelocality and causalevo-

lution in param etertim em ay com e by attem ptsto �nd observablesrelated

to em pty spacetim e. Instead,quantum gravity focuses on the creation of

ever m ore involved geom etric structures for space and tim e, leaving the

qualitative problem ofthegeom etricetheruntouched.W hen an attem ptat

observation isatallm adein quantum gravity,itisonly to link itsproposed,

so farunobservable,geom etricstructureswith som ehypotheticale�ect,but

nevergivingsecond thoughtstotheobservability oftheunderlyinggeom etric

objects| spacetim e pointsam ong them .

In thisrespect,Brans(1999,pp.597{602) arguesthatm uch by way of

unobservable structure is taken for granted in current spacetim e theories,

such asthe existence ofa pointset,a topology,sm oothness,a m etric,etc.

He com pares the vortices and atom s of the m echanicalether with these

unobservablebuilding blocksofspacetim eand,m oreover,with theyetcon-

siderably m orecom plex spacetim estructuresand \superstructures" devised

m orerecently.W einterpretBranstobeasking:willstringsand m em branes,

spin networks and foam s,nodes and links| to nam e but a few| appear a

hundred years from now like the m echanicalether does today? Are the

abovethenew geom etriccounterpartsoftheold m echanicalcontraptions?17

In other words,willthe phrase \quantum theory ofgravity" com e to be

regarded,a hundred yearsfrom now,asweregard today thephrase\vortex-

atom theory ofthe ether"? Itislikely thata new nam e becom e necessary

fora theory that,by being both physicaland faithfulto nature,succeedsin

17Fora recentexplicitrevivaleven ofthe long-dead m echanicalether,see (Jacobson &

Parentani,2005),where the authors ask: \Could spacetim e literally be a kind of uid,

like the etherofpre-Einsteinian physics?" (p.69).


di�erentiating itselffrom thefailed attem ptsthatpreceded it,and thatthe

abused term \quantum gravity" only rem ain as a usefulhandle to denote

the historicaldebacle ofanotherdark age ofphysics.

Be allthis as it m ay,even ifwe turned a blind eye to their physical

uncertainty and assum ed thatsom e ofthese m ore recently proposed struc-

tureswere notto follow such a fate and could eventually be observed,the

observationsrelating to them would atany rategiveevidenceofa new m et-

ric spacetim e structure. Butifthe goalisto advance the understanding of

thegeom etric ether(locality,causality,param etertim e,and �elds)then we

m ustm ove away from geom etry| forwardson a new path.

Foran appraisalofthe di�erentconceptsofetheranalysed,turn to the

sum m ary table attheend ofthischapteron page 118.

9.4 B eyond the geom etric ether

Thelum iniferousetherwassuperseded by strippingitofallitsintrinsicm e-

chanicalpropertiesand rendering them super uous;therewasno lum inifer-

ousetherorany such thing atall.Them etricether,on theotherhand,did

not have its m etric properties rendered super uousbutinstead accounted

foroncetheirconnection with m atterwasfound.Thestory ofthem echan-

icaland inertialethersisrepeating itselftoday in a geom etric,instead ofa

m etricoram echanical,guise| butwhich ofthesetwo fates(denialorobser-

vation),ifany,m ay the futurehold in store forthe geom etric ether? Both,

wehope.W eseek forthegeom etricetherboth a conceptualchangeand the

recognition ofobservableslinked to itsnew nature,a changecom parableto

the shiftfrom the unobservable m echanicalether to the observable m etric

structureofgeneral-relativistic spacetim e.

Asweseek tolearn whatliesbeneath locality,causality,param etrictim e,

and �elds| asweseek to�nd,asitwere,thesource ofthesepassivephysical

propertiesinherited from em pty spacetim e| the geom etric etherm ust,like

its m echanicalancestor,have another layer ofits nature revealed. This is

to bedone,onceagain,by stripping theetherofitsintrinsicproperties,but

thistim ethey aregeom etric ones.W ebelievethat,taking aconceptualstep

beyond the geom etric ether,i.e.beyond geom etry entirely,m ay be the key

to identifying observables related to em pty spacetim e. These hypothetical

observables, however, would not be directly related to spacetim e points,

which aregeom etricobjects,butratherto \m etageom etricthings," justlike

spacetim e curvature has little to do with the m echanicalproperties ofthe

earlier-hypothesized m aterialm edium .

To putitdi�erently,we are here proposing an updated version ofLar-

m or’scentenary words:

W e should notbetem pted towardsexplaining the sim plegroup

ofrelationswhich have been found to de�ne the activity ofthe

aetherby treatingthem asm echanicalconsequencesofconcealed

structure in that m edium ;we should rather rest satis�ed with

9.4 B eyond the geom etric ether 109

having attained to theirexactdynam icalcorrelation,justasge-

om etryexploresorcorrelates,withoutexplaining,thedescriptive

and m etric propertiesofspace.18

Larm or’s statem ent is rem arkable for its correctness,and allthe m ore re-

m arkableforitsincorrectness.Its�rsthalf(up tothesem icolon)isacorrect

testim ony ofwhatsoon would prove to be the way outofthe m echanical-

etherproblem : itsdenialby specialrelativity. Itssecond halfwas| atthe

tim eofitsutterance| acorrectcom parison between thewayLarm orthought

the m echanicalethershould beconsidered and the way geom etry wasthen

regarded, nam ely, not as background for the explanation of phenom ena.

However,16 years later in 1916 the ether would be overtly reinstated asa

dynam icm etricstructure,and thephysicalpropertiesofspacetim ewould be

explained by itin thesenseofa m etricgeom etric substratum .19 Thistrend

ofattem pting to explain phenom ena in term sofgeom etry,and in particular

m etric geom etry,did notstop with generalrelativity but,as we saw,con-

tinueswith the e�orts ofquantum gravity. Therefore,nowadays geom etry

doesexplain,asa substratum ,thepropertiesofspacetim e,20 and thesecond

partofLarm or’sview isno longercorrect.

Theupdated version ofLarm or’swordsproposed herereadsthus:

W e should notbetem pted towardsexplaining the sim ple group

ofrelations(locality,causality,param etrictim e)which havebeen

found to de�ne the activity ofthe geom etric ether (spacetim e

points) by treating them as consequences of concealed struc-

turein thatgeom etric m edium ;weshould ratherseek to explain

the activity ofthe geom etric ether beyond its geom etric nature,

searching for em pty-spacetim e observablesm etageom etrically.

Unlike Larm or,we do notrenounce an explanation ofthe geom etric ether,

butwe do notattem ptto �nd itatthe sam e conceptuallevelasthisether

�ndsitself.

The search for observables related to em pty spacetim e requires taking

a genuine conceptualstep beyond the state ofa�airs as left by Einstein’s

hole argum ent.W e believe thatthe currentphilosophicalliterature on this

problem ,asreviewed e.g.in (Earm an,1989,pp.175{208),hasnotbeen able

to take thisstep by adding som ething physically new to thediscussion.O n

the contrary,the philosophicaldebate appears to function in the spirit of

18Q uoted from (W hittaker,1951,Vol.1,p.303).

19In fact,Newton’sspace had already assum ed thisrole over200 hundred yearsbefore,

and Einstein’s special-relativistic inertialspacetim e 11 years before,butneitherofthem

had been openly considered as substrata with physicalgeom etric properties untilafter

1916.Then Einstein identi�ed �� and g�� (x)with m etricethers.Breaking with history,

wechosetoidentify theinvariantquadraticform sderived therefrom with them etricethers

due to theircloserconnection with actualphysicalobservables.20Friedm an (1983),for exam ple,writes that\the developm entofrelativity appears to

lead...to the view that the geom etricalstructure ofspace or space-tim e is an object of

em piricalinvestigations in just the sam e way as are atom s,m olecules,and the electro-

m agnetic �eld" (p.xi).


Earm an’swords,which m easure the fruitfulnessofa work by asking \How

m any discussions does it engender?" (p.xi). W hat is needed is a new

physicalinsight by m eans ofwhich the present philosophicaldebate (like

every other)m ay berendered inconsequential,m uch like the olderdisputes

astotheshapeand position oftheearth orthenatureoftheheavenly bodies

were only settled by new physicalinvestigations. Indeed,whatisneeded is

a discovery in thesense ofO ppenheim er’swords:

Allhistory teaches us that these questions that we think the

pressing oneswillbetransm uted beforethey areanswered,that

they willbe replaced by others, and that the very process of

discovery willshattertheconceptsthatwetoday useto describe

ourpuzzlem ent.(O ppenheim er,1973,p.642)

To understand how a possible way to go beyond the hole argum ent is

related totheholeargum entitself,wenow re-rehearseitin geom etric,rather

than coordinate,language and fora general�eld f(P ). W e startwith two

points,P and Q ,inside H linked by a di�eom orphism �(P )= Q ,and we

havethesetwopointsbethelocalaspectorlocation of�eldsf(P )and f0(Q ),

each solution to the�eld equation with thesam e source.Thedem and that

f(P )= f0(Q ) (9.8)

(cf.g�� (x)= g0��(x0))reproducesgeom etrically the requirem entthat,after

an arbitrary coordinate transform ation x0= x0(x),a physicalsituation re-

m ainsunchanged. In term s ofan active transform ation,pointP (cf.x)is

dropped in solution f(P ) (cf.g�� (x)),and a new solution is obtained by

displacing P to Q (cf.x0)asfollows:

f(P )! f(Q )= f(�P )= :��[f(P )]; (9.9)

where ��[f(P )]is induced by � in the way shown and is called the push-

forward off(P ).Thenew �eld ��[f(P )]= f(Q )isalso a solution,but

f(Q )6= f0(Q ) (9.10)

(cf.g�� (x0) 6= g0

��(x0)) (Figure 9.3).21 Again,to preserve the causality of

the �eld equation one postulates that displacem ents ofpoints lead to no

observable e�ects and that, therefore, (displacem ents of) points have no

physicalm eaning.

In order to m ove forward and catch a glim pse of what form em pty-

spacetim eobservablesm ay take,wereason heuristically asfollows.W enote

21It would have been m ore naturalto denote Q by P0 and later to perform an active

transform ation dropping P0(cf.x

0) instead ofP (cf.x),in which case one would have

obtained the expressionsf(P )= f0(P

0)instead of(9.8),f

0(P

0)! f

0(P )= f

0(�

�1P

0)= :

��[f0(P 0)]instead of(9.9),and f(P ) 6= f

0(P ) instead of(9.10),in closer parallelwith

Einstein’soriginalcoordinatenotation and with theoneoftheprevioussection.However,

rem ainingfaithfultothisstartingpointwould havecom plicated thenotation and obscured

the intuitive appealofthe investigation thatfollowsatthe end ofthissection.

9.4 B eyond the geom etric ether 111Spa etime{gαβ(P ), Tαβ(P )}

HoleTαβ(P ) = 0

{gαβ(P ), 0}

{φ∗[g′αβ(P ′)], 0}

Figure 9.3: Schem aticrepresentation oftheholeargum entin geom etriclanguage

in tune with the coordinate notation ofthe previoussection. Here ��[g0

��(P 0)]=

g0��(P ). To m atch the geom etric notation just introduced, replace: g�� (P ) by

g0��(Q ),T�� (P )by T

0

��(Q ),and ��[g

0

��(P 0)]by ��[g�� (P )].

that the postulate above appears to go farther in its claim than it really

needsto. The required property ofdi�eom orphism invariance with respect

to an arbitrary displacem ent ofpoints is satis�ed by sim ply having space-

tim e pointsbe allalike,i.e.have no physicalidentity and form a perfectly

hom ogeneous substratum . Thereby,the problem posed by the hole argu-

m entisavoided,since having physically indistinguishable spacetim e points

m akesthetwo solutionsg�� (P )and ��[g0��(P 0)]physically equivalent.22

W hat do we m ean by physically indistinguishable points? W e can see

thisby recognizingthedi�erencebetween m athem aticaland physicalpoints.

M athem aticalpointsm ay wellbelabelled pointsbut,asStachelrem arked,

No m athem aticalcoordinate system is physically distinguished

per se;and without such a distinction there is no justi�cation

for physically identifying the points ofa [m athem atical]m ani-

fold...as physicalevents in space-tim e. Thus,the m athem ati-

cian willalways correctly regard the originaland the dragged-

along �eldsasdistinctfrom each other. Butthe physicistm ust

exam inethisquestion in adi�erentlight...(Stachel,1989,p.75)

Thephysicistm ust,in thiscase,ratherask whetherin a spacetim eem pty of

m atterand m etric structurethereisanything by m eansofwhich itspoints

can be told apart from one another. Finding there are no such m eans,

the physicist is forced to hold spacetim e points physically identicalto one

another.

However,having a m ultitude ofindistinguishablespacetim e pointsdoes

notnecessarily m ean thattheym ustbephysicallym eaninglessin everyother

way. How so? Think,for exam ple,ofhydrogen atom s,which are exact

physicalcopies ofeach other and yet display physicalproperties. W hich

typeofproperty displayed by hydrogen atom sm ay serveasa usefulanalogy

forthecase ofspacetim e points?

22Rewrite ��[g0�� (P

0)]asg0�� (P )and replace P by P 0 to getg0�� (P0)= g�� (P ).


The hydrogen atom s conform ing a gas ofthis elem ent are,like space-

tim epoints,physically identical,butdenying thereality ofhydrogen atom s

on these grounds does not seem at allreasonable.23 Hydrogen atom s,de-

spite being identical,possessa property thatspacetim e pointsdo notseem

to have: they interact with other m atter and correlate with one another

creating bonds,hydrogen m olecules. In em pty spacetim e, there is noth-

ing else forpointsto interact with,and so the interaction ofthe hydrogen

atom s with,for instance,their container walls seem s to be an unworkable

analogy. However, hydrogen atom s also interact am ong them selves| and

thisisthe only analogy with spacetim e pointsthatrem ainsopen forusto

pursue. Could spacetim e points show observable properties| active physi-

cale�ects| dueto m utualcorrelations,justlikehydrogen m oleculesdisplay

observablepropertiesthatgiveevidenceoftheirconstituentidenticalatom s?

Forty yearsago,W heelertouched upon an idea som ewhatsim ilarto this

one within the context ofthe so-called \bucket ofdust." Pondering the

consequencesofchangesin the topology ofspace,hewrote:

Two pointsbetween which any journey waspreviously very long

have suddenly found them selves very close together. However

sudden thechangeisin classicaltheory,in quantum theory there

is a probability am plitude function which falls o� in the classi-

cally forbidden dom ain. In other words,there is som e residual

connection between points which are ostensibly very far apart.

(W heeler,1964,p.498)

However,W heeler did not develop this idea,but only used it in order to

reject the quantum -theoreticalconcept ofnearest neighbour,according to

the m annerin which he had previously de�ned it.

A m ore thorough attem pt in this direction was carried out by Lehto

(1988)and collaborators(Lehto etal.,1986a,1986b).In the contextofthe

geom etrized pregeom etric lattice analysed in Section 6.3.1,they found that

when the principle ofdi�eom orphism invariance wasconsidered in connec-

tion with quantum theory,�eldson thesaid latticedisplayed correlationsof

quantum -m echanicalorigin.

Like the analogy between identicalspacetim e points and hydrogen at-

om s| them selves identicalon account oftheir quantum nature| and like

W heeler’s conjectured consequence ofa change in the topology ofspace in

a quantum -m echanicalworld,these results insinuate the sam e conclusion.

W hen thephysicalidentity ofspacetim epointsisconsidered in conjunction

with quantum theory| the only branch ofnaturalscience,asIsham (1995,

p.65)rem arked,so farforced to confronttheproblem ofexistence| we are

led to results that di�er from the conclusions drawn by classicalgeneral

relativity alone,nam ely,to the concept ofquantum -m echanicalspacetim e

correlations.Could they hold a key to theidenti�cation ofem pty-spacetim e

observables?

23See (Horwich,1978,p.409)and (Friedm an,1983,p.241).


In orderto display theidea ofcorrelationswithin theexisting contextof

theholeargum ent| butm oving,atthesam etim e,beyond itby introducing

a physically new idea into the classicalsetting| we considerthe following.

W e saw that,in an otherwise em pty spacetim e,a �eld f(P )at pointP is

also an acceptable �eld when dragged onto anotherpointQ ,

f(P )! ��[f(P )]= f(Q ): (9.11)

Butwhathappensifwe now pullitback again to get��[f(Q )]? The �eld

could carry propertiespertaining to Q back with it,so thata com parison of

theoriginal�eld f(P )and therestored �eld ��[f(Q )]could yield thatthey

arephysically di�erent.

This is not to say that we expect to �nd new physics by m eans ofa

purely m athem aticaloperation plusitsinverse.Itm eansthe reverse:given

the privately gathered (not logically deduced) insight that indistinguish-

ablepointsm ay nonethelessdisplay physicale�ectsby correlating with each

otherquantum -m echanically,then the above construction constitutesa ge-

om etric m eans to represent the ensuing broken physicalsym m etry ofdif-

feom orphism invariance. M oreover,when we speak of\m oving points," no

system isactually being displaced in thephysicalworld;thephysicalm ean-

ing ofthem athem aticalexpressionsabove(and below)ultim ately fallsback

on spacetim e correlationsofquantum origin,the existence ofwhich ishere

conjectured.

In other words,this heuristic analysis does nottry to prove,by m ath-

em aticalm anipulation ordeduction,thatsuch quantum correlations exist,

like som e (as criticized in Chapter 2) infer the existence ofPlanck black

holes| and,therefore,the relevance ofthe Planck scale| from a heuristic

analysisoftherelationsbetween m ass,energy,and spatiallocalization.W e

only use heuristicsto inspire ourphysicalthoughtand to guide itsm athe-

m aticalexpression,butnotto establish the physicalsubstance ofthe ideas

thusarrived at.Theideaspresented in thissection,then,arepartofan ex-

ploratory approach to theproblem ofthegeom etricether.Itstruesolution,

however,m ustberooted atitsorigin in fam iliarobservationsofthephysical

world;itm ustcom e from sim ple experim entsthathum anscan understand

and a�ectand notfrom disem bodied m athem aticalpush-forwardsand pull-

backs. These ideas should therefore be interpreted from this exploratory

perspective.

G iven f(P ),theappearanceofthepull-back��[f(Q )]restson theneed to

com pare locally the original�eld and the original�eld restored. Thisneed

arises from the fact that physicalexperim ents are not perform ed globally

butlocally.An analogy closeathand isthatofparalleltransportin general

relativity. G iven two (covariant) vector �elds v�(P ) and v�(Q ) in curved

spacetim e,they can only be com pared by com puting �v � at one point P

by parallel-transporting the latter�eld:

�v �(P )= v�(Q )k � v�(P ): (9.12)

The analogy works best for the case ofa sm allclosed loop traced start-

ing at P (u0;v0) by following a worldline x� = x�(u;v0) untilwe reach a


P

Q

du

xα(u, v0)

dv xα(du, v)

xα(−u,dv)

xα(u0,−v)

vα(P )vα(P )‖

dvα(P )

Figure 9.4: Paralleltransportofa vectorv�(P )around a closed loop in curved

spacetim e.

separation du,and then a worldline x� = x�(du;v)untilwe reach a sepa-

ration dv,thusarriving atQ ;we then com e back24 from Q to P following

worldlinex� = x�(� u;dv)untilwereach aseparation du and then worldline

x� = x�(u0;� v) untilwe reach a separation dv. Field v�(P ) is com pared

to itselfafterhaving travelled theloop and restored to itsoriginalposition.

W e �nd that

dv�(P )= R�� (P )v�

�@x�

@udu

� �@x

@vdv

�

; (9.13)

whereR ��

isthe (m ixed)Riem ann tensor,quanti�eshow m uch the com -

ponents,but not the size,ofv�(P ) have been changed by the experience

(Figure 9.4).

Sim ilarly,ifinstead ofthinkingofa�eld asam erevalue,wevisualizeitas

avector f,and f(P )asacom ponentoff,then anew (geom etric)dim ension

to theholeargum entisrevealed:thelocalcom parison of��[f(Q )]and f(P )

quanti�es how m uch the �eld com ponent f(P ), but not the �eld value,

changesdueto thepossiblequantum correlation ofP with Q ;in short,f as

a vector would behave like v�. Thischange would stem from any physical

reality ofquantum -m echanicalorigin thatthe pointsm ay have.Pictorially

speaking,P would behave as ifit rem em bered having interacted with Q ,

rem aining \entangled" with it.

24W e cannot actually com e back to P in the literalsense ofthe word because we can

only traversespacetim eforwards,in thedirection ofthetangentvectorsto theworldlines.

Thatisto say,the separation du alluded to isgiven by the valuesofa param eteru that

increasesm onotonically on the worldline,e.g.a clock reading.W hatwe can in factdo is

travelfrom P to Q following two di�erentroutesand then m athem atically form a loop by

subtracting one change in the transported vector from the other. See Section 10.3.4 for

details.


Let us now push the parallel-transport analogy further. The parallel-

transportprocedureconstitutesthem athem aticalrepresentation ofa phys-

icalproperty ofspacetim e. This property is curvature and is unheard-of

in at spacetim e,which is sterile with respect to active m etric e�ects. A

curved spacetim e,on the other hand,is not. The paralleltransport ofa

�eld around a closed loop uncoverscurvature asa previously hidden active

m etrice�ect.Now,thedescription aboveintendstopointtotheappearance

ofquantum -related spacetim ecorrelationsin acorrespondingm anner.They

area hypotheticalproperty com pletely foreign to thehom ogeneoussubstra-

tum ofspacetim e points,which is sterile with respect to active geom etric

e�ects(di�eom orphism invariance)| butcould such correlationsariseasthe

activee�ectofadi�erentkind ofm edium ? And ifso,whatkind ofm edium ?

W e believe that we should not seek the source ofquantum spacetim e

correlations in extra hidden geom etric properties ofpoints butsom ewhere

else,within a deeperlayerofthenatureofthings:asan active e�ectofthe

quantum -m echanicalconnection between m etageom etric things.Indeed,the

search for observables related to em pty spacetim e should be accom panied

by theconceptualoverthrow ofthegeom etric etheraswell.Ifnothing else,

the factthatthe presentgeom etric description ofspacetim e leadsagain to

the futile problem ofan etherishintenough thatthere m ustbesom ething

am isswith thisform ofdescription.

W hereshallwelook to�nd observablesthatcapturetheabove-described

behaviourof�eld f(P )and thatare,atthe sam e tim e,in principle free of

geom etricdenotation? Thesecannotcertainly bethe�eldsthem selves,since

they are purely geom etric concepts,norare they observableassuch.

Som etim es the answers to the hardestquestions are staring one in the

face.W hatisthem ostfundam entalphysicalconceptin thepresentunder-

standing ofspacetim e? W hat is generalrelativity essentially about? Not

them etric�eld butan observable network ofinvariantintervalsds between

events;itison them easurableintervalsthatthem etricdescription ofspace-

tim e ultim ately rests. The m etric �eld,as we m entioned earlier and will

show in m ore detail in the next chapter, com es after the directly m ea-

surable intervals,and endows them with geom etric m eaning as part ofa

di�erential-geom etrictheoreticaldescription.By picturing in ourm indsthe

m easurableseparation dsbetween twoeventsastheextension ofaspacetim e

vectord~r(x)param etrized with coordinatesx,we �nd

ds2 = �d~r� d~r= �

�@~r

@x�

��Pdx�

�

�

�@~r

@x�

��Pdx�

�

= �g�� (P )dx�dx�: (9.14)

The realization that ds,as a direct observable,takes physicalprecedence

overitsgeom etrictheoreticalinterpretation isusefulto thissearch,because

ithelpsusdisentangleallposteriorgeom etricbaggage(fulldi�erentialgeom -

etry)from thephysically m oreessentialm easurem entresultitself.Recalling

theideasofChapter4,wearethen leftwith ds asa m inim al,baregeom et-

ric notion,nam ely,as the extension ofsom e underlying m etric line. This

isnotyeta m etageom etric realm ,butitplaces usone step closerto it. It


P

P ′

dsPP ′

Q

Q′

dsQQ′

φ

φ

gαβ(P )

φ∗[gαβ(Q)]

gαβ(Q) =φ∗[gαβ(P )]

φ∗

φ∗

Figure 9.5: Push-forward and successive pull-back ofthe m etric �eld g�� (P )in

an otherwise em pty spacetim e. If spacetim e points had any physicalreality of

quantum -m echanicalorigin, long-range correlations would been seen in the line

elem entds.

suggests that we m ay look at the physicalessence behind ds| a classical

link correlating two eventsasphenom ena| to inspire and guide the search

form etageom etric correlations ofquantum origin.

O nce again in geom etric language,we could now explore25 whetherthe

line elem ent

dsP P 0 =

q

�g�� (P )dx�dx� (9.15)

between two points rem ains unchanged after g�� (P ) is pushed forward,

��[g�� (P )],to get

dsQ Q 0 =

q

��[g�� (P )]dx�dx� =

q

�g�� (Q )dx�dx�; (9.16)

and subsequently pulled back,��[g�� (Q )],togetfortheoriginallineelem ent

q

��[g�� (Q )]dx�dx�: (9.17)

Ifspacetim e pointshad any physicalreality ofquantum -m echanicalorigin,

the �nalexpression (9.17)could notbe equalto the initialone (9.15),and

there would be som e long-range correlations seen in the line elem ent ds

(Figure 9.5).

This would not upset generalrelativity,because the envisioned corre-

lations represent a new kind ofe�ect rather than corrections to physical

m agnitudes predicted by the theory. M oreover,at a conceptuallevel,the

di�eom orphism -invariance principle and the hole-argum ent conclusion are

notchallenged either,sincethesebelong in thegeneral-relativistic classical,

geom etric description ofspacetim e,which doesnotconsiderquantum the-

ory atalland,therefore,m ustview quantum -m echanicalcorrelationsasan

elem entforeign to itsfram ework.In particularand furthering the previous

25Here P

0and Q

0are points in the neighbourhoods ofP and Q respectively,and are

linked by a di�eom orphism �(P 0)= Q0,in the sam e way thatP and Q are.


analogy,we m ay say thatthe localcom parison between ��[f(Q )]and f(P )

dealswith changesin thecom ponentsofa �eld,whereastheholeargum ent

refersto (lack of)changes in the value ofa �eld. Thus,the classical,geo-

m etric theory ofgeneralrelativity rem ainsuntouched from thisperspective.

Strictly speaking and consistent with the general-relativistic picture,

then,it is not possible for the correlations envisioned to be displayed by

dsP P 0 as a classicalgeom etric notion,i.e.as the distance between points

ofa non-quantum theory. Building beyond the raw m aterialprovided by

ds with the aid ofm etageom etric thought and quantum -m echanicalideas,

we seek to learn what lies behind it as a m easurem ent value. O ur expec-

tation is forthe classicalgeom etric intervaldsP P 0 to resultas a geom etric

rem nantortrace leftbehind by m etageom etric things. Itisthrough these

thingsbeyond geom etry thattheconceptualoverthrow ofthegeom etricether

can be pursued. In thislight,one should regard pointsthe way one would

W ittgenstein’s ladder: as concepts to be discarded after one \has clim bed

outthrough them ,on them ,overthem " (W ittgenstein,1922,x 6.54).

But the road to this goal is conceptually dem anding; m uch physical

insightneeds�rsttobegained and broughttolightbeforethem etageom etric

path involving ds can be pursued.W hy isds so im portantand,especially,

m oreim portantthan them etric�eld? How isallrelativity to bebuiltupon

it? And whatisitsphysicalm eaning,whatkind ofm easurem entisit? Does

itreally involve theproverbialrodsand clocks? Thesequerieslead usto an

analysisofthe conceptoftim e.

118

Thetragedyoftheether

E ther A ction O bservability A � ectability

M echanical.

M aterialm edium

M edium for the propagation

of light, gravitation, electric-

ity,m agnetism ,heat,etc.

M echanically sterile hom oge-

neoussubstratum .No observ-

able (active)m echanicalprop-

erties(e.g.velocity,density)

Itse�ecton light,gravitation,

electricity, m agnetism , heat,

etc.cannotbechanged

Inertial.

Absolute spaceand absolute

spacetim e

M edia for the determ ination

ofacceleration and spatial,or

spatiotem poral,extension

G eom etrically sterile hom oge-

neous substrata. No observ-

able(active)geom etricproper-

ties

Their e�ect on particles, or

particlesand clocks,cannotbe

changed

G ravitational.

Dynam ic spacetim e

M edium forthedeterm ination

ofacceleration and spatiotem -

poralextension

G eom etrically rich inhom oge-

neoussubstratum .O bservable

geom etric property:curvature

Hasm atterassource.Itse�ect

on particles and clocks can be

changed

G eom etric.

Spacetim e points

M edium forthe localization of

�eldsf(P ).Em bodim entoflo-

cality and causalevolution in

param etric tim e

Q ualitatively hom ogeneous,

quantitatively structureless

geom etric substratum . No

observable (active) geom etric

properties

Its localizing e�ect on �elds

cannot be changed by m oving

the points (f(P )= ��[f(Q )]).

Not even self-interacting due

to geom etric am orphousness

Table 9.1: A sum m ary ofthe di�erent concepts ofthe ether in physics,from the m echanicallum iniferous ether to the geom etric spacetim e

points. O fthe four,only the gravitational\ether" wasbroughtto fullphysicalaccountability: it is observable and open to m anipulation and

control.

C hapter 10

A geom etric analysis oftim e

An analysis ofthe conceptoftim e wasm y solution.

AlbertEinstein,\How Icreated the theory ofrelativity"

Philosophers ofscience thrive in the analysis ofscienti�c concepts. If

we want to draw a lesson on how to m ake a good analysis ofthe physical

conceptoftim e,we should take a look atthe worksof(broadly speaking)

naturalphilosophers| from V b.C.to XXIa.D.| or should we? W hat do

philosophersofscience do when they m ake an analysis?

A conceptualanalysiscan be interpreted variously. Itm ay referto the

m eticulous logicaldissection ofa concept into constituent parts or to the

physicalexam ination ofa conceptregarding itsphysicalorigin and nature.

If,aswesuggest,thesetwo activitiesarenotoneand thesam e,how do they

di�er? And what is their connection with the activities ofscientists and

philosophersofscience?

In hisenlightening book,Reichenbach (1951)�ndsthesourceof\philo-

sophicalerror" in speculative philosophy in \certain extralogicalm otives"

(p.27).Thesearecom pounded in thespeculative philosopher’spsychologi-

calm ake-up,\a problem which deservesm oreattention than isusually paid

to it" (p.36).Reichenbach describesthefallofspeculative(i.e.traditional)

philosophy and the rise ofscienti�c philosophy,a philosophy to be guided

by m odern science and itstools. Unlike speculative philosophy,the goalof

scienti�c philosophy is not a one-person explanation ofthe naturalworld

buttheelucidation ofscienti�c thought;itsobjectofstudy isscience itself.

O r in the words ofReichenbach,\Scienti�c philosophy...leaves the expla-

nation ofthe universe entirely to the scientist;it constructs the theory of

knowledgeby theanalysisoftheresultsofscience" (p.303).Identifying sci-

enti�c philosophy with m odern philosophy ofscience,we can prelim inarily

concludethat,whereastheraw m aterialofscienceisthenaturalworld,the

raw m aterialofphilosophy ofscienceis(asitsnam ecorrectly suggests)the

scienti�c enterprise.

Asdistinctasscienceand philosophy ofscience seem to be,westill�nd

ourselves at a loss to tellwhich of the two disciplines we should pursue

120 A geom etric analysis oftim e

ifwe want to better understand the physicalconcept oftim e. A reading

ofReichenbach suggeststhatwe should em brace philosophy ofscience,be-

cause it is through it,and not norm alscience,that we gain clari�cation

ofscienti�c concepts (p.123). Indeed,Reichenbach traces the rise ofthe

full-blown \professionalphilosopher ofscience" to the incipient attem pts

ofscientists who would unsystem atically ponder the nature oftheir �elds

between the linesoftheirtechnicalworks.Buthasthisevolutionary devel-

opm entfrom enquiring scientistto professionalphilosopherofscience truly

yielded a speciesbetteradapted to the clari�cation ofconceptualscienti�c

puzzles? O r has the speciation product,becom ing overspecialized,shifted

instead to a separate and altogetherdi�erenthabitat,a new place whereit

can notonly survivebuteven thrive?

Ifby \philosophy of science" we understand the naturalextension of

the ordinary work ofthe scientist that com es from asking not only \how

do Icalculate this?" butalso| or instead| \whatis this,and why should

Icalculate it?" then philosophy ofscience is a usefuland integralpart of

science. Progressin the scienti�c worldview would notbe possible without

it. But philosophy ofscience as a discipline in its own right,carried out

\professionally" by philosophersbecause,asReichenbach putit,\scienti�c

research doesnotleaveam an tim eenough todothework oflogicalanalysis,

and...conversely logicalanalysis dem ands a concentration which does not

leave tim e forscienti�c work" (p.123)| thisisanotherm atteraltogether.

Doesthem eticulouslogicaldissection ofscienti�cconceptsproduceuse-

fulresults without lim it? O r is there instead a fuzzy boundary carried

beyond which logicalanalysisstopsclarifying anything atall? Reichenbach

goeson to adm itthatphilosophy ofscience,\aim ing atclari�cation rather

than discovery m ay even im pedescienti�cproductivity." M ay itnotim pede,

when carried too far,notjustscienti�c productivity buteven clarity itself,

thusturning againstitsoriginalgoal?

W ith the professionalization of the conceptual exam ination naturally

falling on the enquiring scientist, the link with the naturalworld as the

m otivator ofscienti�c conceptsand inform er oftheir analysis wassevered

in professionalphilosophy ofscience,for,as Reichenbach put it,the pro-

fessionalphilosopher ofscience has no tim e to dealwith nature. W hat is

then hisworking m ethod? Hisstarting pointisestablished science and his

m ethods scienti�c,but his logicalanalyses roam largely unconstrained by

feedback from the physicalworld. That is to say, philosophy of science

freezesthescienceoftheday and,arm ed with thetoolsofthesam escience,

looksaway from the physicalworld and proceedsto dissectthe conceptsof

sciencein asm anydi�erentways,intoasm anypossibleconstituents,aslogic

and frozen science willallow.Thus,philosophersofsciencebecom e| in the

words ofone ofthem | \professionaldistinction drawers" (Earm an,1989,

p.2),because,withoutfresh feedback from thenaturalworld,progress(ap-

propriately understood)m ustnecessarily be m ade from the di�erentiation

offrozen scienti�c m aterialinto new,abstractform s.

W hether these new form s are usefulto the scienti�c enquiry,whether

121

they actually clarify itor only succeed in m aking ofit a worse conceptual

sham bles,isnotthem easureby which philosophicalfruitfulnessisassayed.

Inevitably,having severed the link with naturein thenam eofprofessional-

ism ,philosophersjudge theirworks,de facto aswellasin the wordsofthe

sam e renowned philosopher ofscience above,\by one ofthe m ost reliable

yardsticks of fruitfulphilosophizing, nam ely, How m any discussions does

it engender? How m any dissertation topics does it spin o�?" (Earm an,

1989,p.xi).To a non-philosopher,thisisan astounding confession,forone

would have thought that philosophers were not just after m ore discussion

butsolutions.

In this way,the discussions ofscienti�c philosophy becom e sim ilar to

thoseofspeculativephilosophy,which itthoughtto havesuperseded.Stuck

in a �eld of hardened scienti�c lava and without any objective, natural

standard to assay their m erits and discard the un�t,the debates ofscien-

ti�c philosophy too are aptto perpetuatethem selvesthrough m illennia,to

last from everlasting to everlasting. To be sure,the debates ofscienti�c

philosophy,unlikethoseoftraditionalphilosophy,arecapableofgrowing in

com plexity on a parwith the developm entofscienti�c tools| butwhether

thisisforbetterorforworse,one doesnotdaresay.

Can weexpecttheresolution ofaconceptualscienti�cproblem outofthe

argum ents ofprofessionalscienti�c philosophy? Hardly,because its yard-

stick ofsuccess inevitably dem ands ofit that it becom e argum ent for ar-

gum ent’ssake (cf.Earm an’srem arks). Scienti�c puzzleshave always been

solved by m oreand betterscienti�cinvestigations,by individualswhodared

to give a fresh new look atthe physicalworld around them .Forthisto be

possible,thelink with thenaturalworld m ustnotbesevered.Theestablish-

m entofnew scienti�c knowledge is then apt to banish som e philosophical

debatesthatlook ratherrusted and obsolete in the lightofthe new knowl-

edge. (Are spheresthe only acceptable shape forthe locusofthe heavenly

bodies? Is the centre ofthe universe the only acceptable position ofthe

earth? How can Achilleseverovertake thetortoise?) Butscienti�c philoso-

phy drinksin the watersofscience| insatiably: forevery banished debate,

an untold num berofotherswillarise on the basisofthe new science,and

so on foreverm ore.

A telltale indication ofscienti�c philosophy asargum entforargum ent’s

sake can be found in itsconspicuously bellicose language,i.e.the language

typically used by philosophers as they, intellectual enem ies, battle each

other| tillthey draw blood (see below) but not to death: it is essential

forthe well-being ofphilosophy thatallargum entsbe keptalive. The sci-

enti�c philosopher’severyday vocabulary isfraughtwith allsortsofwords

allusiveofviolentcon ict,so m uch so thatattim esonegetstheim pression

tobereadingan accountofwar.O nehears1 ofbattlegrounds,whereenem ies,

unfurling m yriad pro-and anti-ism ags,deploy attacks on the opponent’s

position,and defend their own with counterattacks;ofstruggles in which,

am idstthe sm oke ofbattle,rivals seek victory over defeat by dealing each

1Allitalicized wordsfrom genuine textsby professionalphilosophersofscience.


otherblows and underm ining each other’sground.Notwithouta shock,we

catch an epitom icglim pseofthescienti�cphilosopher’sbellicosem indsetin

Earm an’s(1989)book on spacetim e ontology,where he seeksto \revitalize

what has becom e an insular and bloodless philosophicaldiscussion" (p.3,

italicsadded).O nem ay beinclined to excuse such expressionsasm ere �g-

ures ofspeech,butwhen these repeat them selves constantly,one is led to

presum ean underlying reason ofbeing.

Thereason behind thiscustom isnow apparent:locked within theinert

con�nesofa frozen science,yetseeking developm entand progress,thefocus

of scienti�c philosophy shifts in practice to the dynam ic and constantly

evolving views ofother philosophers. As a result,argum ent is built upon

argum entin aviciously upward spiral,which only afresh confrontation with

the bare bones ofnature’s raw m aterial can collapse back to the ground.

A sustained reading ofscienti�c philosophy is thus apt to lead a scientist

(besidesdespair)tothefollowingre ection:philosophy| orphiloversy? The

love ofknowledge,orthe love ofwar/confusion?2

Thissituation rem indsusofW ittgenstein’swords:

A person caughtin a philosophicalconfusion islike a m an in a

room who wantsto getoutbutdoesn’tknow how.He triesthe

window but it is too high. He tries the chim ney but it is too

narrow. And ifhe would only turn around,he would see that

the doorhasbeen open allthetim e!3

The sym bolic dooris in actuality not so easy to �nd,butthe direction in

which we should look to �nd it is clear: the way out ofthe philosophical

houseistheway thatoriginally broughtusin,nam ely,thestudy ofnature.

W ittgenstein’sm etaphorical\turn around" turnsoutto besingularly apt.

Tosum up with an exem plifyingquestion:ifEinstein had been aphiloso-

pher,ashe issom etim escalled,would he have discovered specialrelativity

through hisphysicalanalysisoftim e,orwould hehaveinstead rem ained sci-

enti�cally frozen in an endlesslogicalanalysis ofNewtonian absolute tim e

and space?

Allthisisnotto say thatprofessionalphilosophy ofsciencenever helps

science orclari�esanything.Attim es,in thehandsofexceptional,scientif-

ically inspired philosophers,professionalthough they m ay be,it does;but

on a whole,in thehandsoftheaverageprofessionalphilosopher,itdoesnot.

If the analyses of scienti�c philosophy go astray due to being logical

dissections uninform ed by fresh experience,scientists should know better.

2The etym ological association is particularly apt. W ers-, m eaning \to m ix up, to

lead to confusion," is the original Indo-European root of the m odern English \war."

Cf. G erm an verwirren, m eaning \to confuse, to perplex," originally from the Proto-

G erm anic root werra-. From the �rst reference below we learn that \Cognates sug-

gest the original sense was ‘to bring into confusion.’ There was no com m on G er-

m anic word for ‘war’ at the dawn of historical tim es." (See \war" in O nline Et-

ym ology D ictionary, http://www.etym online.com ; and Am erican Heritage D ictionary,

http://www.bartleby.com /61)3Q uoted from (M alcolm ,1972,p.51).

123

Scientists,afterall,dealwith thephysicalworld.Thisisatany ratetrueof

theidealscientist,foritisnottoday uncom m on to com e acrossthevariety

that �nds m ore inspiration in \self-consistent" m athem aticalabstractions

than in nature,and has thus m ore in com m on with scienti�c philosophers

than scientists. Butwhataboutthe scientistwho investigates the physical

world and worksin thespiritin which physicshasbeen traditionally carried

outform ostofitshistory?4 W hatdoesan analysism ean to him ? To him ,

an analysisdoesnotm ean de�nition,postulation,orlogicaldissection (into

what?) butexam ination. Exam ination ofa concept’sphysicalm eaning,of

itsphysicalrootand nature.Thecontentofhisanalysisisnotto be found

prim arily in hishead,butin the world around him .Thisisnotto say that

an analysis ofthistype doesnotrely on the scientist’s conceptualtoolsof

thought.Itnecessarily does,butonly in conjunction with akeen observation

ofnature: his elem entaltheoreticalconcepts are developed directly from

observations and experim ents| science’s raw m aterial| not o� the top of

hissecluded head.O ne particularadvantage ofproceeding thusisthatthe

scientist in question knows from the start what he is talking about: the

ontology ofhisfoundationalconceptsis,by and large,uncontroversial.

This being said,we proceed to a geom etric analysis ofthe concept of

tim e.W hatcan beexpected from such an analysis? Firstand forem ost,an

understandingofrelativity theory on thebasisoftheraw m aterialofsim ple

m easurem ents and on physical(as opposed to m athem aticalor philosoph-

ical) ideas. Secondly,an understanding ofrelativity built progressively as

a naturalconcatenation ofsteps,where by \natural" we m ean that,atev-

ery roadblock,weletphysics(and,when necessary,psychology)suggestthe

way forward.In particular,weletonly ourphysicalneedssuggestwhatkind

ofm athem aticalconcepts we need,and we develop these naturally,notas

form al,hard-to-swallow de�nitionswithoutany apparentreason ofbeing.

An expected upsideofthisway ofproceeding isa m inim ization ofm ath-

em aticalbaggage,but| asevery upside surely m usthave a downside| the

scepticalreaderm ay ask,areweperhapstradingdepth ofunderstandingfor

m athem aticallightness? Breaking theabove-insinuated conservation law of

upsidesand downsides,we are here unexpectedly confronted by an upside

withouta downside,butinstead conduciveto a furtherupside:itisbecause

m athem atics is used only in its due m easure,as physics dem ands and no

m ore,thatwe end up gaining understanding ofrelativity theory instead of,

asism ore com m on,ending up clouded in confusion. The factthatm athe-

m aticsisin physicsonly a toolcan beignored only atthephysicist’speril.5

4In connection with spaceand tim e,m ostnotably by Riem ann and Einstein.In hisfa-

m ousessay,Riem ann (1873),am athem atician,talksasaphysicistwhen hepointsoutthat

\the propertieswhich distinguish [physical]space from otherconceivable triply extended

m agnitudes are only to be deduced from experience," and is even wary ofassum ing the

validity ofexperience \beyond thelim itsofobservation,on the side both ofthe in�nitely

greatand ofthe in�nitely sm all" (pp.1{2,online ref.).AsforEinstein,hiscontinualuse

ofsim ple thoughtexperim ents during the �rst halfofhis life as a starting point for the

theoreticalconceptsofphysicsiswellknown.5M athem atics as practiced by the m athem atician, i.e.as a form alsystem em pty of


The direconsequencesofthe abuse ofm athem aticalbaggage in physics

have also been noticed by other people pondering these issues. Physicist

RobertG eroch writes:

O nesom etim eshearsexpressed theview thatsom esortofuncer-

tainty principleoperatesin theinteraction between m athem atics

and physics:thegreaterthem athem aticalcareused toform ulate

a concept,the less the physicalinsight to be gained from that

form ulation. Itisnotdi�cultto im agine how such a viewpoint

could com e to be popular. It is often the case that the essen-

tialphysicalideasofa discussion aresm othered by m athem atics

through excessive de�nitions,concern overirrelevantgenerality,

etc.Nonetheless,onecan m akeacasethatm athem aticsasm ath-

em atics,ifused thoughtfully,is alm ost always useful| and oc-

casionally essential| to progressin theoreticalphysics.(G eroch,

1985,p.vii)

In conclusion,for an insightfulphysicalanalysis,m athem atics in due

m easureand professionalphilosophy notatall.

10.1 C lock readings as spatiotem poralbasis

10.1.1 Events and particles as geom etric objects

As we survey the m acroscopic world in connection with classicaland rela-

tivistic theories,two key concepts com e to ourattention: thatofan event

and thatofa particle. An eventis,quite sim ply,a phenom enon,anything

thatweregisterashappening:a ash oflightning,thesound ofa horn,the

touch ofa hand on the shoulder,etc.

Isan eventassuch a geom etric concept? Itcertainly isa hum an-m ade

concept,for what we callan event never appears alone and di�erentiated

from therestofa jum bleofim pressions.Butitcertainly isnota geom etric

concept. W e can ask ofit a pair oftest questions: does an event have a

shape? Doesthe conceptofsize apply to it? Theanswerto both questions

isno.Notbeforewe idealize it.

Theidealization ofan eventm adein physicsisoneofthem ostbasicyet

extrem e cases ofgeom etric idealization. G oing m uch farther than having,

for exam ple,the m ore tangible earth crust idealized as an even spherical

surface,here we have an inde�nite physicalphenom enon squeezed into the

tightestofallgeom etric objects,a point.

Doesnow a point-eventhavea shape,and can wespeak ofitssize? The

�rstquestion is the trickier ofthe two because,ofallgeom etric objects,a

pointisby farthem ostabstract| an extrem ecase.Unlikeallothergeom et-

ricobjects,theshapeofa pointisirrelevantbecauseitssize(understood in

any way) is zero. Thisanswersthe second question above: the concept of

size doesapply to a point,nam ely,itiszero.Thisisnotatallthe sam e as

sem antic m eaning,isanotherm atteraltogetherand notthe subjectofthiscriticism .

10.1 C lock readings as spatiotem poralbasis 125

sayingthatitism eaninglesstospeak ofitssize,asitistospeak ofthesizeof

a raw event.Dueto theirutm ostgeom etricabstractness,pointsarestrictly

speaking im possibleto visualize,although they arenorm ally thoughtofand

depicted asdotsofsom e irrelevantshape(circular,square,spherical,etc.).

How does this translate to the physically inspired point-events? By

geom etrically idealizing physicalphenom ena as points,the form er becom e

endowed with two seem ingly physicalproperties:a point-eventissom ething

thatdoesnottakespaceand doesnothaveduration.Butwith such extrem e

features,how can point-eventsbem easured (orlocalized)assuch? Afterall,

ifthey do nottake space and do nothave duration,they m ustnotexist.

Atthisdi�cultjunction,countingcom estoouraid.Thequality ofbeing

point-like can be characterized analytically by m eans ofnum bers: a point

issim ply a setofrealnum bers. In classicalphysics,a point-event isiden-

ti�ed with the setoffournum bers(x;y;z;t),and the whole ofallpossible

point-eventsform sa four-dim ensionalcontinuum called space-tim e| with a

hyphen,sincetisspecialand given separately.Thesam eidea can betrans-

ferred to relativity theory by loosening the restriction above,according to

which tim etisgiven independently oftheotherthreenum bers.In relativity

theory,apoint-eventisidenti�ed with thesetoffournum bers(x1;x2;x3;x4),

and the whole of allpossible point-events form s a four-dim ensionalcon-

tinuum called spacetim e| without a hyphen,since no elem ent ofthe set

(x1;x2;x3;x4) can be separately singled out as specialand,in particular,

representing tim e.

The sets offour num bers (x;y;z;t) and (x1;x2;x3;x4) are the coordi-

nates of a point-event, but what physicalm eaning shallwe attribute to

them ? Ifcoordinatesareto have any m eaning,itm ustbegiven them oper-

ationally:how arethese num bersattributed to a point-event?

Sincethespace-tim ecoordinatet,sym bolizingabsolutetim ein (x;y;z;t),

hasin factno operationalm eaning,we focusonly on thegeneralspacetim e

coordinates (x1;x2;x3;x4). To localize a spacetim e event,we resort to an

arbitrary operationalm ethod,whereby wespecify theway in which thefour

num bersare m easured.Considerasan exam ple O hanian’s(1975,pp.222{

223)astrogatorcoordinatesused byahypotheticalastronavigator.Todeter-

m ine the spacetim e coordinatesofhisspaceship,he identi�esfourdi�erent

stars (red,yellow,blue,and white),m easures the angles �� between any

four di�erent pairs ofstars,and associates each num ber with a spacetim e

coordinate,�� = :x� (� = 1;2;3;4).

In spacetim e,nocoordinateisdi�erentiated from theothers;each shares

the sam e statuswith the otherthree. Butso thattwo distinctevents(e.g.

two distinctphenom ena inside the spaceship)willbe tagged with di�erent

num bers,atleastoneofthestars(butpossibly allfour)m ustbein relative

m otion with respectto any oftheotherthree.O hanian callsthiscoordinate

the\tim ecoordinate." However,thisnam em ay bem isleading for,ifallthe

starsarem ovingrelativetooneanother,which oneistobethe\tim ecoordi-

nate"? Ifthefourcoordinatesareoperationally equivalentand any ofthem

can be taken as the \tim e coordinate," then why give one a specialnam e


withoutreason? TheghostofNewtonian tim e keepshaunting relativity.

The establishm entofa connection between tim e and a coordinate,say,

x4,isonly possible in specialcases. Forexam ple,ifspacetim e isseparable

into a spatialpartand a tem poralpart,i.e.ifitsquadratic form Q can be

written as

Q = gab(x1;x

2;x

3;x

4)dxadxb+ g44(x1;x

2;x

3)(dx4)2; (10.1)

and ifwe choose com oving coordinatessuch thatx1 = x2 = x3 = constant

(only onem oving starwith respectto theship),then ds2 = constant(dx4)2.

W hen theserequirem entsareful�lled,thedi�erentialdx4 ofthefourth coor-

dinateisdirectly proportionaltotheclock separation ds(seeSections10.1.3

and 10.4.5).

W heneverspacetim e coordinatesare chosen withoutestablishing atthe

sam e tim e theiroperationalm eaning (e.g.sphericalcoordinates),these re-

m ain physically m eaninglessuntila connection isestablished between them

and a m easurableparam etersuch ass.Forexam ple,oncethem etric�eld is

known and thegeodesicequation solved,onecan �nd a relation x� = x�(s).

G iven two setsofcoordinates(x1;x2;x3;x4)and (y1;y2;y3;y4),we can

�nd analytic transform ationsx� = x�(y)and y� = y�(x)thattake one set

ofcoordinate values to another (x represents the whole set ofcoordinate

values,and likewise fory).

The next concept essentialto classicaland relativistic theories is that

ofa (m assive or m assless) particle. Like events but less dram atically so,

particlesare also idealized geom etrically aspoints.Unlike events,however,

particles are taken to be m oving points both in classicaland relativistic

theories.An eventand a particle are connected asfollows:the history ofa

particle isa sequence ofevents.

In classicalphysics,the history ofa particle isgiven by thesetofequa-

tions fx = x(t);y = y(t);z = z(t)g,where thasthe role ofa (coordinate)

param eter. These three equations determ ine a particle’s classicalworld-

line. In relativistic physics,on the other hand,the history ofa particle is

given by the set ofequations x� = x�(u), where u is a one-dim ensional

(non-coordinate) param eter. These four equations determ ine a particle’s

relativistic worldline.Now,ifpossibleand ifso wanted,one could solve for

u (e.g.u = u(x4)) and use this to rewrite the above set offour equations

as a setofthree,xa = xa(x4)(a = 1;2;3),butin so doing the separation

between space and tim e hasnottaken place| there isno reason to take in

generalx4 asa tim e coordinate in relativity theory.

In classicalphysics,aparticleofm assm satis�estheequationsofm otion

md2xa

dt2= F

a; (10.2)

where the classicalforce F a isofthe form F a(x;dx=dt).Thesethree equa-

tionsdeterm ine the history ofa particle,oritsworldline,ifone isgiven (i)

an event (x1;x2;x3;t)that belongs to itand (ii) a direction in space-tim e

atthe said event,i.e.the ratios between the di�erentials dxa and dt. Can


thisidea ofan equation ofm otion also be applied in relativity theory? W e

now dem and thatthe history ofa m assive particle,oritsworldline,be de-

term ined onceoneisgiven (i)an event(x1;x2;x3;x4)thatbelongsto itand

(ii)a direction in spacetim eatthesaid event,i.e.theratiosbetween thedif-

ferentialsdx� and du.Thisdem and can bem etifthedi�erentialequations

fortheworldline ofa particle are ofthe form

d2x�

du2= X

�; (10.3)

whereX � = X �(x;dx=du).Heredu appearsonly asa di�erential.W hatis

X � equalto and whatisitsphysicalm eaning? W ereturn to thisissuelater

on,when thetim e isripeforit.

10.1.2 T he observer as psychologicalagent

W esaw thatphysicalphenom enaaregeom etrically idealized aspoint-events,

and thatm atterisgeom etrically idealized aspoint-particles. In particular,

now a hum an observerisgeom etrically idealized asa m oving point-particle

that m akes observations along its worldline. Conventionally,the so-called

observerdoesnotneed to behum an,butisem bodied by a setofm easuring

devicesm oving along aworldline.However,thereexistsoneessentialsideto

theobserverthatonly a hum an observercan provide,nam ely,itssubjective

experience. According to this experience, events can be laid down in a

certain orderalong the hum an observer’sworldline,who thuscreatesin his

m ind the past,thepresent,and thefuture.

W e saw earlier that none ofthe four coordinates ofa relativistic event

(x1;x2;x3;x4) could in generalbe isolated as tim e. The isolation ofone

ofthese four coordinates as universaltim e would unavoidably lead to an

absolute ordering ofevents asin classicalphysics(classicalabsolute tim e),

butrelativity theory doesnotclaim the existence ofa naturalorderforan

arbitrary pairofpoint-events.W hen atallpossible,apairofeventscan only

beordered with respectto theworldlineofa given observer| heisactually

theonewhodoestheordering.Forthosepairsofeventsthatcannotpossibly

be contained on the observer’s worldline due to restrictions ofhis m otion,

no ordering relation is possible at all. Allthese events belong together in

therelativisticfour-dim ensionalpresent,whereasthosethan can beordered

alongtheobserver’sworldlinebelongin therelativisticfour-dim ensionalpast

orfuture.

Also the classicalworldview can beunderstood from thistem poralper-

spective. Asthe constraintim posed by the three-dim ensionalhypersurface

lim itingthem otion oftheobserverisrelaxed,theforbidden four-dim ensional

presentbecom esan allowed three-dim ensionalpresent,togetherwith awider

four-dim ensionalpastand future(Figure10.1).Classicalphysics,then,ap-

pearsto be a specialcase ofrelativistic physicsasfarasthe dim ensionsof

thepresentareconcerned,but,on theotherhand,relativisticphysicsisalso

m orelim iting than classicalphysics,becausein itthepresentisa forbidden

area. The restrictive nature ofthe relativistic presentisone ofthe reasons


O

A

B

D

C

futurepastpresentpresent physi alboundary

lassi alphysi sA

B

OD

C

futurepastpresentpresent

Figure 10.1: In a relativisticworld,an observerdeterm inesthepast,thepresent,

and the future,ashe orderseventson hisworldline into these three categoriesac-

cording to hissubjectiveexperience.Thepresentisform ed by thesetofallevents

thatlie outside the physicalboundary im posed by restrictionson hism otion.The

classicalconception ofan absolute present ariseswhen the boundary is collapsed

ontoa three-dim ensionalplanet= constant.EventD belongsto both therelativis-

tic and classicalpresent,while event C belongs to the relativistic present but to

the classicalfuture.

why classicalphysicscontinuesto be preferred form ostofourdaily needs,

since in ittheallowed presentisunderstood m ore widely.

10.1.3 C locks and the line elem ent

Thefoundationsofclassicaland relativisticphysicscan belaid down on the

basisofclock m easurem ents. Ideally,a clock should progressata constant

pace;this requirem entis realized in practice by searching for naturalpro-

cesses with respect to which a great m any other naturalprocesses are in

synchronicity,thereby sim plifying theirstudy.Asan idealclock can only be

determ ined by com parison with otherclocks,itsbasicpropertym ay begiven

asfollows:two idealclockscarried along by an observeralong hisworldline

rem ain synchronous| theirtickingratiorem ainsconstant| foranyworldline

C joining any pairofeventsA and B .

Considernow an observerwho carriesan idealclock along hisworldline

C from point-event A to point-event B . At A he records the clock read-

ing sA and on reaching B records the new clock reading sB . These clock

readings(nottim e instants)are the resultsofphysicalm easurem ents. W e

now say thatthe separation between eventsA and B along worldline C is

�s= s B � sA .W hen eventB iscloseto eventA,by which wem ean opera-

tionally thattheobserver\stopshiswatch quickly afterA,"6 agood approx-

im ation to �s is furnished by the separation ds along a straight worldline

elem ent;in sym bols,�s� ds.

Theworldline-elem entseparation dsasherepresented (in term sofclock

6W e m ay also say that A and B are close iftheir coordinate values di�er by a sm all

am ount.


classicalphysics

? ds relativisticphysics

othergeom etrictheories

no geom etry geom etry

geom etry

geom etry

Figure 10.2: Thelineelem entds can actasa starting pointforthedevelopm ent

ofclassicaland relativistictheories,and presum ably forany othergeom etrictheory

ofspace and tim e,when we choose to geom etrize it according to tradition. But

whatkind ofm etageom etrictheory,describing whatphysics,would onearriveatif

onefollowed the opposite,no-geom etry direction?

m easurem ents)givesenough foundation to lead,via di�erentchoices,to the

conceptsofeitherNewton’sclassicalphysics,Einstein’sspecial-and general-

relativistic physics,and presum ably to any othergeom etric theory ofspace

and tim e. Here we shalldealwith classicaland relativistic theorieson the

basisofds.

It is noteworthy,however,that ds,as a tem poralcorrelation between

two events,or two \nows," is not a geom etric concept. There is nothing

geom etric in the (psychological)identi�cation ofthe present(when the ini-

tialand �nalobservationsare m ade),and there isnothing geom etric in an

event,since,aswesaw,thesearenotoriginally pointsbutraw phenom ena.

Theactivity com prised by a clock m easurem entand thevaluesitproduces,

however, can lead to a geom etric representation. Such a geom etrization

and itsm ain resultsare in factthe essence ofthischapter. These observa-

tionslead then to a provoking thought:whatkind oftheory m ay bearrived

at by following an opposite,no-geom etry direction from ds? W hat kind

ofm etageom etric theory,describing what physics oftim e,would this be?

(Figure 10.2).

A geom etric picture em erges when events becom e point-events,an ob-

servera point-particle,an observer’shistory a world-line,�s the length of

theworldlinebetween thejoined point-events,and ds thelength ofthe dif-

ferentialarrow dx between point-eventsA atx and B atx + dx,i.e.when

westartto view ds in term sofcoordinatepointsand coordinatearrows.In

fact,thegeom etrization ofthe di�erentialds leadsus| naturally enough|

to di�erentialgeom etry. W hat else do we know about ds? How does it

depend on thecoordinatesofpoint-eventsA and B ?

To m ove forward,som e choices m ustbe m ade. The m ain stopoverson

theroad to Einstein’sgeom etric theory ofgravitation have,in fact,already

been called into attention in two publiclectures:by Riem ann in 1854,\O n

thefoundationswhich lieatthebasesofgeom etry," and by Einstein in 1922,

\How Icreated the theory ofrelativity"| it isthrough theirstepsthatwe


shallprogress.

The �rst clue is older than relativity itselfand com es from Riem ann’s

fam ous1854 lecture. Ashe setshim selfthe task ofdeterm ining the length

ofa line in a m anifold,he writes:

The problem consists then in establishing a m athem aticalex-

pression for the length ofa line...Ishallcon�ne m yselfin the

�rstplaceto linesin which theratiosofthequantitiesdx ofthe

respective variables vary continuously. W e m ay then conceive

these lines broken up into elem ents,within which the ratios of

thequantitiesdx m ay beregarded asconstant;and theproblem

isthen reduced toestablishingforeach pointageneralexpression

forthelinearelem entdsstarting from thatpoint,an expression

which willthuscontain thequantitiesx and thequantitiesdx.I

shallsuppose,secondly,thatthelength ofthelinearelem ent,to

the�rstorder,isunaltered when allthepointsofthiselem entun-

dergo the sam e in�nitesim aldisplacem ent,which im pliesatthe

sam etim ethatifallthequantitiesdx areincreased in thesam e

ratio,thelinearelem entwillalsovaryin thesam eratio.O n these

assum ptions,the linear elem entm ay be any hom ogeneous func-

tion ofthe �rst degree ofthe quantities dx...(Riem ann,1873,

p.5,online ref.,italicsadded)

Following Riem ann,wesetthelineelem entds to bea function ofthecoor-

dinatesofA and B ,

ds= f(x;dx); (10.4)

wheref(x;dx)isa hom ogeneousfunction of�rstdegree in the di�erentials

dx ,i.e.f(x;kdx)= kf(x;dx)(k > 0 can be a function ofx).7 W hen the

equation ofworldline C is given by x� = x�(u),where u is a param eter

whosevalue growsfrom thepastto the future,we can then write

ds= f

�

x;dx

dudu

�

= f

�

x;dx

du

�

du; (10.5)

and therefore theseparation �s between A and B isgiven by

�s=

Z B

AC

ds=

Z uB

uA

f

�

x;dx

du

�

du: (10.6)

10.1.4 Invariance ofthe line elem ent

Talking abouthispersonalstruggletounderstand gravitation geom etrically,

Einstein (1982) relates during his K yoto lecture discussing with M arcel

G rossm ann whether \the problem could be solved using Riem ann theory,

in otherwords,by using theconceptofthe invariance ofthe lineelem ents"

(p.47).Thisisa key issue,whosestudy now allowsusto m ove on.

7W e have found that also Synge (1964) has understood relativity chronom etrically

based on the Riem annian hypothesis.


Event separation �s and its di�erentialapproxim ation ds are the re-

sultsofphysicalm easurem entswith a clock m adeby theobserveralong his

worldline;ds is an absolute spacetim e property ofthis worldline,nam ely,

its extension between events A and B . Because ds is here m eant only in

thisclassicalsense,itsinvarianceisa sim plem atter.8 Having proposed that

thelineelem entisa function ofthecoordinates,ds= f(x;dx),thephysical

raw m aterialencoded asds now dem andsthatitsm athem aticalexpression

f(x;dx)bean invariantin them athem aticalsenseoftheword.Thatis,the

value off(x;dx)m ust be independentofthe set ofcoordinates chosen by

theobserveron hisworldline,i.e.f(x;dx)= f0(y;dy),wherey� = y�(x).

The need to dealwith physicalinvariance leads usto search form ath-

em aticalobjects that are m eaningfulregardless ofthe coordinate system

in which they m ay be expressed. Tensor equations (not just tensors) are

such objects;ifa tensorequation holdsin onecoordinatesystem ,itholdsin

any coordinate system .Tensorequationshave thisproperty becausetensor

transform ationsbetween coordinate system sare linearand hom ogeneous.

Events A at x and B at x + dx determ ine a di�erentialdisplacem ent

vectorwhose com ponentsare dx� and whose tailis�xed atA. In another

coordinate system ,the com ponents ofthis di�erentialdisplacem ent vector

are dy� and can be obtained by di�erentiation ofthe coordinate transfor-

m ation y� = y�(x).W e �nd

dy� =@y�

@x�dx�; (10.7)

which is a linear and hom ogeneous coordinate transform ation of the dif-

ferentialwith coe�cients @y �=@x�. The set ofdi�erentials dx�,together

with the way they transform from one coordinate system to another,gives

us a prototype for a �rst type of vector (and, in general, tensor) called

contravariant.9

Following thecontravariantvectorprototype,wecan build othersetsof

m agnitudesattached to A thattransform in thesam eway.Them agnitudes

T�� ,transform ing linearly and hom ogeneously in T�� according to

T0�� =

@x0�

@x

@x0�

@x�T �; (10.8)

8In thepresentcontext,thephysicalscale(seconds,m inutes,hours)ordim ension used

to expressdsisinconsequentialto itsinvariance.A m easurem entresultexpressed as60 s,

1 m in,etc.,orin a di�erentdim ension,say 2� rad,m akesreference to thesam e objective

physicalextension| this is som ething we can m etaphorically point at,in the sam e way

that\book" and \Buch" referto thesam e physicalobject.(Cf.com m entsby Eakinsand

Jaroszkiewicz (2004,p.10)). Beyond the tentative,exploratory rem arks at the end of

Chapter9,a quantum -m echanicaldiscussion ofinvariance (in particular,thatofds)falls

outside the scope ofthis thesis. For a discussion ofquantum experim ents involving two

di�erentinertialfram es,see (Jaroszkiewicz,2007,pp.50{54).9Although contravariant vectors and tensors are inspired in di�erentialdisplacem ent

vectors(theircom ponentsaresm all),thecom ponentsofcontravariantvectorsand tensors

do notin generalneed to besm all.Forexam ple,thecom ponentsdx�=du ofthederivative

along a worldline representa unbounded contravariantvector.


constitute a second-order contravariant tensor. And sim ilarly for higher-

ordercontravarianttensors.

A zeroth-ordertensorisascalarthattransform saccordingtotheidentity

relation,T0= T,i.e.itisan invariant.Ifwenow takea zeroth-ordertensor

to be a function ofthe coordinatesf(x),we m usthave thatf(x)= f0(x0).

How doesthegradientoff(x)transform when expressed in anothercoordi-

nate system ? Using the chain rule,we have

@f0(x0)

@x0�=@f(x)

@x0�=

@x�

@x0�

@f(x)

@x�: (10.9)

It turns out that a gradient transform s in a m anner opposite to that of

contravariant vectors (and tensors). The gradient then serves as a proto-

type for a second type ofvector (and,in general,tensor)called covariant.

Historically,thisnam e actually com esfrom a shortening and reshu�ing of

\variation co-gradient," i.e.variation in the sam e m anneras the gradient,

and the nam e contravariant com es from \variation contra-gradient," i.e.

variation againstthe m annerofthe gradient.

Like before, following the covariant-gradient prototype, we can have

other sets ofm agnitudes attached to A that transform in the sam e way.

The m agnitudes T�� ,transform ing linearly and hom ogeneously in T�� ac-

cording to

T0�� =

@x

@x0�

@x�

@x0�T �; (10.10)

constitute a second-order covarianttensor. And sim ilarly for higher-order

covarianttensors,and alsoform ixturesofboth types.Finally,when atensor

isattached notonly to a single event,butto every eventalong a worldline

or to every event in the four-dim ensionalspacetim e continuum ,we have a

spacetim e �eld called tensor �eld.

10.1.5 T he N ew tonian line elem ent

So far,weknow two chiefpropertiesofds.O neisthatitrepresentsa classi-

calm easurem entofan absolutespacetim eproperty and isthereforea phys-

icalinvariant;thism eans thatf(x;dx)m ustbe a m athem aticalinvariant.

The other is that f(x;dx)is hom ogeneous with respectto allcom ponents

dx�.W ithin theserequirem entsthereisstillfreedom to choose theform of

f(x;dx),butwhatchoices lead to physically relevantresults? W e proceed

in whatone m ay calla naturalway.Letustry to separate the dependence

off(x;dx)on x and dx usingthem athem aticalinvarianceand hom ogeneity

ofthisfunction.Both requirem entscan beful�lled by considering an inner

product.

W eproposeasa�rstcasetobeanalysed an innerproductoftwovectors,

a linear form

ds= f(x;dx)= g�(x)dx�: (10.11)

Thisfunction isclearly hom ogeneousof�rstdegreewith respectto dx� and

producesan invariantnum ber.M oreover,becauseg�(x)dx� isinvariantand


dx� isageneralcontravariantvector,wem usthavethatg�(x)isacovariant

vector.10 Butwhatisthe physicalm eaning ofg�(x)?

Thesim plestoption isto choosethecovariantvectorafterthecovariant

prototype,

g�(x)=@t(x)

@x�; (10.12)

where t(x) is an invariant function ofthe spacetim e coordinates x. The

separation between eventsA and B becom es

sB � sA =

Z B

A

ds=

Z B

AC

@t(x)

@x�dx� =

Z B

AC

dt(x)= tB � tA : (10.13)

In otherwords,becausedt(x)isan exactdi�erential,theseparation between

eventsA and B doesnotdepend on the worldlineC travelled.Because we

know experim entally that this result is false,we can conclude that there

m ustbe som ething wrong with our choice ofEq.(10.12),and presum ably

even with the linearform (10.11). O n the positive side,we learn thatthis

choice leads to the interpretation ofs as absolute classicaltim e t. Thisis

one ofthe essentialdi�erencesbetween classicaland relativistic physics;in

the latter,ds is not an exact di�erentialand the separation �s between

eventsdependson the worldlinetravelled.

10.1.6 T he R iem ann-Einstein line elem ent

Thesecond sim plestchoiceforf(x;dx)satisfyingthesam etworequirem ents

setaboveconsistsofan innerproductofasecond-ordercovarianttensorand

two di�erentialdisplacem entvectors,thatis,a quadratic form

ds= f(x;dx)=

q

� g�� (x)dx�dx�: (10.14)

Thesquarerootisnecessary to m eettherequirem entofhom ogeneity of�rst

degreewith respectto thedi�erentialdisplacem ents,whereasthereason for

them inussign willbecom e apparentlateron.

O n the basis ofa sim ilar procedure as above (see previous footnote),

we can now deduce som ething about the tensorialnature ofg�� (x), but

not as m uch as before. Although dx� is a generalcontravariant tensor,

sinceitappearsin thesym m etricproductdx�dx�,wecan now only deduce

that g�� + g�� m ust be a second-order covariant tensor, but we cannot

conclude anything about the tensorialnature ofg�� itselfon the basis of

the invariance ofthe innerproduct. However,ifwe assum e thatg�� (x)is

sym m etric| and this we shalldo| then we can conclude that g�� (x) is a

second-ordercovarianttensor.11

10Theinvarianceoftheinnerproductm eansthatg0� dx0� = g�dx

�,and thefactthatdx�

is a contravariant vector m eans that it transform s as dx0�

= (@x0�=@x

�)dx

�. Together

these two relations lead to [(@x0�=@x

�)g

0� � g�]dx

�= 0. O n account ofdx

�being an

arbitrary contravariantvector,the bracketed expression m ustbe null,which m eans that

g� transform sasa covariantvector.11Einstein considered the possibility ofan asym m etric m etric tensorin the latterpart

ofhislife in the contextofattem ptsata uni�ed theory ofelectrom agnetism and gravity.


G iven then a sym m etric second-order covariant tensor g�� (x) that is,

m oreover,non-singular(det(g�� )6= 0),the invariantquadratic form

Q = g�� (x)dx�dx� (10.15)

determ ines a four-dim ensionalcontinuum called Riem ann-Einstein space-

tim e. The line elem ent (10.14) of Riem ann-Einstein spacetim e, which is

based on thisquadratic form ,isthe foundation ofrelativity theory.

A challenging question,however,rem ains:whatisthephysicalm eaning

ofg�� (x)? To sketch a solution,we now need to take a look atthefounda-

tions ofRiem annian m etric geom etry. In connection with the search fora

new description ofgravitation,thenecessity ofthisstopoverwasrelated by

Einstein during hisK yoto lecture:

W hatshould welook forto describeourproblem ? Thisproblem

wasunsolved until1912,whenIhitupontheideathatthesurface

theory ofK arlFriedrich G aussm ightbethekey to thism ystery.

I found that G auss’surface coordinates were very m eaningful

forunderstanding thisproblem .Untilthen Idid notknow that

Bernhard Riem ann [who wasa studentofG auss]had discussed

the foundation ofgeom etry deeply...I found that the founda-

tions ofgeom etry had deep physicalm eaning. (Einstein,1982,

p.47)

10.1.7 From G aussian surfaces to R iem annian space

M etric geom etry| geom etry that deals essentially with m agnitudes| has

length as its basic concept. In Euclidean m etric geom etry, this concept

always refers to the �nite distance between two space points. In Riem an-

nian m etric geom etry,on the otherhand,itrefersto the distance between

two very (not necessarily in�nitesim ally) close points. In other words,the

passagefrom �niteEuclidean geom etry todi�erentialRiem annian geom etry

isachieved by approxim ating the�nitedistance�lbetween two pointswith

the di�erentialdistance dl.

W e start by exam ining Cartesian coordinates (straight lines m eeting

at right angles) in a three-dim ensionalEuclidean space. O n the basis of

Pythagoras’theorem ,the squared distance between the points (y1;y2;y3)

and (y1 + dy1;y2 + dy2;y3 + dy3)isgiven by

dl2 = (dy1)2 + (dy2)2 + (dy3)2: (10.16)

In term s ofgeneralcoordinates x1;x2;x3 (curved lines m eeting at oblique

angles),becausedyi= (@yi=@xa)dxa (i;a = 1;2;3),we obtain

dl2 = hab(x)dxadxb; (10.17)

where

hab(x)=

3X

i= 1

@yi

@xa

@yi

@xb(10.18)


issym m etric with respectto a and b by inspection. Again,because hab(x)

isassuredly sym m etric,thesquared length dl2 isan invariant,and dxa isa

generalvector,wecan conclude(based on theresultsoftheprevioussection)

that hab(x) is a (sym m etric) second-order covariant tensor.12 This tensor

determ inesthe m etric structure,orm etric,ofthree-dim ensionalEuclidean

space,and istherefore called the m etric tensor.

The next step towards Riem annian space is to consider with G auss a

two-dim ensionalsurface yi = yi(x1;x2) (i= 1;2;3),where x1 and x2 are

param eters, em bedded in a three-dim ensionalEuclidean space. Now the

squared distance between neighbouring pointson thesurfaceisgiven by

dl2 = (dy1)2 + (dy2)2 + (dy3)2 = ab(x)dxadxb; (10.19)

where

ab(x)=

3X

i= 1

@yi

@xa

@yi

@xb(a;b= 1;2): (10.20)

Thissym m etric second-ordercovariant tensoriscalled the surface-induced

m etric, i.e.the m etric that the surrounding three-dim ensionalEuclidean

space induces on the surface. W hat happens ifwe now rem ove the space

surrounding thesurface altogether?

Although it was G auss who �rst developed the idea ofa surface em -

bedded in a higher-dim ensionalEuclidean space,itwasRiem ann who took

theseem ingly nonsensicalstep ofrem oving theem bedding spaceand taking

what rem ained seriously: a two-dim ensionalnon-Euclidean space, a two-

dim ensionalcontinuum determ ined by theinvariantquadratic form

Q = ab(x)dxadxb (a;b= 1;2); (10.21)

wherex1 and x2 areG auss’ssurfacecoordinatesand ab(x)isasecond-order

covarianttensor.W e assum e thisnew tensorinheritsitsm etric role in this

new spacefrom G auss’scase,and thatit,too,issym m etric.W hatrem ains,

then,is a two-dim ensionalRiem annian space,a space one can inhabitbut

notlook atfrom theoutsidebecauseithasno outside.Forexam ple,whatis

norm ally thoughtasa sphericalsurface is,in fact,only a G aussian picture

(em bedded in Euclidean space) ofwhat is truly a non-Euclidean elliptical

plane.

If ab(x)isthe(sym m etric)m etrictensoroftwo-dim ensionalRiem annian

space,does this m ean that its quadratic form gives a m eaningfulidea of

the squared distance between two neighbouring pointsofthisspace? Asit

stands,itdoesnot,becausethereisno reason to assum ethatthequadratic

form (10.21)ispositivede�nite,13 asitwasforEuclidean spaceand G aussian

surfacesem bedded in it. In Riem annian space,the quadratic form can be

12Ifhab(x)= 0 when a 6= b,the coordinate lines m eet at right angles. If,in addition,

hab(x)= 1 when a = b,the coordinatesare Cartesian.13The quadratic form is positive de�nite if it is always positive except when allthe

com ponents of the di�erentialdisplacem ent dxaare null, i.e. when the two points in

question coincide with each other.


negative for som e di�erentialdisplacem ents dxa (i.e.in certain directions

from a point),positive for others,and nullfor others;in consequence,the

conceptofdistance m ustbe introduced separately. Thiscan be done with

the help ofan indicator�,which can have the values+ 1 or� 1,such that

dl2 = � ab(x)dxadxb � 0 (10.22)

foralldxa.Theequality holdsfora nulldisplacem ent,butcan also hold for

a non-nullone,i.e.the distance between two pointscan be nulleven when

they arenotcoincident.

Indicators can also be used in a di�erentway. The coordinates can be

so chosen that the quadratic form ab(x)dxadxb can be brought into the

sim plerone

Q = �1(dx01)2 + �2(dx

02)2; (10.23)

atone pointata tim e butnotglobally.14 For a Riem annian space with a

positive-de�nite quadratic form ,we have the case �1 = �2 = + 1,leading

to a locally Cartesian quadratic form ;itdescribesa space whose geom etry

is locally Euclidean. For a Riem annian space with an inde�nite quadratic

form ,wehavetherestofthecases,leading to a pseudo-Cartesian quadratic

form ; they describe a space whose geom etry is locally pseudo-Euclidean.

The pair (�1;�2) is called the space signature and it is an invariant,i.e.it

cannotbechanged by m eansofa (real)coordinate transform ation.

W ith the introduction of a stand-alone two-dim ensional Riem annian

space,wehavebanished three-dim ensionalEuclidean space.Butaswenext

add a dim ension to takeinto accounta three-dim ensionalspace,should Eu-

clidean space be resurrected again? Ifwe want to conserve the chance of

describingnew physicsthrough theideaofRiem annian space,weshould not.

W e should now considera three-dim ensionalRiem annian space (i)which is

notem bedded in any higher-dim ensionalEuclidean space,(ii)whosegeom -

etry isby naturenon-Euclidean,and (iii)which isdeterm ined by theinvari-

antinde�nite quadratic form hab(x)dxadxb (a;b= 1;2;3). Here hab(x)isa

sym m etric second-ordercovariantm etric tensor,and the quadratic form in

question islocally Cartesian orlocally pseudo-Cartesian,i.e.locally ofthe

form

Q = �1(dx1)2 + �2(dx

2)2 + �3(dx3)2; (10.24)

with �1;�2;�3 each being + 1 or � 1. Finally,the concept ofdistance is at-

tached to thisthree-dim ensionalspacevia theuseofindicator� asbefore:15

dl2 = �hab(x)dxadxb: (10.25)

14To do this,write ab(x)dx

adx

b(a = 1;2)in explicitform ,com pletesquares,and m ake

the change ofvariablesx01 =p�1 11x

1 + ( 12=p�1 11x

2),x02 =p

�2[ 22 � ( 12)2= 11]x2

to obtain �1(dx01)2+ �2(dx

02)2.The quadraticform thusobtained isvalid only ata point

ata tim e,because forthism ethod to work ab(x)m ustbe keptconstant.The indicators

should be chosen so as to avoid the appearance of im aginary roots, thus keeping the

transform ationsreal.15In physicalspacetim e,dl

2isnotan invariant,butthespatialpartoftheinvariantds

2

(when itispossible to m ake the space-tim e split).

10.2 R iem ann-Einstein physicalspacetim e 137

10.2 R iem ann-Einstein physicalspacetim e

10.2.1 Tow ards localphysicalgeom etry

Having studied G aussian surfacesand Riem annian space to suitourneeds,

wenow turn to physicalspacetim e,resum ing wherewe leftearlier.

W eproposed theform ofthelineelem entdsofRiem ann-Einstein space-

tim e in Eq.(10.14)to be

ds=

q

� g�� (x)dx�dx�:

W ecan now m akephysicalsenseofthisexpression,including an elucidation

ofthe m inussign,by approaching the line elem entfrom the perspective of

Riem annian geom etry.Becausedsrepresentsclock-reading separationsand

these are positive realnum bers,ds is always realand positive. Its square

ds2 m ustthen also berealand positive.16 O n the otherhand,we saw that

the quadratic form g�� (x)dx�dx� ofa Riem annian (now four-dim ensional)

spaceisnotpositive de�nite.However,welearnthow to solve thisproblem

(in trying to geta non-negative length dl2)using an indicator�.Itisthen

naturalto propose

ds2 = �g�� (x)dx�dx� (10.26)

in orderto ensurethatds2 isnevernegative.

Asan aside,wealso �nd thegeom etricm eaning ofg�� (x):g�� (x)isthe

m etric tensor,or m etric,ofRiem ann-Einstein four-dim ensionalspacetim e.

Asa Riem annian m etric,g�� linkscontravariantand covariantvectors,T�

and T�,which have otherwise no connection;the invariant inner product

between two such vectorsT�T� isthesam eastheinvariantquadraticform

g�� T�T�,whereby we obtain theidenti�cation

g�� T� = :T�: (10.27)

W e conclude that,from the point ofview ofRiem annian geom etry, con-

travariant and covariant vectors (and,in general, tensors) are essentially

thesam e.

This interpretation ofg�� (x) com es as an anticlim actic answer to the

challenging question we posed earlier,for at the back ofour m inds we al-

ready knew that g�� (x) was the m etric ofspacetim e. This m athem atical,

geom etricm eaning,then,isnotreally whatwewerepursuingwhen weasked

forthem eaning ofg�� (x);itisa m orephysicalm eaning,ifany,thatweare

stillafter.

As we now com pare Eq.(10.14) with Eq.(10.26), we m ust conclude

that for any observer m oving along his worldline and actually m easuring

the positive real-num ber separation between two events with his clock, �

m ust be equalto � 1. In other words,an observer with a clock can only

m ove in directions determ ined by di�erentialdisplacem ents dx� such that

g�� (x)dx�dx� isnegative;otherdirectionsare physically forbidden.

16The case ofa photon isdealtwith later.


Signature �1 �2 �3 �4

(+ ;+ ;+ ;+ ) + 1 + 1 + 1 + 1

(+ ;+ ;+ ;� ) + 1 + 1 + 1 � 1

(+ ;+ ;� ;� ) + 1 + 1 � 1 � 1

(+ ;� ;� ;� ) + 1 � 1 � 1 � 1

(� ;� ;� ;� ) � 1 � 1 � 1 � 1

Table 10.1: Five di�erent possibilities for the signature of Riem ann-Einstein

spacetim e. Each ofthese choices leads to a di�erent localphysicalgeom etry for

spacetim e.

According to the direction of a di�erentialdisplacem ent dx�, and in

keeping with theusualnam es,we can now say that17

(i) dx� istim elike (i.e.allowed)ifg�� (x)dx�dx� < 0 (� = � 1),

(ii) dx� isspacelike (i.e.forbidden)ifg�� (x)dx�dx� > 0 (� = + 1),and

(iii) dx� islightlike (i.e.allowed only forphotons)ifg�� (x)dx�dx� = 0.

Regarding item (iii),in relativity theory the hypothesis is m ade that the

di�erentialdisplacem entscharacterizing the history ofa photon satisfy the

relation ds= 0.Strictly speaking,thishypothesisisoutsidetheboundaries

ofthe operationalm eaning ofds,since the clock to be carried along by a

photon to m akethenecessary m easurem entsisa m aterialobjectwhich can

neverreach the speed oflight.O nem ay,however,understand thisphoton-

history hypothesis as the lim iting case ofincreasing velocities ofm aterial

bodies(see Section 10.3.2).

Because Riem ann-Einstein spacetim e islocally pseudo-Cartesian,coor-

dinatescan befound such thatitsinde�nitequadratic form takestheform

Q = �1(dx1)2 + �2(dx

2)2 + �3(dx3)2 + �4(dx

4)2 (10.28)

atone pointata tim e. W hatare allthe possible di�erentcom binationsof

�i? Because no separation between three space coordinates and one tim e

coordinate has as yet been m ade| and any such separation would at this

pointbearti�cial| allfourcoordinatesstand on an equalfooting.Itisthen

im m aterialthesignsofwhich �i arebeing changed in any onecom bination.

The�ve di�erentpossiblecom binationsare listed in Table 10.1.

W hich ofthe�vepossibilitiesshould wechoose? And on whatgrounds?

No answerm akes itselfnaturally available to us| we have reached a road-

block.Thisisa criticalstage ofthis analysis.Itistim e to recapitulate.

17The sam e classi�cation can be generalized to include any contravariant vector T

�.

An exam ple ofanother tim elike (i.e.allowed) contravariant vector is the tangent vector

dx� =du to a worldline.


10.2.2 Localphysicalgeom etry from psychology

Thepresentanalysisoftim ehasso farbeen based,�rstand forem ost,on a

physicalcom ponent consisting ofclock m easurem ents ds;and secondly,on

theothersideoftheequation,on am athem aticalcom ponentconsistingofthe

geom etric description ofspacetim e via itscharacteristic invariantquadratic

form g�� (x)dx�dx�. Is there anything m issing from this analysis oftim e?

Should we consider som ething else besides physics and m athem atics? W e

m ust;ifwe are to m ove forwards in a natural,non-axiom atic way,we are

forced to also take into account a psychologicalcom ponent regarding the

conception of tim e: the hum an notions of past, present (or being), and

future.

The past and the future include allthe allowed directions that an ob-

servercan follow,and so,recalling thatwecan only m easurepositivevalues

ofds and thatEq.(10.14)carriesa m inussign,we have that

(i) g�� (x)dx�dx� < 0 (� = � 1)forthe pastand the future,

(ii) g�� (x)dx�dx� > 0 (� = + 1)forthe present,and

(iii) g�� (x)dx�dx� = 0 forthe boundary surface between the presentand

thepastand future,i.e.thelightcone (see below).

W hat choice ofindicators �i do the conceptions ofpast,present,and

future lead to? Choices (+ ;+ ;+ ;+ ) and (� ;� ;� ;� ) can be discarded as

soon aswe notice thatthey lead to quadratic form softhetype

Q = � [(dx1)2 + (dx2)2 + (dx3)2 + (dx4)2]; (10.29)

which are always positive and do not allow the conceptions of the past

and the future,or always negative and do notallow the conception ofthe

present.O ftherem aining options,which should we choose? And again,on

whatgrounds?

W hile the existence ofthe present on the one hand,and the past and

future on the other, acted as a �rst guideline, it is now the distinction

between the past and the future that acts as the next one. W e perceive

thepastbecom ing thefutureasa serial,unidirectional,orone-dim ensional

a�airthrough the present,and so itappearsthatthisprocessofbecom ing

should bepictured locally by m eansoftheprogression ofa singlecoordinate

param eter and not of severalof them . G iven one coordinate param eter

x that we m ay locally associate with the psychologicalconception ofthe

unidirectional ow oftim e,we m ay then connectthe pastwith decreasing

valuesofx and thefuturewith increasing valuesofx.

Because we associated the past and the future with negative values of

theinvariantquadraticform ,wem ay now discard options(+ ;+ ;� ;� )and

(+ ;� ;� ;� ),which introduce,respectively,two and three (coordinate)pa-

ram eterslocally associated with thepastand thefuture(pre�xed with neg-

ative signs)| a so-called em barrassm entofriches. W hy? Because there do

not exist two or three di�erent ways in which we feelit is possible to go


from the past to the future| e.g.is there a way through the present but

perhapsalso anotherthatshortcutsit,asifpastand future shared one or

m ore direct connections? Thisidea doesnot conform with the seriality of

the psychologicalconception oftim e,and we therefore discard these two

options.W e are leftwith option (+ ;+ ;+ ;� ),which we adopt.18

The choice ofthe invariant quadratic form with signature (+ ;+ ;+ ;� )

leadsto the following classi�cation:

(i) (dx1)2 + (dx2)2 + (dx3)2 � (dx4)2 < 0 forthe pastand future,

(ii) (dx1)2 + (dx2)2 + (dx3)2 � (dx4)2 > 0 forthe present,and

(iii) (dx1)2 + (dx2)2 + (dx3)2 � (dx4)2 = 0 forthe lightcone.

M oreover,through theincreaseand decreaseofcoordinatex4,wecan locally

di�erentiate the pastfrom the future,thus:

(i-a) (dx1)2 + (dx2)2 + (dx3)2 � (dx4)2 < 0 & dx4 < 0 forthepast,and

(i-b) (dx1)2 + (dx2)2 + (dx3)2 � (dx4)2 < 0 & dx4 > 0 forthe future.

In addition,wecan also identify

(iii-a) (dx1)2 + (dx2)2 + (dx3)2 � (dx4)2 = 0 & dx4 < 0 with the branch

ofthe lightcone thatdi�erentiatesthe pastfrom the present,and

(iii-b) (dx1)2 + (dx2)2 + (dx3)2 � (dx4)2 = 0 & dx4 > 0 with the branch

ofthe lightcone thatdi�erentiatesthe futurefrom the present.

Now, why should physics take into account these aspects ofthe psy-

chologicalnotion oftim e? W hy should physics be partly based on them ?

Should physicsalso take into accountthe propertiesofotherpsychological

conceptions,such ashunger,happiness,pain,and disdain? Notreally,not

in any speci�cway.W hatisthen specialabouttim e? Itisthis:thepsycho-

logicalnotion oftim easa stream thatdragsusfrom pastto futurethrough

theubiquitouspresentisthedeepestrooted featureofhum an consciousness

and,as a result,physics has since its beginnings described the changing

world guided by thisform ofhum an experience,yetwithoutsucceeding (or

even trying) to explain it. From G alilean dynam ics to quantum gravity,a

picture oftim e as \t"is allphysicshasso farm anaged: an abstractm ath-

em aticalparam eter that by itselfcannot tellus why the past is gone,the

presentall-pervading,and the futureunknown and outofreach.

In classicalphysics,the m athem aticaldepiction oftim e as\t" issim ple

enough forthe psychologicalproperty ofthings-in- ow to besuperim posed

on itin a straightforward way: past,present,and future are taken asuni-

versalnotions,each associated with di�erent positions along the absolute

18Ifwe had not pre�xed Eq.(10.14) with a m inussign,we would have been forced to

choose option (+ ;� ;� ;� ),with the positive coordinate now being associated with the

pastand thefuture.Thiswould havebeen a workablealternativetoo,although wewould

now have to carry along three m inussignsinstead ofone.


param etric tim eline. Thisisnotto say thatthe hum an experience oftim e

was explained by t,butthat it was easy enough to project ourselves onto

that tim eline and im agine ourselves to ow along it. Classically,then,as

wellasin conventionalexpositionsofrelativity theorywith itsblock-universe

picture,thehum an categoriesofpast,present,and futurerem ain physically

ignored.

In this presentation ofrelativity, however,we notice that in this the-

ory an allowance m ustbem adeforhum an tem poralnotionsin a som ewhat

m ore explicit way. O n the one hand,from the subjective point ofview of

theorderingofeventsalong an observer’sworldline,weagain �nd thatpast,

present,and future are ignored by physics,ifwe now replace the absolute

param etric tim eline given by tby the relative param etric tim eline given by

clock readings s. O n the other hand,however,relativistic physics is not

justaboutthehistory ofone observer’sworldline;itisa description ofany

and every worldline. In other words,relativistic physics is about absolute

spacetim e properties,and we �nd that in the construction ofthe absolute

property ofspacetim esignature(+ ;+ ;+ ;� ),itneedsto explicitly acknowl-

edge,though m inim ally,thepsychologicalconceptionsofpast,present,and

futureforthe problem to beunlocked in a m eaningfulway.

Thescepticalreader,having heard thatphysicsoughtto beindependent

ofhum ansand externalto them ,m ay ask: butifthe conceptions ofpast,

present,and future are psychological,why m ake them a part ofphysics?

O r then,ifthey are part ofphysics,why stress they are ofpsychological

origin? Because we wantpsychology,where relevant,to inform physics,so

thatphysics,where possible,willbereconciled with psychology.Thepursuit

ofthistwofold connection,which one m ay callpsychophysics,requiresthat

weseriouslyacknowledgethefactualnessofconsciousexperienceand thatwe

identify thosefeaturesofitthatarerelevantto physics.In orderto do this,

we m uststartby taking ourselvesseriously and notdism iss,in the pursuit

ofthe idealof\objectivity," hum an experience asan irrelevantdelusion.

W hen it com es to tim e,that we should take ourselves seriously m eans

thatweshould notconsiderourselvesthevictim sofan illusion in which we

feelthattim e owsirreversibly,with the pastknown and irretrievable,the

presenta perm anentand all-pervading \now," and thefutureunknown and

outofreach,whilein facttim eisreversiblebecausethedeterm inisticequa-

tionsofphysicsm akeno dynam icaldistinction between tand � t,and past,

present,and futureareallon an equalfootingbecausespacetim etheorieslay

them outasa hom ogeneousblock.Ifphysicssaysthe earthly carouseland

the heavenly bodiesin theirorbitscan equally wellrun backwardsin tim e,

butourraw experienceoftim esaysthatthisisim possible,which should we

trust? Isthe physicsoftim e correctasitisand we delusional,orcould we

learn a betterphysicsoftim e by taking ourselvesm oreseriously?

That we should do so is not a forgone conclusion but an attitude we

propose. Against such an attitude,one m ay observe with Dennett (1991)

that\naturedoesn’tbuild epistem icengines" (p.382).Thatis,why should

wetrustourm indsto capturesom ething essentialaboutthenatureoftim e


that physicaltheories have not yet grasped,ifour m inds did not arise to

know theworld butonly to im proveourchancesofstaying alive? Regarding

colour perception, for exam ple, the context in which Dennett m akes his

observation,thisisquite true;the perception ofred doesnotrepresentany

basic property ofthe world.Butwhen itcom esto tim e,we are inclined to

give them ind the bene�tofthedoubt.Unlike any otherfeeling,tim e isan

inalienable experientialproductofhum an consciousnessand,while physics

desperately needssom e conceptoftim e,sofarithasonly m anaged too�era

caricatureofit,graspingbarelyitsserialaspectin am athem aticalparam eter

and nom ore.Becausetheconception oftim eisaconsciousness-related idea,

to understand it,we need to pay heed to ourselves.

This presentation of relativity takes a sm allstep in a positive direc-

tion. Through it we learn that,despite appearances,relativity does not

straightout hold the spatiotem poraluniverse to be a com pletely undi�er-

entiated block,because in its choice oflocalphysicalgeom etry (or global

signature)there isa m ild recognition that(i)pastand future are di�erent

from the present(�rstdiscard ofsignatures),and (ii)echoing \t" in classi-

calphysics,that past and future are serially connected (second discard of

signatures).

However,thisunderstandingofrelativity doesnotachievenearlyenough.

How are past,present,and future qualitatively di�erent from each other?

W hy is past-present-future a serialchain? And,m ore signi�cantly,what

about the present,that perm anent now in which we constantly �nd our-

selves? How doesphysicsexplain the presentm om ent? Can itsingle itas

specialin any way? O ur consciousness does,and so distinctly so that we

feelwe are nothing butthe present.Ifwe decide to heed ourselveswhen it

com esto tim e,we should considerthisan enigm a.An enigm a neitherclas-

sicalnorrelativistic physics can solve. Butwhataboutquantum physics?

Can it succeed where its predecessor theories fail? Can quantum physics

do betterin accounting forwhatisotherwise the hum an delusion thatthe

presentis param ount? And whatis the role ofgeom etric tools ofthought

in thissearch? W e return to thisissuein Chapter12.

10.2.3 Spacelike separation and orthogonality from clocks

Up to now,we have used an idealclock to m easure separations between

eventsthatcan bereached by an observertravelling along hisworldline,i.e.

the separationsm easured have always been in directionsgiven by tim elike

displacem entvectorsdx�.Can wealso m easureseparationsbetween events

along directions given by spacelike displacem ent vectors? And ifso,what

kind ofexperim entalsetup do weneed to do this?

W e exam ine two neighbouring events A and B joined by a spacelike

displacem ent vector dxA B ,with A situated between events P and Q and

allthree events on the observer’s tim elike worldline. Events P and Q are

determ ined experim entally by the application ofthe radar m ethod: P is

an eventsuch thata lightsignal(a photon)em itted from itisre ected at

B and received by the observer at Q ;both tim elike separations dsP A and


P

A

Q

B

dxPA

dxAQdxAB

dxBQ

dxPB

worldlineFigure 10.3: M easurem entofa spacelike separation by m eans ofa clock and a

re ectedphoton.Displacem entvectorscanbedetached from theiroriginalpositions

dueto the factthatthey aresm all.

dsA Q are experim entally determ ined with the observer’sclock. Looking at

thediagram ofFigure10.3,wecan write down thesystem ofequations8<

:

dxA Q = kdxP A (k > 0)

dxP B � dxP A = dxA B

dxA Q � dxA B = dxB Q

: (10.30)

Using the lasttwo equationsto expressthe lightlike separationsds2P B = 0

and ds2B Q

= 0 in term softhe two tim elike and one spacelike displacem ent

vectors,and subsequently using the �rstequation,we �nd

ds2A B = dsP AdsA Q :19 (10.31)

Spacelikeseparationscan then bem easured with thesim pleaddition ofem is-

sion,re ection,and reception ofphotonsto the originalclock-on-worldline

type ofm easurem ents,i.e.spacelike separations can be expressed in term s

oftim elike ones.

W e�nd,m oreover,thatifeventsA,B ,P ,and Q can beexperim entally

so arranged asto havedsP A = dsA Q ,then thek-factorisunity and dxA Q =

dxP A.In thatcase,ds2B Q = 0 becom esthe bilinearform

g�� dx�A B dx

�

A Q= 0; (10.32)

wheredx�A B

isspacelikeand dx�A Q

istim elike.W hatisthegeom etricm ean-

ing ofthisresult?

W e know that, in Euclidean space, whose quadratic form is positive

de�nite,thecosineoftheanglebetween vectors��!AB and

�!AQ isgiven by the

dotproductofthe corresponding unitvectors:

cos(�)= uA B � uA Q = u1A B u

1A Q + u

2A B u

2A Q + u

3A B u

3A Q : (10.33)

19In m ore detail,we have (1) ds

2P B = � ds

2P A + ds

2A B + 2g�� dx

�P A dx

�

A B= 0 and (2)

ds2B Q = � k

2ds

2P A + ds

2A B � 2g�� (kdx

�P A )dx

�

A B= 0.Now k(1)+ (2)givesds

2A B = kds

2P A ,

from which followsthe desired result.


A

Q

BdxAQ

dxAB

dxBQ

A

Q

BdsAQ

dsAB

ds = 0

Figure 10.4: Experim entaltest to determ ine the orthogonality of a spacelike

vectordxA B and atim elikevectordxA Q .ThetwovectorsareorthogonalifdsA B =

dsA Q ,whereeventsB and Q arejoined by the worldlineofa photon.

G eneralizing foraRiem annian spacewith apositive-de�nite quadraticform ,

we have

cos(�)= hab

�dxaA B

dlA B

� dxb

A Q

dlA Q

!

: (10.34)

In consequence,alsoin Riem annian spacewesaythattwovectorsareorthog-

onalwhen theirinnerproductisnull.Although in the case ofan inde�nite

quadratic form it is not sensible to de�ne an acute or obtuse angle with

a cosine-type relation,we do nevertheless accept the concept ofa straight

angle,i.e.oforthogonality,from the positive de�nite case,and character-

izetwo orthogonalvectorsin Riem ann-Einstein spacetim eby therelation of

Eq.(10.32).Asaresult,theexperim entaldeterm ination ofspacelikesepara-

tionssuppliesam ethod fortheexperim entaldeterm ination oforthogonality

between displacem ent vectors: Eq.(10.32) m eans that spacelike displace-

m entvectordx�A B and tim elike displacem entvectordx�

A Qareorthogonalin

an experim entaltest where dsA B = dsA Q and where events B and Q are

joined by theworldlineofa photon (Figure 10.4).Thisconceptoforthogo-

nality can begeneralized forany pairofcontravariantvectorsT� and U �.

Can now two spacelike displacem entvectorsdxA B and dxA C beorthog-

onal? And ifso,underwhat conditions? By m eans ofwhat experim ental

testcan we verify theirorthogonality? Because now we have two spacelike

displacem entvectors,we use the radarm ethod twice to m easure ds2A B and

ds2A C in term s oftim elike separations. Noting thatdxB C = dxA C � dxA B

(Figure 10.5),we �nd

g�� dx�B C dx

�

B C� ds2A C � ds2A B = � 2g�� dx

�A C dx

�

A B: (10.35)

Theleft-hand sidecan only benulland orthogonality possiblewhen dxB C is

spacelike and thusg�� dx�B C

dx�

B C= ds2

B C> 0. In thatcase,the condition

forthe orthogonality ofdxA B and dxA C is

ds2B C = ds2A C + ds2A B : (10.36)

ThisisPythagoras’theorem in spacetim e!

Two tim elike vectors,however,can neverbeorthogonal.Thisfactleads

to a counterintuitive (from a Euclidean point ofview) triangle inequality

in spacetim e,with consequences reaching farther than one would expect.


A

B

C

dxAB dxBC

dxACA

B

C

dsAB dsBC

dsAC

Figure 10.5: Experim entaltest to determ ine the orthogonality oftwo spacelike

vectorsdxA B and dxA C . The two vectorsare orthogonalifthey are joined by a

spacelikevectordxB C and ds2B C = ds2A C + ds2A B ,thatis,when dxB C ,dxA C ,and

dxA B satisfy Pythagoras’theorem in spacetim e.

To show �rst that two tim elike vectors T � and U � cannot be orthogonal,

we investigate their behaviour at a point (chosen to be the origin) after

recasting them in pseudosphericalcoordinatesr;�;�;� (r> 0):

8>><

>>:

T1 = rsin(�)cos(�)sinh(�)

T2 = rsin(�)sin(�)sinh(�)

T3 = rcos(�)sinh(�)

T4 = rcosh(�)

(10.37)

and 8>><

>>:

U 1 = ~rsin(~�)cos(~�)sinh(~�)

U 2 = ~rsin(~�)sin(~�)sinh(~�)

U 3 = ~rcos(~�)sinh(~�)

U 4 = ~rcosh(~�)

(10.38)

Thequadraticform ofT� atthe origin gives

(T1)2 + (T2)2 + (T3)2 � (T4)2 = � r2 < 0; (10.39)

which assures us that T� is tim elike at that event;and sim ilarly for U �.

After using severaltrigonom etric-and hyperbolic-function identities,20 we

�nd thatthe innerproductofthese two vectorsattheorigin is

T1U1 + T

2U2 + T

3U3 � T

4U4 = � r~r[cosh(� � ~�)cos2(!=2)�

cosh(� + ~�)sin2(!=2)]� � r~r< 0; (10.40)

where

cos(!)= cos(�)cos(~�)+ sin(�)sin(~�)cos(� � ~�): (10.41)

Thism eansthat

g�� T�U� � � r~r6= 0; (10.42)

i.e.tim elike vectorsT� and U � cannotbeorthogonal.

20The trigonom etric identities needed are 2sin(x)sin(y) = cos(x � y)� cos(x + y),

2cos(x)cos(y)= cos(x� y)+ cos(x+ y),and cos(2x)= cos2(x)� sin

2(x).Thehyperbolic

identitiesneeded are2sinh(x)sinh(y)= cosh(x+ y)� cosh(x� y)and 2cosh(x)cosh(y)=

cosh(x + y)+ cosh(x � y).


Thetriangleinequality in spacetim eresultsnow asfollows.In Euclidean

space,the triangle inequality forthe sidesofa triangle with verticesA,B ,

and C readsj�!AC j� j

��!AB j+ j

��!B C j,which saysthatthesum ofthelengthsof

any two sidesofa triangle isgreaterthan the length ofthe rem aining side.

In spacetim e,thingsare otherwise.

LetT�A B

,T�B C

,and T�A C

bethreetim elikevectorsform ingatrianglewith

vertices at events A,B ,and C . W e assum e that the three vectors point

towards the future and that the vectors characterizing the displacem ents

are sm all.O n thebasisofFigure 10.6,we have

T�A C = T

�A B + T

�B C : (10.43)

W e expressthesevectorsusing theirnorm s

T =

q

�g�� T�T� (� = � 1) (10.44)

and unitvectors u to get

TA C u�A C = TA B u

�A B + TB C u

�B C : (10.45)

Because theunitvectorsare tim elike,wehave

g�� u�A B u

�

A B= g�� u

�B C u

�

B C= � 1 and g�� u

�A B u

�

B C� � 1: (10.46)

Asa result,we �nd

T2A C = T

2A B + T

2B C � 2TA B TB C g�� u

�A B u

�

B C; (10.47)

from where

T2A C � (TA B + TB C )

2: (10.48)

Itfollowsthat

TA C � TA B + TB C (10.49)

isthetriangleinequality in spacetim e,with equalityholdingwhenT�A C

= T�A B

orT�A C = T�

B C (no actualtriangle).

Thetriangleinequality fortim elikevectorshasconsequencesthatextend

beyond thecharacterofthegeom etricstructureofspacetim e.Notably,ithas

a bearing on thefeaturesofm aterialprocessesthattakeplacein spacetim e,

such asthe spontaneousdecay orinduced �ssion ofa m assive particle into

decay or�ssion productswith an ensuing release ofenergy.

In classicalphysics,the collision ofm aterialparticles is understood in

term s ofthe conservation oftwo quantities,the totalquantity ofm otionP

im ivi and the totalkinetic energyP

im iv2i=2 ofthe system ofparticles.

From a strictly m echanicalpoint ofview,this is only true ofelastic colli-

sions,in which no energy islostby thecolliding bodiesin theform ofheat.

Classicalm echanics needs to be com plem ented by therm odynam ics in or-

derto m akefullsenseofenergy-dissipating (i.e.inelastic)collisions.O n the

otherhand,relativity theory isin thisrespectself-contained;every aspectof


A

B

C

TαAB

TαBC

TαAC

Figure 10.6: Thecounterintuitivetriangleinequality TA C � TA B + TB C forthree

tim elikevectorsin spacetim e.

theinteraction ofm assiveparticlescan beexplained based on thegeom etric

structureofspacetim e wheretheprocessesoccur.

Taking now the naturalstep ofusing clock readings s as a worldline

param eter,we recast the worldline equation asx� = x�(s). Because ds is

alwayspositive,thecovariantvectordx�=dspointsin thesam edirection as

dx�,and sodx�

ds= :V � (10.50)

is a good representation of the worldline tangent vector, which is called

the four-velocity. The four-velocity V � is,in fact,a unit vector. W riting

dx� = V �ds in the quadraticform fords2,we�nd

V2 = �g�� V

�V� = 1: (10.51)

Because the worldline isthatofa m aterialparticle,then � = � 1,and so

g�� V�V� = � 1: (10.52)

Finally,we attach the m ass ofthe particle m (an invariant scalar) to the

four-velocity V � to getthefour-m om entum

P� = m V

�: (10.53)

In the event ofa particle decaying into two subproducts,in relativity

theory we only require the conservation ofthe four-m om entum ofthe total

system ,nam ely,

P�1 = P

�2 + P

�3 ; (10.54)

which we can graphically representwith three tim elike vectorsin the form

P�A C = P

�A B + P

�B C ; (10.55)

asshown in Figure 10.7. Because V � isa unitvector,m actsasthe norm

ofP �,and we can apply the resultsofthe triangle inequality we found for

T�,to get

m 1 > m 2 + m 3: (10.56)


A

B

C

Pα2

Pα3

Pα1

Figure 10.7: Particle decay (A) into two subproducts (B and C ). Energy is

released in these processesasa resultofthe spacetim etriangleinequality.

W e �nd that the conservation of the four-m om entum in the decay of a

particleinvolvesthereleaseofenergy,m 1c2 > m 2c

2+ m 3c2,asan inescapable

consequence ofthe non-Euclidean geom etric structure ofRiem ann-Einstein

spacetim e,nam ely,the non-orthogonality oftim elike vectors.Thispositive

energy de�citm 1c2 � (m 2c

2 + m 3c2)isnorm ally denoted asQ in textbooks

on the topic,butitsorigin isleftunexplained.

10.3 G ravitation as spacetim e curvature

In hisK yoto lecture,Einstein describesthetrain ofthoughtleading him to

the generaltheory ofrelativity:

Ifa m an falls freely, he would not feelhis weight...A falling

m an is accelerated...In the accelerated fram e ofreference,we

need a new gravitational�eld...[I]n thefram eofreference with

acceleration Euclidean geom etry cannot be applied. (Einstein,

1982,p.47)

Ifa falling m an does notfeelhisweight yet is accelerated,it seem s he

m ustnotbesubjected to any gravitationalforce or�eld buttosom eform of

interaction (whose e�ects we see in hisacceleration) nonetheless| he m ust

be subjected to a \new gravitational�eld." W hat is this new interaction

that induces gravitational e�ects yet is not a force? An answer can be

sketched as follows. It has been long since realized that the gravitational

force is a specialinteraction. G alileiand Newton observed that inertial

m assesm i and gravitationalm assesm g werethesam e(Newton’sprinciple)

and that,therefore,allm assivebodiesin freefallin avacuum su�erthesam e

acceleration (G alilei’sprinciple)regardlessoftheirconstitution.Thispulling

on allbodiesand lightin likefashion insinuatesthatthe\new gravitational

�eld" we are after could be som e property ofspace or spacetim e in which

m assive bodiesand lightm ove.

Could gravitation,then,beconceived asaproperty ofspace? Intuitively,

asRindler(1977,p.115)explains,thiscannotbejustso becausethefuture

10.3 G ravitation as spacetim e curvature 149

m otion ofabodyundergravitationalinteraction dependson itsinitialveloc-

ity,butthisinform ation isnotincluded in thedescription ofaspace geodesic,

which is determ ined only by initialposition and direction.21 However,an

accountofallnecessary initialconditionsisincluded in the speci�cation of

a spacetim e geodesic. The validity ofthis heuristic reasoning is strength-

ened when we observe with Einstein that,in the m an’s accelerated fram e

ofreference,Euclidean geom etry cannotbeapplied beyond a point-event,22

and note atthe sam e tim e the essentialrole thatthe geom etry ofthe four-

dim ensional,non-Euclidean Riem ann-Einstein spacetim e hashad so farin

thedevelopm entofrelativity theory.

G iven the geom etric setting ofrelativity theory,we suspect then that

the \new gravitational�eld" m ust be som ehow included in the geom etry

of Riem ann-Einstein spacetim e. G ravitation, m oreover, m ust be an ab-

solute property ofspacetim e,in the sense that either gravitationale�ects

are present or they are not. As has been often rem arked,when it com es

to spacetim e properties,there is nothing relative in relativity theory. The

prim eexam pleofthistruth isthespacetim e intervalds.In absoluteclassi-

calphysics,on theotherhand,gravitation is,in fact,a relative concept:an

observeron earth deem sitto bea forceacting on thefalling m an,whilethe

falling m an deem sitasno force atall. Einstein’sclassicalthoughtexperi-

m entwould teach usnothing were we notwilling to also take into account

therelative pointofview ofthem an on earth.

W hatspacetim eproperty shallwenow considerto representgravitation

as an absolute interaction? And what is the role ofm easurem ents in this

search? O n accountofalongacquaintancewith it(and perhapsalsobecause

ofthesuggestive notation),one’s�rstintention isto take them etrictensor

g�� (x)toplay thevacantroleof\new gravitational�eld," whileatthesam e

tim e �nally giving an answer to the still-standing question ofits physical

m eaning. However,the m etric tensor is a rather bad candidate to act as

\gravitation," theabsolutespacetim eproperty weareseeking.For,iftruth

be told,there is nothing absolute in g�� (x): it is a tensor �eld,and that

is allvery well,but what absolute spacetim e property does it represent?

In one coordinate system ,g�� (x) takes certain values;in another,it takes

di�erentones.

W hen we introduced tensors at the beginning ofthis chapter,we ob-

served thattheirreason ofbeingwastoprovidem athem aticalobjectsm ean-

ingfulin allcoordinate system s.W e then identi�ed such m athem aticalob-

jects,not with the tensors them selves,but with tensor equations. Here is

then thekey tothissearch.Isthereany tensorequation thatg�� (x)satis�es

21Forexam ple,a freeparticlecon�ned to a surfacefollowsa geodesicthatdependsonly

on itsinitialposition and direction,butisindependentofitsinitialvelocity.22Thism eansthat,becausethevelocity ofthe(rigid)fram echangesconstantly,wem ust

attach a di�erent inertialsystem to each spacetim e point or event. However,Einstein

(1983b) som etim es reached the sam e conclusion in a m ore intuitive way by looking at

a rotating fram e; here an inhom ogeneous distortion of\rigid bodies" (m easuring rods)

ensuesin am oreeasily visualizableway,aseven each space pointbecom esan own,separate

inertialfram e \on accountofthe Lorentz contraction" (p.33).


assuch? So farthe only absolute equation connected with the m etric �eld

has been ds2 � �g�� (x)dx�dx� = 0,but this is an equation for the whole

quadratic form and for the clock separation ds. Evidently,this elem ental

relation istightly connected with clock m easurem entsbuthasno straight-

forward connection with gravitation.Itistim e to give up the illusion that,

in relativity theory,itissim ply them etric�eld thathastaken up theroleof

the\new gravitational�eld." Ifa nam ewith a strong physicalconnotation

m uststillbe given to the m etric �eld,one should then perhapscallg�� (x)

the chronom etric �eld| and stop atthat.

10.3.1 Einstein’s geodesic hypothesis

Newton’s�rstlaw ofm otion,orthelaw ofinertia,statesthata freem assive

particlefollowsa uniform and rectilineartrajectory.Here\free" m eansfree

ofallexternalinteractions,including thegravitationalforce.

In relativity,weproceed to includea description ofgravitation by m eans

ofan extension ofNewton’s �rst law ofm otion called Einstein’s geodesic

hypothesis. According to it,a free m assive particle has a tim elike world-

line that is a geodetic curve,or geodesic,in Riem ann-Einstein spacetim e,

where by \free" we now m ean free ofallexternalforces butnotofgravita-

tion,which wewantto describenotasan externalforcebutasan absolute

geom etric property ofspacetim e. Assuch,itisim possible to be free ofits

e�ects.Einstein’sgeodesichypothesisalsoappliestoafreephoton,in which

case itsaysthata free photon followsa nullgeodesic in Riem ann-Einstein

spacetim e.

A geodesic isa line whose extension (length)between two given points

isstationary,i.e.hasan extrem alvalue.Thiscan bea m axim um ora m ini-

m um .W hen weconsidertwo pointsin three-dim ensionalEuclidean space,a

geodesicrefersto a spatialpath ofm inim allength,i.e.theshortestdistance

between thetwo points.In Riem ann-Einstein spacetim e,on theotherhand,

our Euclidean intuition fails again: where we expect less we get m ore (cf.

triangle inequality and energy surplus in particle decay). In spacetim e,a

geodesicrepresentsaworldlineofm axim alseparation asm easured byclocks.

Incidentally,thisisnot,strictly speaking,am axim allength in any literal

sense, although our geom etric intuition leads us to think of it that way.

Insightfully,in thisregard,Syngewrites:

Itisindeed a Riem annian chronom etry ratherthan a geom etry,

and the word geom etry,with its dangerous suggestion that we

should go about m easuring lengths with yardsticks,m ight well

be abandoned altogether in the present connection were it not

forthefactthatthecrudeliteralm eaning oftheword geom etry

hasbeen transm uted into theabstractm athem aticalconceptof

a \space" with a \m etric" in it.(Synge,1964,p.109)

Hereisyetanotherinstanceofarulewehaverepeatedly observed:them ind

turnseverything itbeholdsinto itsgolden standard| geom etry.23

23W hatm ythicalcharacterdoesthisbehaviourrem ind usof? (See Epilogue.)


In orderto �nd theequation x� = x�(u)satis�ed by thegeodeticworld-

linewith extrem esateventsA and B ,wecalculatethevariation oftheevent

separation �s and equate itto zero,

�

ZB

A

ds= �

ZuB

uA

f

�

x;dx

du

�

du = 0: (10.57)

Taking thevariationsofx� and x0� = :dx�=du,we �nd

ZuB

uA

�@f

@x��x

� +@f

@x0��x

0�

�

du = 0: (10.58)

Rewriting �x0� = d(�x�)=du and integrating the second term by parts

(~u� = @f=@x0�,d~v = d(�x�)),we get

��uB

uA

@f

@x0��x

�(u)+

Z uB

uA

�@f

@x��

d

du

�@f

@x0�

��

�x�du = 0: (10.59)

Because the �rst term is nullon account of �x(uB ) = �x(uA ) = 0 (the

extrem esare�xed),and because�x� isan arbitrary sm allvariation,we�nd

that,atevery pointofthegeodesic,f(x;x0)m ustsatisfy theEuler-Lagrange

equationsd

du

�@f

@x0�

�

�@f

@x�= 0; (10.60)

where f(x;x0)=p� g�� x

0�x0�. To avoid having to di�erentiate a square-

rootexpression,we rewriteEq.(10.60)in theequivalentform

d

du

�@f2

@x0�

�

�@f2

@x��

1

2f2df2

du

@f2

@x0�= 0: (10.61)

Replacing u by thenaturalworldlineparam eters,we �nd

f2(x;x0)= � g�� V

�V� = + 1 and

df2

ds= 0; (10.62)

to get

d

ds

�@f2

@x0�

�

�@f2

@x�= 0: (10.63)

Replacing f2(x;x0)by � g�� x0�x0�,we obtain the geodesic equation in the

m oreexplicitform

g��dx0�

ds+ [��;�]x0�x0� = 0; (10.64)

where

[��;�]=1

2

�@g��

@x�+@g��

@x��@g��

@x�

�

(10.65)

is called Christo�el’s sym bolofthe �rstkind,which is sym m etric with re-

spect to its �rst two indices. Finally,m ultiplying Eq.(10.64) by g �,we

obtain a yetm oreexplicitform ofthe geodesic equation:

d2x

ds2+�

��

dx�

ds

dx�

ds= 0; (10.66)


where�

��

= g �[��;�]iscalled Christo�el’ssym bolofthesecond kind.In

addition,the tim elike condition

g��dx�

ds

dx�

ds= � 1 (10.67)

holdsforalleventson thegeodesic.Thesolution x� = x�(s)to thissecond-

orderdi�erentialequation can be determ ined unequivocally when we know

x�(s)and dx� atsom e event on the tim elike geodesic,i.e.forsom e clock-

reading values.

The geodesic equation (10.66) gives us now the answer to a question

we asked earlier on. O n page 127,we had expected that the equation of

m otion (10.3)ofa particlewould beoftheform d2x�=du2 = X �(x;dx=du).

Now we have found thatthisisindeed the case,with

X�

�

x;dx

ds

�

= ��

��

dx�

ds

dx

ds; (10.68)

wherethe x-dependenceiscontained in Christo�el’ssym bol.

As an exam ple, we can apply the geodesic equation to the case ofa

spacetim e whosequadratic form is

Q = (dx1)2 + (dx2)2 + (dx3)2 � (dx4)2 (10.69)

everywhere and its m etric,therefore,g�� = diag(1;1;1;� 1) = �� . As a

result,allChristo�elsym bolsarenulland thegeodesic equation reducesto

d2x�

ds2= 0; (10.70)

with tim elike condition

�dx1

ds

� 2

+

�dx2

ds

� 2

+

�dx3

ds

� 2

�

�dx4

ds

� 2

= � 1: (10.71)

Thesolutionsare straightlinesin spacetim e

x�(s)= A

�s+ B

�; (10.72)

whereA � and B � areconstants,and A � satis�estim elikecondition (10.71).

Excepting thislastrestriction,theequation ofm otion and itssolution have

the sam e form as those for a free particle in classical physics, nam ely,

d2xa=dt2 = 0 with solution xa(t) = A at+ B a. The correspondence sug-

geststhataspacetim ewhosequadraticform ispseudo-Cartesian everywhere

resem bles atEuclidean space.

So far we have dealt with geodesics of m aterial particles, for which

ds> 0. How shallwe dealwith geodesics ofphotons,for which ds = 0?

In this case, the use of s as a geodesic param eter is im possible,and we

m ust use a di�erent param eter u together with its corresponding lightlike

condition

f2(x;x0)= g�� U

�U� = 0; (10.73)


where U � = dx�=du. O n account of it, the sim pli�cation df2=ds = 0

introduced earlier now becom esdf2=du = 0. From here on,the procedure

isanalogousto the one fora tim elike geodesic. The equation ofa lightlike

geodesic isthen

d2x

du2+�

��

dx�

du

dx�

du= 0; (10.74)

with lightlike condition

g��dx�

du

dx�

du= 0: (10.75)

This way of proceeding, however, begs the question as to what kind of

param eteru is.How generalcan itbeand whatisitsrelation tos? W edefer

theanswerto thisquestion untilwe have studied theabsolute derivative in

connection with the generalized law ofinertia ofrelativity theory.

The geodesic equation for m aterialparticles and photons can also be

derived by m eans ofa di�erent m ethod. W e return to the originalEuler-

Lagrange equation (10.60)and proceed to �nd its�rstintegral.In orderto

avoid confusion with the preceding analysis,we now renam e f as� in this

equation.

M ultiplying Eq.(10.60)by x0� = dx�=du in the sense ofan innerprod-

uct,we �nd

x0� d

du

�@�

@x0�

�

� x0� @�

@x�= 0; (10.76)

which can berewritten as

d

du

�

x0� @�

@x0�

�

�

�@�

@x�

dx�

du+

@�

@x0�

dx0�

du

�

= 0: (10.77)

Because the second term isd�=du,the whole expression can berecastasa

totalderivative,

d

du

�

x0� @�

@x0��

�

= 0: (10.78)

Integrating once with respectto u,we get

x0� @�

@x0�� = C (constant): (10.79)

Proposing thesolution � = g�� x0�x0�,we�nd � = C .Choosing u = s leads

to the earlierresultsfora tim elike geodesic;in thatcase,C = � 1 (tim elike

condition).W hen C = + 1,we �nd thespacelike condition g�� x0�x0� = + 1,

and we are led to a \spacelike geodesic," a m athem aticalconceptthathas

no correspondentin the physicalworld and we can forthwith forget.W hen

C = 0,we �nd the lightlike condition g�� x0�x0� = 0,and we are led to a

nullgeodesic,the worldline followed by a free photon. Finally,because we

now have � = � f2(x;x0),24 the earlier derivation ofthe geodesic equation

leading to Eq.(10.74)rem ainsthesam e.

24The extra m inussign doesnotchange the �nalresult.


10.3.2 T he absolute derivative

G iven thegeodesicequation (10.74)(in principleapplicabletoboth m aterial

particles and photons),let usperform a generalparam eter transform ation

u 7! v(u)and rewrite the geodesic equation in term softhe new param eter

v.Using thechain ruleto go from u to v,we �nd

d2x

dv+�

��

dx�

dv

dx�

dv= �

d2v

du2

�dv

du

� �2dx

dv: (10.80)

Thisresulttellsustwo things.First,thatthe geodesic equation retainsits

sim ple form only forlinear (also called a�ne)param etric transform ations.

Returningtothepossiblerelation between thenullparam eteru and tim elike

param eters,welearn that,given sisanaturallyvalid param eter,in orderfor

thegeodesicequation ofaphoton toconserveitssim plestform ,wem usthave

thatu = as+ b,where a and b are yetphysically undeterm ined constants.

Second,ittellsusthatbecausetheright-hand sideisa contravariantvector

U (v)oftheform f(v)dx =dv,then som ustbetheleft-hand side,despitethe

fact that neither its �rstterm nor its second term in isolation are vectors

them selves (their sum m ust be). As we see next,the second observation

opens up a natural(i.e.physically inspired) road towards the concept of

absolute derivative. W e pursue the im plications of the �rst observation

lateron.

W hat we wish to do is �nd a form for the geodesic equation that is

explicitly tensorial;toachieveit,weneed torem ovefrom sightboth thetotal

second derivativeofthecoordinates(which isnota tensor)and Christo�el’s

sym bol(which isnota tensoreither)and recastthe equation in term sofa

single,explicitly contravariantvector.

Returning to a notation in term sofu,in a changeofcoordinatesx 7! x0

we should then have U 0 (u)= (@x0 =@x�)U �;explicitly,thisis

d2x0

du+�

��

0dx0�

du

dx0�

du=@x0

@x�

d2x�

du+�

��

@x0

@x�

dx�

du

dx�

du: (10.81)

Applyingthechain rulerepeatedly toexpresstheright-hand sidein term sof

partialderivativeswith respectto x0and totalderivativesofx0with respect

to u,and subsequently com paring with the left-hand side,we get

��

��

0��

��

@x0

@x�

@x�

@x0�

@x�

@x0��@x0

@x�

@2x�

@x0�@x0�

�dx0�

du

dx0�

du= 0: (10.82)

Therefore,Christo�el’ssym bolofthe second kind transform sasfollows:

�

��

0=�

��

@x0

@x�

@x�

@x0�

@x�

@x0�+@x0

@x�

@2x�

@x0�@x0�; (10.83)

theappearanceofthesecond term on theright-hand sideshowsthatChristof-

fel’ssym bolofthesecond kind isnota tensor.25

25W ith a sim ilarprocedure,onecan also show thatChristo�el’ssym bolofthe�rstkind

isnota tensoreither.


Next,wetakeageneralcontravariantvectorT�,which isattached toev-

ery pointofaworldlinex�(u);thisis,in otherwords,avector�eld T �[x(u)],

although itsbeing attached only to a worldline is,atthe m om ent,a m ore

realistic(i.e.physical)situation than itsbeing attached to thewholespace-

tim econtinuum .Inspired by thepresentform ofthegeodesic equation and

ourgoalofexpressing ittensorially,we proposea derivative extension ofit

ofthe form�T�

�u= :

dT�

du+�

��

T� dx

du: (10.84)

Isthisderivative extension ofT� a contravariantvector?

Tocheck thepossibletensorialnatureof�T�=�u,wem ustverify whether

�T0�

�u�@x0�

@x�

�T�

�u

isnull.W ritten explicitly,the expression becom es

dT0�

du�@x0�

@x�

dT�

du+�

��

0T0� dx

0

du�@x0�

@x�

��

�

T dx

�

du: (10.85)

The�rsttwo term staken togetherreduce to

@2x0�

@x�@x T� dx

du6= 0; (10.86)

showingthatdT�=du aloneisnota vector.UsingEq.(10.83),thethird and

fourth term staken togetherreduceto

@2x�

@x0�@x0�

@x0�

@x�

@x0�

@x�

@x0�

@x T� dx

du6= 0; (10.87)

showing that the second term ofthe derivative extension (10.84) is not a

vectoreither.However,when both resultsaretaken together,we �nd

�@2x0�

@x�@x +

@2x�

@x0�@x0�

@x0�

@x�

@x0�

@x�

@x0�

@x

�

T� dx

du= 0 (10.88)

on accountofthe parenthesized expression being null.26 W e conclude that

�T�=�u isan explicitlycontravariantvectorand callittheabsolutederivative

ofT�.

W e can now put the absolute derivative ofa vector to good use. W e

observe that the condition �T�=�u = 0 pictures the absolute constancy of

thevectorT�;T� issaid to beparallel-transported along a worldline.These

nam es are inspired in the Euclidean picture, shown in Figure 10.8, that

resultsfrom visualizing thissituation:a curvein spaceand an arrow vector~V attached to two pointsP and Q on the curve,such that ~V (P )pointsin

thesam edirection as~V (Q ).ThisEuclidean im ageisgiven by thecondition

d~V=du = 0,which is only a specialcase of�T�=�u = 0. The im age also

26Thiscan be proved by di�erentiating both sidesof(@x

0�=@x

)(@x

�=@x

0�)= �

� with

respectto x�.


z

y

x

P

~V (uP )

Q

~V (uQ)

xa = xa(u)

Figure 10.8: Euclidean vizualization via spacevector ~V ofthe paralleltransport

ofa spacetim e vector T � along a worldline. The parallel-transportcondition on~V is given by the nullordinary derivative,d~V=du = 0,at each point ofa space

trajectory,while the sam e condition on T � isgiven by a nullabsolute derivative,

�T�=�u = 0,ateach pointofthe wordline.

holdsunderconditions ofpseudo-Euclidicity as in specialrelativity,where

theChristo�elsym bolisnulland theabsolutederivativereducestotheusual

totalderivative,�T�=�u = dT�=du. The nullabsolute derivative pictures

the paralleltransport ofa vector along a worldline in a spacetim e that is

notin general at.

Because the form ofthe absolute derivative wasinspired in thatofthe

geodesic equation,we should not be surprised to �nd that,when parallel-

transported, T� satis�es an equation rem arkably sim ilar to the geodesic

equation,nam ely,

�T�

�u= :

dT�

du+�

��

T� dx

du= 0: (10.89)

Itisnow onlyasm allstep torecastthegeodesicequation in theform wewere

seeking,one thatisexplicitly covariant| and notonly thatbut,m oreover,

one thathasboth an elegantform and interpretation:

�

�u

dx�

du= 0: (10.90)

W hile the elegantform isself-evident,the elegantinterpretation isthe fol-

lowing:a geodesictangentvectorprogressesalong itsworldlinein thedirec-

tion which ititselfpointsto,and so,necessarily,withoutchanging direction.

This is a generalization ofthe case ofa geodesic in the Euclidean plane,

which,being a straight line,is traced by the unidirectional,unperturbed

progression ofits tangent vector. In consequence,a geodesic in Riem ann-

Einstein spacetim eisastraightlinewhich progresses,asitwere,byfollowing

thenaturalintrinsic shapeofspacetim e.TheusualEuclidean visualization

of this situation is, however, som ewhat m isleading, because by picturing

Riem ann-Einstein spacetim easan extrinsically curved surface(e.g.asphere

em bedded in three-dim ensionalEuclidean space),a geodesic tangentvector


(thetangenttoam axim alcircle)sticksoutofthesurfaceintotheem bedding

space,thusgiving the im pression thatitdoesnotprogressin the direction

thatit points to butthat,rather,som e externalpressure forces it to turn

with thesphere.

W e �nd, furtherm ore, that the above m etaphorical allusions to \un-

perturbed progression" ofand \externalpressure" on a geodesic tangent

vectorare particularly aptto describe itsbehaviour,since in question here

is a (new) physicalpicture ofgravitation. A free particle (in the general-

relativistic sense,i.e.not gravitation-free) follows a geodesic in spacetim e.

O urclassicalintuition doesnot m islead uswhen we expect the free parti-

cle to continue along itsworldline unperturbed,since there are no external

forces to change the direction orrate ofits progression;and so itdoes,as

itiscarried along in the intrinsic spacetim e direction ofleastresistance on

accountofitsinertia.

This is a generalization ofNewton’s �rst law ofm otion. Because the

m ass m ofa particle is a constant,we can rewrite Eq.(10.90) in term s of

itsfour-m om entum P � = m V � and tim elike param eters to get

�P�

�s= 0: (10.91)

This m eans that the four-m om entum ofa free m assive particle progresses

unperturbed (i.e.is parallel-transported) in spacetim e,ofwhich a special

case isd~p

dt= ~0; (10.92)

nam ely,Newton’s�rstlaw ofm otion,orthelaw ofinertia (with theproviso

thatweinterprets asabsolute tim e t).

Finally, we exam ine the specialcase ofa photon. This is a m assless

particle whose four-m om entum P � is nevertheless not null, and so for a

photon we m ay sim ply write27

P� =

dx�

du: (10.93)

Com paring with the four-m om entum P � = m dx�=ds ofa m assive particle

leadsto the suggestion thata photon can be understood asa lim iting case

ofa m assive particlewhosem assm tendsto zero astheintervalseparation

ds m easured by a clock on itsworldlinetendsto zero with it,such thatthe

quotientofdsto m convergesto anon-nulllim it;thislim itisdu.Thelinear

connection between u and s envisioned earlieristhus

u =1

ms: (10.94)

So far,the absolute derivative was developed only for a contravariant

vector,but because in the context ofRiem annian geom etry contravariant

27Choosing P

�= �dx

�=du,we �nd that� m ustbe absolutely constant,��=�u = 0,on

account of�P�=�u = 0 and �U

�=�u = 0,where U

�= dx

�=du. The sim plest choice is

� = 1.In general,Eq.(10.94)becom esu = (�=m )s.


and covariantvectorsarefundam entally thesam e,itisreasonableto expect

thatthe absolute derivative can also be a derivative extension ofcovariant

vectors.In orderto �nd itsform ,weresortoncem ore to the conceptofin-

variance.LetT� beacovariantvectorattached totheworldlinex� = x�(u),

and letU � bean arbitrary contravariantvectorwhich isparallel-transported

along thesam e worldline,and therefore

dU �

du= �

��

U� dx

du: (10.95)

Com paring theordinary rate ofchange oftheinvariantoverlap between T�

and U �,afterapplying the chain ruleand Eq.(10.95),we �nd

d(T�U�)

du=

�dT�

du��

��

T�

dx

du

�

U�: (10.96)

Because the left-hand side isan invariantand U � on the right-hand side is

an arbitrary contravariantvector,theparenthesized expression,which isan

extension ofvectorT�,m ustthen bea covariantvector.W e call

�T�

�u= :

dT�

du��

��

T�

dx

du(10.97)

the absolute derivative ofT�.

The sam e heuristic process that led us to the absolute derivative ofa

covariant vector can also lead us to the absolute derivative ofa tensor of

any type and order. Forexam ple,by setting up the ordinary derivative of

the invariant T�� U�W �,where U� and W � are parallel-transported along

the worldline,we obtain,proceeding asbefore,the absolute derivative ofa

second-ordercontravarianttensor

�T��

�u=dT��

du+�

� �

T � dx

�

du+�

�

�

T� dx

�

du: (10.98)

The m annerin which thisresultisobtained allowsusto conclude thatthe

absolute derivative ofa generaltensorisbuiltoutofitsordinary derivative

plusonepositiveterm foreach ofitscontravariantindicesand onenegative

term foreach ofitscovariantindices,each index being sum m ed overoneat

a tim e.

From its generalform follows that the absolute derivative satis�es the

linear-com bination ruleand Leibniz’schain rule.Anotherusefulresultalso

follows,nam ely,that the absolute derivative ofthe m etric �eld is nullin

both itscovariantand contravariantversions,28

�g��

�u= 0 and

�g��

�u= 0: (10.99)

This m eans that,although the m etric �eld is not a constant in the usual

sense,itisa constantin theabsolutesenseofRiem ann-Einstein spacetim e.

28Thiscan beshown by rewriting dg�� =du as(@g�� =@x

�)(dx

�=du)and considering the

identity [��;�]+ [��;�]= 0 in the expression for the absolute derivative ofthe m etric

tensor.


Finally,because an invariant scalar has no indices,we add the convention

thatitsabsolute derivative correspondsto itsordinary derivative,

�T

�u= :

dT

du: (10.100)

At this point,the reader m ight object that our way ofproceeding in

order to �nd the absolute derivative as an extension ofvectors and gen-

eraltensors has no realjusti�cation behind it. W hy should it be what it

is and not otherwise? To this we reply sim ply that there is no certi�ed,

royalroad to m athem aticalphysics.By pretending thatthere isand de�n-

ing m athem atical concepts in an axiom atic spirit, we trade learning the

inform al,freewheeling,and privately conceived physicalm otivation behind

m athem aticalconcepts for the illusion ofsecurity o�ered by authoritative

disem bodied de�nitions. W ould de�ning the absolute derivative in a dry

axiom ,while leaving out allm ention ofthe heuristic m ethod by which we

obtained it,have m ade thisconceptclearerorm ore obscure in the present

physicalcontext? Let usleave rigorously stated de�nitions and axiom s to

purem athem aticians;thesearetheirrightfulturf| butphysicsisnotm ath-

em aticsnorshould itstrive to be.

10.3.3 T he covariant derivative

Absolute di�erentiation appliesto tensorsthatare attached to a worldline

in Riem ann-Einstein spacetim e.And thisisallvery wellbecausetheobject

ofourexam ination has so farbeen m aterialparticles and photons m oving

along geodesicsin spacetim e.However,because the geodesic equation does

notgive usknowledge ofthe spacetim e geom etry butonly inform sushow

particleswillm ovegiven them etrictensor,itneedsthepartnership ofa�eld

equation. A �eld equation can �llthis gap by providing knowledge ofthe

m etrictensorgiven thedistribution ofm atterin spacetim e,which thusacts

asthesourceofspacetim egeom etry,akin to theway chargeand currentact

asthe sourcesoftheelectric and m agnetic forces.

A �eld equation linking m atterdistribution to curvaturedistribution in

spacetim e di�ers from the geodesic equation in that the tensors involved

in the form er need to be attached to whole regions ofspacetim e and not

just to the worldlines ofparticles. The absolute derivative,then,willnot

do asa physically m eaningfulderivative extension of�elds,since the term

dx�=du appearing in it does not have any m eaning in the absence of a

worldline.Thisisenough m otivation forthedevelopm entofa di�erentkind

ofextension ofa tensor| for a spacetim e derivative| as closely related to

theabsolute derivative aspossible,yetwithoutreference to a worldline.

W hat this spacetim e derivative should be is insinuated after a sim ple

rewriting oftheabsolute derivative asfollows:

�T�

�u=

�@T�

@x +�

��

T�

�dx

du: (10.101)

Again, because the absolute derivative of T� on the left-hand side is a

contravariant vector and the worldline derivative on the right-hand side


is as well, the parenthesized expression| which does not depend on the

worldline| m ust now be a second-order once-covariant once-contravariant

tensor.Asthisexpression satis�esallwe asked from a derivative extension

applicable to �eldsin spacetim e,we take

T�; = :

@T�

@x +�

��

T� (10.102)

as the spacetim e derivative we seek. W e see that,just like the absolute

derivative was a generalization ofthe totalderivative in at space or at

spacetim e, so is the spacetim e derivative a generalization of the partial

derivative in at space or at spacetim e. From now on,we callT �; the

covariantderivative ofT� in line with a long physicaltradition.Asjusti�-

cation forthisnam e| lessdescriptiveofitsphysicalfunction than thenam e

spacetim ederivative| itisnoted thatthisderivativealwayshasthee�ectof

addingoneextra covariantindex to thecovariantly di�erentiated tensor.As

before,because the covariantderivative com esfrom the absolute derivative

in a straightforward way,thecovariantderivativeofa tensorofgeneraltype

follows the sam e form as that ofits absolute derivative. Finally,also the

covariantderivative satis�esthe linear-com bination ruleand Leibniz’srule,

and them etric tensorisalso covariantly constant,g��; = g��

; = 0.

O nce the covariantderivative hasbeen developed,we can recreate with

itgeneralizations ofwell-known concepts from vector analysis to be useful

in the following pages:

(i) The covariantgradient T;� ofan invariant scalar T is by convention

(because ithasno indices)taken to correspond with itsordinary gra-

dient(ordinary partialderivative),

T;� = :@T

@x�= :T;�; (10.103)

wherefrom now on we also denote partialderivativeswith a com m a.

(ii) Thecovariantdivergence T�;� ofa contravariantvectorT� is29

T�;� = T

�;� +

1

2g� @g�

@x�T�: (10.104)

(iii) ThecovariantcurlT�;� � T�;� ofa covariantvectorT isitsordinary

curl,

T�;� � T�;� = T�;� � T�;� (10.105)

on account ofthe sym m etry property ofthe Christo�elsym bol. As

a consequence, when T� is som e invariant gradient U;�,we recover

the spacetim e (i.e.covariant) version ofthe well-known resultthata

gradientisa conservative vector �eld,

U;�;� � U;�;� = 0; (10.106)

29Thisisaconsequenceofthesim pli�ed form oftheappearing Christo�elsym bol

˘

��

¯

,

when thisform ism ade explicit.


since ordinary partialderivativescom m ute. From now on,we denote

consecutiveordinarypartialderivativesoftensorswith asinglecom m a;

e.g.U;�;� = :U;�� .

10.3.4 T he curvature tensor

W esaw on page152thatthegeodesicsofafreeparticlein aspacetim ewhose

quadraticform iseverywherepseudo-Cartesian arestraightlinesresem bling

those of at Euclidean space ofclassicalm echanics. This m otivates us to

characterize at spacetim e by the condition that it be possible to choose

coordinatessuch thatthe quadratic form be everywhere pseudo-Cartesian,

i.e.Q = (dx1)2 + (dx2)2 + (dx3)2 � (dx4)2;in otherwords,thatg�� can be

transform ed into �� = diag(1;1;1;� 1) at every point. Ifthis were not

possible,we take spacetim e to becurved.

Thischaracterization of atand curved spacetim eisclearand intuitive,

butisitpractical? G iven a quadratic form ,how shallwe proceed to show

thatitcan orcannotberecastasan everywherepseudo-Cartesian one? Can

wedevisean easierway oftestingwhetherspacetim eis atorcurved? Todo

this,wetakeadvantageoftheintuitiveim agegiven by thespacetim e-related

conceptsinherited from ordinary vector calculus,and think in term sofan

analogy. Itm ay be objected thatthe analogy isfar-fetched,butwe should

notworry aboutthisso long asitnaturally guidesusto thesolution ofour

problem .

W etakeacovariantvector�eld T� and useitssecond covariantderivative

T�;� = :(T�;� ); = T�;�; ��

��

T�;� �

��

T�;� (10.107)

to construct a kind ofcovariantcurl(third-order tensor) T�;� � T�; � of

its �rst covariant derivative T�;� (second-order tensor). This curlcan be

written outexplicitly to �nd

T�;� � T�; � = T�;�; ��

��

T�;� �

��

T�;� � T�; ;� +

��

T�; +

��

T�;�: (10.108)

The third and sixth term s on the right-hand side cancelout. W riting out

therem aining covariantderivatives,we �nd

T�;� � T�; � = (T�;� ��

��

T�); �

��

(T�;� �

��

T�)�

(T�; ��

��

T�);� +

��

(T�; �

��

T�): (10.109)

The�rstand �fth term s,partofthesecond term and theseventh term ,and

the third term and partofthe sixth term canceloutin pairs. Finally,we

�nd

T�;� � T�; � = R�� T�; (10.110)

where

R�� =

��

;��

��

; +�

��

��

��

��

��

(10.111)


m ustbea fourth-orderm ixed tensor,sincetheleft-hand sideofEq.(10.110)

isa third-ordercovarianttensorand T� on theright-hand sideisa covariant

vector. W e callR ��

the m ixed curvature tensor or the m ixed Riem ann

tensor. In general,R ��

T� 6= 0,suggesting that the discrepancy arising

from the orderofspacetim e directionsin which a vectorisdi�erentiated is

theresultofanon-conservativeproperty ofspacetim e captured by thetensor

�eld R ��

.Butwhatisthe connection ofallthiswith curvature?

Ifspacetim e is at,there existcoordinatessuch thatthe m etric �eld is

�� = diag(1;1;1;� 1) atevery point. In thatcase,allChristo�elsym bols

and theirordinary partialderivativesin Eq.(10.111)are null,and so

R�� T�(x)� 0 (10.112)

at every spacetim e point x. Since this is a tensor equation,its validity is

independent ofthe coordinate system used; ifit holds in one coordinate

system ,it holds in allofthem . Now,while this is a necessary condition

for the atness ofspacetim e,is it also a su�cient condition,such that we

m ay say that there is a one-to-one correspondence between the curvature

tensorand curvature? A reading ofEinstein’soriginalpaperon thegeneral

theory ofrelativity is not auspicious regarding the levelofcom plexity re-

quired to answer this question. Einstein (1952a) prefersnotto tackle this

problem and,in a footnote,writes:\Them athem aticianshave proved that

thisisalso a su�cientcondition" (p.141).Letusnonethelessattem ptthis

dem onstration| the physicist’sway.

W estartwith an arbitrary covariantvectorT�(~x)atan arbitrary space-

tim epoint,and parallel-transportit�rstalong a worldlinex� = x�(u;v0)a

separation du,and subsequently along anotherworldline x� = x�(du;v)a

separation dv,whereu0 and v0 areconstants,thusreaching spacetim epoint

x (see Figure 9.4 on page 114). Alternatively,we can reach x from ~x by

parallel-transporting T�(~x)�rstalong worldlinex� = x�(u0;v)a separation

dv,and subsequently along worldline x� = x�(u;dv)a separation du. The

two resulting vectorsatx arethe sam e only ifthedi�erence

�2T�

�v�u��2T�

�u�v=

�

T�;�@x�

@u

�

;

@x

@v�

�

T�;�@x�

@v

�

;

@x

@u(10.113)

isnull.Because @x�=@u and @x�=@v on the right-hand sidearethevectors

tangenttotheworldlines,they areabsolutely constantand can betaken out

ofthecovariantderivative.W e �nd

�2T�

�v�u��2T�

�u�v= T�;�

@x�

@u

@x

@v� T�;�

@x�

@v

@x

@u

= (T�;� � T�; � )@x�

@u

@x

@v; (10.114)

and we now recognize the Riem ann tensorin theparenthesized expression:

�2T�

�v�u��2T�

�u�v= R

�� T�

@x�

@u

@x

@v; (10.115)


wheretheworldlinesarearbitrary and so isT�.

Therefore,if the Riem ann tensor is identically null,then the result of

parallel-transporting T�(~x)to x isa unequivocalvectorT�(x)independently

ofthe worldline followed. But then,its covariant derivative m ust be null,

T�;� (x)= 0.Sincespacetim epointx isarbitrary,given R ��

� 0(su�cient

condition),30 wecan createaspacetim evector�eld atany spacetim epointx

via parallel-transportofan originalvector.Based on ourintuition,we now

expectthattheunequivocalexistence ofT�(x)willim ply thatspacetim e is

at. Forexam ple,in the intuitively curved case ofa sphericalsurface,the

parallel-transportofa vector from the equator to the north pole produces

di�erent vectors at the north pole depending on the path chosen,butthe

sam evectoron a atsurface.

W enow takethecovariantvector�eld T�(x)to bethegradientS;�(x)of

an arbitraryinvariantscalar�eld S(x).W etake,m oreover,fourindependent

scalars �elds,which we choose to be the coordinates x01(x);x02(x);x03(x),

and x04(x).Now T�;� = 0 translatesinto x0

;�;�(x)= 0,which,in turn,gives

@2x0

@x�@x�=�

��

@x0

@x�: (10.116)

W enow show thatin theprim edcoordinatesobtained through paralleltrans-

port,them etric tensorg0��(x0)isconstant.

Because

g�� =@x0

@x�

@x0�

@x�g0 �; (10.117)

we�nd

@g��

@x�=

@2x0

@x�@x�

@x0�

@x�g0 � +

@x0

@x�

@2x0�

@x�@x�g0 � +

@x0

@x�

@x0�

@x�

@g0 �

@x�; (10.118)

which,on accountofEq.(10.116),becom es

@g��

@x�=�

��

@x0

@x�

@x0�

@x�g0 � +

��

��

@x0

@x�

@x0�

@x�g0 � +

@x0

@x�

@x0�

@x�

@g0 �

@x�: (10.119)

The�rstand second term scan besim pli�ed to get

@g��

@x�=�

��

g�� +

��

��

g�� +

@x0

@x�

@x0�

@x�

@g0 �

@x�

= [��;�]+ [��;�]+@x0

@x�

@x0�

@x�

@g0 �

@x�; (10.120)

�nally giving

@g��

@x�=@g��

@x�+@x0

@x�

@x0�

@x�

@g0 �

@x�: (10.121)

30This is also a necessary condition. The system ofequations T�;� =

˘

��

¯

T� is inte-

grable only ifT�;� = T�; � .Thisgives`˘

��

¯

T�´

; =

`˘

��

¯

T�´

;�,im plying R �

�� 0.


Itfollowsthat@g0 �=@x� = 0,and therefore @g0

�=@x0� = 0.In conclusion,if

the Riem ann curvature tensor is identically null,then coordinates x0 exist

such thatg0��(x0)isconstanteverywhere in spacetim e.31

The m ixed Riem ann tensor R ��

,its covariant version R �� ,and its

contractions,R

�� and R �

�,satisfy severalusefulidentities. W e can see

straightfrom Eq.(10.111)thatthem ixed Riem ann tensorisantisym m etric

on itslasttwo covariantindices,

R�� = � R �

� � ; (10.122)

and thatthesum ofitscyclic perm utationsisnull,

R�� + R

� �� + R

�� = 0: (10.123)

ThecovariantRiem ann tensorcan beobtained from them ixed onelowering

the contravariantindex,R �� = g��R��

,which gives32

R �� = [��;�]; � [� ;�];� + g��([��;�][� ;�]� [� ;�][��;�]): (10.124)

A m ore usefulexpression ofthe covariantRiem ann tensorcan be obtained

by writing the Christo�elsym bolsofthe �rstkind in term softhose ofthe

second kind,the m etric tensor,and itsderivatives.W e �nd

R �� =1

2(g��;� + g� ;�� g� ;�� g��;� )+

g��

��

��

�

��

��

��

��

�

: (10.125)

By directinspection,we see thatthe covariantRiem ann tensorisantisym -

m etric on its�rsttwo and lasttwo indices,

R �� = � R �� and R �� = � R �� ; (10.126)

and itissym m etric with respectto a swap ofthe �rstand second pairsof

indicestaken asa unit,

R �� = R �� : (10.127)

Applying alltheseresultstogether,wealso �nd thatthecovariantRiem ann

tensorissym m etric with respectto an oppositereordering ofitsindices,

R �� = R � �� : (10.128)

In addition,the covariant derivatives ofthe m ixed and covariant Rie-

m ann tensorseach satisfy an identity called theBianchiidentity,which says

31W e have notproved,however,thatg

0�� (x

0)= �� .

32Expressg��

˘

��

¯

; asthederivativeoftheproductm inusan extra term ,and rewrite

both in term sofChristo�elsym bolsofthe �rstkind and theirderivatives.


thatthe sum ofthe cyclic perm utationson the lastthree covariantindices

ofthe derivative tensorisnull.Forthe m ixed Riem ann tensor,itreads33

R�� ;� + R

��; + R

�� ;� = 0: (10.129)

The corresponding Bianchiidentity for the covariant Riem ann tensor can

beobtained from the form erby lowering thecontravariantindex;itreads

R �� ;� + R �� ;� + R ��; = 0: (10.130)

TheRiccitensor R �� isa usefultensorderived from the covariantRie-

m ann tensorR �� by contracting the �rstand lastindices,

g �R �� = R

�� = :R �� : (10.131)

The Riccitensorissym m etric34 and hasonly 10 independentcom ponents,

which is the sam e am ountofindependentcom ponentsas that ofthe m et-

ric tensor. This property becom es essentialwhen we form Einstein’s �eld

equation linking gravitation with its sources. Anotherusefulquantity,the

invariantcurvature scalar R,can be derived from the Riccitensorby con-

tracting itstwo indices,

g� R � = R

�� = :R: (10.132)

DotheRiccitensorand curvaturescalarhaveanynoteworthyproperties?

Taking thesecond Bianchiidentity (10.130)and contracting itsindicesin a

suitable way,we�nd

g� g��R �� ;� + g

��g� R �� ;� + g

� g��R ��; =

g� R � ;� � g

��R ��;� � g

� R ��; = R ;�� 2R �

�;� = 0: (10.133)

M ultiplying by g��,we get

g��R ;�� 2R

��

;�=

�

R��

1

2g��R

�

;�

= 0: (10.134)

W e have found thatthesym m etric tensor

G�� = :R ��

1

2g��R; (10.135)

called theEinstein tensor,hasthespecialproperty thatitscovariantdiver-

gence isnull,

G��

;�= 0: (10.136)

33To provethe Bianchiidentity,calculate the\covariantcurl" ofthethird-ordertensor

(A �B �); to�nd (A � B �); �� (A �B �);� = A �R�� B �+ B �R

�� A �.Nextidentify A � B �

with T�;� and form the sum ofitsthree cyclic perm utationsof�, ,and �. Expressthis

sum ,on theonehand,in term softhem ixed Riem ann tensorand �rstcovariantderivatives

ofT� and,on theotherhand,in term sofcovariantderivativesoftheproductofthem ixed

Riem ann tensorand T� .Use R�� + R

� �� + R

�� = 0 and sim plify.

34To show thisresult,write the Riccitensorin term softhe covariantRiem ann tensor

and use the sym m etry propertiesofthe latter.


Finally,noticing thattheEinstein tensorG �� consistsofa second-order

tensor R �� and a scalar R,we consider whetherany further,stillsim pler,

term could be added to it such that its covariant divergence continues to

be null. Thinking in purely m athem aticalterm s,a term ofthe form �g�� ,

where� issom econstantunrelated to theRiccitensor,satis�es�g��

;�= 0.

Addingitto theEinstein tensorfornow only with thispurely m athem atical

m otivation,we getthegeneralized Einstein tensor

G�� = :R ��

1

2g��R + �g��

; (10.137)

where

G��

;�= 0: (10.138)

Istheinclusion of� in theEinstein tensoralso justi�ed on physicalgrounds

by thedisplay ofrelevantobservablee�ects? And ifso,whatisitsphysical

interpretation? Not dark and m ysterious,one should hope. W e return to

thisissue lateron.

10.3.5 G eodesic deviation and the rise ofrod length

How do we observe the active e�ects ofthe curvature tensor? In a curved

spacetim e,weexpectthattwo particlesthatfollow at�rstparallelgeodesics

willnot rem ain on parallelspacetim e tracks but willstart to converge or

diverge,i.e.two test particles released at rest willeither m ove towards or

away from each other (discounting theirm utualgravitationalinteraction).

Thistwo-geodesic m ethod ofprobing curvature soundsprom ising,because

the testin question involvesonly localinform ation.

Toinspectthebehaviouroftheseparation between twoparticlegeodesics

in relation to curvature,we exam ine two close-lying geodesics x� = x�(u)

and x0� = x0�(u)(notitsderivative)thatare nearly parallelto each other,

as shown in Figure 10.9 for the case oftwo m assive particles. The sm all

di�erence��(u)= x0�(u)� x�(u)isa di�erentialvectorthatconnectsthose

pointsin the geodesicsthathave the sam e value ofparam eteru.Since the

two geodesicsarenearly parallel,therateofchanged��=du oftheirm utual

separation isalso very sm all.

In orderto obtain an equation for��,wesubtractthegeodesicequation

ofx�(u) from that ofx0�(u). Using the �rst-order approxim ation to the

prim ed Christo�elsym bol,

��

0=�

��

+�

��

;��; (10.139)

on accountofx0�(u)= x�(u)+ ��(u),we �nd

d2��

du2+�

��

;�

dx�

du

dx

du�� +

��

dx�

du

d�

du+�

��

d��

du

dx

du= 0: (10.140)


xα(s) x

′α(s)

s

sη

α(s)

Figure 10.9: Deviation oftwo tim elikegeodesicsasa localactivee�ectofspace-

tim e curvature. W hen spacetim e is at,two geodesics deviate from each other

linearly. W hen the size � of�� doesnotdepend on s,d�=ds= 0,�� actslike an

idealized m easuring rod.

Introducingthederivativeofa sum and subtracting two extra term s,weget

d

du

�d��

du+�

��

�� dx

du

�

��

��

;�� dx

du

dx�

du��

��

�� d

2x

du2+

��

;�

dx�

du

dx

du�� +

��

dx�

du

d�

du= 0: (10.141)

Finally,using the geodesic equation to rewrite d2x =du2,adding and sub-

tracting a term (fourth and lastbelow)to form an extra ��=�u,and rear-

ranging the indices,we �nd

d

du

�d��

du+�

��

�� dx

du

�

+��

�d��

du+�

��

�� dx

du

�dx�

du+

��

;��

��

;�+�

� �

��

��

��

��

�

�� dx

du

dx�

du= 0: (10.142)

In the�rsttwo parenthesized expressions,werecognizetheabsolutederiva-

tive ofthe geodesic separation and,in the last one,the Riem ann tensor.

Theequation satis�ed by thegeodesicseparation,which connectsspacetim e

curvature with the way the two geodesics deviate from one another,can

then beexpressed in sim plerterm sas

�2��

�u2+ R

� ��

� dx

du

dx�

du= 0: (10.143)

The solution is com pletely determ ined when we know �� and d��=du for

som e value of the param eter u. In at spacetim e, where the curvature

tensorisidentically null,the equation reducesto

d2��

du2= 0: (10.144)

Itssolutionsare��(u)= A �u+ B �,which m eansthatin atspacetim ethe

separation vector between two geodesics changes linearly with u| the two


geodesicsarestraightlinesin spacetim e and,therefore,converge ordiverge

linearly.

The separation between tim elike spacetim e geodesics gives us now the

possibility to recovertheconceptofrod distance � from chronom etricfoun-

dations.In relativity theory,the conceptoflength between two particlesis

notstraightforward because,in Riem ann-Einstein spacetim e,the present|

thatis,when the length m easurem entshould be m ade| istied to each ob-

server’sworldline;in classicalphysics,on the otherhand,the presentisan

absolute notion. This di�culty notwithstanding,the rod length between

two particlescan bedeterm ined forspecialcasesin relativity theory too,as

follows.

Thegeodesic separation ��(s)isa spacelike vector,whosesize

�(s)= :ds�(s) =

q

g�� [x(s)]��(s)��(s) (10.145)

depends on the value of the geodesic param eter s. This size, indicating

theseparation between two neighbouringfreem assiveparticles,ism easured

with a clock and a photon;showing the dependenceon s explicitly,itis

�(s)= dssP sA dssA sQ ; (10.146)

whereP and Q aretheeventsofphoton em ission and reception and A m arks

the tailof��(s)(cf.Figure 10.3 on page 143). If�(s)doesnotdepend on

s,the spacelike separation m ustbeconstant,

d�

ds= 0: (10.147)

In that case,we can identify the two particles joined by �� with the end-

pointsofan idealized rigid m easuring rod and � with itslength.In practice,

thiscondition isobtained when allthephotonssentoutfrom the�rstparticle

and bouncingbackfrom thesecond particlearriveatconstantclockintervals.

In thisroundaboutm anner,theconceptofrod length arisesasa particular

case ofm easurem entsofspacetim e intervalswith clocksand photons.

10.4 T he sources ofgravitation

10.4.1 T he vacuum

W e start the search for the di�erent instances ofEinstein’s �eld equation,

which connectsspacetim ecurvaturewith sourcesin spacetim e,in each case

looking forguidancein Newton’sclassicaltheory ofgravitation.Attheout-

setwith thecaseofthevacuum ,Newtonian theory helpsusrealizethatthere

is no straightforward general-relativistic counterpart ofthe classicalgravi-

tationalpotential�(~r),but it also provides a usefulanalogicalpicture on

which to basethesearch forthedi�erentinstancesofthegeneral-relativistic

�eld equation.

10.4 T he sources ofgravitation 169

In the vacuum outside a spherically sym m etric m ass distribution,the

Newtonian gravitationalpotential�(~r),connected with the universalforce

ofgravitation according to ~r �(~r)= �~g(~r),satis�esLaplace’sequation

r 2�(~r)= 0: (10.148)

Thisfollowsfrom the factthat~g(~r)isdivergence-free,~r � ~g(~r)= 0,where

~g(~r)= � G M r=r2.In Cartesian coordinates,Laplace’sequation becom es

3X

i= 1

@2�(~r)

(@xi)2= 0: (10.149)

In specialrelativity,an equation corresponding to the classicalLaplace

equation is

� �(x)= :�� ;�� (x)= 0: (10.150)

Thisequation hasthe form ofthe K lein-G ordon equation fora scalar�eld

associated with am asslessparticle;expressed in theform ofawaveequation,

itis3X

i= 1

@2�(x)

(@xi)2�@2�(x)

(@x4)2= 0: (10.151)

Could we now generalize Laplace’s equation for the general-relativistic

case to �nd the equation ofgravitation in the vacuum ? Should we per-

haps,in analogy,take our equation to be g�� ;�� (x) = 0? But what is

now the m eaning of�eld �(x)? Having com e this far in the form ulation

ofgeneralrelativity,we know it for a fact that its foundation is given by

thequadratic form ds2 = �g�� (x)dx�dx�,so why should allofa sudden an

extraneous �eld like �(x) be needed? How about g�� g �;�� (x) = 0 then?

Notreally,becausethecovariantderivativeofthem etrictensorisidentically

null,g �;�(x)� 0,and so no realinform ation can beextracted from such an

equation.O nceagain,itseem sthatsim ply g�� (x)isnotenough to serveas

the\new gravitational�eld."

Aswe suspected earlier,gravitation should instead be related to space-

tim ecurvature,and weknow thatcurvatureischaracterized bytheRiem ann

tensor. Buthow should one take advantage ofthisknowledge to build the

�eld equation in vacuum ? Ifwe say that \spacetim e curvature is nullin

vacuum ," would thism ean thatspacetim eis at,and thusgravitation-free,

in vacuum even when thereism atterelsewhereand notnecessarily very far

away? Such an interpretation goes against our intuition,because gravita-

tionale�ects exist in the vacuum near m assive objects;spacetim e cannot

thereforebe atthere.

W hen wesay,aswewill,thatspacetim ecurvatureisnullin vacuum ,we

take the m eaning ofthis assertion in direct analogy with Newtonian the-

ory,by identifying a nullcurvature tensor with the nullLaplacian ofthe

Newtonian potential. And so,just like in classicalphysics the nullLapla-

cian can give the solution to the non-nullgravitational potentialoutside

a m aterialdistribution, so can a nullcurvature tensor give the non-null


curvature solution outside a m aterialdistribution. O nly when the vacuum

extendsthoroughly everywherearegravitationale�ectsnonexistentand,in

relativistic physics,we can say thatspacetim e is at.

Sofarwehavedeveloped severalversionsofthecurvaturetensor.W hich

of the following �ve alternative �eld equations: (i) R �� (x) = 0, (ii)

R ��

(x)= 0,(iii)R �� (x)= 0,(iv)R(x)= 0,and (v)G �� (x)= 0 should

wenow chooseastheanalogueofr 2�(~r)= 0? The�eld equation wesearch

should becapableofunequivocally determ ining the10 independentcom po-

nentsg�� ofthem etrictensor,and so wecan narrow down thepossibilities

to expressions(iii)and (v)| a nullRicciorEinstein tensor,respectively|

theonly onessuitablefrom thisperspective,becauseonly they havenom ore

norlessthan 10 independentcom ponents.Butwhich ofthe two should we

choose?

Forthepurposesofthisand thenexttwo sections,thatis,forthestudy

ofthe vacuum ,which ofthe two tensorswe choose isirrelevant,because in

thevacuum ,asweseebelow,theEinstein tensorreducestotheRiccitensor.

However,when welateron considersolutionsinsidea distribution ofm atter

(i.e.m atter as an explicit source),only the choice ofthe Einstein tensor

willbe suitable,because only it is capable ofleading us to a law for the

localconservation ofthem om entum and energy ofthem attersource,i.e.to

thegeneralization ofthe classicallaw ofcontinuity ofm atter.Thisism ade

possible by the extra m athem aticalproperty ofEq.(10.136) seen earlier,

nam ely,thatthe Einstein tensorisdivergence-free (in the covariantsense).

W hen,after establishing a connection between curvature and m atter,we

dem and thatthem attersourcebedivergence-freeforconservation purposes,

thism athem aticalproperty ofG �� becom esphysically relevant.

In general,weshould then choose the �eld equation in vacuum to be

G �� (x)= 0: (10.152)

In practice,however,thecurvaturescalarR isnullin vacuum ,35 and in that

case wecan useR �� (x)= 0 forsim plicity.

The sim plestsolution to the �eld equation in vacuum isobtained when

the vacuum extends everywhere. Then the curvature tensor is identically

null,R �� (x) � 0,and coordinates can be found such that the solution to

the �eld equation isg�� (x)= �� ,thatis, atspacetim e.

Anotherexactsolution to the spacetim e m etric can be found outside a

spherically sym m etric m atter distribution ofm ass M . This solution was

found by K arlSchwarzschild in 1916.Considerationsofsphericalsym m etry

lead to constraintsin thepossibleform thatthem etric�eld can take.Using

sphericalcoordinates,the m etric can only beofthe form

g�� (x)= diag

�

e�(x 1)

;(x1)2;(x1)2sin2(x2);� e�(x1)�

; (10.153)

with asym ptotic atnessatin�nity,i.e.with boundaryconditions�(x1)! 0

and �(x1)! 0 when x1 ! 1 .

35Express G �� (x)= 0 in explicit form and contract its indices with the m etric tensor

to �nd � R = 0.


Thecalculation ofthe13 non-nullChristo�elsym bolsgives

�111

=1

2�0;

�122

= � x1e�� ;

�133

= � x1sin2(x2)e�� ;

�144

=1

2e��

�0;

�212

=�

221

=

1

x1;

�233

= � sin(x2)cos(x2); (10.154)

�313

=�

331

=

1

x1;

�323

=�

332

= cot(x2);

�414

=�

441

=1

2�0:

Asa result,the Riccitensorisdiagonal,with com ponentsgiven by

R 11 =1

2

�

�00�

1

2�0�0+

1

2�02 �

2

x1�0

�

; (10.155)

R 22 = � 1+ e��

�

1+1

2x1(�0� �

0)

�

; (10.156)

R 33 = sin2(x2)R 22; (10.157)

R 44 = �1

2e��

�

�00�

1

2�0�0+

1

2�02 +

2

x1�0

�

: (10.158)

To solve the system offour equations R �� = 0,we consider �rstR 11 = 0

and R 44 = 0 together with the condition ofasym ptotic atness. W e �nd

straightforwardly that�(x1)= � �(x1). Nextwe use thisresultto sim plify

and solve R 22 = 0. G rouping the variables x1 and � with their respective

di�erentialsand integrating once,we �nd

e� =

�

1�C

x1

� �1

(10.159)

and,therefore,

e� = 1�

C

x1: (10.160)

Thevalueoftheintegration constantC isfound by com paring with the

Newtonian lim it(expressed in naturalunits).Because36

�

�

1�C

x1

�

= g44c= � (1+ 2�)= �

�

1�2G M

c2r

�

; (10.161)

then C = 2G M =c2,wherex1 can beidenti�ed with theEuclidean distancer

only when x1 ! 1 .The quadratic form in the vacuum outside a spherical

m atterdistribution then is

ds2 = �

( �

1�2G M

c2x1

� �1

(dx1)2 + (x1)2[(dx2)2 + sin2(x2)(dx3)2]�

�

1�2G M

c2x1

�

(dx4)2

)

; (10.162)

36See nextsection,Eq.(10.185).


where 0 � x2 � �,0 � x3 � 2�,and � 1 < x4 < 1 .Thissolution isvalid

eitherfor(i)x1B< x1 < 1 ,where x1

Bisthe radiusofthe m aterialedge of

the m atter distribution or (ii) x1S< x1 < 1 ,where x1

S= 2G M =c2 is the

Schwarzschild radiusofthe m atterdistribution,depending on which ofthe

two values is reached �rst as we approach from in�nity. Ifx1S= 2G M =c2

is reached �rst,the spacetim e region in question is called a (non-rotating)

black hole.

10.4.2 Trem ors in the vacuum

Ifnotin thegeneralway described (and rejected)atthebeginningofthepre-

vioussection,do the classicalgravitationalpotential�(~r)and the Laplace

equation r 2�(~r) = 0 have som e kind ofrelativistic counterpart in som e

specialcase?

To �nd out,we consider physically weak trem ors (from the spacetim e

perspective)induced by a sm alldeviation from m etric atness,

g�� (x)= �� + �� ; (10.163)

whereall�� and theirderivativesarevery sm all.In thiscase,itisaccurate

enough to solve the equation R �� (x)= 0 to �rstorderin �� . W hatkind

ofweak \gravitational�eld" �� (x)resultsfrom thisapproxim ation?

To �rstorder(� )in �� ,wehave thatthe inverse m etric tensoris

g�� (x)= �

�� (10.164)

because

g��g� � (��

�� )(�� )= �� + �

� � �

� = �

� : (10.165)

In thefollowing,them etrictensorg�� istheonlytensorwhoseindiceswillbe

raised and lowered with itself;forany other�eld,we usethe at-spacetim e

m etric �� or�� .

Thenextstep isto�nd the�rst-orderapproxim ation totheRiccitensor.

From expression (10.111) ofthe m ixed Riem ann tensor,we know that the

exactRiccitensoris

R �� =�

�

;��

��

; +�

��

�

��

��

��

� �

: (10.166)

Because to �rstorderthe Christo�elsym bolis

�

��

�1

2� �(��;� + ��;� � ��;� ); (10.167)

we then have�

��

; �1

2(�

� ;� + �

� ;� � � �� ) (10.168)

and�

�

;��1

2�;�� : (10.169)


To �rst order, disregarding products of �� or its derivatives, the Ricci

tensoris

R ��

�

;��

��

; �1

2(�;��

� ;� � �

� ;� + � �� ): (10.170)

Becausewe�nd ourselvesoutsideany m aterialdistribution,wesetR �� = 0

to get

� �� + �;��

� ;� � �

� ;� = 0:37 (10.171)

This is the linearized general-relativistic equation for a perturbation

�� (x) ofthe m etric �eld. Its form coincides with the Bargm ann-W igner

equation fora �eld in atspacetim e associated with a m asslessspin-2 par-

ticle. O n accountofitsform alcorrespondence with the linearized general-

relativisticequation forthem etricperturbation �� (x),thisparticleissom e-

tim escalled a \graviton." However,the correspondenceappearsto beonly

form al,becausetheBargm ann-W igner�eld equation wasderived fora �eld

in M inkowskispacetim e,which is atand hastherefore nothing to do with

gravitation.Ashard asonem ighttry to establish theresem blancebetween

a \graviton" and other quantum particles,a \graviton" doesnotstand on

the sam e conceptualfooting asquantum particles,because the �eld associ-

ated with itisnotan ordinary �eld on atspacetim e butisitself(partof)

them etric �eld of spacetim e.

To seewhat�� (x)m ay haveto do with theNewtonian potential�(~r),

we start by considering a \slow particle" such that the com ponents V a

(a = 1;2;3)ofitsfour-velocity V � = dx�=ds are very sm alland,therefore,

g�� V�V � = � 1 becom es

g44(V4)2 � � 1; (10.172)

from where

V4 �

1p� g44

: (10.173)

W hen the slow-particle condition is applied to the geodesic equation for

� = a,we �nddV a

ds� �

�a44

(V 4)2: (10.174)

ButbecausedV a

ds=dV a

dx�V� �

dV a

dx4V4; (10.175)

wegetdV a

dx4� �

�a44

V4: (10.176)

Theleft-hand sideofthisequation iscom parablewith theNewtonian accel-

eration dva=dt.

37Taking advantage ofthe gauge freedom ofthis equation,it can be recast sim ply as

� �� = 0 using thegauge� � ; = 0,where�� = �� =2.In som em oredetail,the

linearized equation is invariant with respect to the gauge transform ation �� ! ~�� =

�� + (��;� + ��;� )=2,where �� (x)isan arbitrary vector�eld.W hen � �� = 2��

� �,the

gauge condition is satis�ed (i.e. ~� � ; = 0) and the linearized equation sim pli�ed with

respectto thenew �eld ~�� .Thisisthestarting pointofthestudy ofgravitationalwaves.


Assum ing nextthatthe fullspacetim e m etric is\quasistatic," i.e.that

g��;4 ism uch sm allerthan g��;a ,weget

�a44

� �

1

2ga� @g44

@x��

1

2gab@g44

@xb: (10.177)

Using thisresult,we�nd

dV a

dx4�1

2gab@g44

@xbV4: (10.178)

Using Eq.(10.173),we com e to the expression

dV a

dx4� � gab

@p� g44

@xb: (10.179)

Finally,because we aredealing only with trem orsin the vacuum ,

g44 = 1� �

44 (10.180)

on accountofEq.(10.164).Thecounterpartofthe Newtonian acceleration

thusbecom es

dV a

dx4� � (�ab� �

ab)@(1� �44=2)

@xb�1

2�ab@�44

@xb; (10.181)

thatisdVa

dx4�1

2

@�44

@xa: (10.182)

Com paring thisexpression with theNewtonian acceleration

dva

dt= ga(~r)= �

@�(~r)

@xa; (10.183)

we discover the correspondences (valid only in the context ofthe approxi-

m ationsm ade)

�c= �

1

2�44 (10.184)

and

g44c= � (1+ 2�): (10.185)

A rough estim ate ofhow � com pareswith unity in Eq.(10.185) atthe

surfaceofthe earth,where�(r)= � G M =c2r,38 gives

�� 10�9 and@�

@xa� 10�16 m �1

: (10.186)

From this,the acceleration atthe surface ofthe earth turnsoutto be the

fam iliarresult

c2 @�

@xa= 10 m s�2 : (10.187)

38In naturalunits,as used so far,� is a dim ensionless num ber. O n the other hand,

� G M =r hasSIunitsL2T�2.To converta quantity from SIunitsto naturalunits(c= 1),

we m ust divide it by c2as we just did;to convert a quantity from naturalunits to SI

units,we m ustm ultiply itby c2.


Therelativesizeofthegravitationaltrem orswith respecttothe at-spacetim e

background iscom parablewith ripplesrising 1 centim etreabovethesurface

ofan (unearthly)lake 10,000 kilom etresin depth.

W e�nd,m oreover,thatin thiscorrespondencetheLaplacian r 2�ofthe

Newtonian potentialisthe 44-com ponentoftheRiccitensor,

R 44 �1

2(�;44 � 2�

4 ;4 + � �44)�1

2��44;��

1

2�ab�44;ab =

1

2

3X

a= 1

�44;aa =1

2r 2

�44c= � r 2�: (10.188)

Thus,theLaplaceequation r 2�= 0fortheNewtonian potentialin vacuum

correspondsto R 44 = 0.

The specialrole ofcoordinate x4 in the way the correspondence with

classical physics was established deserves specialm ention. Early on, x4

played a crucialpartin the localcharacterization ofthe past,the present,

and thefuturein relativity theory.Now itseem sthat,in thisapproxim ation,

x4 hasbecom etheabsolutetim etofclassicalphysics.Thelikelihood ofthis

conclusion is strengthened by a look at Einstein’s (1952a) originalpaper,

where,on m aking this approxim ation,he writes that \we have set ds =

dx4 = dt" (p.158). Butisthistrue to �rstorder in �? Itisnot,because

wehave �s= �tin theNewtonian case,but

dx4

ds= V

4 �1

p1� �44

� 1+1

2�44

c= 1� �6= 1: (10.189)

W econcludethattheidenti�cation ofthefourth coordinatex4 with absolute

tim e tis not possible even in the relativistic approxim ation to Newtonian

physics. If,in spite ofthis,the association x4 = ctstillwantsto be m ade,

one m ust rem em ber that in this case tis just a notation| the m isleading

nam eofa spacetim e coordinate,notabsolute tim e.

10.4.3 T he cosm ic vacuum

So farwe have studied the �eld equationsG �� = 0 orR �� = 0,where the

constant� isnull.In neglecting thisconstant,we have assum ed thatj�jis

sm allenough,but now we would like to m ake sure that this is correct by

setting an upperbound forj�j.

W e startby noting that,because

g��

G�� = R � 2R + 4� = 0; (10.190)

then R = 4�,and the�eld equation reducesto

R �� g�� = 0: (10.191)

Using again the correspondence R 44c= � r 2� in the contextofthe Newto-

nian approxim ation,to �rstorderin � we have

�g44c= � �(1+ 2�)� � �: (10.192)


Ifwe now take �g44 as if it was a source ofcurvature (instead ofbeing a

curvatureterm itself),wecan write thePoisson equation

r 2�=4�G �

c2+ � (10.193)

forthe Newtonian potentialinside a m edium ofdensity � + �c2=4�G (ex-

pressed in SIunits).Here� standsfortheordinary density ofm atter,while

�c2=4�G stands for the m ass density ofa new source ofgravitation per-

vading allspace in the m anner ofan ubiquitous ether,present also inside

m atter.

W e seefrom Eq.(10.193)that,iftheclassicaldynam icsinsidea system

ofdensity � is welldescribed by Newtonian theory,then j�j� 4�G �=c2,

obtaining an upperbound forthisnew constant.Them oredilutea system

insideofwhich Newtonian theory holds,them orestringenttheupperbound

seton j�jisand,consequently,thesm alleritm ustbe.

Fora conservativeestim ate,wetakeourgalaxy asa system insidewhich

Newtonian m echanics can be applied successfully. G iven that the average

density ofthe M ilky W ay is approxim ately � = 1:7 � 10�21 kg=m3,39 we

�nd,j�j� 1:4� 10 �30 s�2 . To expressthisbound in a m ore fam iliarway

in term sofnaturalunits,we divide by c2 to getj�j� 1:6� 10 �47 m �2 .

It has been conjectured by other m ethods that a m ore accurate but

lessconservative estim ate isj�j� 10�35 s�2 or,m ore com m only in natural

units,j�j� 10�52 m �2 .O n accountofEq.(10.193),thegravitationale�ect

ofa constant� ofthism agnitude would begin to be feltinside a m aterial

m edium ofdensity � � 1:7 � 10�26 kg=m3,i.e.10 hydrogen-atom m asses

percubic m etre. Thisvalue correspondsto the density ofthe intergalactic

m edium .Thus,in orderfor�tohavenon-negligiblee�ects,wem ustconsider

cosm ologicaldistances| hence itsnam e,the cosm ologicalconstant.

To seewhatkind ofgravitationale�ectsarecaused by thecosm ological

constant| whose so far enigm atic source,we recall,pervadesallspace| in

theabsenceofany otherm aterialm edium (i.e.� = 0),weconsidera point-

like40 m assM surrounded by a perfectvacuum in theordinary senseofthe

word.In theNewtonian approxim ation underconsideration,weshould then

solve Eq.(10.193).

Since � is constant and the centralm ass point-like (or spherical),we

solve for�(r)in a spherically sym m etric context:

1

r2

d

dr

�

r2d�

dr

�

= �: (10.194)

Integrating twice and absorbing oneconstantinto the potential,we �nd

�(r)= �C 0

r+1

6�r2; (10.195)

39Expressed di�erently,� = 106� 1:7� 10�27 kg=m3,thatis,106 hydrogen-atom m asses

percubic m eter.Thisisone m illion tim esthe density ofthe intergalactic \vacuum ."40O rspherical,in which case we are only interested to �nd the gravitationalpotential

in the vacuum outside it.


which leadsto a gravitationalforce on a particle ofm assm ofthe type

~Fg = � md�

drr= � m

�C 0

r2+1

3�r

�

r; (10.196)

where C 0= G M =c2.If� ispositive,itgeneratesan extra attractive gravi-

tationalforcebetween thecentralm assand a testm ass.M oreinterestingly,

ifitisnegative,itgeneratesan extra repulsive gravitationalforce between

thetwo m asses.In both cases,theextra forceincreaseslinearly aswerecede

from thecentralm ass.Fornegative�,thereexiststhen a distanceatwhich

theattraction from thecentreisbalanced by therepulsion caused by theso

farenigm atic source ofthe cosm ologicalconstant. G iven the upperbound

j�j� 10�52 m �2 ,wecan getafeelforthedistanceinvolved when thecentral

m assisthe sun (M � 2� 1030 kg). W e �nd that,in an otherwise perfect

universalvacuum ,the cosm ologicalconstant is so sm allthat its repulsive

force cancelsouttheattraction ofthe sun ata distance

r=3

r3G M

�c2� 3:5� 1018 m � 375 ly: (10.197)

Thevalueofthisresultisonly pedagogical.Nothingneartheperfectuniver-

salvacuum here assum ed existseitherin the solarsystem orin the galaxy.

W eshould goasfaroutasintergalacticspaceto�nd am edium oflow enough

density thatthecosm ologicalconstantcan exertitsin uenceatall.Thisis

a distance ofm illions or tens ofm illions oflight-years| four to �ve orders

ofm agnitude greaterthan the m ere375 light-yearsjustobtained.

Leavingclassicaltheory behind,wecan now consider�and �nd an exact

solution to the generalized Einstein �eld equation G�� = 0,which,outside

an ordinary m aterialdistribution,wesaw reducestoEq.(10.191).Following

essentially thesam em ethod asfortheSchwarzschild solution with theadded

extra sourceterm s,41 weget

e� = e

�� =

�

1�C

x1+1

3�(x 1)2

�

: (10.198)

To consider a cosm os everywhere void ofm atter,we set C = 0,since the

origin ofthisconstantisa localized m atterdistribution.Asa result,weget

ds2 = �

( �

1+1

3�(x 1)2

��1

(dx1)2 + (x1)2[(dx2)2 + sin2(x2)(dx3)2]�

�

1+1

3�(x 1)2

�

(dx4)2

)

: (10.199)

Thissolution wasfound by W illem de Sitterin 1917. For� < 0,the value

x1H =p� 3=� characterizestheradiusoftheeventhorizon ofthespacetim e

in question,beyond which no inform ation can reach an observer.

41To solve the resulting di�erentialequation by integration,rewrite (1+ x

1�0)e

�asthe

derivative ofa product.


W hen allissaid and done,whatareweto m akeofthecosm ologicalcon-

stant? W hatphysicalm eaning areweto attach to it? W hen thegeneralized

�eld equation is written in the form R �� = �g�� ,one gets the im pression

thatthe cosm ologicalconstantisa m aterialsource ofspacetim e curvature.

Butsince itcertainly isnotrelated to m atter,we appearto be confronted

by a m ysteriouskind ofstu�thatisall-pervading and whoseorigin weknow

nothing of, but which has active physicale�ects| a dem iether. Evoking

this sense ofwonder,the presum ed physicalstu� behind the cosm ological

constanthasbeen nam ed \dark energy." Butisin question herea physical

m ystery or only a m ysterious physicalinterpretation? The power ofpsy-

chologicalinducem entperhapsdeservesm oreattention thatisusually paid

to it.Ifthe generalized �eld equation isinstead sim ply written in theform

R �� g�� = 0,the im pression vanishes,and one is free to see the cos-

m ologicalconstantsim ply asan extra term ofspacetim e curvature possibly

relevantin a cosm ic near-vacuum .

In the context ofpresent theories,consideration ofa negative cosm o-

logicalconstant is deem ed justi�ed by the observed cosm ologicalredshift

ofdistantgalaxies. Ifthisinterpretation ofthe cosm ologicalredshiftstood

the testoftim e in the lightofhypotheticalbettercosm ologicaltheoriesto

com e,and theneed fora cosm ologicalconstantbecam ethusbettersecured,

we could then,in the lightofthe previousparagraph,regard itasnothing

m ore than a new fundam entalconstant related to the curvature ofspace-

tim e. Ifwe do notworry and try to explain the origin ofthe gravitational

constant G ,the speed oflight c,and Planck’s constant h,why should we

any m oreworry abouttheorigin ofthecurvature constant �?

10.4.4 T he dust-�lled cosm os

The next and �nal step is to generalize the �eld equation for the case

ofa non-em pty cosm os. Now,instead ofthe hom ogeneous �eld equation

G �� (x)= 0 in vacuum ,weexpectan inhom ogeneousequation with a source

term characterizing the m atter distribution in analogy with the classical

Poisson equation

r 2�= 4�G �: (10.200)

The source term isusually expressed as�T�� ,where � isa constant. Fur-

ther,because G �� is sym m etric and divergence-free (a choice based on a

m athem aticalproperty so farwithoutphysicalrelevance),T�� should beso

too.

A sim ple case isthatofa cosm os�lled with m atterresem bling an ideal

uid whosepressureisnull(orapproxim ately null)and itsconstituentpar-

ticles are free,interacting with each other only gravitationally. This kind

of uid is usually called dust. W hen we identify the uid particles with

galaxies, we obtain a usefulapplication ofthis idea for the study ofthe

cosm os as a whole. Since we are dealing with a continuous m aterialdis-

tribution,instead ofconsidering itsfour-m om entum P �(x)= m V �(x),itis

m oreappropriateto characterize this uid by m eansofitsinvariantm atter


density �(x)and four-velocity V�(x).Togetherthey form a�eld �(x)V �(x),

called thefour-m om entum -density �eld,thenaturalcounterpartofthefour-

m om entum �eld m V �(x).

However,�V� isnotyeta viablechoiceforthesourceterm T�� weseek,

because this should be a second-order tensor. Since we are looking for a

sym m etric second-order tensor and the only �elds characterizing the ideal

pressureless uid areitsdensity and itsfour-velocity,wedonotseem tohave

any otheroptionsbutto proposethat

T�� = �V

�V� (10.201)

is the source term we are looking for. Ifit is a bad choice,it should next

lead usto physically unpleasantresults.

Because T�� should bedivergence-free,wehave

T��

;�= (�V�);�V

� + �V�V�;� = 0: (10.202)

M ultiplying by V� on both sides,we get

(�V�);�V�V� + �V

�V�V

�;� = 0: (10.203)

The second term is null42 and,because V �V� = � 1,the equation reduces

to

(�V�);� = 0: (10.204)

Thisisa conservation equation forthefour-m om entum density,and itisthe

general-relativistic generalization ofthe classicalcontinuity equation

~r � (�~v)+@�

@t= 0; (10.205)

expressing the localconservation ofm atter. This is a physically pleasant

result.Furtherm ore,itjusti�esthechoice ofG �� overR �� from a physical

pointofview.

Returning to Eq.(10.202),since the �rstterm isnull,itfollowsthatso

m ustbethesecond term .Becauseneither� norV� arein generalnull,then

V�;� = 0: (10.206)

In words, the free particles com posing the idealpressureless uid follow

spacetim egeodesics.And thisisasitshould be| anotherphysicallypleasant

result.O urpresum ption abouttheform ofT�� isnow con�rm ed.

In thisideal- uid picture,the�eld equation isthen

R��

1

2g��R = ��V

�V�: (10.207)

42W riting V� V

�;� = (V� V

�);� � V

�V�;� ,we �nd that the �rst term is nullbecause

V�V� = � 1 and thatthe second is� g� V V�;� = � V V

;�.Therefore,V� V

�;� = 0.


Contracting the indiceswith g�� ,we �nd thatR = ��. The �eld equation

isthussim pli�ed to

R�� = ��

�

V�V� +

1

2g��

�

: (10.208)

To determ ine� weresortagain to thecorrespondencewith Newtonian the-

ory.The44-com ponentoftheRiccitensorgives

� r 2�c= R

44 = ��

�

(V 4)2 +1

2g44

�

: (10.209)

To the lowestnon-trivialorderin ��,(V4)2 � 1 and g44 � � 1,and so

r 2�= �1

2��: (10.210)

Com parison with Poisson’s equation r 2� = 4�G � leads �nally to the re-

sult43

� = � 8�G : (10.211)

In the pressureless- uid picture here considered,the �eld equation �nally

becom es

G�� = � 8�G �V�V �

: (10.212)

In hisoriginalpaper,Einstein (1952a)holdsthatthegeneralexpression

ofthisequation,nam ely,

G�� = � 8�G T�� (10.213)

is indeed the �eld equation describing \the in uence ofthe gravitational

�eld on allprocesses,withoutourhaving to introduce any new hypothesis

whatever" (pp.151{152). This is to say that the equation describes the

gravitationalinteraction ofall�eldsrelated tom atterand radiation (velocity

�eld,m ass-density �eld,pressure �eld,electrom agnetic �eld,etc.),so long

as they are described by a m atter-radiation tensor T�� that isdivergence-

free.Finally,when we considerthe cosm ologicalconstantaswell,the m ost

general�eld equation forgravitationalinteractionsis

G�� = � 8�G T�� : (10.214)

The curvature ofspacetim e becom esin thisway connected to the distribu-

tion ofm atterand radiation in it.

10.4.5 T he problem ofcosm ology

\The problem ofcosm ology," said Einstein (1982)in hisK yoto lecture,\is

related to the geom etry ofthe universe and to tim e" (p.47). W e �nish

thischapterwith a briefanalysisoftheinsightcontained in thisdeceptively

sim plestatem ent.

43In SIunits,the constantofproportionality is� = � 8�G =c4.


In orderforthe study ofthe cosm osasa whole to be practically possi-

ble,standard cosm ology isforced to m ake severalassum ptions.Firstly,the

assum ption ism adethatgalaxies,conceived asm aterialparticles,arein free

fall,interacting with each other only gravitationally (cf.previous section).

Nextfollowsa m uch lessinnocuoussupposition,according to which cosm ic

spaceislocally isotropic,i.e.looksthesam ein alldirectionsfrom anywhere

atany one tim e. Againstthissupposition,itm ightbe argued thatan ap-

proxim ation in which thecosm osism aterially sm ooth istoounrealistic,and

that,m oreover,the cosm osm ay only look approxim ately sm ooth from our

own vantage point,which isboth severely restricted spatially to the M ilky

W ay and tem porally to a few thousand yearsofastronom icalobservations.

M artin (1996),forexam ple,readily adm itsthat\we have no way oftelling

thattheUniverseisatallhom ogeneousand isotropicwhen seen from other

placesoratothertim es" (p.143).

Indeed, all this and m ore could be argued against the cosm ological

prem ise of localisotropy. But it is the unspoken presum ption lying be-

neath thiscosm ologicalprem isethatm ustberegarded with a m uch higher

degree ofsuspicion. For what is \cosm ic space at one tim e" supposed to

m ean in generalrelativity? How shallweseparatethewholeuniversalfour-

dim ensionalspacetim einto universalspacesata given tim e,when thiscan-

not even be done for sm aller spacetim e regions? W hat kind oftim e is it

thattheexistence ofcosm ic space im plies?

In cosm ology,theseparation ofspacetim einto a spatialpartand a tem -

poralpartistranslated into geom etric language by expressing the line ele-

m entin theform

ds2 = �

h

gab(x)dxadxb� (dx4)2

i

: (10.215)

O ncethissplitism ade,the localisotropy ofcosm ic spaceisexpressed geo-

m etrically by recasting the(non-invariant)spatialpartQ s = :gab(x)dxadxb

ofthisequation in sphericalcoordinatesas

Q s = eA (x4;x1)

�(dx1)2 + (x1)2(dx2)2 + (x1)2sin2(x2)(dx3)2

�: (10.216)

How are the coordinates interpreted in this space-tim e setting? Coor-

dinate x4 istaken to represent\cosm ic tim e" and denoted x4 = c�. \Cos-

m ic tim e" � works as a \universaltim e," equalfor all\cosm ic observers"

travelling along galactic worldlines. This m eans that \cosm ic observers"

throughoutthecosm osm easure thesam e absolute \cosm ic separation"

ds2 = c2d�2: (10.217)

Cosm ic space is then nothing but the three-dim ensionalhypersurfaces or-

thogonal to the galactic worldlines at � = �now ; i.e.cosm ic space is an

instantaneousslice ofspacetim e atabsolute \cosm ic tim e" �now .Butisthe

assum ption of\cosm ic tim e" realistic in a general-relativistic world?

As for x1,it is interpreted as the radialdistance to an event from a

cosm ic observer and denoted x1 = �. The supposition is then m ade that


thefunction A(�;�)can beseparated into a partB (�)connected only with

\cosm ic tim e" and to be related to the expansion ofcosm ic space,and a

partC (�)connected only with m atterand to berelated to thecurvatureof

cosm ic space,i.e.

A(�;�)= B (�)+ C (�): (10.218)

Since the cosm osisspatially hom ogeneousata given \cosm ic instant," the

spatialpartofthe tensordescribing itsm atterand radiation contentm ust

beofthe form Tab= constant�a

b.

O n allthese assum ptions and with suitable integration constants,the

�eld equation leadsto the solution

ds2 = �

"

eB (�)

�

1+k�2

4�20

� �2

(d�2 + �2d�2 + �

2sin2(�)d�2)� c2d�2

#

;

(10.219)

wherek = 0;1 or� 1.An equivalentsolution wasfound by Howard Robert-

son in 1935and by ArthurW alkerin 1936,and itisthegeom etricfoundation

ofrelativisticcosm ology.Itisinterpreted topicturea at,positively,orneg-

atively curved locally isotropicspace| pictured by exp[C (�)]| \expanding"

in \cosm ic tim e"| pictured by exp[B (�)].

A sim ilarinterpretation isusually m adeofthedeSittersolution (10.199)

forthecosm icvacuum wesaw earlier.By m eansofa changeofcoordinates,

(x1 = �ec�

p��=3

x4 = c� + 1

2

p� 3=�ln

h

1+ (�=3)�2e2c�p��=3

i ; (10.220)

thedeSitterlineelem entcan beexpressed asa spatially isotropicpartthat,

for� < 0,\expands" in \cosm ic tim e" �:

ds2 = �

n

e2c�p��=3 [d�2 + �

2d�2 + �2sin2(�)d�2]� c

2d�2o

: (10.221)

Thispictures,as it were,an expanding em pty cosm os or \m otion without

m atter." Itstandsin contrastto Einstein’sproposed solution ofa m atter-

�lled static universe or\m atterwithoutm otion."

W e can now see thatwhen Einstein deem ed the problem ofcosm ology

to be tied to the geom etry ofthe universe (i.e.ofspace) and to tim e,he

was dead right. M oreover,realizing now thatthe form erproblem depends

on thelatter,werecognize thattheproblem ofcosm ology really boilsdown

to itspresum ptionsaboutthe nature oftim e.

Theproblem ofspacereferstothequestion ofthespatialcurvatureofthe

locally isotropiccosm os.But,aswejustsaw,in ordertobeabletoeven talk

aboutspatialproperties,spacetim e should be separable into a spatialpart

gab(x)dxadxb and a tem poralpartg44(x

1;x2;x3)(dx4)2.Thisisfeasible,for

exam ple,for atspacetim eorforthespacetim earound an isolated spherical

m ass distribution,but it is not a generalproperty ofspacetim e. In fact,

it seem s least circum spect to assum e the validity of the space-tim e split

in the cosm ologicalsetting,since we are here dealing with the totality of

spatiotem poralevents,which istherefore the m ostgeneralcase there is.


This unwarranted space-tim e split ofcosm ology signals a regress to a

Newtonian worldview: an absolute cosm ic space whose existence requires

that ofabsolute tim e. Through \cosm ic tim e," the universe becom es en-

dowed with an absolutepresent(absolutespace)and a history.Thevalidity

of the space-tim e split in cosm ology, however, runs counter to the basic

lessonsofgeneralrelativity,and itisa presum ption whosetruth can only be

hoped forand guessed at.The psychologically suggestive notation x4 = c�

doesnotby itselfturn � into a physically m eaningfulabsolute tim e.

W hatm otivates thiscosm ologicalregressto the classicalview oftim e?

Ultim ately,only ourhum an questforunderstanding. The problem ofcos-

m ology liesessentially in thesuppositionsitm ustm akeaboutthenatureof

tim e in orderto create a theoreticalpicture ofsom ething asvast,com plex,

and rem ote as the cosm os sim ple enough to be hum anly m anageable and

understandable. It lies in the reversed logic ofa situation in which we �t

the great wild cosm os to our existing,fam iliar tools ofthought instead of

tryingtodevelop new toolsofthoughtthatsuitand dojusticetothecosm os.

It seem s ironic| although on second thoughts only natural| that the

m ore we have learnt,the deeper and farther we have probed the natural

world in pursuitofthe disclosure ofone m ore ofits secrets,the closer we

have com e to experience the de�ciencies ofour own cognitive tools. As if

the deeper we sink in an oceanic expedition,the m ore the dim m ing light

and building pressure interfere with our intended objective study of the

ocean’s fauna| until, in the utm ost darkness of the inhospitable depths,

we becom e oblivious to allfauna,only rem aining aware ofthe feeble light

ourtorch dispensesand the low oxygen supply in ourtank.Dim m ing light

leadsto virtualblindnessand oxygen de�ciency to brain dam age.The�rst

sym ptom softheseconditionsareeasy foradivertoself-diagnose.Thesam e,

regrettably,cannotbesaid oftheircounterpartsin thehum an expedition to

theintellectually inhospitabledepthsofthe cosm os.

In an often-quoted passage,Eddington captured,with rare insight,this

hum an dilem m a. Concluding his book on spacetim e and gravitation, he

wrote:

[W ]e have found thatwherescience hasprogressed thefarthest,

them ind hasbutregained from naturethatwhich them ind has

putinto nature.

W e have found a strange foot-print on the shores of the un-

known. W e have devised profound theories,one after another,

to account for its origin. At last,we have succeeded in recon-

structing the creature that m ade the foot-print. And Lo! it is

ourown.(Eddington,1920,pp.200{201)

The geom etric analysis of clock tim e, leading us to the classicaland

relativistic worldviews,isnow com plete.

W hatnext?

C hapter 11

Interlude: Shallw e

quantize tim e?

A picture held us captive. And we could notgetoutside it,

for itlay in our language and language seem ed to repeatit

to usinexorably.

Ludwig W ittgenstein,Philosophicalinvestigations

Them ain lesson to bedrawn from thepreviouschapteristheidenti�ca-

tion ofclock tim easthekey physicalconceptin relativity theory.O urgoal

being a \quantum theory ofgravity," thisseem san excellentopportunity to

�nd it. Having uncovered the confusion thatholdsthe m etric �eld g�� (x)

asthephysicalessenceofrelativity,and having identi�ed theworldlinesep-

aration ds in itsstead astherightfulholderofthisplace,whatelseisthere

leftforusto do than quantizenotg�� (x)butds? W hatelseisthereleftfor

usto do than quantize tim e?

Rightattheoutset,wecan considertwo alternatives.Ifwebelievethat

thesm ooth background ofRiem ann-Einstein spacetim eisasuitablestarting

point,we m ay quantize ds and introduce a quantum �eld ofthe form cds.

However,ifforsom e reason we preferto give up the com fortofdi�erential

geom etry and return to �niteEuclidean geom etry,wem ay quantize�sand

introduce a quantum �eld ofthe form c�s. In the following, we use the

di�erentialversion cds;which version we choose,however,is irrelevant to

ourulterioraim .

Is cds an observable? W e know thatclock readings ds are positive real

num bers;they are the results ofthe experim ents m ade with clocks as we

m easure the intervalbetween two events,A and B . It is then naturalto

assum ethatdsaretheeigenvaluesofcds.O n thiscondition,cdsisindeed an

observable,i.e.Herm itian.In e�ect,introducingaHilbertspaceofquantum -

m echanicaleigenvectorsj ii,we have that

(cdsj ii = dsj iij ii

h ijcdsy

= dsj iih ij: (11.1)

185

The second equation is obtained as the dualcorrespondent ofthe �rston

accountofds� = ds. M ultiplying the �rstequation by h ijon the leftand

thesecond by j iion the right,we �nd

h ijcdsj ii= h ijcdsyj ii: (11.2)

Sincej iiisan arbitrary eigenvectorofcds,itfollowsthatcds isHerm itian,

cds= cdsy: (11.3)

Fora generalstate vectorj i,wealso �nd thattheexpression h jcdsj i

can beattributed a specialm eaning.Rewriting itin term softheorthonor-

m aleigenstatesj iiofcds,we�nd

h jcdsj i=X

i;j

h j iih ijcdsj jih jj i=X

i

jh ij ij2dsj ii; (11.4)

which suggests that jh ij ij2 can be interpreted as the probability that,

upon m easuring with a clock the spacetim e intervalbetween events A and

B ,thevaluedsj iiwillbeobtained.Them easurem entofthisvalueis,atthe

sam e tim e,linked with an underlying quantum -m echanicaltransition from

spacetim e state j ito spacetim e state j ii. Thus,h jcdsj i is the average

obtained after a series ofm easurem ents ofthe separation operator cds in

state j i;in otherwords,theexpected value.

These ideas could be developed further,buteven at this early stage it

isappropriate to stop,think,and ask: is thisthe physicalfoundation ofa

quantum theory ofgravity? O rperhapsthefoundation ofa quantum theory

oftim e? Hardly.Atbest,theaboveonly teachesusa lesson abouthow one

should notgo aboutdoing new physics,and abouthow easy itisto beheld

captive ofthe conventionalpictures ofa preestablished physicaltradition.

Letussee why.

The only positive thing to be said aboutthe quantization ofthe space-

tim e intervalis that ds is indeed a realnum ber resulting from a sim ple

physicalm easurem entthatweknow how to perform :stand ateventA,take

a clock,read itsdial,record the displayed num ber,travelto eventB ,read

the dial,and record the displayed num ber. Butassoon aswe enquire into

the operationalm eaning ofthe quantum -theoreticalfram ework introduced

around ds,weare in deep trouble.

Essentialtothequantum -theoreticalpicturedeveloped isthestatevector

j i. W hen we introduced it,we did so non-com m ittally;when we lateron

referred to it,we called it a quantum -m echanicalspacetim e state. Thisis

indeed what j i m ust denote in theory,because the intervals ds between

spacetim e events are determ ined by the geom etric structure ofspacetim e,

but is there in practice a denotation behind the words \spacetim e state"

and itsm athem aticalsym bol\j i"?

Talk about the expected value h jcdsj i ofobservable cds in state j i

im pliesthatwecan m akea(largeenough)setofm easurem entsofcdsstarting

186 Interlude: Shallw e quantize tim e?

each tim ewith asystem in thesam e\spacetim estate"j i,and then proceed

to calculate the average ofthe results. Butwhatis here the system to be

prepared into state j i? The quantum -m echanicalsystem m ust surely be

spacetim e,buthow arewesupposed to prepareitinto state j i? W hatare

the hum anly feasible operationsinvolved? W hatisa spacetim e state,after

all? How do we gain accessto it,a�ectit,m odify it?

The sam e problem a�ectsthe conceptofprobability.W hatoperational

m eaning do we attribute to the probability ofobtaining value dsj ii upon

a m easurem entofds when spacetim e isin state j i,when we do notknow

what a spacetim e state m eans physically? And,further,how plausible is

it for a clock m easurem ent to exert an in uence and m odify the state of

spacetim e,inducing thetransition ofj iinto j ii?

In this light, the quantization of the spacetim e intervalappears as a

m im icry| acaricature| ofthequantum m echanicsofm aterialsystem s.The

operationally well-determ ined notion ofquantum -m echanicalstate1 ofam a-

terialsystem (such asa particleshower)isuncritically transferred to space-

tim e,arriving attheidea ofa quantum -m echanicalstate ofspacetim e.But

justbecause we know how to prepare m aterialparticlesinto states,itdoes

notfollow by them agicofanalogy thatweknow how,oreven whatitm eans,

to prepare spacetim e into states.In the contextofthe quantum m echanics

ofm aterialsystem s,the state ofa particle isinduced to change by the di-

rectaction ofan externaldeviceupon theparticle,e.g.by them agnetic�eld

which in a Stern-G erlach devicechangesthespin stateofa particleshower.

Ifthiswereto translateto thequantum m echanicsofspacetim e,them aking

ofaclock m easurem entshould havethecapacity todirectly m odify thestate

ofspacetim e.Unlikely? Far-fetched?

Thesam ecriticism appliesto theuncriticalim portation ofotheropera-

tionally well-determ ined key conceptsofthequantum m echanicsofm aterial

particles,such as expected values and probabilities,that rely on the idea

ofan accessible,m odi�able,and reproducible quantum -m echanicalstateofa

system to have physicalm eaning.

Som e ofthese and othersim ilarproblem sbesetan attem ptalong these

lines carried out by Isham ,K ubyshin,and Renteln (1991). A noteworthy

rem ark by these authorsdeservescom m ent. Regarding how theirproposal

should beinterpreted,they wrote:

Since the m etric [distance function d(x;y); cf. �s]...is non-

negative it is not clear how the pseudo-Riem annian structure

ofspacetim e ofgeneralrelativity and the notion oftim e could

em erge in quantum m etric topology. Thusthe approach can be

considered eitherasa generalization oftheEuclidean version of

gravity,orasa description ofthespatialpartofphysicalspace-

tim e. In the latter case tim e rem ains an additionalparam eter

labelling the m etric in the quantum m etric topology approach.

(Isham etal.,1991,p.163)

1Butnotstate vector;see nextchapter.

187

Firstofall,there is,in fact,no problem here| only a long-standing confu-

sion.Aswesaw in thepreviouschapter,thenon-negativity of�sisperfectly

com patiblewith thepositive,negative,ornullquadraticform g�� (x)dx�dx�,

becausetheirconnection takesplace via the � indicator,

�s=

Z sB

sA

r

�g�� (x)dx�

ds

dx�

dsds; (11.5)

where� = � 1.Fortim elikedisplacem ents,� = � 1 and �srem ainspositive,

as it should. But what is here rem arkable is not the confusion| which is

com m on| butthe attitude taken in the face ofseem ing di�culties. Ifthe

fram ework chosen willnotdescribe spacetim e and gravitation asintended,

then itm ightaswelldescribesom ething else.In the face ofadversity,even

the essentialsacri�ce is m ade to leave the description oftim e out ofthe

originally intended quantum -theoreticalpicture ofspacetim e,so asto thus

rescueand continueto cling to an idea thatstarted o� with thewrong foot.

Thisisonly an allegoricalexam pleofthewidespread workingphilosophy

offrontiertheoreticalphysics.Ifthem athem aticswecom eup with willnot

conform to the physicalworld,then the physicalworld had betterconform

to them athem aticswecom eup with.Isthisthepath ofphysicaldiscovery?

Isthis the road to quantum gravity?

W hat spacetim e m ay be quantum -m echanically| ifanything at allbe-

neath the classicalgeom etric structure so faruncovered| we do notknow.

Butourignorancein thisregard should notjustifythedesperateinvention of

quantum -theoreticalfram eworkswithouta shred ofphysicalm eaning.O ne

m ay play with thetransm uted m athem aticalsym bolsborrowed from a suc-

cessfulphysicaltradition,butto presum e thatthis,too,isphysicsbecause

ofm ere resem blance isto presum etoo m uch.

Physicsm ustbe done from the bottom up,building upon nature’sraw

m aterial,explicitly,in term softhingsthathum an beingshaveaccessto and

can a�ect.In thecontextofthequantum m echanicsofparticles,weturn to

such thingsnext.

C hapter 12

T he m etageom etric nature of

quantum -m echanicalthings

and the rise oftim e

Groping, uncertain, I at last found m y identity, and after

seeing m y thoughts and feelings repeated in others,Igradu-

ally constructed m y world ofm en and God. As I read and

study,I�nd thatthis is whatthe restofthe race has done.

M an lookswithin him selfand in tim e �ndsthe m easure and

the m eaning ofthe universe.

Helen K eller,The world Ilive in

Passagesfrom fam ousphysicistsaboutthedi�cultiesinvolved in under-

standing quantum m echanicsabound.Speaking ofit,Erwin Schr�odingeris

reported to have said,\Ido notlike it,and Iam sorry Ieverhad anything

to do with it." Niels Bohr,for his part,stated that \those who are not

shocked when they �rst com e across quantum m echanics cannot possibly

have understood it." M urray G ell-M ann described quantum m echanics as

\thatm ysterious,confusing discipline which none ofusreally understands

butwhich we know how to use." And Richard Feynm an decreed,\Ithink

itissafe to say thatno oneunderstandsquantum m echanics."

Although theseviewsweresom etim esexpressed only lightheartedly,one

can yet grasp the gist ofthese physicists’com plaints| and the seriousness

thereof.Itisnotonly thatquantum -theoreticaldescriptionscannotbem e-

chanically visualized;m ore worrying isthe fact thatthe foundationalcon-

ceptsofquantum theoryhavephysicallyunexplained origins.In otherwords,

it is not justthe m echanical-how but,m ore im portantly,the physical-why

ofquantum m echanics that rem ains elusive. O n the other hand,the the-

ory a�ords us a geom etric visualization of the processes it describes; its

geom etric-how is readily available| but m ay this be m ore ofa hindrance

than a help?

Am ong quantum theory’slong-standing enigm asarethefollowing:what

is the physicalm eaning ofthe arrow state vector j i? W hy are vectors

189

|pi〉 |ui〉

|uj〉 |ψ〉

〈pi|pi〉

Figure 12.1: Q uantum -m echanicalprobability asthe squared length ofthe pro-

jection jpiiofj ion juii.

needed atallto form ulate quantum m echanics? How and why do they col-

lapsewhen am easurem entorstatepreparation occurs? W hy isitpossibleto

picturesuch a collapse through a projection by m eansofan inner product?

W hy isthe probability ofan outcom e the squared length jhuij ij2 = hpijpii

ofthecom plex-valued projection vector(cf.collapse)jpii= huij ijuiiofan

initialstate vector j i onto another juii(Figure 12.1)? W hy do probabil-

ities arise geom etrically from Pythagoras’theorem ? W hat is the physical

m eaning ofcom plex num bersand com plex conjugation? W hy is it always

possibleto �nd Herm itian operators(with realeigenvalues)forany observ-

able? And why isitalwayspossibleto attach orthogonaleigenvectorsto an

observable?

These enigm as, som e m ore pressing and disquieting than others, are

directly related to thepostulatesofquantum theory;and allofthem are,in

turn,founded on the geom etric rendering ofthe theory (cf.italized words

above),which weanalysed in Section 4.2.Furtherthought,however,reveals

that it is precisely in these unquestioned (and unquestionable) geom etry-

ridden postulates that our problem s com prehending the physicalroots of

quantum m echanicslie.

Indeed,geom etry resem bles a double-edge sword. W hereas it greatly

aidsat�rstourcom prehension ofthe world,iteventually bringsup prob-

lem sofitsown,i.e.problem sstem m ing from theparticularway in which it

forcesusto view thestatesofa�airsitdescribes.O n theonehand,itallows

fora geom etrically intuitive form ulation ofquantum theory thatis consis-

tent with observations but, on the other, its very geom etric form ulation

turnsoutto bethebearerofitsleastunderstood concepts.In otherwords,

although we only seem to be able to form ulate and understand quantum

m echanicsgeom etrically,itisprecisely itsgeom etric notionswhosephysical

underpinningsare elusive.

The need to understand the physicalorigin ofthe basic,yetcontrover-

sial,geom etricconceptsofquantum m echanics,and therebyclarifytheabove

enigm as,suggestsitselfastheim m ediatem otivation to go beyond geom etry

in the form ulation ofthistheory.Butthe m otivation ofthe m etageom etric

190 T he m etageom etric nature ofquantum -m echanicalthings

analysis that follows does not end at this,as it also has a less im m ediate

goal.Thisistolay thefoundation foram etageom etricquantum -m echanical

pictureoftim e,which weintend toconnectwith theclock-based form ulation

ofgeneralrelativity ofChapter10.Toachievethisgoal,wewillseek inspira-

tion forthepresentanalysisofquantum m echanicsin sofarunaccounted-for

aspectsoftim e asa consciousexperience.

12.1 Q uasigeom etry and m etageom etry

As explained in Chapter 3,we equate geom etry,not with geom etric exis-

tence,butwith geom etric thoughtand language.Neithergeom etricobjects

(shape) nor geom etric m agnitudes (size) are present in nature,but are a

conceptualtoolby m eansofwhich hum anspicture nature.In thischapter,

we introduce the notion ofquasigeom etry and,by exam ple,elucidate the

m eaning ofthe anticipated notion ofm etageom etry.

By quasigeom etry we refer to a realm between m etageom etry and ge-

om etry thathelpsbridgethegap between thesetwo in connection with the

foundations ofquantum theory. The word \quasigeom etry"1 m eans \hav-

ing som e resem blance to geom etry butnotbeing really geom etric," and is

attributed to this realm due to its sim ilarity and connection with geom e-

try. W e shall�nd the m ain quasigeom etric entities to be the fam iliar real

and com plex num bers. How they appearand help bridge the gap between

m etageom etric quantum -m echanicalthings and the usualgeom etric repre-

sentationswillbeone oftheissuesofthischapter.

Realnum berscan bethoughtasextended arrowslaid along a graduated

one-dim ensionalline. In spite ofthis,they are neither genuine geom etric

objectsnorm agnitudesforthe elem entary reason thatthey are them selves

the very carriers ofgeom etric size. Aswe saw in Chapter3,realnum bers

act,in a geom etric setting,as the go-betweens between geom etric quality

and geom etric quantity.In otherwords,a realnum berisnotsize butgives

size.Ifthoughtofasarrows,they can,in particular,give size to them selves

(thatwhich they inheritfrom theirm odulijaj),in which case they becom e

degenerate2 geom etric concepts.Due to theirdegenerate geom etric nature,

we give thenam e quasigeom etry to therealm they characterize.

Com plexnum bers,too,can bethoughtofasarrowswith ade�nitelength

only now laid on the (Argand) plane. In this case,however,the situation

is di�erent. Now itis the realm odulusjcjofa com plex num berthatrep-

resents its length,while the com plex num ber c itselfcan be viewed as a

non-degenerate geom etric object (arrow). For this reason,ifwe want to

usecom plex num bersasquasigeom etric concepts| conceptstruly devoid of

1Q uasi,from Latin quam \asm uch as" + si\if":asif.

2D egenerate,from Latin degenerare,de-\down from ,o�" + genus \birth,race,kind":

to depart from one’s kind, fall from ancestral quality (O nline Etym ology D ictionary,

http://www.etym online.com );having declined or becom e less specialized from an ances-

tralor form er state;having sunk to a condition below that which is norm alto a type

(M erriam -W ebsterO nline D ictionary,http://www.m -w.com ).

12.1 Q uasigeom etry and m etageom etry 191ImRe

c

a−a︸︷︷︸

a︸︷︷︸

a

︸

︷︷

︸

|c|

Figure 12.2: G eom etric visualization ofrealand com plex num bers. W hen real

num bersa,originallythecarriersofsize,arethoughtofasarrowsoflength jaj,they

becom e degenerate geom etric concepts. Com plex num bers c,on the other hand,

havea genuinegeom etricinterpretation asarrowsoflength jcj,butto conserveour

chancesofa deeperunderstanding ofquantum ontology,itisessentialto stay clear

ofthisgeom etrization ofthe com plex num ber.

geom etric m eaning| we m ustbe carefulneverto regard them asextended

arrows. W e m ust eschew allgeom etric visualization ofcom plex num bers

and,instead,seek their physicalm eaning other than as m erely conducive

to probability as the size (jcj2) ofa geom etric object (arrow c): we m ust

strive to �nd the physicalm eaning ofc itself,which doesnotassuch refer

to the shape or length ofanything. To conserve our chances ofa deeper

understanding ofquantum ontology,itis essentialto stay clear ofthis ge-

om etrization ofthe com plex num ber.Asweprogressin thischapter,so far

the only known role ofcom plex num bersasdescriptive ofsom ething phys-

icalwillbe uncovered,nam ely,the characterization ofthe physicalprocess

ofa directionalquantum -m echanicaltransition.Thiswillgiveusa di�erent

reason to callcom plex num bersquasigeom etric (Figure 12.2).

By m etageom etry3 wem ean \beyond geom etry," and thereforereferto a

theoreticalpicture| ahum an scienti�clanguage| thatdoesnotusegeom et-

ric conceptsofany sort. Thisisthe sense in which one should understand

thereferenceto the\m etageom etric natureofquantum -m echanicalthings"

in the title ofthis chapter. That is,not as hinting at the m etageom etric

essenceofthephysicalworld,asifto say that\G od isa m etageom etrician"

in neo-Platonic fashion,orthat\the Book ofNature iswritten in the lan-

3M eta,from G reek preposition m eta:in them idstof,am ong,with,after;from Proto-

Indo-European *m e-: in the m iddle. The original sense of the G reek word is \after"

ratherthan \beyond," butitwasm isinterpreted when Aristotle’streatiseswerediscussed

by Latin writers.From the reference below,we learn that,originally,m etaphysika m eant

\ ‘the (works)afterthe Physics’[in the]title ofthe 13 treatiseswhich traditionally were

arranged after those on physics and naturalsciences in Aristotle’s writings. The nam e

wasgiven circa 70 B.C.E.by AndronicusofRhodes,and wasa referenceto thecustom ary

ordering ofthe books,butitwasm isinterpreted by Latin writersasm eaning ‘the science

ofwhatis beyond the physical.’" (See \m eta" and \m etaphysics" in O nline Etym ology

D ictionary,http://www.etym online.com )


guage ofm etageom etry" in neo-G alilean fashion,butrather m eaning that

som e aspects ofnature| quantum -m echanicalones,in this case| m ay not

be open to hum an understanding through geom etric tools ofthought. W e

seethatgeom etricm odesofthinkingand talkinghavenotonly been capable

ofleading to m yriad physicaltrium phs ranging from Newton to Einstein,

but also to possible philosophicaland physicalblind alleys; for exam ple,

spacetim eontology,quantum ontology,and thegeom etricroadsto quantum

gravity.

W e suggest that, if progress concerning the conceptual di�culties of

quantum theory wants to be m ade,the theory should be written after a

m etageom etric fashion, i.e. in term s of m etageom etric things. Now this

m ay seem like a long shot indeed. G iven that hum ans by nature think

geom etrically,and thatthusallphysicaltheoriesso farspeak the language

of geom etry, what chance is there that any m etageom etric theory could

be understood| even less developed| by us? W ould not a m etageom etric

theory be hopelessly abstruse? W ould notm etageom etric things,devoid of

allshapeand size,behopelessly unim aginable,m eaningless,and abstract?

Notso.W eshallseethatm etageom etric thingscan befound such that,

farfrom abstruse,areactually based on physicalexperim entsand arerather

close to hum an experience. Indeed,we shall�nd that,besides the well-

established,unquestionably geom etric aspects ofour practicalthinking,a

deeper realm ofhum an conscious experience| closely connected with the

feeling oftim e| isnaturally captured m etageom etrically.By identifying its

constituentsand theirm utualrelations,and by havingthesebeinspirational

sourcesoftheexperim ent-based physicalthingsofam etageom etricquantum

theory,wewilllay thefoundation fora theory thatistransparentto hum an

cognition,isopen to hum an accessand control,and providesa setting fora

new understanding oftim e.

12.2 Towardsm etageom etricquantum -m echanical

things

O ur search for a m etageom etric,but at the sam e tim e fam iliar and com -

prehensible,foundation ofquantum m echanics is guided by the following

working principle:

A profound and trustworthy knowledgeoftheontology ofquan-

tum m echanicscan only be obtained from the resultsofthe ac-

tivitiesofM an,from such thingswhose originsM an can under-

stand,have access to,and a�ect. O ur description ofquantum

m echanics m ustbe expressible with the help ofsim ple physical

things because,otherwise,the workings ofquantum m echanics

arenotknowableorunderstandable.Thesethingsm ustbesuch

thatwe know thephysicalm ethodsthrough which they arise.

Thism eansthatwedo notstartthesearch form etageom etricquantum -

m echanicalthings in an im plicit way,e.g.with wave functions as in wave

12.2 Tow ards m etageom etric quantum -m echanicalthings 193

m echanics, with canonical m atrix pairs as in m atrix m echanics, or with

state vectors as in Dirac’s algebra. O n the contrary,ifquantum theory is

to be laid plain before our eyes and its enigm atic features com prehended,

we should startthe m etageom etric search explicitly,with conceptsclose to

both hum an psychologicaland physicalexperience.

The pursuit ofthe psychologicalconnection m eans that we would like

quantum -m echanicalthingsto beclosely connected with therealm ofexpe-

rienced thoughtsand feelings.Thisisnotto say thattheintention hereisto

develop a new theory ofconsciousness;thisrequirem entratheraim sto start

with a setting closely connected with the raw feeling oftim e,in which the

presentm om entisparam ount,becausein thisway thequantum -m echanical

theory willa�ord the possibility to picture tim e in a physically new way.

In turn,the pursuitofthe physicalconnection m eansthatwe require that

quantum -m echanicalthingsbebased upon an experim entally explicitbasis,

becausethisistheonly way tom akesurethattheelem entalontologicalcon-

stituents ofthe resulting theory willbe observationally concrete,i.e.wide

open to hum an m anipulation and control.

The discovery,within a m etageom etric theoreticalsetting,ofquantum -

m echanicalthingsthatare related to consciousnessand are experim entally

explicit is the task of what m ay be called psychophysics. In the rest of

thissection,wepursuethepath ofpsychophysicsalong two com plem entary

directions. The �rst involves the investigation ofpossible phenom enologi-

calsim ilarities between conscious and quantum -m echanicalphenom ena in

search for an experim entallink between psychology and physics,whereas

the second involves the investigation ofthe m etageom etric quality ofcon-

sciousphenom ena and theirrelationsin search foran extra m etageom etric

link between psychology and physics.In Section 12.3,these two pathscon-

verge on the sought-for m etageom etric psychophysicalthing,4 on which to

build thefoundationsofan experim entally overtm etageom etricform ulation

ofquantum theory thatcan throw new lighton quantum ontology and the

notion oftim e.

12.2.1 Experim entalfacet ofpsychophysics

Do quantum -m echanicalphenom ena shareany sim ilaritieswith phenom ena

related toconsciousnessthatm ayserveasinspiration in thisphysicalsearch?

4In hisbook,Isham (1995,pp.63{67)pondersthequestion,\W hatisa thing?" from a

perspectivethattakesintoaccountontology and epistem ology| theexistenceofthingsand

ourknowledge ofthings| and the degree to which ontology m ay be the projection ofour

own epistem ologicalshadows. For us,a thing willbe essentially a sim ple psychological

abstraction ofa physicalexperim ent; insofar as this is so,the ontology ofthings is as

uncontroversial(orcontroversial,accordingtophilosophicaltaste)asthephysicalexistence

ofourm indsand theexperim entitself,whiletheepistem ology ofthingsbecom escon ated

with theirontology,forourknowledgeofthingsistantam ountto our�rst-hand experience

ofthingsthem selves.Asforthem etageom etriclanguagein which weshallpicturephysical

things,into itgo,naturally and unavoidably,ourm ethodologicalbiasesand prejudices.


Dennett’s(1991)5 theory ofconsciousness,with itsunorthodox approach to

consciousphenom ena,can help usgive a positive answerto thisquestion.

A paradigm aticexperim entthatisaptto revealseveralpeculiarfeatures

ofhum an consciousnessisthecolour-phiphenom enon studied by K olersand

von G r�unau (1976),but originally studied by W ertheim er (1912) nearly a

hundred yearsago in itssingle-colourversion.A red spotoflightis ashed

on ascreen and 200 m illisecondslateragreen oneis ashed within 4degrees

ofangle from the �rstspot. Subjectsreportnotonly thatthe originalred

spotfollowsa continuoustrajectory6 butalso thatitchangescolour(from

red to green) m idway between the endpoints ofthe continuous trajectory.

How should wem ake senseofthisphenom enologicalobservation,and what

doesitrevealaboutthe m echanism sofconsciousness?

Theexplanation thatcom esm ostreadily to m ind isthat,sincein order

forthe red spotto change colourseem ingly before the green spothaseven

been ashed,there m ust be an intrinsic delay in consciousness itselfofat

least200 m illiseconds.However,there isenough evidence to thee�ectthat

conscious responses(e.g.button pressing)to stim uliare notassluggish as

this hypothesis requires (pp.121{122,150). Ifconscious experience is not

being puttem porarily on hold (certainly notan evolutionary advantage),is

thecolour-phiphenom enon,then,an indication thatconsciouseventsviolate

causality in tim e and space? This alternative seem s even less palatable.

W e seem to be trapped,here as in frontier physics,in a deep conceptual

confusion.

To help disentangle it,Dennetturgesusto rid ourselvesofthe natural

idea that to becom e conscious ofsom ething is tantam ount to certain (e.g.

electrochem ical) processesoccurring atsom e speci�c pointin the brain;to

reject the notion that what we experience as the stream ofconsciousness

hasan exactanalogueatsom esingularlocation in thebrain.Dennettasks,

\W here doesitallcom e together?" and replies,\Nowhere...[T]here isno

placein thebrain through which allthesecausaltrainsm ustpassin orderto

deposittheircontent‘in consciousness’" (pp.134{135).O n theassum ption

ofa brain singularity| and withoutthe bene�toflong delaysorviolations

ofcausality| the colour-phiphenom enon de�esexplanation.

Itseem srem arkable that,in a �eld ofscience asrem oved from cosm ol-

ogy and spacetim e physics as can be,one should also be forced to com e

to term s with a singularity| a point at which our understanding becom es

feebleand powerless| if unableto refrain from theapplication ofgeom etric

thinking beyond its ostensible capabilities. O n second thoughts,however,

there is nothing rem arkable abouthitting upon a new singularity,as both

psychologicaland cosm ologicaltheoriesarem adeby hum answith thesam e

geom etric tools ofthought. Dennett,also noticing the rem iniscence with

singularities in physics,criticized the seem ingly naturalgeom etric m ethod

5Consult this reference (especially,pp.101{137, 209{226, 275{282) for m ore details

on the consciousness-related topicsdiscussed below.Page num bersin parenthesesin this

section and the nextreferto thisreference unlessotherwise stated.6This is the result ofW ertheim er’s experim ent with a single-colour spot and is the

working principle behind m otion pictures.


by which one arrivesatthebeliefin a singularity in thebrain:

Doesn’t it follow as a m atter of geom etric necessity that our

consciousm indsarelocated attheterm ination ofalltheinbound

processes,justbeforetheinitiation ofalltheoutbound processes

thatim plem entouractions? (Dennett,1991,p.108)

He subsequently argued thatthe geom etric force ofthiscontention ism is-

leading: inbound processesneed notend at,and outbound processes need

notleave from ,a physiologicalpoint:thebrain workslargely asa unit.

The neuronalactivity giving rise to consciousness is extended over the

brain asa whole. Dennettsuggests that,asa result,we cannotsay where

orwhen a consciousexperience occurs(p.107);thatalthough

Every event in your brain has a de�nite spatio-tem poralloca-

tion...asking \Exactly when do you becom e conscious of the

stim ulus?" assum esthatsom eoneoftheseeventsis,oram ounts

to,yourbecom ing consciousofthe stim ulus. [...] Since cogni-

tion andcontrol| and henceconsciousness| isdistributed around

in the brain,no m om ent[orplace]can countasthe precise m o-

m ent[orplace]atwhich each consciouseventhappens.(Dennett,

1991,pp.168{169)

Thus,the colour-phiphenom enon (and other tests to the sam e e�ect)

leadsto theneed to di�erentiatebetween a representation| consciousness|

and the m edium of the representation| the spatiotem poralbrain events:

thebrain doesnotrepresentspace(e.g.theim ageofa house)with spacein

the brain,and,to a lim ited extent,neitherdoesitrepresenttim e (e.g.the

sequentialm otion in the colour-phitest)with tim e in the brain.Viewed in

thisway,consciousnessisa representationalabstraction oftheinternaland

externalworld,whoseconveying m edium isthespatiotem poraleventsin the

brain.Itis,am ong otherthings,a representation of space and tim e,butit

isnotitselflocated anywherein thespaceand operatingtim eofthebrain in

any literalsense.Consciousnessis,in thissense,a spatially and tem porally

delocalized,global,and locally irreducible property ofthe brain.How com e,

then,that conscious experience resem bles so closely a linear sequence of

spatiotem poralevents?

Dennettelaboratesasfollows.Neuronswork togetherm ultifunctionally

and globally within the space and tim e of the brain to produce num er-

ous,parallel,undecanted stream sofconsciousness,which hecalls\m ultiple

drafts." It is parts ofthese drafts that can project them selves as a single

serialsequence ofnarrative. This is achieved by m eans ofprobessuch as

self-exam ination and externalquestioning,which actto decantcertain draft

contentsofconsciousnessto a di�erentlevel. These probesare asessential

to thedeterm ination ofwhatone isconsciousofasare theexternalstim uli

them selves (pp.113,135{136,169). Thisloosely suggests also an indeter-

m inistic picture ofconsciousness,since there isno single,actualstream of


consciousness unequivocally determ ined by externalstim uliindependently

oftherandom probesthatoriginate it.

According to Dennett,itisthedecanted stream ofconsciousnessthatwe

experience as a serialchain ofevents and readily associate with conscious

experience. Itisthisstream thatprovidesa subjective worldline ofevents

as reported by the conscious subject. This worldline can be subsequently

com pared with otherworldlinesofobjectiveeventsaselem entsoftheexter-

nalworld (p.136).Thecom parison oftwo such worldlinesgivesm eaning to

the idea ofthe projection ofconsciousness,as a sequentialstream outside

spacetim e,onto thespatiotem poralworld.

A better understanding ofthe colour-phiphenom enon is now closer at

hand. Ifthe phenom enon is thought ofas occurring in consciousness in a

m irror-like,linearsequence asthe lightstim ulioccurin spacetim e,itleads

to them istaken beliefthatthebrain m usthousea consciousnesssingularity

capableofprojecting certain events(green ash)backwardsto tim esearlier

than theiractualoccurrence in the brain (p.127). However,if,asDennett

proposes,thecolour-phiphenom enon isthoughtofasoccurringin conscious-

ness as a m ultifaceted,higher-order representation ofstim uli-in-spacetim e

thatdoesnotneverthelessuse space in the brain to picture space nortim e

in the brain to picture tim e (p.131), a new conception arises. Because

consciousnessonly projectsitselfonto spacetim e in the way justexplained,

thereisnoquestion ofthelaterappearanceofthegreen spotbeingprojected

backwardsin tim eoranythingofthesort,butratherofa m ultitudeofbrain

eventsworkingin paralleltoyield abiased representation oftheouterworld.

Theprojection ofthisrepresentation ontospacetim eisfuzzy within thetim e

scale (oftheorderofhalfa second)ofthe said brain workings.

The phenom enologicalpropertiesofconsciousnessjustrevealed display

som e sim ilarity with quantum -m echanical ideas| in particular, with the

preparation ofphysicalsystem sinto quantum -m echanical\states." Butbe-

fore proceeding, a warning m ust be sounded. So far as we can see, the

resem blancesto beuncovered between thetwo phenom enologicalrealm sare

super�cial;orto usetheprecisely suited wordsofK lausG ottstein (2003)in

a letterto CarlW eizs�ackerdealing with a di�erentissue,\Itappearsto m e

that this is not a realconnection,but rather a kind ofparallelism " (p.2,

onlineref.) W hatisthen thevalueoftheparallelism weintend to draw? If

nothing else,itisitsheuristic power,asitisaptto guide thoughttowards

physically usefulconcepts.Ideas,afterall,m ustcom e from som ewhere.As

such,the following parallelism stakesno claim otherthan to bepartofour

private train ofthought; the reader m ight as wellignore it ifpersonally

unconcerned with theprivate aspectofpublicscience.7

W estartby notingarem iniscenceofthe�rst,erroneousinterpretation of

7In particular,the parallelism to be drawn doesnotm ean to suggestany m icroscopic

role forquantum m echanicsin thebrain,norany connectionsbetween them echanism sof

consciousnessand state-vectorcollapse.In fact,the conceptsofstate vectorand collapse

willoccupy no place at allin the m etageom etric form ulation. This should have been

already clear| but,in an atm osphere asprone to m isunderstanding asisthatofscience,

bettersafe than sorry.


thecolour-phiphenom enon with equally m isleading interpretationsofquan-

tum phenom ena.Forexam ple,likethephenom enon ofconsciousness,sohas

thephenom enon ofquantum correlations(orquantum entanglem ent)global,

non-causal,non-reductionistic,and indeterm inistic characteristics,leading

som etim es to talk about particles or inform ation travelling backwards in

tim e.

The�rsttoproposethisideawasCosta deBeauregard (1977),originally

in 1953;talking ofa correlated,widely-separated system oftwo photons,he

wrote:\Einstein’sprohibition to ‘telegraph into the past’doesnot hold at

thelevelofthe quantalstochastic event(the wave collapse)" (p.43).Davi-

don (1976) also expressed sim ilar views: \In this form ulation ofquantum

physics,e�ects ofinteractions with a previously closed system ‘propagate

backwards in tim e’" (p.39);and so did Stapp (1975): \This inform ation

owsfrom an eventboth forward to itspotentialsuccessorsand backward

to itsantecedents..." (p.275). These interpretations were devised on the

erroneousassum ption thatthequantum -m echanicalpreparation ofsystem s

isa spatiotem poralprocessand that,therefore,itgainsaysspacetim ecausal

structure.These interpretations,then,stem from a desirenotto relinquish

thislocal,causalspacetim e structure,and so trade itfora new,anom alous

way in which causalconnectionscan stillbeachieved locally.AsStapp put

it:

Bohr has insistently m aintained that quantum phenom ena are

incom patible with causalspace-tim e description. The m odelof

reality proposed hereconform stoBohr’sdicta,in sofarasclassi-

calcausalspace-tim e description isconcerned,butcircum vents

it by introducing a nonclassicalspace-tim e structure ofcausal

connection.(Stapp,1975,p.276)

How do the colour-phiphenom enon and that ofquantum correlations

m easureup to each otherwithin thetroubled fram ework ofthislocalspace-

tim e picture? O n the one hand,the consciousness singularity in the brain

needsto know thata green ash hasoccurred beforeithashad tim e to ac-

quireit(beforethe green ash hashad tim e to travelpastthe singularity).

This is solved by thinking that the green ash travels backwards in tim e

and insertsitselfatan earliertim e ofthe stream ofconsciousness. O n the

otherhand,a quantum -m echanicalsubsystem S2 needsto know the result

ofan experim entE 1 perform ed on subsystem S1 before ithashad tim e to

acquireit.Thisissolved by thinkingthatinform ation abouttheresultofE 1

travelsbackwardsin tim eto a tim ewhen S1 and S2 weretogetherand gets

transferred to S2.Both problem sarecreated by thinking locally aboutcon-

sciousnessand quantum -m echanicalinform ation,nam ely,(i)consciousness

isa localprocess,and (ii)the transferofquantum -m echanicalinform ation

is a localprocess. However, by regarding consciousness and the transfer

ofquantum -m echanicalinform ation asintrinsically globalphenom ena that

areontologically priorto spacetim e,neitherneedsto challengethenotion of

causalspacetim e structureand thuslead to far-fetched explanations.


Finally,when we picture the transfer ofquantum -m echanicalinform a-

tion in the usualway,nam ely,asthe preparation ofa quantum -m echanical

system into a certain \state" (not a state vector),the parallelism can be

broughteven further.W e �nd thatpreparation,a globaland spatiotem po-

rally delocalized phenom enon,has,likeconsciousness,theability to im press

itselfontoparticularphysicalsystem s(i.e.m atter).Itcan thuscreateevents

that are localized in spacetim e,providing a worldline in analogy with the

stream ofconsciousness.

The experim entalapproach to psychophysics,understood asa heuristic

discipline,hasled tothephysicalnotion ofquantum -m echanicalpreparation.

Is this an adequate or usefulstarting point for a m etageom etric quantum

theory?

12.2.2 M etageom etric facet ofpsychophysics

Despite our�rst-hand acquaintance with the subjective quality ofourcon-

sciousexperience,asystem aticaccountofthisrealm | atheory ofconscious-

ness| islacking. Notonly doesconsciousexperience presently elude quan-

titative buteven qualitative scienti�c description;in otherwords,notonly

do we failto grasp the size ofthoughts and feelings,but also failto pic-

ture their form s,shapes,outlines,or structures in any sensible way. O n

the other hand,however,thoughts and feelings can be about objects with

shape and size. In particular,as we just learnt,conscious events have no

properspatiallocation,butthey can beaboutspatialobjects.In thislight,

Pylyshyn (1979)hasm adethefurtherpointthatitshould strikeusascuri-

ousto \speak so freely oftheduration ofa m entalevent," when we\would

notproperly say ofa thought(orim age)thatitwaslarge orred" (p.278).

Thus,in thesam etoken,consciouseventshaveno tem porallocation either,

although they can beabouttem poralhappenings.

W hile thisfailureto picture consciouseventsgeom etrically need notbe

aproblem in itself,itisneverthelessperceived asaproblem in thecontextof

ourtypically geom etric waysofthinking| scienti�cally orotherwise.To us,

the factthatthe readily available ideasofgeom etric objectsand geom etric

m agnitudesare inapplicable to consciousevents,instead ofsuggesting that

consciousnessisa dead-end problem ,pointsto a world ofpossibilities:m ay

the phenom ena ofconsciousness be properly captured in a m etageom etric

theoreticalpicture?

Rem arkably,in ascienti�ccontextin which geom etry istaken forgranted

as the G od-given essence ofthe physicalworld,Foss (2000) held the geo-

m etric non-representability ofconsciousexperience responsible forthe fact

that consciousness has rem ained a scienti�c m ystery up untilnow. \Since

science deals only with geom etric properties," he wrote,\and the sensual

qualities are not geom etric,they cannot even be expressed,let alone ex-

plained,in scienti�c term s...Flavors have no edges,tingleshave no curva-

ture" (p.59). Fossfurtherm ore m ade an enlightening historicalanalysisof


the process that,through what he called the Pythagorean Intuition8 that

\reality is geom etricalat its heart" and the G alilean Intuition that \the

sensuousqualitiesareessentially non-geom etric"9 (pp.40,66),banned con-

sciousnessfrom a paradigm atically geom etric-m inded science. Foss argued

thatthisway ofthinking stem sfrom a confusion between thegeom etric vs.

the non-geom etric essence ofthings| which,understood in a m etaphysical

sense,sciencedoesnotconcern itselfwith| and thegeom etric picturesthat

science m akesofthe phenom ena itstudies.

Foss’sinsightaboutthegeom etricnatureofscienti�cdescription sim ply

asasuccessfultoolisatruegem ,valuableasitisrare,butdid hem anageto

stareatthetabooon thoughtaboutgeom etry beingalim ited toolstraightin

the eye? Notquite,because hisassessm entofthe \riddle ofconsciousness"

com esdown to this:

Science doesnotrequirethatthethingsitm odelsbe geom etric,

but only that they can be m odeled geom etrically. And there

is no reason to think,no im m ovable intuition indicating,that

consciousness,including itsm any-colored qualia,cannotbejust

as successfully m odeled by science as any other naturalphe-

nom enon.(Foss,2000,pp.94{95)

Pictured by science| how? G eom etrically?

W edonotseeany reason why thestudy ofconsciousnessin thebosom of

a geom etric theory,asim plied by Foss,should notbe possible.Buthaving

looked throughout this work at the reigning thought-taboo on geom etric

lim itationsin the eye,we are interested in a di�erentline ofthought.M ay

thesefactsbesayingthatconsciousnessalsohasontologicalaspectsthatcan

only becaptured by going beyond geom etry asa toolofthoughtin science?

Based on the heuristic parallelism with quantum -m echanical phenom ena

drawn earlier, we should be inclined to at least consider the possibility.

W hy? Because itison the otherhand the case with quantum phenom ena

thatgeom etric m ethodssucceed in capturing theirsuper�cialfeatures,but

thatadescription oftheirheartand naturecontinuestoeludesuch m ethods;

because the heart and nature of quantum phenom ena, as we see in this

chapter,can in factbedisclosed beyond the boundsofgeom etric thought.

Understood from this m etageom etric point ofview,Foss’s description

ofthe current clim ate ofthought that \it is intuitively obvious that the

8Asnoted in Chapter7,\Platonic Intuition" m ay be m ore accurate.

9Forexam ple,G alilei(1957)wrote:\Now Isay thatwheneverIconceiveany m aterial

or corporealsubstance,I im m ediately feelthe need to think ofit as bounded,and as

having thisorthatshape;asbeing large orsm allin relation to otherthings,and in som e

speci�cplaceatany given tim e;asbeing in m otion oratrest;astouching ornottouching

som e otherbody;and asbeing one in num ber,orfew,orm any.From these conditionsI

cannotseparate such a substance by any stretch ofm y im agination. Butthatitm ustbe

white orred,bitterorsweet,noisy orsilent,and ofsweetorfoulodor,m y m ind doesnot

feelcom pelled to bring in asnecessary accom panim ents. [...]Hence Ithink thattastes,

odors,colors,and so on are no m ore than m ere nam es so far as the object in which we

place them isconcerned,and thatthey reside only in the consciousness" (p.274).


sensuous qualities cannot be conceived in geom etric term s,hence cannot

be convincingly m odeled by science, therefore cannot �nd a place in its

ontology" (p.66) dem ands what seem s to be an unforeseen quali�cation:

the sensuousqualitiescannot�nd a place in the ontology ofscience unless

thisontology wasexpanded beyond the geom etric realm .Understood thus,

the following wordsfrom Dennettcould begiven unsuspected m eaning:

[W e should notbedissuaded]from asking...whetherm ind stu�

m ight indeed be som ething above and beyond the atom s and

m oleculesthatcom posethebrain,butstillascienti�cally investi-

gatablekind ofm atter...Perhapssom ebasicenlargem entofthe

ontology ofthephysicalsciencesiscalled forin orderto account

forthephenom ena ofconsciousness.(Dennett,1991,p.36)

Ifwetakethegeom etricnon-representability ofconsciousphenom ena in

earnest,how can it help to develop a m etageom etric form ulation ofquan-

tum theory and tim e based on the previously arrived-at experim entalno-

tion ofquantum -m echanicalpreparation? Perhapswe should leave the de-

tails of that m yriad of sensations, perceptions, em otions, thoughts, and

general m entalactivities that we callconscious feelings alone. Perhaps,

instead ofnearsightedly enquiring into their inner structure (even worse,

into theirinnergeom etric structure)afterthe prevailing windsofa science

thatprogressesthrough dissection into sm aller(butnotnecessarily sim pler)

constituents,one should take conscious feelings as prim itive notions,and

enquirethen into theirm utualrelationships.

Atthispoint,partofCli�ord’swords,quoted and analysed in Chapter8,

on theroleoftherelationsbetween feelingsin sciencestrikearesonantchord:

These elem ents offeeling have relationsofnextnessorcontigu-

ity in space,which are exem pli�ed by the sight-perceptions of

contiguouspoints;and relationsofsuccession in tim e which are

exem pli�ed by allperceptions.O utofthesetwo relationsthefu-

turetheoristhasto build up theworld asbesthem ay.(Cli�ord,

1886b,p.173)

G iven our purposes,we do not wish to take Cli�ord’s account in a literal

sense;thatis,we do notwantto concern ourselves with relations between

feelings(i.e.between m etageom etric things)in space and tim e. Butwe do

nevertheless relate to Cli�ord’s generalexhortation to the future theorist,

nam ely,tobuild up theworld asbesthem ayoutofthetworelationsbetween

feelings, nextness and succession,when the two relations are understood

outside a geom etric context.

W einterpretnextnessbetween consciousfeelingsto picturethecapacity

to recognize,distinguish,ordiscrim inate am ong the contents ofconscious-

ness, to the degree that it is possible to tell one conscious feeling from

another. According to Dennett (pp.173{175),the ulterior reason behind

this is evolutionary: our unicellular ancestors,in order to survive,had to


distinguish theirown bodiesfrom the restofthe world,developing a prim -

itive form ofsel�shness. Nextness,then,does not picture the geom etric

idea ofhow close(in spaceorelsewhere)two consciousfeelingsare,buttells

whetherornotthey are distinctfrom each other.

In turn,we take succession between conscious feelings to picture the

stream ofconsciousness,which we experience asa linear,narrative-like se-

quence ofconscious feelings. The inspirationalrole ofthe stream ofcon-

sciousness in the search for knowledge and understanding is not unprece-

dented.Fam ously,Turing (1950)inspired him selfin hisown stream ofcon-

sciousnessto understand how a m achine could solve a m athem aticalprob-

lem .In Dennett’swords,Turing

took theim portantstep oftrying to break down thesequence of

hism entalactsintotheirprim itivecom ponents.\W hatdoIdo,"

he m ust have asked him self,\when I perform a com putation?

W ell,�rstIask m yselfwhich rule applies,and then Iapply the

rule,and then writedown theresult,and then Iask m yselfwhat

to do next,and..." (Dennett,1991,p.212)

Succession,then,doesnotpicture the geom etric idea ofhow close (in tim e

orelsewhere)two consciousfeelingsare,buttellswhetherornotonefollows

theother.

These two relations between conscious feelings are by nature m etageo-

m etric. Neither nextness nor succession involve any geom etric objects or

m agnitudesin theirdepiction;theattribution ofshapesand sizesisutterly

foreign to these ideas, and we should not hesitate to eschew their intro-

duction ifan ill-advised futuretheoristwere to attem ptit| such geom etric

tam pering would undoubtedly result in his psychologicalsatisfaction,but

m ostlikely notin any gain in physicalunderstanding.

Nextnessand succession offeelingsform theessenceofthepsychological

conception oftim e. Tim e is linked to the sensitivity ofconsciousness to

changes;itarisesasthe serialchaining (succession)ofsim ilarfeelingsthat

areyetdistinctenough tobediscrim inated (nextness)from each other.This

serial chaining of discrim inated feelings| psychological tim e| constitutes

theperm anentcontentofourconsciousness,i.e.the perm anentpresent.

Has psychophysics as heuristics served its purpose? It has. By �rst

bringing to our attention som e basic features ofour psychologicalexperi-

ence,itled to thephysicalidea ofquantum -m echanicalpreparation;and by

alerting usnow to the m etageom etric nature and m utualrelationsbetween

feelings,itsuggests the physicalnotion ofpreparation as a m etageom etric

physicalconcept. Iteven givesusa clue aboutwhatusefulm etageom etric

relations m ay hold am ong such physicalthings,and gives us hope that a

new quantum -m echanicalpicture oftim e m ay be achievable on theirbasis.

The developm ent ofquantum theory along these lines begins in the next

m ain section.

Incidentally,wenow seethatif,aswesuggested in Chapter6,Riem ann

isthefatherofdiscretespacetim eand W heeleristhefatherofpregeom etry,


then Cli�ord isthefatherofpsychophysics.

12.2.3 A ppraisal

How necessary is,in general,a psychologicalapproach to physics? At the

end ofChapter10,we criticized cosm ology forfalling victim to the earthly

necessitiesofhum an cognitivetools,evidenced especially in itstreatm entof

theconceptoftim e.Butwearewho weare,them akersofourown toolsof

thought,so how could the essentiallim itation im posed by hum an cognitive

toolsbeovercom e? Strictly speaking,itcannot;the\objective study ofthe

ocean’s fauna" referred to in that chapter is practically unattainable. A

theory,any one theory,is (and m ust necessarily be)a biased �lter.10 But

the exploration ofour cognitive tools,not in search for absolute ones but

betterones,is open to enquiry,and itisin itthatthechanceofprogressin

science lies.

The clich�e that \to know the world we m ust �rst know ourselves" be-

com es especially relevant at the edge ofour theoreticalpicturing abilities.

A sign ofarrivalat this lim it can be recognized by the endless,repetitive

application ofthe sam e traditionalconceptsto the �eldsofhum an enquiry

regardlessofthe resultsobtained,i.e.by the onsetofwhatLem hascalled

an age ofepigonism (see Epilogue). If,as is our contention, science has

reached today such a state with itsgeom etric worldview,to overcom e itwe

should look within ourselves in search forbettertools ofthe m ind. Aswe

do so,and furthering the previousparable,we m ay discover,in the utm ost

darkness and pressure ofthe inhospitable oceanic depths,that we have a

budding ability for photolum inescence,the vestiges ofa swim bladder,or

even the tracesofgills.

No-onehasbeen forced to �rstlook within herselfbeforeshecould m ake

sense ofher surroundingsm ore than the extraordinary deaf-blind wom an,

Helen K eller.Ifwe heed herrem arkable words,perhapsthen we shallcon-

serve a greater-than-nilchance to grow hum anly wiser:

G roping,uncertain,Iatlastfound m y identity,and afterseeing

m y thoughts and feelings repeated in others,I gradually con-

structed m y world ofm en and G od.AsIread and study,I�nd

thatthisiswhattherestoftheracehasdone.M an lookswithin

him selfand in tim e �nds the m easure and the m eaning ofthe

universe.(K eller,1933,pp.88{89)

12.3 M etageom etric realm

12.3.1 Prem easurem ent things

W etakepreparation orprem easurem entasthephysicalbasisofa m etageo-

m etric thing.

10Cf.com m entsabouta \theory ofeverything" in Chapter1.

12.3 M etageom etric realm 203

oven y

z

slits SNmagnets lower s reenupper s reen

NS

x

z

~Binhom

Figure 12.3: Schem atic setup ofthe Stern-G erlach experim ent. Side view (top).

Frontview (bottom ). For a non-nullprem easurem ent,at leastone ofthe screens

(upperorlower)m ustbe rem oved.

W e picture a quantum -m echanical,m easurable m agnitude by m eansof

an observableA (notan operator).Thepossibleresultsofm easuring A are

a1;a2;a3;:::,such that ai is a realnum ber;these constitute the spectrum

faigofA.LikewiseforotherobservablesB ,C ,etc.11 Now aprem easurem ent

refers to a selective process which consists ofpreparation and �ltering of

parts ofa physicalsystem ,such that,after it,only certain m easurem ent

resultsare possible.

Thebasicideaofaprem easurem entisexem pli�ed by(butnotlim ited to)

aStern-G erlach experim entalsetup,shown in Figure12.3.Ifa(lower)screen

isplaced blocking only those silveratom swith spin value s2 = � ~=2,while

thosesilveratom swith spin values1 = ~=2areallowed topassthrough,then

thisconstitutesaprem easurem entofspin m agnitudes1 (norm allyreferred to

as\spin up").Anotherexam pleofa quantum -m echanicalprem easurem ent

is the following. Ifa beam oflight is m ade to pass through a polarizer,

thephotonsoftheoutgoing beam willallbepolarized perpendicularto the

polarizer’sopticalaxis,whileallotherswillbeblocked.

Prem easurem entisheretaken asaprim itiveand irreducible| butunder-

standable| conceptin quantum ontology on which tobuild them etageom et-

11In thisinvestigation,the spectrum ofobservablesneed notbe discrim inated into dis-

crete and continuous; allthat m atters is that the m easurem ent result ai exist. Both

possibilities willbe taken into account sim ultaneously by the m etageom etric form alism ,

although thetraditionaldistinction can,ifso wished,bem adeoncewehaveproceeded to

the geom etric realm lateron.

204 T he m etageom etric nature ofquantum -m echanicalthingsraw systemsA

ai

︸︷︷︸preparationaj ∀j 6= i

︸︷︷︸filteringFigure 12.4: Schem aticrepresentation ofa quantum -m echanicalprem easurem ent

thing P(ai).

ricedi�ceofquantum theory.In analogy with aconsciousfeeling,wepicture

the basic physicalidea in question by m eans ofa m etageom etric prem ea-

surem entthing P(a),characterizing a prem easurem entassociated with an

observable A.Thisconsistsin the �ltering ofallpartsofa physicalsystem

(e.g.a beam ofparticles)exceptthose thatsucceed in being prepared into

a-system s,i.e.system son which theresultofa possible,laterm easurem ent

ofA givestherealvalue a (Figure 12.4).

A quantum -m echanicalthing is inherently global;in resem blance with

conscious events produced by the brain,itisinadequate to ask where and

when within the prem easurem ent device the prem easurem ent takes place.

The globalontology in question here does not respect the spatiotem poral

lawsofcausalprocesses. Prem easurem entthings,however,can create pre-

pared physicalsystem s(m atter)thatare localized in spacetim e.

Furtherm ore,likea consciousfeeling,a prem easurem entthing callsfora

m etageom etric depiction. The reason why we should regard a prem easure-

m entthingP(a)asa m etageom etric conceptis,quitesim ply,thatitisnot

a geom etric objectofany sort| itreally hasno shapeforourm ind’seye to

grab onto| and has no geom etric m agnitudes attached,such as length or

area or volum e;neither do prem easurem ent things have m utualgeom etric

relationships such as overlaps or distances de�ned between them ;expres-

sionssuch as\isP(a)round orsquare?",aswellaskP(a)k,hP(a)jP(b)i,

and dist[P(a);P(b)]arem eaningless and utterly foreign to them .Thereis

nothing in the idea ofprem easurem entthatrequiresany form ofgeom etric

portrayal;and weshould,in fact,rejectitsgeom etrization ifanyonewereto

attem ptitto satiate hisgeom etric instinct.

Now,what relationships hold am ong di�erent prem easurem ent things,

and whatm eaning should we attach to them ? Cli�ord’s ideas ofnextness

and succession from the previoussection can now help usuncoverthese re-

lationships,leading tothem athem aticalalgebraobeyed by prem easurem ent

things,while atthe sam e tim e pointing atthe explicitphysicalm eaning of

the operationsofthisalgebra.12

12The experim entalfoundation behind what we have called a prem easurem ent (and

later also a transition)thing hasbeen discussed by Schwinger(1959) underthe nam e of

m easurem ent sym bols,including the algebraic relationships holding between them . The

m etageom etric interpretation ofprem easurem ent things,their relationships to conscious

experience and psychologicaltim e,the elucidation ofthe need forcom plex num bers,the


A

ai, aj

ak∀k 6= i, j

Figure 12.5: Schem aticrepresentation ofthesum P(ai)+ P(aj)oftwo prem ea-

surem entthings.

A

ai

ak∀k 6= iA

aj

al∀l 6= j

Figure 12.6: Schem atic representation ofthe productP(aj)P(ai) oftwo pre-

m easurem entthings.

First,we centre ourattention on the relation ofnextnessbetween feel-

ings. Ifwe take the di�erent prem easurem ent things,P(ai),P(aj),etc.,

associated with the sam e observable A to be related by the m etageom et-

ric relation ofnextness,we are led to the idea ofadjustable discrim ination

(within a single and indivisible prem easurem entprocess)am ong the di�er-

entm easurem entvaluesai,aj,etc.W eem body such a discrim ination in the

m athem aticaloperation ofa sum ofprem easurem entthingsasfollows.The

sum P(ai)+ P(aj)oftwo prem easurem entthingsdescribesa selectivepre-

m easurem entprocessthatdiscrim inatesai-and aj-system s(taken together)

from the totality ofallpossible system sassociated with A. Itworksby al-

lowing only ai-or13 aj-system sto passthrough asprepared system s,while

blocking allother ak-system s,k 6= i;j (Figure 12.5). Therefore,the m ore

term s in the sum ,the less discrim inatory the indivisible prem easurem ent

pictured by thewhole sum is.

Second,wefocuson therelation ofsuccession between feelings.Itserves

as straightforward inspiration for the idea oflinear succession or consecu-

tiveness ofprem easurem entthings,which we em body in the m athem atical

operation ofa productofprem easurem entthings as follows. The product

P(aj)P(ai)oftwo prem easurem entthingsdescribesthe succession oftwo

prem easurem ents,such that som e fraction ofphysicalsystem s is �rstpre-

pared into ai-system s,and theseareconsecutively fed into prem easurem ent

P(aj)(Figure 12.6).

Having established thisphysicalfoundation,we proceed to develop the

di�erentiation between m eta-,quasi-,and geom etricrealm s,theorigin ofthestatevector,

and the rise ofm etageom etric and preparation tim e,however,are our originalcontribu-

tions.Thischapter,then,m ay be seen asa fresh,physically and psychologically inspired

conceptualelaboration on Schwinger’spreviousideas.13Understood asthe logicaloperation O R.


A

a1, a2, a3, . . .

︸︷︷︸not fun tioningFigure 12.7: Schem atic representation ofthe identity prem easurem entthing I .

m etageom etric theory. W e start by uncovering the properties ofthe sum

and theproductofprem easurem entthings.

Itisstraightforward to see thatthe sum oftwo prem easurem entthings

iscom m utative,

P(ai)+ P(aj)= P(aj)+ P(ai); (12.1)

because in the prem easurem ent P(ai)+ P(aj), ai- or aj-system s pass

through together irrespective ofthe order in which they appear. Further,

the sum ofthreeprem easurem entthingsisassociative,

[P(ai)+ P(aj)]+ P(ak)= P(ai)+ [P(aj)+ P(ak)]; (12.2)

forsim ilarreasons.

The identity thing I appears now naturally as the prem easurem ent

which allows all(prepared)system sto passthrough (Figure 12.7). There-

fore,the identity thing m ustbe physically equalto the sum ofallpossible

prem easurem entthingsfora given observable:X

i

P(ai)= I : (12.3)

W e callthis the (m etageom etric) com pleteness relation. By \allpossible

prem easurem entthings," we m ean asm any asthere are di�erentm easure-

m ent results ai. Thus,the above sumP

i includes as m any term sP(ai),

with asclosely spaced valuesai,asm adenecessary by observations.Itisof

no consequence here whetherthe possible valuesresulting from a m easure-

m entare believed to form a continuousspectrum ornot,i.e.whetherthey

m ustberealorm ay only berational.Itisenough forusto beassured that,

no m atter what value ai results from a m easurem ent,its prem easurem ent

appearsexpressed via P(ai).

Sim ilarly,the nullthing O describesthe situation in which allsystem s

are �ltered and nonepassthrough (Figure 12.8).

In the productP(aj)P(ai) oftwo prem easurem ent things associated

with thesingle observableA,thelatterprem easurem entP(aj)takesin ai-

system sand �ltersallexceptaj-system s,and so allsystem swillbe�ltered

unlessi= j.Thisproductcan then be understood asallowing the passage

ofai-and14 aj-system s.Thus,we obtain theresults

P(aj)P(ai)= O fori6= j (12.4)

14Understood asthe logicaloperation AND .


Aa1, a2, a3, . . .

Figure 12.8: Schem aticrepresentation ofthenullprem easurem entthing O.

and

P(aj)P(ai)= P(ai) fori= j; (12.5)

representing the factthatthe repeated succession ofthe sam e prem easure-

m ent is physically (and therefore m athem atically) equivalent to itself,i.e.

P(ai) is an idem potent thing. The two results (12.4) and (12.5) can be

connected with the help ofthe delta thingDij:

P(aj)P(ai)= DijP(ai); (12.6)

where

Dij =

�O fori6= j

I fori= j: (12.7)

From this result follows (now even without recourse to physicalthinking)

that the product ofprem easurem ent things associated with the sam e ob-

servableisboth com m utative and associative:15

P(aj)P(ai)= P(ai)P(aj); (12.8)

[P(ak)P(aj)]P(ai)= P(ak)[P(aj)P(ai)]: (12.9)

W e note �nally that I and O behave as if they were num bers with

respectto the sum and theproduct:

I + O = I ;

P(a)+ O = P(a);

I I = I ;

OO = O; (12.10)

I P(a)= P(a)= P(a)I ;

OP(a)= O = P(a)O;

I O = O = OI :

Could,underappropriate circum stances,the thingsO and I be replaced

by thenum bers0 and 1?

15The product is associative in general,i.e.[P(c)P(b)]P(a)= P(c)[P(b)P(a)]. To

show this,the furtherresults(12.15)and (12.20)are needed.


A

a

B

b

Figure 12.9: Schem aticrepresentation ofa quantum -m echanicaltransition thing

P(bja) for the case of di�erent observables A and B . The transition thing is

represented by the region enclosed by the dashed line (the wholeoftransition box

B isincluded butnotits�ltering action).

12.3.2 Transition things

W e exam ine nextthe productsP(b)P(a) and P(a)P(b) associated with

two di�erentobservablesA and B .In general,

P(b)P(a)6= P(a)P(b) (12.11)

becausethe�rstproductpreparesa-system sintob-system s,whilethesecond

productpreparesb-system s into a-system s. Both situations are equivalent

only ifA and B are com patible observables (see Section 12.5.2 for details

on com patibility).

The productofprem easurem entthingsgivesm otivation fora new type

ofm etageom etricthing,nam ely,transition things.A transition thingP(bja)

representsapurequantum -m echanicaltransition from a-system stob-system s

(Figure 12.9).W e notice thatP(bja)isnotequalto P(b)P(a)buta part

ofit;a transition thingP(bja)m akesreferenceto apuretransition in which

every a-system becom esa b-system withoutany �ltering loss,whilea prod-

uctP(b)P(a)ofprem easurem entsinvolves(i)thepreparation ofsom eraw

system sintoa-system swith thelossofsom e,(ii)thetransform ation ofsom e

a-system s into b-system s,and (iii) the �ltering ofa-system s that failed to

becom e b-system s. W hatwe identify asa pure transition (and m ark o� in

dashed linesin thepictures),then,isitem (ii)within thethree-partprocess

constituting a productofprem easurem ents.16

The�rstfeatureto noticeabouta transition thing isthatitisin general

asym m etric,

P(ajb)6= P(bja); (12.12)

since a transition from b-system s to a-system s is physically di�erent from

a transition from a-system sto b-system s. (W hen A and B are com patible

observables,we shallsee thatP(ajb) and P(bja) com m ute.) Asym m etry

holdsin particularfortransition thingsassociated with a singleobservable,

P(aijaj)6= P(ajjai); (12.13)

since ai-and aj-system sare necessarily m utually exclusive (fori6= j)(Fig-

ure12.10).

16The exactcorrespondencebetween thepictorialrepresentation ofa transition aspart

ofthe productofprem easurem ents and itsphysico-m athem aticalrepresentation willbe-

com e cleareraswe m ove along;see Eq.(12.21).


ai

A

aj

ak ∀k 6= j

Figure 12.10:Schem aticrepresentation ofaquantum -m echanicaltransition thing

P(ajjai)forthe caseofa singleobservableA.Thedouble box representsthefact

thatsuch a transition thing cannotbe im plem ented with a singleprem easurem ent

device (see p.211). The transition thing is represented by the dashed-o� region

(thewholedoublebox A isincluded);the�ltering action ofthebox isnota proper

partofit.

ai

A

aj ak

A

al

Figure 12.11: Schem aticrepresentation ofthe productP(aljak)P(ajjai)oftwo

quantum -m echanicaltransition things for the case ofa single observable A. The

�ltering action notbelonging to the transition properhasnotbeen drawn.

M ore detailed knowledge about the nature oftransition things can be

obtained by analysing theproductP(aljak)P(ajjai)forthecaseofa single

observableA (Figure12.11).AfterSchwinger(1959,p.1544),sincea prod-

uctdescribestheoutcom eofthe�rstprocessbeing fed into thesecond,and

the system sassociated with a single observable are m utually exclusive,the

only way fortheaboveproductto benon-nullisto havetheoutcom eofthe

�rsttransition m atch the inputofthe second:

P(aljak)P(ajjai)= DjkP(aljai): (12.14)

From thisresultfollowsthattheproductoftransition thingsassociated with

a singleobservableisnotcom m utative exceptin them athem atically trivial

casesj6= k and i6= l(productsreduceto thenullthing),and i= j= k = l

(productsreduceto the sam e transition thing).

Physicalreasoningcan also convinceusatthisearly stagethattheprod-

uctP(djc)P(bja)oftransition thingsassociated with di�erentobservables

is not com m utative either. Despite a lack ofproper tools at the m om ent,

one m ightwish to speculate on the nature ofthisproduct. In principle,it

should work akin to P(dja),although in generalP(djc)P(bja)6= P(dja).

To �nd the m anner in which they are connected,we m ust know to what

degree b-and c-system s are com patible (cf.the case ofa single observable

A,wheredi�erentsystem sare totally incom patible).

Finally,reasoning on the basis ofa schem atic picture is apt to lead us

to a noteworthy connection between a prem easurem ent thing P(a) and a

transition thingP(aja),nam ely,

P(a)= P(aja): (12.15)


A

a

A

a(i) (ii)Figure 12.12: Schem atic representation ofthe productP(a)P(a) ofthe sam e

prem easurem entthing in succession for observable A. The overallprocessis pre-

sented asthesuccession oftwo parts:(i)theprem easurem entofa-system sand (ii)

the super uoustransition ofa-system sinto a-system s.

In e�ect, consider the representation of the product P(a)P(a) (Figure

12.12), and analyse it in two parts: (i) the prem easurem ent of raw sys-

tem sinto a-system sim plem ented via preparation and �ltering,and (ii)the

subsequent preparation ofa-system s into a-system s. W e can see that the

�rstpartactsasa regularprem easurem entP(a),while the second part|

im plem ented with an identicaldevice as the �rst| acts as a super uous

transition P(aja)thatoutputsitsfullinput;itsim plem entation requiresno

�ltering action,although the�lterispresent.From thiswelearn intuitively

that

P(a)P(a)= P(aja)P(a) (12.16)

holds,i.e.thatonce a-system saregiven,P(a)and P(aja)are equivalent.

O n theotherhand,asim ilarreasoningstartingfrom therepresentation of

P(b)P(a)(seeFigure12.9)doesnotlead to a sim ilarconclusion.From the

picture,wehaveP(b)P(a)6= P(bja)P(a),becausein theim plem entation

ofP(bja)(second box)an extra �ltering m echanism isneeded to getrid of

som e a-system s.

W hataboutthen those situationsin which we do notenjoy the bene�t

ofhaving a-system s as given? Can P(a) and P(aja) be equivalent also

then? The equivalence continues to seem plausible. A prem easurem ent

thingP(a)preparessom eraw system sinto a-system s,whiletherestofraw

system s are �ltered out. O n the other hand,the specialtransition thing

P(aja) im plem ents a super uous transition from a-system s to a-system s.

Thephysicalprocessesrepresented bythesetwothingsarenotnow identical,

butthey appearto beequivalent:to preparea-system sfrom raw oneswith

�ltering lossisequivalentto having a-system sand super uously preparing

them into a-system s without any �ltering loss. Both processes am ount to

the sam e thing: they have the sam e output and are im plem ented with the

sam e device| with respecttoP(aja),the�ltering action ofP(a)balances

orneutralizeswhatisin excessin itsraw input.

The m eaning of\the sam e output with the sam e device" can be illus-

trated asfollows.Im agine we wantto perform P(s1)and P(s1js1)with a

Stern-G erlach setup.Both can beperform ed using onesingleStern-G erlach

device setto �lter outalls2-system s by m eans ofan appropriately placed

screen;further,although theinputsystem softheprem easurem entand tran-


y

s2

S

N

s1x

s2x

x

z

s2x

y′

z SN

s1

s2

Figure 12.13: Realistic representation ofthe Stern-G erlach setup for the per-

form ance of a transition P(s1js2) corresponding to a single spin observable S.

Sym bolized asa double box in schem aticdiagram ssuch asFigure12.10.

sition thingsaredi�erent(araw beam vs.aprepared beam ),theoutputpro-

duced by the sam e device isthe sam e. O n the otherhand,itisim possible

to perform P(s1js2)with a single sam e device because s1-and s2-system s

are m utually incom patible: given an inputofs2-system s,the above device

would outputno system satall,i.e.itwould actasthe nullthing O.W hat

isrequired to perform P(s1js2)isa com pound device (drawn as a double

box);forexam ple,a succession oftwo Stern-G erlach devices,with the �rst

one oriented along an axisperpendicularto thatofthe second such thatit

willoutput,say,s1x-and s2x-system sin equalquantities.Subsequently,ei-

therofthese,say,s1x-system s,can be�ltered and theoutputofs2x-system s

transform ed into s1-and s2-system s by a second device (oriented perpen-

dicularly to the �rst),which could �lter outthe unwanted s2-system s and

output only s1-system s17 (Figure 12.13). Therefore,P(s1js1) is the only

single-observable18 transition that can be perform ed with the sam e device

used forP(s1).

W e decide to trustthisphysicalintuition and acceptEq.(12.15) asan

equivalence valid in generalforany situation.W e rem ain alert,however,to

thepossibility thatitm ay lead toinconsistencieslateron in thedevelopm ent

of the theory. W hat we shall�nd is that, after the introduction of the

transition postulate in the nextsection,the origin ofa-system sbecom esin

each and every case irrelevant,and thatany situation can be recastin the

spiritofEq.(12.16).

W ith this,the m etageom etric foundation ofquantum theory is ready.

Ahead lies the task ofshowing how a description ofquantum phenom ena

arises from it,thereby revealing the m etageom etric origin oftheir charac-

teristic features. However,quantum theory cannotbe derived on the basis

17Because transition thingsrepresentpure transitions,allsystem s �ltered outin their

im plem entation should notbe considered partofthe transition thing itself.18Transitions involving di�erent observables can be perform ed with the sam e devices

used forprem easurem entssince,in thiscase,the degree ofcom patibility isnoteitherall

ornothing.


ofm etageom etric things and their sum sand productsalone: where would

num bers com e from given that quantum -m echanicalthings have nothing

to do with them ? The introduction ofa m ain postulate willtherefore be

needed,butitsbeing a postulate doesnotm ean thatitsphysicalm eaning

m ustrem ain unclear.

12.4 Q uasigeom etric realm

12.4.1 T he transition postulate and the transition function

W e take asa starting pointrelation (12.14),since itappearsthatthe only

way in which num berscould beintroduced in a physically reasonableway is

via thedelta thing.Letusreplacethedelta thingDjk by K ronecker’sdelta

�jk to get

P(aljak)P(ajjai)= �jkP(aljai); (12.17)

thusstarting with thetwo m ostnaturalnum berstherecan be,0 and 1.The

form ernum berrepresentsthecom pleteabsenceofan overalltransition (to-

talincom patibility between di�erentsystem sassociated with A),while the

latter representsthe existence ofan overalltransition in which no system s

are lost.19

Thisreplacem entisquitesigni�cantsinceherea thing isbeing replaced

by a num ber,which isan objectofa totally di�erentnature.However,the

replacem entisacceptable and doesnotlead to inconsistenciesbecause the

delta thingalwaysappearsm ultiplying otherthings;therefore,thenatureof

allequationsispreserved when weconsiderthat,in allcases,thingscontinue

to beequated to things:

0P(aljai) = O; (12.18)

1P(aljai) = P(aljai): (12.19)

A question,however,rem ains. W here could allothernum berscom e from ?

Thisisa criticalstage ofthisinvestigation.

G eneralizing the idea behind Eq.(12.17),we postulate afterSchwinger

(1959,p.1545)that

P(djc)P(bja)= hcjbiP(dja); (12.20)

19As we saw in Chapter 8, Eddington’s (1920, p.188) own quest for the nature of

things led him to an analysis ofspacetim e point-eventsand the intervalsbetween them .

He argued that the intervals m ight have non-quantitative aspects; that the individual

intervals probably cannot be captured by rulers and clocks,but that they could sim ply

exist(1)ornot(0).Thus,Eddington’sm annerofintroducingnum bersintoan outspokenly

non-quantitativefram ework is,atleastsuper�cially,akin to thepresentone.Eddington’s

eitherexistentornon-existentprim itiveintervalsareanalogousto thetransition functions

hajjaii = �ij (to be introduced) for the case ofa single observable A. These transition

functionstellwhethersom ethingoccursornot| in aprem easurem entofA,itsown system s

are accepted com pletely (1)ornotatall(0).

12.4 Q uasigeom etric realm 213

where the notation hcjbi represents a num ber. W e callthis the transition

postulate and nam e hcjbi the transition function (or transform ation func-

tion).20 Ifwe are to throw new light on quantum ontology,we m ust not

interpretthe notation hcjbiasan innerproduct| notjustyetand without

furtherquali�cation| butsim ply asa m annerofm arking a physicalaction

involving b-and c-system s.Thetransition function hcjbiisa num berthatis

taken to com m ute with prem easurem entand transition things,since these

areobjectsofcom pletely di�erentnatures.

Butwhatkind ofnum beristhe transition function? Natural,rational,

real,com plex,hypercom plex? Thisisnotsom ething thatwe can postulate

withouta reason;to say whatkind ofnum berthe transition function m ay

be,we m ustletthe physicsinvolved naturally lead to an answer.W e m ake

and justify thischoice in Section 12.4.3.

Physically,the transition postulate describeswhatpartofthe group of

b-system sassociated with observable B isaccepted in the prem easurem ent

associated with observableC ,succeedingtobecom ec-system s.Thisfraction

ofsystem s,described by a num ber,istheshareofsystem sthatprem easure-

m entsP(b)and P(c)havein com m on.O necan seefrom thism oregeneral

perspective that the earlier result that ai-and aj-system s are either fully

com patible orfully incom patible followsnow asthe specialcase ofa single

observablein the form hajjaii= �ij.

W e rem ark that the characterization in question here is qualitative|

\part" is a num ber. W e would only be forced to callthis characterization

quantitative ifthe transition function turned outto be a non-negative real

(or naturalor rational) num berbut not otherwise,because other kinds of

num bers,e.g.com plex num bers,do notpicturesize| only theirm odulido.

Theabove wording \have in com m on" suggeststhe possibility ofestab-

lishingaconnection| ifsowewished todo| between thetransition function

and theinnerproductoftwo arrow-likevectors.Theway in which thisgeo-

m etricinterpretation ofthetransition function ariseswillteach ushow state

vectorsare born| and revealthem asphysically insubstantialconcepts.At

this point,however,we have not yet introduced any geom etric concepts,

and the transition function m ust be understood sim ply as a concept| a

num ber| in the quasigeom etric realm that serves to bridge the chasm be-

tween a genuinely m etageom etric realm and a genuinely geom etric one.

Is now the transition postulate in harm ony with the equality we in-

troduced in the previous section,nam ely,P(a) = P(aja)? It is indeed,

because,on the basis ofthe transition postulate,the m ethod by which a-

system sare�rstobtained becom esnow irrelevantforanysituation.In e�ect,

using both Eq.(12.15)and Eq.(12.20),wehaveP(bja)= P(bja)P(a)be-

causeP(a)= P(aja)and hajai= 1.Asa result,wecan now write

P(b)P(a)= hbjaiP(bja)P(a); (12.21)

20Function,from Latin fungi:to perform ,to executean action.Thetransition function

isa function in thisetym ologicalsense;itisnotto be interpreted asan explicitfunction

f(b;c)ofthe realnum bersband c.


whereby weobtain acom pletecorrespondence,elem entby elem ent,between

thepictorialand thephysico-m athem aticalrepresentation oftheproductof

two di�erent prem easurem ents. Com paring with Figure 12.9,we see that

each elem ent on the right-hand side ofEq.(12.21) is pictured in it: (i) a

prem easurem entP(a)ofa-system sincluding �ltering,(ii)a puretransition

P(bja)from a-system stob-system s,and (iii)aquali�cation hbjaiofthepart

ofa-system s that m anage to becom e b-system s sym bolized by the second

�ltering action.Thepictorial-algebraic parallelism isnow com plete.

Beforeconcluding thissection,wem ention two noteworthy propertiesof

the transition function.The�rstoneresultsasfollows.O n the one hand,

P(b)P(a)= P(bjb)P(aja)= hbjaiP(bja); (12.22)

on the otherhand,introducing the identity thingP

iP(ci),weget

P(b)

"X

i

P(ci)

#

P(a)= P(bjb)

"X

i

P(cijci)

#

P(aja)

=X

i

hbjciihcijaiP(bja): (12.23)

Therefore,

hbjai=X

i

hbjciihcijai: (12.24)

W ecallthisthe(quasigeom etric)decom position relation.Ittellsusthatthe

partofa-system sthatsucceedsin being transform ed into b-system scan be

expressed via transitionsto and from allpossible ci-system s.21

A sim ilarprocedureyieldsa second property,nam ely,

P(bja)=X

i;j

hdjjbihajciiP(djjci); (12.25)

which tells that a transition from b-system s to a-system s can be built out

oftransitionsfrom ci-system sto dj-system s,so long aswe know whatpart

ofci-system s is com patible with a-system s and what part ofb-system s is

com patible with dj-system s. W e see, then, how the transition functions

havetheroleofbuildingalinearcom bination oftransition things;transition

functionstellwhatqualitative weighteach transition thing possessesin its

com bination with othersuch things,depending on whatpartevery pairof

physicalsystem shave in com m on.

21Thisproofassum esthatthesum ofthingsisdistributivewith respectto theproduct.

Sincetheexpression P(ci)+ P(cj)isirreducible,thisresultcannotbeproved m athem ati-

cally;however,wedo no �nd thisproblem aticatall.Thatthesum isdistributivewith the

productcan be seen to hold by considering the m eaning ofthe expression physically.W e

should notdem and anything else from a physicaltheory. Alternatively,one m ay choose

to im pose thisasa m athem aticalaxiom ,butweconsiderthisto be a pointlessprocedure

in the presentphysicalcontext.


12.4.2 A lead on probability

W e found previously that,in the specialcase ofa single observable,hajjaii

is either 0 or 1. O ne m ay be tem pted by this fact and sim ple-m indedly

interpret the transition function to be the probability for the occurrence

ofa quantum -m echanicaltransition,with each hbijaiassum ed to be a real

num berbetween 0 and 1,andP

ihbijai= 1.Thisinterpretation,however,is

notbased on any physicalreasoning and is,in fact,com pletely m isguided.

In order to obtain a sensible interpretation ofthe transition function,

and onethatisalso in harm ony with property (12.24),wem ustnotblindly

assum eaphysicalm eaningforit,butrather�nd itasitnaturally arisesfrom

a furtheranalysis ofprem easurem entthings| especially,ofa succession of

them afterSchwinger(1959,p.1547).

In the physicalsituation described by the productP(b)P(a),prem ea-

surem entP(b)in generaldestroysa partofthe inform ation obtained from

prem easurem entP(a) because allthose a-system s that failto perform a

transition into b-system s are lost. The partofsystem s that is notlost,or

the degree ofcom patibility between a-and b-system s,is described by the

transition function hbjai.Theexpression

P(b)P(a)= hbjaiP(bja) (12.26)

tells us precisely this,nam ely,that the succession ofthe two prem easure-

m entsP(a) and P(b)is equivalent to a (pure)transition P(bja) from a-

system sto b-system squali�ed by hbjai,thedegreeofcom patibility between

thesystem sin question.

Ifnow we perform prem easurem entP(a) again in order to regain a-

system s(Figure12.14),wecan seewhatkind ofdisturbanceprem easurem ent

P(b)hascaused;we �nd

P(a)P(b)P(a)= hajbihbjaiP(a): (12.27)

If hajbihbjaiwasa non-negativerealnum ber,thisresultcould bephysically

interpreted to quantify the disturbance caused by P(b),i.e.to be telling

us what the chances are that a given,individuala-system willsurvive its

becom ingab-system and berecovered again asan a-system .In otherwords,

hajbihbjai could quantify the equivalence between a prem easurem ent ofa-

system shaving passed through b-system s,and a single prem easurem entof

a-system s. These two di�erent situations are physically the sam e only in

thosecaseswhen thedisturbanceintroduced byP(b)iscleared successfully.

In the specialcase ofa single observable,the results obtained are in

harm ony with thispossibleinterpretation.In e�ect,

P(ai)P(aj)P(ai)= haijajihajjaiiP(ai)= �ijP(ai) (12.28)

would tellusthatthechancesofan ai-system survivinga disturbanceintro-

duced by P(aj)is nullunlessi= j,in which case the chances ofsurvival

areguaranteed.


A

a

B

b

A

a

Figure 12.14: Schem aticrepresentation ofthesuccessionP(a)P(b)P(a)ofpre-

m easurem entthingssuggesting the conceptofquantum -m echanicalprobability.

The role ofthe product ofm etageom etric things as a lead on transi-

tion probabilitiescannotbeoverem phasized.Itisourintuitive grasp ofthe

situation represented by P(a)P(b)P(a) as a recovery operation,and our

intuition that the recovery ofan a-system willsom etim es failand som e-

tim essucceed,thatgivesusreason to think that,perhaps,hajbihbjaican be

interpreted asa probability.

The analysis of the role of the sum ofprem easurem ent things in the

possible birth ofprobability adds m ore justi�cation to this interpretation.

W hatresultswould we obtain ifwe allowed forthe possibility ofa-system s

being disturbed through allpossible (m utually exclusive) bk-system s? W e

�nd that the supposed probability ofan a-system surviving this least se-

lective disturbance introduced byP

k P(bk) adds up to unity. In e�ect,

because wehaveP

kP(bk)= I and P(a)isidem potent,we �nd

P(a)= P(a)

"X

k

P(bk)

#

P(a)=X

k

hajbkihbkjaiP(a); (12.29)

so that X

k

hajbkihbkjai= 1: (12.30)

Thisresultisconsistentwith property (12.24).In thism anner,theproduct

ofm etageom etricthingsiscom patiblewith theidea ofprobability forsingle

transitions,and the sum ofproductswith the idea ofprobability fora set

ofm utually exclusive transitions. Unexpectedly,the interpretation ofthe

product and sum ofthings agrees with the interpretation ofthe product

and sum ofnum bers in probability theory.

Ifthisinterpretation isworkableand itslead on probability correct,then

thereisnoneed to postulatecontroversialstatevectors| with theirm ysteri-

ouscollapses| here.Probability would arisenotm ysteriously butnaturally,

from the very physicalm eaning ofthesituation underexam ination.

12.4.3 T he transition-function postulate

Thehurdlesthatm uststillbeovercom eon theway to probability are(i)to

revealand justify thenatureofthenum berrepresentingthetransition func-

tion hbjaiguided notonly by thepictureoftherecovery ofan a-system and

its lead on probability,butalso by the physicalm eaning ofthe transition

function and a deeper insight into the signi�cance and utility ofnum bers


in physics;and (ii) to m ake sure that the product hajbihbjai we have hy-

pothetically identi�ed with a probability is indeed a realnum ber between

0 and 1.

To m ove forwards,we ask whetherany kind ofrelation could existbe-

tween the two num bers hbjai and hajbi. In other words,considering that

they both picturerelated processes,nam ely,a transition from a-system sto

b-system sand viceversa,could one be obtained from the other? And ifso,

on whatgrounds?

Considerthefollowing reasoningwithin therealm ofrealnum bers.Letr

bearealnum ber,and letr? = � rdenotethedualcounterpartofr.Nextwe

ask (i)whatisthephysicalm eaning ofoperation \?"? and (ii)whatspecial

properties does it inheritfrom its physicalm eaning? The �rstquestion is

straightforward to answer.O peration \?" describesthereversalofa sim ple

physicalprocess.Forexam ple,ifr representsgaining r apples,r? represents

losing or owing r apples;ifr sym bolizes a force ofintensity r pulling in a

certain direction,r? sym bolizes a force ofthe sam e intensity acting in the

opposite direction;etc. As we can see,the physicalm eaning ofoperation

\?" isvery close to everyday experience.

Since \?" representsthe reversalofa physicalprocess,and since to re-

verse a physicalprocess twice brings us back to the starting point, this

operation m ust be its own inverse,r?? = r (recovery oforiginaltransac-

tion). M oreover,the con ation ofa process in one direction with another

in theopposite direction m ustproduceno physicalprocessatall;this�nds

expression in the property r?r = 0 (nullnet transaction),22 where 0 is a

specialrealnum berwith the property 0? = 0 because,roughly speaking,0

pointsin no direction.

The previously unearthed physicalm eaning ofhbjaiasgiving the direc-

tion ofa quantum -m echanicaltransition from a-to b-system sgivesusnow

an essentialclue,becausethetransition function,through itsexplicitexpres-

sion,likewisetellsusthathajbirepresentsa quantum -m echanicaltransition

in a direction opposite to thatofhbjai.

This �ts the m eaning of the ? operation, and so we ask: could the

transition function,then,bea realnum berand \?" therelation wearelook-

ing for? W e see im m ediately that it cannot,because we would then have

thathajbihbjai= � r+ r � 0,i.e.always null.23 Returning to the product

P(a)P(b)P(a),this would m ean that the originala-system is always ir-

recoverableforarbitrary observablesA and B .Butthisrunscounterto our

intuition ofthe originalphysicalsituation. And so would a sim ilar use of

naturaland rationalnum bers.W e m usttry harder.

A clue to this problem com es from realizing that the kind ofphysical

processrepresented by the transition function is farther rem oved from ev-

eryday experiencethan thosecharacterized by realnum bers.Thequantum -

m echanicalprocessin question sim ply cannotbe grasped by realnum bers,

which have been designed and introduced into physics to picture classical

22Expression r

?r isto be read � r+ r.

23W e read hbjai?hbjaias� hbjai+ hbjaiaccording to the m eaning ofthe ?-operation.


phenom ena,phenom ena whose physicalnature is m ore intuitive than that

ofthequantum world.Realnum bershavebeen designed to picturegaining

and losing,com ing and going,pulling and pushing,and othersuch classical

transactions. Q uantum transitions have nothing to do with these actions;

theyaretransactionsofadi�erentkind,which m ust,accordingly,begrasped

in term sofa di�erentkind ofnum bers.Ifsim plernum bersalso fallshortof

thetask athand,then how aboutthenextkind ofnum berin line,com plex

num bers?

Save foran increm entin the com plexity ofthe picturing num ber,letus

conjecture thatthe underlying logic ofquantum -m echanicaltransactionsis

nonethelessthesam easthatofclassicalonesand seewhereitleads.Letthen

the transition function be a com plex num ber,and leteach pairofcom plex

num bersrepresenting oppositetransitionsbedualcounterparts,linked by a

new operation ofdualcorrespondence \� ," hajbi = hbjai�. O peration \� ,"

as before,m ust sym bolize the reversalofthe physicalprocess in question,

and m ust therefore share its properties with \?." A double reversalofa

quantum -m echanicaltransition m ustbringusback totheoriginaltransition,

hbjai�� = hbjai,i.e.\� " isitsown inverse;and thecon ation ofa quantum -

m echanicaltransition from a-tob-system swith anotherfrom b-toa-system s

m ustproduceno transition atall,hbjai�hbjai= r2 R.

W hy a realnum ber? Because,like 0 satis�es0? = 0 and failsto picture

a classical transaction, realnum bers have the property r� = r, as they

failto represent any quantum -m echanicaltransition. W e know this fact

from two separate sources. W e justlearnt heuristically that realnum bers

are inadequate to representquantum -m echanicaltransactions;and we also

know from theearlierresulthajjaii= �ij 2 R thattransition functionsthat

reduce to realnum bers represent no proper transition: in those cases,ai-

system seitherrem ained ai-system s(1)orthey failed to becom eaj-system s

(0) (Figure 12.15). M oreover,because hajjaii�hajjaii= 0 (cf.Eq.(12.28))

represents the extrem e case in which the system s involved in the recovery

operation have nothing in com m on (i6= j),we conjecture that this is the

lower bound ofthe m ore generalrealnum ber hbjai�hbjai,characterizing a

recovery operation where the system sinvolved are partially com patible;in

sym bols,hbjai�hbjai� 0.

W ith theirunderlying physicalm eaning clari�ed,com plex num bersand

com plex conjugation lose their aura ofm ystery,while at the sam e tim e a

new dim ension isadded totheconceptofisolated com plex num bers,nam ely,

thatevery m eaningfulcom plex num berhasan equally physically m eaning-

fuldualcounterpart characterizing an opposite quantum -m echanicaltran-

sition.24 Furtherm ore,with this,thequasigeom etric realm ofthetransition

function re-earns its nam e,since it introduces the notion ofdirection| a

look-alike ofgeom etricthought| withoutyet,however,introducing any no-

24W e saw in Section 4.2 thatPenrose understood com plex conjugation asreversing the

direction in tim e ofthe ow ofquantum -m echanicalinform ation.Thepresent�ndingsgo

deeperintothenatureofcom plex conjugation,asthey do notrely on any idea oftim e,nor

on theinessentialgeom etricdecom position| and ensuing geom etrization asan elem entof

Argand’splane| ofa com plex num berhbjaiinto x + iy.


0r⋆r

rr⋆

⋆

⋆

R

〈b|a〉∗〈b|a〉

〈b|a〉〈b|a〉∗

∗

∗

Figure 12.15: The operation \?" generates the dualcounterpart r? ofa real

num ber r (left). The operation \� " generates the dualcounterpart hbjai� of a

com plex num berhbjai(right).Them eaning ofboth isto reversea physicalaction;

theform era classical,thelattera quantum action.Both are,asthey should,their

own inverses,and the con ation ofa physicalaction with itsreverse resultsin no

action atall.

tionsofshapeorsize.

Now how aboutnum bersm ore com plicated than com plex ones? Could

these be suitable or even m ore suitable? Taking hbjai to be a m ore com -

plicated num ber such as a quaternion,25 would not lead to di�culties as

farasthe rise ofprobability isconcerned,because quaternionsbehave like

com plex num bers with respect to com plex conjugation: qq� = q�q � 0.

However,their adoption leads to di�culties in other areas. At the m eta-

and quasigeom etric level,the productofthree prem easurem entthingsfails

to be associative,26 contrary to the results established in Section 12.3.1.

And,although inessentialto our m ain concerns,quaternions do not work

at the traditionalgeom etric leveleither because,their productbeing non-

com m utative,theydonotform a�eld and cannotthereforeactasthescalars

ofa(Hilbert)vectorspace.Thechoiceofotherhypercom plex num bers,such

asbiquaternions,octonions,hypercom plex num bersproper,etc.,isequally

troublesom e.

However,thisisnotsay thatthe need forcom plex num bers,to the ex-

clusion ofany other types ofm ore generalnum ber,rem ains thus proved.

25A quaternion isa num berqoftheform x0+ ix1+ jx2+ kx3,wherei

2= j

2= k

2= � 1,

ij= � ji= k,jk = � kj= i,and ki= � ik = j.26Taking thecom m utation oftransition functions(in thiscase,quaternions)and things

to be a reasonable assum ption,we �nd:

[P(c)P(b)]P(a)= hcjbiP(cjb)P(a)= hcjbihbjaiP(cja)6=

P(c)[P(b)P(a)]= P(c)hbjaiP(bja)= hbjaihcjbiP(cja);

since quaternions do not com m ute. At this point, one m ight want to invent an extra

\chainingrule"statingthattheonly correctorderofm ultiplication fortransition functions

is one in which the left-hand system on the right m atches the right-hand system on the

left (cf.b in �rst product). This �x appears arti�cialand,although it does not con ict

with m ostofthe forthcom ing calculations atthe m eta-and quasigeom etric level,itdoes

m ake the �ndingsofSection 12.5.3 im possible to obtain.


These �ndingsrathersuggestthatonly com plex num bersare adequate for

thesim ple fram ework ofthism etageom etricform ulation ofquantum theory,

as wellas for that ofthe traditionalform ulation. O ne suspects that ifei-

theroftheseform ulationscould bem odi�ed into m orecom plex alternatives,

quaternions(orotherhypercom plex num bers)m ightbesuitablefora quan-

tum theory justaswell.Thedi�culty liesin fathom ing outsuch a theory;

and thequestion rem ainsasto why attem ptthisfeatwhen a sim plertheory

thatusescom plex num bersalready achievesthedesired explanatory goals.

Postulates and hypotheses should not be feared in physics| so long as

we understand their physicalorigin and m eaning (cf.Einstein’s geodesic

hypothesis).Resting assured thatwenow understand theunderlyingreason

ofbeing ofcom plex num bers,wepostulate thatthetransition function isa

com plex num ber,and thatthe relation between hajbiand hbjaiis

hbjai� = hajbi: (12.31)

W e callthisthetransition-function postulate.Itattributesto com plex con-

jugation (lateron:dualcorrespondencein general)theroleofreversing the

direction ofa quantum -m echanicaltransition.27

Theinterpretation ofhajbihbjaiasaprobability isnow fully justi�ed.W e

have that

Pr(bja)= hajbihbjai= hbjai�hbjai= :jhbjaij2 � 0;28 (12.32)

wherethe probability ofeach individualtransition issym m etric:

Pr(bja)= Pr(ajb); (12.33)

i.e.theprobability ofan a-system becom ing a b-system isequalto thatofa

b-system becom ingan a-system .M oreover,foran exhaustivesetofm utually

27D r.Jaroszkiewicz hasbroughtup the issue ofthe role ofthe observer;in particular,

in relation to reversals ofquantum processes: ifthe observer is an inalienable part ofa

quantum system (involvingalsosystem sunderobservation and apparatuses),how could he

(orhisstate),too,be said to be reversed? From the m etageom etric standpoint,however,

the role ofhum an beingsisofan epistem ologicalratherthan ontologicalnature.Hum an

beings are essentialto quantum physicsbecause quantum theory m ustbe tailored to do

justicetoboth quantum phenom enaand hum an cognition and m ethodologicalcapabilities.

Q uantum theory isthen to bean explicitexplanation ofquantum phenom ena by hum ans

for hum ansbut,in the m etageom etric setting,their role does notextend to an essential

m odi�cation ofm icroscopic system svia \actsofobservation." Herehum ansdo notactas

observersin theusualquantum -m echanicalsenseoftheword;thatis,a hum an looking or

notlooking hasno e�ectupon outcom es| only theirpreparing m icroscopic system swith

m acroscopic devicesdoes.28W e have now identi�ed the realproduct ofa transition function and its dualcoun-

terpart with the squared m odulus ofa com plex num ber,turning it into an arrow. This

isdone atthislate stage partly fornotationalconvenience,and partly to give a fam iliar

look to thetraditionalprobability results.So farasthissection isconcerned,nothing else

should be read into this m ere notation: we initially set out to avoid the geom etrization

ofcom plex num bers| and so we did,with positive new results. This geom etric-ridden

notation,however,becom esessentialin thenextsection,because itisitthatinspiresthe

rise ofthe geom etric state vector.


A

a

B

b

C

c

︸︷︷︸fun tioning ︸︷︷︸fun tioningFigure 12.16: Schem atic representation ofthe succession P(c)P(b)P(a)

of prem easurem ent things transform ing a-system s into c-system s passing

through a certain b-system .

exclusiveoutcom es| a singlea-system can transform into any ofallpossible

bi-system s| we recoverfrom Eq.(12.30)the usualresult

X

i

Pr(bija)=X

i

jhbijaij2 = 1: (12.34)

This,togetherwith Eq.(12.32),�nally im plies

0� Pr(bja)� 1: (12.35)

Before closing this section and m oving onto the geom etric realm ,it is

worthwhileto analysesom epropertiesofprobabilities,seen asdirectconse-

quencesoftheproductofprem easurem entthings.W e study threedi�erent

cases:

1. Theseriesofprem easurem entsP(c)P(b)P(a)(herethe�lterassoci-

ated with P(b) is on)involves a transform ation ofa-system s into c-

system sgoing through a determ ined b-system (Figure12.16).Because

the transition from a-system sto b-system sisindependentofthe sec-

ond transition from b-system s to c-system s,the probability Pr(cjbja)

ofthe whole eventcan beexpressed asPr(cjb)Pr(bja).Thisgives

Pr(cjbja)= hbjcihcjbihajbihbjai= jhcjbihbjaij2: (12.36)

2. Theseriesofprem easurem entsP(c)I P(a)(herethe�lterassociated

with I is o�) involves a transform ation ofa-system s into c-system s

going through eitheroneofallpossiblebi-system s(Figure12.17).Be-

cause the di�erentpossibilitiesofpassing through each bi-system are

m utually exclusive,we �nd

X

i

Pr(cjbija)=X

i

hbijcihcjbiihajbiihbijai

=X

i

jhcjbiihbijaij2: (12.37)

3. The seriesofprem easurem entsP(a)P(c)(here the prem easurem ent

associated with B istotally absent)involvesa straighttransform ation


A

a

B

b1, b2, b3, . . .

C

c

︸︷︷︸fun tioning ︸︷︷︸not fun tioningFigure 12.17: Schem atic representation of the succession P(c)I P(a)

of prem easurem ent things transform ing a-system s into c-system s passing

through allpossible bi-system s.

A

a

B C

c

︸︷︷︸not fun tioning ︸︷︷︸not fun tioningFigure 12.18:Schem aticrepresentation ofthesuccessionP(a)P(c)ofpre-

m easurem entthingstransform ing a-system sinto c-system swithoutpassing

through any bi-system .

of a-system s into c-system s, apparently without going through any

bi-system (Figure 12.18). However,using the quasigeom etric decom -

position relation (12.24),we �nd thatthe probability

Pr(cja)= hajcihcjai (12.38)

isalso equalto

Pr(cja) =X

i

X

j

hajbiihbijcihcjbjihbjjai

=X

i

hajbiihbijcihcjbiihbijai+X

i

X

j6= i

hajbiihbijcihcjbjihbjjai

=X

i

Pr(cjbija)+ interference term s: (12.39)

The extra crossterm sthatappearwith respectto situation (2.) represent

the interference ofallpossible waysofpassing through bi-system staken in

pairs. This is as ifthe system s,besides being able to pass through any

bi-system separately, were also able to pass through allof them at once

when no prem easurem entassociated with observable B ispresent. Thisis

a phenom enon that is classically forbidden. This com parison tells us that

even the identity prem easurem ent thing I (which has the weakest action

asfarasprem easurem entisconcerned)hasphysicalconsequences,sincethe

m ediation ofI associated with B in case(2.) leadstodi�erentprobabilistic

predictions for the m easurem ent ofc-system s than in case (3.),where no

prem easurem entassociated with B ispresentatall.Thisisa positiveresult

12.5 G eom etric realm 223

as far as the experim entalconcreteness of quantum -m echanicalthings is

concerned.

This concludes the characterization ofthe quasigeom etric realm m ade

possibleby theintroduction ofnum bers,m idway between them etageom et-

ric and geom etric realm s. How do the traditionalgeom etric concepts of

quantum theory arise from the m eta- and quasigeom etric foundations we

have justlaid?

12.5 G eom etric realm

12.5.1 State vectors from geom etric decom position

W hereastraditionally statevectorsconstitutethestarting pointofquantum

theory,they arise here as derived| and physically insubstantial| objects.

The quasigeom etric probability relationP

ijhbijaij

2 = 1 is the key to un-

derstand how the introduction ofstate vectors and the geom etrization of

quantum theory takesplace.29

Letususean analogy and exam inea realn-dim ensionalEuclidean space

to which belongthegeneralunitvector~a and thevectors~bi(i= 1;2;:::;n),

which form an orthonorm albasisofthisspace:~bi�~bj = �ij (Figure 12.19).

Thus,if�i isthe angle between ~a and ~bi,by virtue ofPythagoras’theorem

in n dim ensions,we haveX

i

cos2(�i)=X

i

(~bi� ~a)2 = 1; (12.40)

an analogicalgeom etric version ofquasigeom etric property (12.30). This

resulttellsusthatthetransition function hbjaican be geom etrically decom -

posed and thought as the inner product,or overlap,hbj� jai oftwo di�er-

ent arrow \state vectors" to yield now the quantum -m echanicalgeom etric

version ofproperty (12.30) (Figure 12.20). Because,as we have seen,the

physically m ore prim itive concept is that ofthe transition function, and

because its geom etric decom position is only a hum an prerogative| only a

connotation30| statevectorsneed notbeassociated with any concrete,gen-

uinely physicalentities. In this light,state vectors appearas the arti�cial

constructions ofour m inds| which cling to the intuitive picture given by

arrow-likevectorsoutoftheirbuilt-in habitto seek visualization in term sof

fam iliargeom etric concepts. In the lecture course on which thischapteris

based,Dr.Lehto described thepredicam entin which wehaveputourselves

asa resultofthishabitin thisway:

29See Schwinger’s(1960) follow-up article for hisway ofintroducing state vectors. He

then suggests that \the geom etry ofstates" thus introduced is m ore fundam entalthan

\the m easurem ent algebra," which \can be derived" from the form er (p.260). This is

truefrom a m athem aticalbutnotphysicalpointofview,whereobservationsalwayscom e

�rst;we keep m etageom etry asontologically prim ary.30Connotation:som ethingsuggested (twooverlappingarrows,hbj� jai)byaword orthing

(\hbjai")apartfrom thething itexplicitly nam esordescribes(com m on partbetween the

players in a quantum -m echanicaltransition). (Adapted from M erriam -W ebster O nline

D ictionary,http://www.m -w.com )


(~b1 · ~a)~b1

(~b2 · ~a)~b2

~a

θ2

θ1

|~b1 · ~a|2

= cos2(θ1)

|~b2 · ~a|2

= cos2(θ2)

~a · ~a = 1

Figure 12.19:Two-dim ensionalrepresentation ofan n-dim ensionalrealEuclidean

vectorspace,in which a unit arrow vector~a can be decom posed into projections

along orthonorm alaxes~bi with com ponents~bi� ~a satisfyingP

i(~bi� ~a)2 = j~aj2 = 1

by virtue ofPythagoras’theorem in n dim ensions.

Philosophicalproblem sin quantum physicsarise from oursim -

plifying m entalpictures,which wecreateoutofourhabitto use

fam iliar concepts and words. These problem scannotbe solved

by increasing ourknowledge| wehaveim prisoned ourselvesin a

labyrinth ofourown m aking,and wecannot�nd theway out.31

This�ndingabouttheorigin ofthestatevectorisrem iniscentofAuyang’s

(2001)view regarding whatwem ightcalltheconnotative decom position of

physical�elds f(P ) into a localaspect (\spacetim e" P ) and a qualitative

aspect (\quality" f). Auyang argued,in a m anner sim ilar to the present

one regarding the transition function,that �elds are physically indivisible

entities,and thatitisonly ourprerogativeto decom posethem into two con-

stituentparts. However,she m aintained,spacetim e (points)need nothave

any independentphysicalm eaning justbecause we can abstractthem from

the �eld| \our thinking does not create things ofindependentexistence;"

she wrote,\we are not G od" (p.214). W e notice then that,also in the

quantum -m echanicalcase,ourm entalactivities work towards the creation

oforiginally absentgeom etric conceptswith unclearphysicalm eaning.

Itisalso noteworthy thatthe resultaboutthe transition function sug-

gests,contrary to com m on belief,that from a physical (not logical) point

ofview them etric(probability)com es�rstand vectorsariseonly after the

m etric,when thisissplitup in ordertosatisfy thehum an need forgeom etric

visualization.32 This idea can be related to sim ilar concepts in spacetim e

31Physical,m athem atical,and philosophicalfoundationsofquantum theory (2005),Lec-

ture notes.32In Article I(see Related publications),we presented an idea concerning the possible

m etageom etric foundationsofquantum theory which iscontradicted by the present�nd-

ings. W e speculated then thatthe geom etric form ulation ofquantum theory could arise


〈b1|a〉|b1〉

〈b2|a〉|b2〉

|a〉

θ2

θ1

|〈b1|a〉|2

= cos2(θ1)

|〈b2|a〉|2

= cos2(θ2)

〈a|a〉 = 1

Figure 12.20:Riseofquantum -m echanicalarrow statevectorsfrom thegeom etric

decom position ofquasigeom etricpropertyP

ijhbijaij

2 = 1.

physics.M ay itbe the case that,from a physicalstandpoint,the worldline

separation ds com es�rstand thatspacetim epointsfollow only asm entally

helpfulappendages?

From theirrole asgeom etric partsofthe transition function,we gather

thatketsjaicould be associated with incom ing system s,and brashajwith

outgoing system s. By analogy with the transition-function postulate,we

can say thatstate vectorsare linked by som e form ofdualcorrespondence,

jai� haj. Thisis,however,only a transm ogri�cation ofthe physicaldual

correspondence between com plex num bers(transition functions),and need

notcarry with itan autonom ousphysicalm eaning.

From the m etageom etric perspective,the following two geom etric prop-

ertiesfollow autom atically.Interpretinghbjai� = hajbigeom etrically,and as-

sociatingketjaiwith~aand brahajwith~a�,werecoverawell-known property

ofcom plexvectors,nam ely,thattheirinnerproductsatis�es(~b� � ~a)� = ~a� �~b.

Alsotheorthonorm ality condition hajjaii= �ij forvectorsfollowsnow auto-

m atically from earlierconsiderationsofcom patibility ofai-and aj-system s.

12.5.2 D erivative geom etric connotations

Ifthegeom etriccounterpartofthetransition function hbjaiistheinnerprod-

uctofarrow vectorshbj� jai,whatm ay bethegeom etriccounterpartsofthe

from the\overlap ofgeom etry-freethingsj iand theirdualcounterpartsh j"(p.437).By

thiswe m eantthat,ifm etric-free vectorscould befound to bephysically m eaningful,the

geom etric concepts ofprojection and probability would com e second as a consequence.

Here the sense of vectors as non-geom etric concepts is the following. Vectors are not

geom etric concepts in general,but becom e geom etric objects with associated geom etric

m agnitudesafterbeing endowed with a m etric structure.In the lightofthe present�nd-

ings,statevectorsaregeom etricconceptsfrom thestartbecausethey arisefrom geom etric

decom position ofthe transition function;at no tim e are there vectors without a m etric.

Forthisreason,we now leave the olderidea behind.


|a〉B

〈b| or 〈a|

︸︷︷︸

〈b|a〉

|b〉

︸︷︷︸

|b〉〈a|

Figure 12.21: G eom etric representation ofthe prem easurem entprocessin term s

ofthe innerand outerproducts,hbjaiand jbihaj,ofarrow vectorsjaiand jbi.

m etageom etric prem easurem ent and transition things P(a) and P(bja)?

To answer this question,we proceed as we did with the transition func-

tion,nam ely,by �ndinga correspondencewith thegeom etricrealm through

geom etric decom position.

A transition thing P(bja) has the capacity to cleanly turn incom ing

a-system sinto b-system s. W e notice that,geom etrically,thisaction isdis-

played bytheoperatorjbi� haj,which takesin \a-system s"jaiand turnsthem

into \b-system s" jbi,thus:(jbi� haj)jai= jbi.Theouterproduct,then,arises

from the geom etric decom position ofa transition thing into the productof

two separate state vectors,a bra and a ket.W e denotethiscorrespondence

asfollows:

P(bja)g= jbihaj: (12.41)

In particularand in agreem entwith earlierrem arks,

P(a)g= jaihaj (12.42)

representsno transition atall,since (jaihaj)jai= jai.

W hen thetransition function isinterpreted asan innerproductofarrow

vectors,and prem easurem entand transition thingsas an outerproductof

arrow vectors,thewholeprem easurem entprocesscan bereinterpreted geo-

m etrically in two parts:an incom ing one related to the inner productand

an outgoing one related to the outer product. Itis notessentialto do so,

butit is com forting to be able to recover the traditionalgeom etric results

from m etageom etric foundations.

In the incom ing part,we start with system s in state jai before their

preparation into b-system s,and with no system sin the preparation box it-

self. Som e system s are then \absorbed" in the preparation box into state

hbj.In theoutgoing part,wehavesystem sin thepreparation box \em itted"

from state hajinto state jbi,which leave the preparation box| now em pty

ofsystem s| behind. In this geom etric setting,it is then naturalto con-

necttheinnerproducthbjaiwith theinternalrearrangem ent(\absorbtion")

ofsystem s at preparation,and the outer product jbihajwith the external

rearrangem ent(\em ission")ofsystem satpreparation (Figure 12.21).

At this point ofthe geom etric developm ent,and ifso wished,the cor-

respondence between prem easurem entand transition thingsand the hypo-

theticalcontinuum ofm easurem ent results a can be introduced. This can


be done by assum ing that an in�nitesim ally sm allvariation �a ofa m ea-

surem entresulta isphysically m eaningful,and thusthatitissigni�cantto

write

P(a)g=

Za+ �a

a��a

ja0iha0jda0; (12.43)

aswellas

P(bja)g=

Za+ �a

a��a

Zb+ �b

b��b

jb0iha0jdb0da0: (12.44)

However,geom etricexpressionssuch asRa2a1jaihajda haveno m etageom etric

counterpartsatall.

As expected, the transition postulate (12.20) also holds in geom etric

form :

P(djc)P(bja)g= (jdihcj)(jbihaj)= hcjbijdihaj

g= hcjbiP(dja): (12.45)

Theouterproductsjaihajand jbihaj,geom etric connotationsofthepre-

m easurem ent and transition thingsP(a) and P(bja),transform a vector

into anothervector. Therefore,they are (linear)operators.Allotheroper-

ators com e from these basic operators,which are,in turn,connected with

m easurem entvalues.

W e notice next that Herm itian conjugation generalizes the concept of

com plex conjugation in a genuinely physicalway because,unlike the case

ofdualcorrespondence between state vectors,itretainsthe previousphys-

icalm eaning. In e�ect,ifwe com pare the relation hbjai� = hajbi for the

innerproductwith the relation (jbihaj)y = jaihbjforthe outerproduct(op-

erator),we see that in both cases the action of\� " and \y," respectively,

consists in inverting the direction of the quantum -m echanical transition:

(jbihaj)jai= jbiand (jbihaj)yjbi= jai.Further,ifwe acceptthe notation

P(bja)yg= (jbihaj)y; (12.46)

wethen have the m etageom etric relation

P(bja)y = P(ajb); (12.47)

and,in particular,

P(a)y = P(a): (12.48)

Also thislatterresultisphysically reasonable,considering thatthere isno

transition involved in a prem easurem ent.Ifwe take Herm itian conjugation

(including com plex conjugation) as a form of dualcorrespondence, then

the physicalm eaning of dualcorrespondence is to invert the direction of

a quantum -m echanicaltransition. This interpretation applies to m etageo-

m etric things,quasigeom etric com plex num bers (transition function),and

geom etric operators,butnotto isolated state vectors.

W e now have enough knowledge to analyse the com patibility ofobserv-

ables. So far little has been said about the possibility ofP(a) and P(b)


com m uting;whatphysicalresultsdo we obtain in such a case? Using the

transition postulate,we �nd

P(b)P(a) = P(a)P(b)

P(bjb)P(aja) = P(aja)P(bjb)

hbjaiP(bja) = hajbiP(ajb) (12.49)

Using Eq.(12.47) to obtain the Herm itian conjugate ofEq.(12.49), and

m ultiplying term by term ,the transition functionsvanish to get

P(bja)P(ajb)= P(ajb)P(bja); (12.50)

orequivalently

P(b)= P(a): (12.51)

Furtherm ore,we can now useEq.(12.51)to �nd

P(a)P(b)P(a)= P(a)= jhbjaij2P(a); (12.52)

so that

Pr(bja)= Pr(ajb)= jhbjaij2 = 1: (12.53)

To sum up, ifP(a) and P(b) com m ute, transition things P(bja) and

P(ajb)also com m ute,P(a)and P(b)are them selvesequalto each other,

and in transitions involving observables A and B ,any b-system becom es

with certainty an a-system and viceversa. In other words,A and B are

com patible observables.

Finally,since the geom etric counterpart ofthe identity thing I m ust

be the identity operator 1,Ig= 1,and since

P

iP(ai)g=P

ijaiihaij,we

recoverthe geom etric com pletenessrelation

X

i

jaiihaij= 1: (12.54)

W ith itshelp,a generalvectorj iand a generaloperator X can bebuiltin

term sofa vectorbasisfjaiig oroperatorbasisfjaiihaijg,respectively.

12.5.3 G eom etric traces

Thenextnoteworthy conceptisthatofthetrace.Foran operator,thetrace

consistsofthesum ofitsdiagonalelem ents,

Tr(X )=X

i

haijX jaii; (12.55)

which isin generala com plex num ber.Thetrace,togetherwith thedecom -

position relation (12.24)and thecom m utativity ofcom plex num bers,33 now

allows usto �nd enlightening connections between the m etageom etric and

geom etric realm s:

33Cf.non-com m utativity ofquaternions.


1. Thegeom etrictraceleftbehind by them etageom etrictransition thing

P(bja)isthe innerproduct(geom etrized transition function)hajbi:

Tr[P(bja)]g= Tr(jbihaj)=

X

i

hcijbihajcii= hajbi: (12.56)

2. In particular,

Tr[P(ajjai)]g= Tr(jajihaij)= haijaji= �ij: (12.57)

3. Thegeom etrictraceleftbehind bythem etageom etricprem easurem ent

thingP(a)is

Tr[P(a)]g= Tr(jaihaj)= hajai= 1: (12.58)

4. Thegeom etrictraceleftbehind by theproductofm etageom etrictran-

sition thingsP(djc)and P(bja)is

Tr[P(bja)P(djc)]= Tr[P(djc)P(bja)]

g= Tr[(jdihcj)(jbihaj)]= hcjbihajdi: (12.59)

From thisfollowsa noteworthy result:

5. The geom etric trace left behind by the product ofprem easurem ent

thingsP(b)andP(a)istheprobability ofatransition from a-system s

to b-system sorviceversa:

Tr[P(a)P(b)]= Tr[P(b)P(a)]g= hbjaihajbi= Pr(bja): (12.60)

In otherwords,the geom etric �ngerprintleftby m etageom etric things

isquantum -m echanicalprobability.

12.5.4 O bservables

W e �nish this section| and com plete the geom etrization of the theory|

with the geom etric and m etageom etric interpretation ofobservables. W e

startwith theweighted average ofm easurem entresultsfbig,

X

i

Pr(bija)bi=X

i

hajbiihbijaibi= haj

X

i

bijbiihbij

!

jai; (12.61)

wherewegeom etrically identify observableB with an operator B such that

Bg= B =

X

i

bijbiihbij: (12.62)

Then

hajB jai= :hB ijai (12.63)


istheexpected value ofoperator B in statejai(i.e.when m easurem entsare

perform ed on a sam ple ofa-system s),and h(� B )2ijai isitsvariance,where

� B = B � hB i1.

From the geom etric representation (12.62) ofan observable,the eigen-

value equation followsautom atically:

B jbii=

0

@X

j

bjjbjihbjj

1

A jbii=X

j

bjjbji�ij = bijbii: (12.64)

In contrast,forageneraloperatorX =P

ijxijjbjihbijwhich isnotan observ-

able,theeigenvalueequation doesnotfollow.TheHerm iticity ofobservables

also followsautom atically from Eq.(12.62):

By =

X

i

b�i(jbiihbij)

y = B : (12.65)

Thus,an observable B isalways represented geom etrically by a Herm itian

operator, and we do not need to postulate this separately. Particularly,

theeigenvaluesofoperatorsassociated with observables,com ing from m ea-

surem ent results bi,are autom atically real;and di�erent eigenvalues auto-

m atically have orthogonaleigenvectors associated with them (hbjjbii = 0,

i 6= j) since bi-system s and bj-system s are m utually incom patible unless

i= j. These are the bene�ts ofbasing physics on observations instead of

disem bodied m athem atics.

Itisnow only a sm allstep to �nd thata pure observable,i.e.the m eta-

geom etric counterpartofB ,is

B =X

i

biP(bi): (12.66)

How does it di�er from B ? W hen we began building the theory,we in-

troduced observables A;B ;C ,etc., which we assum ed to represent phys-

icalm agnitudes. But we saw in Chapter 4 that allphysicalm agnitudes

are underlain by geom etric associations to becom e geom etric m agnitudes.

These are now pictured by geom etric operator B .34 O bservable B ,on the

other hand,is a linear com bination ofm etageom etric things. It involves

not m easurem ents| activities producing physicalvalues| but prem easure-

m ents| activities producing prepared system s. In fact, this is a helpful

34W e m ay com plem ent the geom etric picture underlying B ,and discover som e m ore

connections,by �nding outwhatgeom etric object representsthisoperator and how size

isattached to it.O perator B isassociated with a m atrix related to a linearcom bination

ofouterproductsofarrow vectors. In general,ifthe m atrix operator X = jZihY jcom es

from the outer product ofjY i=P

iyijyii and jZi=

P

jzjjzji, X is taken as an arrow

vectorwith com ponentsxij = yiz�j and length

kX k =

s

X

ij

jxijj2 =

q

Tr(X yX ): (12.67)

A geom etric observable B isthen pictured asan arrow vectorwith one com ponentbijbii

12.6 T im e 231

distinction to visualize how the starting pointsofa geom etric and a m eta-

geom etric theory di�er.

W e callthe geom etry-free m easurable observable represented by B a

quantum -m echanical,purephysicalproperty.W e can then say thata quan-

tum -m echanicalobservableisbynaturem etageom etric.In otherwords,pure

m etageom etricphysicalpropertieslieattheontologicalrootofthegeom etric

physicalm agnitudeswe are used to dealing with in everyday physics.

Finally,what geom etric traces do m etageom etric observables,i.e.pure

physicalproperties,leave behind? Thetrace oftheproductofB and P(a)

is

Tr[BP(a)]g= hB ijai; (12.69)

which tellsusthatthe geom etric �ngerprintleftby a m etageom etric observ-

able isthe quantum -m echanicalexpected value,i.e.statistics.

12.6 T im e

Do m etageom etricthingsand theconstructionsderived therefrom also have

som ethingtotellusabouttim easaconceptinherentin quantum -m echanical

system s? O ristim e,afterall,justan externalparam eteralso according to

quantum m echanics? And how can the originalpsychophysicalparallelism

between quantum -m echanicalthings and conscious feelings guide the an-

swersto these questions?

In this section,we study the rise ofm etageom etric tim e| itselfnot a

coordinateorparam eterofany kind| purely from m etageom etricquantum -

m echanicalthings. M etageom etric tim e, in turn,can lead to the rise of

coordinatetim eintervals�tand coordinatetim etappearingin theequations

ofm otion ofclassicalphysicsand non-relativistic quantum m echanics,thus

revealing itsm etageom etric origin.

Besides m etageom etric tim e, we have also found another intrinsically

quantum -m echanicalconceptoftim e butnow atthe geom etric level. This

tim e isliked to the preparation ofquantum -m echanicalsystem sand called

preparation tim e.W e introduce itseparately in Appendix B because ithas

no direct application in this thesis but m ay nonetheless prove usefulfor

futureresearch.

foreach m easurable value bi,and itslength isgiven by

kB k =

s

X

i

b2i =

q

Tr(B 2): (12.68)

W enow noticehow thetraceplaystheroleofan innerproductin orderto producethesize

ofB . W e also learn thatthe geom etric tracesofthe previoussection are generalizations

(because they do not necessarily produce a non-negative or even realnum ber) of the

norm ofm atrix operators. Cases(3.) and (5.),however,can be reinterpreted assquared

operatornorm sifrewritten asTr[P(a)yP(a)]and Trf[P(b)P(a)]yP(b)P(a)g.


12.6.1 M etageom etric tim e

W e are interested in a description of tim e that is connected to physics

through quantum m echanics but that,at the sam e tim e,captures so far

disregarded aspects ofconsciousexperience| m ostnotably,the perm anent

present. Q uantum -m echanical things were introduced with this goal in

m ind. They are globaland belong outside conventionaltim e and space;

theirproductpicturessuccession (yetnotconventionaltem poralevolution)

and theirsum picturesdiscrim ination,both in analogy with the succession

and discrim ination ofconscious feelings form ing the basis ofpsychological

tim e asan experience in a perm anentnow.

Q uantum -m echanicalthings have the potentialforthe new description

oftim e we seek,butnotsim ply assuch.Instead ofa prem easurem entora

transition thing alone,wewould likea sim pleconstruction outofthem that

evokessom e propertiesofthe tim e ofconsciousexperience som ewhatm ore

closely. For exam ple,we would like,�rst and forem ost,a m etageom etric

tim e that doesnotevolve in the conventionalsense ofphysics,thatdeter-

m inesno pastorfuture nor,therefore,a presentin the custom ary sense of

a tim einstant.W ewould also likea m etageom etric tim ein which theevent

ofno (quantum -m echanical)change translatesinto an em pty,orbetter,in-

e�ectual\elem entoftim e," in analogy with thelack offeeling oftim ewhen

consciousnessisinsensitiveto change(e.g.during a dream lesssleep).In ad-

dition,m etageom etrictim eshould also allow fora succession ofitselem ents

to be reversed and thus recover its beginning,in analogy with the exact

repetition ofa conscious feeling. And �nally,a m etageom etric tim e whose

building blockssucceed each othersm oothly,in analogy with thecontinuity

ofconsciousexperience.

W earriveatam etageom etricconceptoftim ehaving alltheseproperties

by searching for an arrangem ent ofquantum -m echanicalthings such that

(i)only apuretransition beinvolved and thusleavenoplaceforany �ltering

action (cf.preparation tim e),and such that (ii) each and every ai-system

associated with an observableA betransform ed into a bi-system associated

with anotherobservableB in a one-to-one correspondence.

Thetransition thingP(bijai)isalm ostwhatwewant.Itsatis�esthe�rst

requirem entbecause,aswesaw earlieron,itinvolvesonly a puretransition

and doesnotcontain any �ltering action.However,thefollowing geom etric

testshowsthatthesaid one-to-onecorrespondencedoesnothold.In e�ect,

(jbiihaij)jaji= �ijjbii6= jbji; (12.70)

which m eansthatany aj-system where j 6= iisleftwithouta counterpart

into which to undergo a transition.Now a sm allm odi�cation givesuswhat

we seek,asthe sumP

iP(bijai)oftransition thingssatis�esboth the said

requirem ents.Itdoesnotcontain any�lteringaction,and itattachesacoun-

terpartto each and every ai-system ,asguaranteed by the sam e geom etric

testasbefore: X

i

jbiihaij

!

jaji=X

i

�ijjbii= jbji: (12.71)

12.6 T im e 233

a1, a2, a3, . . .

B

b1, b2, b3, . . .

Figure 12.22: Schem atic representation of a m etageom etric tim e elem entP

iP(bijai).

W e callP

iP(bijai)a m etageom etric tim e elem ent

35 (Figure 12.22).

W e can justify this provocative nam e from two di�erent perspectives.

From a psychologicalpointofview,wecan do so through theconsciousness-

related properties it displays,shown im m ediately below. From a physical

pointofview,we can retrospectively justifyP

iP(bijai)asm etageom etric

tim e by noticing thatitcan explain the origin oftim e asan absolute para-

m etric (coordinate)interval�tasused in physics. However,we would still

like to justify it as an expression ofquantum -m echanicaltim e by linking

itto the relativistic separation ds,directly connected with clock tim e asa

genuinely m easurablem agnitude(absolutetim eintervals�tarenot).Find-

ing the link between clock tim e ds and m etageom etric tim eP

iP(bijai)is

neverthelessatask dem andingenough tofalloutsidethescopeofthisthesis;

wem ust,forthatreason,leaveitforthefuture,and restcontentwith giving

an outlook on how to approach thisproblem attheend ofthenextand last

chapter.

A m etageom etric tim e elem entsatis�esfouressentialproperties:

1. Two successivem etageom etrictim eelem entslead to onem etageom et-

rictim e elem ent(Figure 12.23):

X

j

P(bjjcj)X

i

P(cijai)=X

ij

hcjjciiP(bjjai)=X

i

P(bijai):

(12.72)

Thissaysthat,in theevolution ofm etageom etric tim e,thesuccession

oftwo tim e elem ents is one tim e elem ent. In other words,from the

perspectiveofcoordinate tim e,m etageom etric tim e doesnotevolve|

ithappens,so to speak,alwaysand never.(Cf.perm anentpresentof

consciousexperience.)

35Here \m etageom etric tim e elem ent" is favoured over \m etageom etric tim e instant"

because the latter refers to an in�nitesim alintervaloftim e,whereas the form er is free

from any geom etric connotations. A m etageom etric tim e elem ent is a m etageom etric

concept,to which itism eaninglessto apply thegeom etric conceptsofpointorlength (be

itnull,in�nitesim al,or�nite).


a1, a2, a3, . . .

C

c1, c2, c3, . . . c1, c2, c3, . . .

B

b1, b2, b3, . . .

a1, a2, a3, . . .

B

b1, b2, b3, . . .

︸︷︷︸

Figure 12.23: Schem atic representation of a physically possible successionP

iP(bjjcj)

P

iP(cijai)oftwo m etageom etrictim eelem entsresulting in onetim e

elem entP

iP(bijai).

2. Thesuccession ofthreem etageom etric tim e elem entsisassociative:

X

k

P(bkjdk)

2

4X

j

P(djjcj)X

i

P(cijai)

3

5 =

2

4X

k

P(bkjdk)X

j

P(djjcj)

3

5X

i

P(cijai)=X

i

P(bijai): (12.73)

3. There exists a m etageom etric tim e elem entI =P

iP(xijxi),with

xiassociated with any suitable observable,which doesnotchangeany

preceding orsucceeding m etageom etric tim e elem ent:

X

j

P(bjjbj)X

i

P(bijai)=X

i

P(bijai)=X

i

P(bijai)X

j

P(ajjaj):

(12.74)

(Cf.insensitivity ofconsciousnessto change.)

4. Any m etageom etrictim eelem entcan beturned backwards,cancelled,

or undone with another preceding or succeeding m etageom etric tim e

elem ent:

X

j

P(bjjaj)X

i

P(aijbi)= I =X

i

P(aijbi)X

j

P(bjjaj): (12.75)

(Cf.repetition ofconsciousfeeling orexperience.)

W e can im m ediately see that the above four properties constitute the

fouraxiom sthatde�nea group G with theproductasthegroup operation.

Thefourgroup axiom sare:

1. Closure:ifA;B belong to G ,AB belongsto G ;

2. Associativity:ifA;B ;C belong to G ,A(B C )= (AB )C ;

12.6 T im e 235

c1, c2, c3, . . .

B

b1, b2, b3, . . .

≀a1, a2, a3, . . .

C

c1, c2, c3, . . .

Figure 12.24: Schem atic representation of a physically im possible successionP

iP(cijai)

P

jP(bjjcj)oftwo m etageom etrictim e elem ents.

3. Existence ofan identity: ifA belongs to G ,there exists I such that

AI = A = IA;

4. Existence ofan inverse:ifA belongsto G ,thereexistsA �1 such that

AA �1 = I = A �1 A.

However,we notice thatthere existsone signi�cantdi�erence between this

traditionalgroup structure and that ofthe m etageom etric tim e elem ents

when it com es to the group operation. O n the one hand,the elem ents of

the traditionalgroup are,so to speak,structureless,while,on the other,

thegroup ofm etageom etrictim eelem entspossessesinternalconstitution:a

m etageom etric tim e elem entconsistsofa sum oftransitionsP(bijai)from

certain ai-system s to certain bi-system s (cf.elem ent A). As a result,the

group operation consisting oftheproductofsuccessive tim eelem entsisnot

free butinstead physically restricted:from a physicalstandpoint,itisonly

possible to have products in which the outgoing system in the factor on

the rightisassociated with the sam e observable asthe incom ing system in

the factoron the left.W e callthe group ofm etageom etric tim e elem entsa

physically restricted group.36

In particular,as a consequence ofits inner structure,the question of

whetherthe physically restricted group ofm etageom etric tim e elem ents is

Abelian ornon-Abelian isnow nota reasonablequestion to ask.In general,

whenevera succession oftwo elem entsisphysically possible,e.g.

X

j

P(bjjcj)X

i

P(cijai); (12.76)

thesuccession determ ined by the inverse order,

X

i

P(cijai)X

j

P(bjjcj); (12.77)

doesnotm akephysicalsenseasa m etageom etric tim eelem ent,asshown in

Figure 12.24.(Cf.continuity ofconsciousexperience.)

36In a conventionalgroup,the identity isunique.Assum ing I1 and I2 are two di�erent

identities,we �nd I1I2 = I1 and I1I2 = I2,and therefore I1 = I2,which contradicts the

hypothesis. In this respect,the existence ofm yriads ofdi�erent identities in the group

ofm etageom etric tim e elem entsdoesnotlead to any inconsistenciesbecause thephysical

restriction on what is a possible product only allows the sam e identity to be m ultiplied

by itself.


Farfrom �ndingtheseincongruitieswith thetraditionalconceptofgroup

distressing,we believe that the distinctiveness displayed by the group of

m etageom etrictim eelem entsm akesthem even m oreappealingfrom aphysi-

calpointofview.Herewewitnessyetanotherexam pleofphysicsgivingrise,

foritsown independentreasons,to a new bitofm athem atics| m athem atics

doesnotrulephysicsbutisonly itscom plianttool.

Next we pursue the geom etric interpretation of m etageom etric tim e,

leading to an understandingoftim eintervals�tand,in general,tim et.The

key to thispursuitwilllie in the physicalrestriction im posed on the group

structure ofthe m etageom etric tim e elem ents and in the circum stances of

itsdisappearance.

12.6.2 T he rise ofcoordinate tim e

Coordinate tim e intervals�tand coordinate tim e tcan be understood asa

particular outgrowth ofm etageom etric tim e. To achieve this,we start by

�ndingthegeom etriccounterpartofa m etageom etrictim eelem ent,nam ely,

X

i

P(bijai)g=X

i

jbiihaij= :Tba: (12.78)

W ecalloperator Tba thetim e-elem entoperator.Thewell-known claim that

in quantum m echanicsthereexistsno universaltim eoperator37 istherefore

true for in�nitesim alintervals,or instants,�t ofcoordinate tim e,but not

fortim e elem ents.O perator Tba,however,isnotHerm itian in any case.In

fact,we �nd

Ty

ba=X

i

(jbiihaij)y =

X

i

jaiihbij= Tab: (12.79)

By interpreting geom etrically allearlierresultsapplyingto m etageom et-

rictim e,we�nd thatwhatholdsform etageom etrictim eelem entsalsoholds

forgeom etric tim e-elem entoperators.The geom etric evolution oftim e ele-

m ents isgiven by the product TbcTca. Based on this,we can now see that

the properties (1.){(4.) above for m etageom etric tim e elem ents hold for

tim e-elem entoperators:

1. Closure: TbcTca = Tba;

2. Associativity: Tbd(TdcTca)= (TbdTdc)Tca;

3. Existence ofan identity Txx: TbbTba = Tba = TbaTaa;

37The widespread beliefthatthe tim e coordinate constitutesa specialkind ofproblem

in quantum m echanics,afterPauli’sfam ousargum ent,hasbeen uprooted,forexam ple,by

Hilgevoord (2002).Heaccountsforthisfalseim pression asa confusion between spacetim e

coordinatesx;tand thedynam icvariables q;� ofsystem s:whereasin quantum m echanics

coordinate tim e tis a param eter exactly as in Newtonian m echanics| and should there-

fore not be attem pted to turn into an operator| particular system s (clocks) m ay adm it

particular dynam ic tim e variables �,in analogy with position q. The properties of�,or

whether it exists at all,are not universalbut dependent on the system ofwhich � is a

variable.

12.6 T im e 237

4. Existence ofan inverse: TabTba = 1 = TbaTab.

Therefore,tim e-elem ent operators also form a physically restricted group.

M oreover,from property (4.) and Eq.(12.79) follows the new result that

tim e-elem entoperators Tba are unitary:

Ty

baTba = 1 = TbaT

y

ba: (12.80)

K nowing,aswedo,thatthephysicalm eaningofHerm itian conjugation isto

reverse the direction ofa transition,thisresultistelling ussom ething that

is perfectly reasonable,nam ely,that to undo a tim e elem ent (T�1

ba) is to

reverse the transitions itinvolves(Ty

ba).Here,then,aretheseedsofunitary

tim e evolution in coordinate tim e tin quantum m echanics.

How should we proceed to recover coordinate tim e from m etageom et-

ric tim e? The �rst thing to note is that this recovery,leading to allthe

usualequations ofquantum m echanics,is not a necessary but a possible

consequence.Thisisenough and allthatwe require.

Coordinate tim e can be recovered if the physically restricted group of

unitary tim e-elem ent operators Tba is replaced by a new group oflikewise

unitaryoperatorsU ,in which thediscreteinnerstructure(ba)ofthetim eel-

em entsislostand replaced by som esetofcontinuouscoordinateparam eters

x1;x2;x3;:::on which U depends,U (x1;x2;x3;:::).How m any coordinate

param etersshould we choose,and on whatphysicalgrounds? Nothing pre-

vents us from choosing,for exam ple,x1;x2,and x3 and associating them

with thethreespatialcoordinates.Buton whatgroundswould wedo this?

W hy choose threecoordinate param etersand notfour,�ve,orjustone?

The only physicalidea behind the operators U left to guide us is that

they originate from the operators Tba,which are connected with quantum -

m echanicaltransitions.W eproposed thatthesedo notbelong in spacetim e,

butthatatpreparation they projectthem selvesonto spacetim e asa linear

sequenceofeventsin analogy with theprojection ofthestream ofconscious-

ness.Theheuristicvalueofthispsychophysicalparallelism isnow asfollows.

As a geom etric spatiotem poralconcept (as we now exam ine),a transition

takes place instantaneously in an in�nitesim alinstant of(unidim ensional)

tim e.Thissuggeststhatweshould choose U to depend only on onecoordi-

nateparam eter,callitan in�nitesim al(cf.instantaneity)coordinateinterval

�t,and associate itwith the usualphysicalconceptoftim e instant.

W e call the unitary continuous one-param eter operators U (�t) tim e-

instantoperators,also known astim e-evolution operators.Thereplacem ent

leading from tim e-elem entoperatorsto tim e-instantoperators,schem atized

in Figure 12.25,is seem ingly sm all,but in fact it is dram atic. Seem ingly

sm allbecause a tim e instant(assm allaswe wish yetnon-null)hassubsti-

tuted an extensionless tim e elem ent;in fact dram atic because in question

isnotthe di�erence between an in�nitesim alintervaland an extensionless

point,butbetween theattributeoftem poralextension (beitnull,in�nites-

im al,or�nite)and the m eaninglessnessofthe application ofthisattribute

to tim e elem ents. In thisdram atic passage,the innerstructure ofthe tim e


Tba U(δt)time element time instant

Figure 12.25: Replacem ent leading to the obtention ofevolution in coordinate

tim e from tim e-elem ent operators,in turn originating from m etageom etric tim e

elem ents.

elem ents is lost. As a result,the operators U (�ti) becom e free ofphysical

restrictionsconcerning m ultiplication and form now an unrestricted group.

To recover from this basis the equations ofm otion ofnon-relativistic

quantum m echanics, we start by noting that the closure property ofthe

unrestricted group ofunitary operators U (�ti)dem andsthat

U (�t1)U (�t2)= U (�t1 + �t2): (12.81)

Furtherm ore,assum ing evolution in coordinate tim e to be continuous,we

have

U (�t)= 1+ �U (�t): (12.82)

W hatisnow the form of�U (�t)? BecauseU isunitary,then

1 = UyU = (1+ �U

y)(1+ �U )= 1+ �Uy+ �U + O [(�U )2]; (12.83)

which im plies

�Uy = � �U: (12.84)

Therefore,�U isanti-Herm itian and we can write �U = � i�V ,where�V is

Herm itian.W e obtain

U (�t)= 1� i�V (�t): (12.85)

The factthatthe group property (12.81)holdsfortim e-evolution oper-

ators U (�t),togetherwith (12.82),now requiresthat

�V (�t1)+ �V (�t2)= �V (�t1 + �t2): (12.86)

Thism eansthatitispossible forusto take �V (�t)to be ofthe form �t!,

where ! isHerm itian,doesnotdepend on �tanym ore,and isnotitselfan

in�nitesim al,although itcan depend on t.W e obtain

U (�t)= 1� i�t!: (12.87)

In view thatourcontextispurely quantum -m echanical,wecan justi�ably38

usedim ensionalanalysisto �nd that[!]= [tim e]�1 = [energy][~]�1 ,so that

we can take ! = H =~,where the Ham iltonian operator H works as the

generatoroftim e translations.Finally,we obtain

U (�t)= 1�i

~�tH ; (12.88)

38W hatcannotbejusti�ed withoutpreviousknowledgeofquantum theory isthechoice

of~ overh.

12.6 T im e 239

where H isHerm itian and can depend on t.

Alltheequationsofm otion ofquantum m echanicsfollow from Eq.(12.88).

Because

U (t+ �t)� U (t)= [U (�t)� 1]U (t)= �i

~�tH U (t); (12.89)

weget

U (t+ �t)� U (t)

�t= :

dU (t)

dt= �

i

~H U (t); (12.90)

which isSchr�odinger’s equation ofm otion forthetim e evolution operator:

i~dU (t)

dt= H U (t): (12.91)

Notice thatthe linearity ofthisdi�erentialequation need notbe assum ed,

but follows directly from the continuous group structure and unitarity of

theevolution operators U (�ti).

Schr�odinger’s equation for a geom etric state vector results as follows.

Sinceja;t+ �ti= U (�t)ja;ti,we have

ja;t+ �ti� ja;ti= [U (�t)� 1]ja;ti= �i

~�tH ja;ti: (12.92)

Dividingon both sidesbythein�nitesim al�t,we�nd Schr�odinger’sequation

ofm otion fora state vector,nam ely,

i~dja;ti

dt= H ja;ti: (12.93)

Finally,weobtain Heisengberg’sequation ofm otion fora tim e-indepen-

dentoperator A (in the Schr�odingerpicture). Because we can use U (t)to

m ake a ket evolve in coordinate tim e,and therefore U y(t) to m ake a bra

evolve in coordinate tim e,we have

hb;tjA Sjb;ti= hb;0jU y(t)A S U (t)jb;0i; (12.94)

whereweidentify

Uy(t)A S U (t)= :A H (t) (12.95)

with theHeisenberg operator A H (t),where A S = A H (0).Using Eq.(12.91)

to di�erentiate Eq.(12.95), we �nd Heisenberg’s equation of m otion for

A H (t):

i~dA H (t)

dt= � [H ;A H (t)]: (12.96)

Thus,allm ajor traditionalequations ofm otion in coordinate tim e can be

explained on the basisofm etageom etric tim e elem entswhich,in turn,rest

solely on m etageom etric things.

Severallayers ofthe veilobscuring the physicalorigin and m eaning of

quantum physicshave now been lifted from in frontoureyes. W e can now

seem oreclearly into theheartofquantum physics| and aswestaretheold


quantum m ysteries in the eye with the fresh vision m etageom etry a�ords

us,we see com plexity and philosophicalconfusion turn to sim plicity and

physicalunderstanding.

For a conceptualappraisalof the m etageom etric picture oftim e pre-

sented here,we turn to thenextand �nalchapter.

C hapter 13

T im e,the last taboo

Eltiem po es la sustancia de que estoy hecho. Eltiem po

esun r��o que m e arrebata,pero yo soy elr��o;esun tigre

que m e destroza,pero yo soy eltigre;esun fuego que m e

consum e,pero yo soy elfuego.1

Jorge LuisBorges,Nueva refutaci�on deltiem po

13.1 T he enigm a oftim e

Them etageom etric idea oftim easthesuccession ofcom plexesofquantum -

m echanicalthingsoutside conventionalspacetim e thatwe arrived atin the

previouschapter isinspired in an essentialaspectofraw consciousexperi-

ence.Atthesam etim e,however,itrunscounterto both ourbeliefofwhat

tim eis,could be,orshould be,aswellaswhatphysicstellsusthattim eis.

W e cam e to a m etageom etric quantum -m echanicaltim e that does not

ow,that\happensalways and never"| the physicalcorrespondentofthe

perm anent present of hum an experience. And yet, we cannot let go of

the feeling that also the past,which we rem em ber,and the future,which

we anticipate, are as realand basic as the present. This beliefis partly

innate,stem m ing from the fact that we have m em ory and the ability to

plan and anticipate,and partly im bibed,stem m ingfrom thecom plem entary

factthatweareeducated (perhapsagainstexperience)to regard tim easan

equipartition past-present-future.M oreover,aswe look to the highercourt

ofphysicsforan answer,weonly getfrom itan indi�erentreply:tim eisonly

a param eter,t,and allthree,past,present,and future,arem ere illusions.

Like Dennettsaid,the m ind isnotan epistem ic engine. Thisisclearly

true. W hat is not so clear is what practicalvalue to attach to this truth.

Shallwe letitguide usnegatively,by distrusting allourintuitions,orshall

1\Tim e is the substance Iam m ade of. Tim e is a river that grabs m e,butIam the

river;a tigerthattearsm e apart,butIam the tiger;a �re thatconsum esm e,butIam

the �re." (Jorge LuisBorges,A new refutation oftim e)

242 T im e,the last taboo

we letitguide uspositively,by distrusting only som e ofourintuitionsbut

notallofthem ?

The �rstalternative isunfeasible.To doubtthe reality ofeverything in

ourconsciousexperienceand dism issitall,2 likeaDescartesreborn,can lead

us nowhere useful. O n what basis would we then build and test scienti�c

theories? Thesecond isa m orereasonable and m easured option.Ittellsus

thatthem ind isnotbuiltto grasp theontologicalessenceoftheworld,but

thatneitherm ustitbeconstantly m isleading usforthatreason.

Now,whatconsciousintuitionsshould we take seriously aswe attem pt

to picture the world,and which should we cast aside? O urperceptions of

colour and ofhot and cold,for exam ple,belong to the latter group;these

are prim itive and sim ple hum an notionsevolved to keep usalive,butthey

arecom plicated constructionsviewed in term sofbasicphysicalconstituents:

com binationsofwavelength,surface re ectivity,lighting conditions,etc.in

the form er case;com binations oftem perature,conductivity,speci�c heat,

heat ow,etc.in thelatter.3

O n the other hand,ourperceptions ofshape,ofdepth (close and far),

and oflight and heavy m atch m ore closely, under norm alcircum stances,

the scienti�c conceptsofgeom etric object,distance,and weightinspired in

them . Thatis,science hasnotyet seen any need to relegate these hum an

feelingsto secondary roles| asithasdone with red,hot,and the sound of

violins. W hat we perceive as straight,the straight-edges ofphysics verify

(save forspecially fram ed opticalillusions);whatwe perceive asclose,the

m easuring rodsofphysicstellusisa shortdistanceaway;whatweperceive

asheavy,thescalesofphysicstelluspossessesconsiderableweight.In these

cases,them ind doesa good job asan epistem ic engine.

W hatabouttim e? Isita consciousexperiencethatweshould dism issin

theform ulation ofaphysicaltheory becauseitisostensibly withoutasim ple

counterpartin the world,orisitan intuition ofsom ething basic enough to

bephysically relevantand useful? W ith thiswecom eback fullcircleto the

early question ofthePreface:doestim ehavean own structure,oritisonly

a hum an sensation like the feelings\happy," \red," and \hot"?

W e do not know. Butat least this m uch can be said forpsychological

tim e.Asm entioned in Chapter10,thisform ofexperienceisso ingrained in

hum an thoughtthat,sinceitsbeginnings,physicshasdescribed thechanging

world guided by it.Thefeeling ofserialchangeispictured| albeitpoorly|

by a background m athem aticalparam etertalong which we im agine things

to ow; a param eter in term s of which we write equations of things in

change,butforwhich physicshasno equation describing its own properties

and behaviour. W e see,then,that in a sense physics takes psychological

tim ein earnestbut,unableto pictureitin any depth,rem ainscontentwith

a m inim alistic account ofit as \t." As a result ofthis m ixture of close

acquaintanceship and im potence for furtherelucidation,rare is the theory

2Excepting,thatis,that\there isa thought." Notm uch help in that,though.

3See (D ennett,1991,pp.375{383) for colour and (Foss,2000,pp.80{82) forhotand

cold.

13.1 T he enigm a oftim e 243

in which the innocent-looking sym bol\t" doesnotappear,butjustasrare

is encounter with a questioner who dem ands,\W hat does ‘t’stand for?"

W e are left to gather its m eaning from our psychologicalexperience and,

physicsbeingunableto say m ore,wearekindly asked to refrain from asking

awkward questions.

Besides this scienti�c denialofany substance to any aspect oftim e as

psychologically experienced,there exists another sense in which even our

com m on-sensebeliefofwhattim eisdi�ersfrom ourfeeling ofit.Perhapsas

aresultofthephysicalworldview,weareeducated toregard tim easan equal

division into past,present,and future,butthe conscious feeling oftim e is

notprecisely alongtheselines.W ecertainly rem em berthepast,wecertainly

anticipate the future,butdoesnotthe present| thisconspicuous\now" in

which we exist| atany rate feelin�nitely m ore especialand distinctthan

thepastwerem em ber(in the present)and the futurewe anticipate (in the

present)? W hy is the \now" a specialtim e? Is this a sign ofsom ething

physically essentialorjusta psychologicalaccident?

Naturalscience issilenton these m ysteries| on the enigm a oftim e. Is

thatbecausephysicsdoesnotneed to explain tim e? Because,perhaps,tim e

isonly a hum an illusion withoutany bearing on thefoundationsofphysics?

Ba�ed by theelusivenatureofthispsychologicalconcept,som ehaveopted

to outrightdeny the relevance oftim e to elem entalphysics,telling usthat

\In quantum gravity,[they]seeno reason to expecta fundam entalnotion of

tim e to play any role" despite the factthat\the nostalgia for tim e ishard

to resist;" they tellus that \At the fundam entallevel,we should,sim ply,

forgettim e"(Rovelli,2000,p.114).4 Ifallthereistotim eisnostalgiabutno

physics,then physicsshould atleastbecapableoftelling usthatthe\now,"

thepast,and thefuturearesuch and such com plicated constructionsofsuch

and such elem entalentities,like it tells us that \hot" is a com bination of

tem perature,conductivity,speci�cheat,and heat ow.O neway oranother,

the enigm a oftim e rem ains.

O therphysicists,however,farfrom dism issingtim esolightly,havetaken

hum an psychologicalexperiencein earnestasa path towardsbetterphysics.

Buccheri,Jaroszkiewicz,Saniga,and othershavevowed foran endophysical

perspective| i.e.taking into account hum an experience| on what has so

farbeen an exophysics| i.e.physicsbuiltsolely upon an \objective reality"

outsideand independentofhum an experience(Buccheri& Buccheri,2005).

They,too,have deem ed the dichotom y between psychologicaland physical

tim e nota failure ofpsychology butofphysics,and have expected that,if

tim e as an experientialproduct is not an illusion,then physics should be

able to accountforit(Buccheri& Saniga,2003,p.3,online ref.).

In particular,and indirectly m aking a connection with theidea ofm eta-

geom etry,Buccheriand Saniga (2003)criticized norm alphysicsforstriving

exclusively towards a quantitative account ofnature,and deem ed it \nec-

essary to go beyond quantifying" (p.4)to grasp the qualitative aspectsof

4See also (Rovelli,2004,p.84)and (Rovelli,2006,p.4).


psychologicaltim e.5 Thispointhad been previously broughtup by Shallis

(1982,pp.153{154),who wrote that

thefactthattheexperienceoftim eisnotquanti�ableputsitinto

an arena ofhum an perceptionsthatareatoncericherand m ore

m eaningfulthan thosethingsthatarem erely quanti�able...The

lack ofquanti�cation oftem poralexperiences is notsom ething

thatshould stand them in low stead,to bedism issed asnothing

m ore than eeting perceptions or as m erely anecdotal; rather

that lack should be seen as their strength. It is because the

experienceoftim eisnotquanti�ableand notsubjectto num er-

icalcom parison that m akes it som ething ofquality,som ething

containing the essence ofbeing...(italicsadded)6

To getan edge on thisproblem and break free from the restricted con-

ception oftim e only as a background param eter,we took the perm anent-

now aspectofpsychologicalexperiencein earnest.M etageom etricquantum -

m echanicaltim eisa step in thisdirection.Com ing to gripswith a new view

oftim e,however,isa tallorder,because the currentview oftim e isdeeply

ingrained in the scienti�c and com m on-sense viewsofthe world.A look at

an abnorm alm ind,however,m ay help ussee tim e in thisnew light.7

Tulving (1989)reportson thecaseofa m an called K .C.,who,upon suf-

fering brain dam age in a m otorcycle accident,becam e am nesiac in a rather

uniqueway:K .C.knowsthepastbutdoesnotrem em berit.W hatdoesthis

m ean? K .C.hasim personalknowledgeabouthim selfand theworld,buthe

does not rem em ber any episode from his past. K .C.can drive a car and

play chess,buthasno m em ory ofever driving a car orplaying chess with

anyone.W hatism ore,K .C.notonly lacksconception ofthe pastbutalso

ofthe future;as far as he is concerned,there is only the present. Tulving

writes:

He cannotconjure up im agesabouthisfuturein hism ind’seye

any m orethan hecan do so abouthispast.W ithouttheability

torem em berwhathehasdoneortocontem platewhatthefuture

m ightbring,K .C.isdestined to spend the rem ainderofhislife

in a perm anentpresent.(Tulving,1989,p.364)

Thissoundstragic,butisitnotthere| in theperm anentpresent| wherein

a lessdram atic sensewe allspend ourlivesafterall?

5Echoing J.Sanfey,Buccheri(2002,pp.2{3,online ref.) expressed that the physical

problem ofpsychologicaltim e m ay even be related to a so far elusive understanding of

consciousness. Extending this idea, closing the 2002 NATO W orkshop, Jaroszkiewicz

pondered the future of physics in historical perspective with these thought-provoking

words: \Ifthe eighteenth century m ay be called the century ofthe classicalparticle,the

nineteenth century the century ofthe classical�eld,the twentieth century the century of

the quantum ,then the twenty-�rstcentury m ay turn outto be the century ofthe m ind"

(http://www.pa.itd.cnr.it/buccheri).6Q uoted from (Buccheri& Saniga,2003,pp.5{6).

7For a review of anom alous psychologicalexperiences concerning tim e (and space),

though notnecessarily arising from brain dam age,see (Saniga,2005).

13.2 T he taboo oftim e 245

In the context of the physical enigm a of tim e, K .C.’s case raises an

intriguing question: what would physics look like ifnorm alhum an beings

knew the pastbutdid notrem em beritand neitherhad conception ofthe

future? In principle,since we would know the past as a set ofrecorded

facts,physicscould continueto work in thesam elocaland causalway aswe

know it. Butwould we,in practice,withoutthe suggestive encouragem ent

ofour episodic m em ory and our anticipation ofthe future,have com e to

develop the sam e theories? W ould then any rolesbe attributed to locality

and causality? W ould \t" bea partofphysics? W ould there bedi�erential

equationsforthingschanging in tim e? Ifwe were alllike K .C.,how would

wehave m anaged? O necan only wonder.

Allthis is not to say that the past and future of com m on sense are

nothing butinsubstantialdelusions. The purpose ofthis probing is rather

to �nd a clue,an edgewecan grab onto,fora deeperunderstandingoftim e.

Thequestion we intend,the seed ofdoubtwe m ean to plant,isthis:isthe

presentm ore basic than the pastand futurefrom a physicalpointofview?

And is this hypotheticalphysicalfact actually being m irrored by aspects

ofthe way our brains capture the world? Pondering K .C.’s case,Tulving

hazardsaprom isinginterpretation in thisregard.Hewrites:\Rem em bering

one’s past is a di�erent and perhaps m ore advanced achievem ent of the

brain than sim ply knowing aboutit" (p.361,italicsadded).Ifthiswereso,

the psychologicalpresent would be m ore elem entalthan the psychological

pastand future. And ifthe m ind hasany role in inform ing physicsin this

connection,then perhapsa kind ofperm anentpresentiswhatphysicaltim e

ism adeof.

13.2 T he taboo oftim e

Aswepresented theenigm a oftim e,werem arked thata conception oftim e

otherthan asa featurelessparam eterrunscounterto physicalcustom ,and

thata conception oftim eotherthan asan equipartition past-present-future

runscounterto ourpartly innate,partly im bibed beliefofwhattim e is.In

the form ercase,we violate scienti�c custom ;in the latter,we provoke the

em otionalaversion ofcom m on sense.

W erem arked thatthequestion,\W hatistim e?" isnotposed in physics,

as ifthe tools ofscience were helpless to answer it;we behave as ifonly

speculative philosophers had anything (useless) to say about it. To ask

the question ofa professionalphysicist, it is m ost likely to be m et with

annoyance.\W ell,tim eistim e,ofcourse!" repliesthescholar,\aparam eter

along which things ow.W hatelse could itbe?"

An annoyed reaction to theprobing question,\W hatistim e?" isa sign

thatwe have trodden upon a taboo.O fthe varying de�nitionsoftaboo,in

thiscontexta taboo isbestdescribed by the �rstreference in the footnote

below as\a ban oran inhibition resulting from socialcustom orem otional

aversion."8

8Taboo,from Tongan adjective tabu: under prohibition. The English word �rst ap-


Hardin (1973,pp.xi{xiii),a self-professed stalkeroftaboos,noted that,

while we like to believe that only prim itive cultures have taboos,we too

in sophisticated societieshave ourown taboos.He explainsthattabooson

objectsand actions,although present,donoto�eranyintellectualchallenges

because,although we observe them ,we can see right through them | they

are rulesofgood taste,notsacred culturalelem ents. W ord-and thought-

taboos,however,are quitea di�erentm atter.

A word-taboo m akesreference to a subjectthatisoutside the range of

topics we are ready or willing to discuss. Two excellent, com plem entary

exam ples ofword-taboos in currentsociety are \overpopulation" and \the

desirability ofeconom icalgrowth." These are tabooed topics whose m ere

m ention iso�-bounds.Theylieattherootofm ankind’sm ountingecological

woes(e.g.poverty;land,air,water,and sound pollution;tra�ccongestion;

deforestation;nuclear waste;species extinction;globalwarm ing;etc.),but

nowherebesidesin ignored publicationsarethesewordsdared tobeuttered.

Instead,wedodgetaboo by believing thattechnology willsolveourecologi-

calproblem stoo,butin a �niteworld with unrestrained population growth,

thishopecan only beillusive.9

From a taboo on expression to a taboo on thought,i.e.from word-to

thought-taboo,itisonly a sm allstep.Thelesswe heara word orquestion

uttered, the less capable we becom e to think about the problem ,falling

victim ofa thought-taboo: \A word-taboo held inviolate for a long tim e

becom esa taboo on thinking itself(forhow can we think ofthingswe hear

no wordsfor?)" (p.xi).

In physics, the question about the nature of tim e, in the sense that

it could be anything else than an unexplainable background param eter,is

taboo.But,we believe,the nature ofthisthought-taboo on tim e isspecial

and unlike any otherwe m ay be victim sof.Asthe provocative title ofthis

chapterdeclares,tim e isthe lasttaboo we m ustdealwith| lastbecause of

being rooted deepest,to di�erent degrees,in both scienti�c and com m on-

sense thoughtand language;lastbecause the received view oftim e isknit

into thefabricofscienceand society,and to changeitisto changeboth the

scienti�c and com m on-sense viewsoftheworld.10

peared in Captain Jam es Cook’s logbook \A voyage to the Paci�c ocean" from 1777.

D uring his stay in the Polynesian island ofTonga,he wrote that the word \has a very

com prehensive m eaning;but,in general,signi�es that a thing is forbidden...W hen any

thing isforbidden to beeat,orm adeuseof,they say,thatitistaboo." (TheFreeD ictio-

nary,http://www.tfd.com )In thissense,the W estern word originatesfrom Tongan tabu,

but sim ilar words also exist in other Paci�c languages: \[L]inguists in the Paci�c have

reconstructed an irreducible Proto-Polynesian *tapu,from Proto-O ceanic *tabu ‘sacred,

forbidden’(cf.Hawaiian kapu ‘taboo, prohibition, sacred, holy, consecrated;’ Tahitian

tapu ‘restriction,sacred;’M aoritapu ‘be underritualrestriction,prohibited’).The noun

and verb are English innovations �rst recorded in Cook’s book." (O nline Etym ology

D ictionary,http://www.etym online.com )9See (Hardin,1968,1972).

10Tim e as last taboo in the sense ofone com ing at the end ofa series ofrecognizable

taboos we m ay be in need ofconfronting,but not,however,in any way im plying that

with itsoverthrow \theseries[oftaboos]iscom pleted orstopped" (Cf.\last" in M erriam -

W ebster O nline D ictionary,http://www.m -w.com ). W hat new taboos the scienti�c and

13.3 O verview and outlook 247

How shalloneproceed to beingto break thespelloftaboo? Notwithout

a hintofhum our,Hardin suggestsasfollows:

Violating aword-tabooisbad taste,buthewhoisan eccentric|

i.e.,onewhoisoutofthecenterofthings| m ayultim ately heara

callto break thetaboo lesthissilenceperm itthe�nalm etam or-

phosisofword-taboointothought-taboo| ofrepressed stateinto

non-existence. W hen the vocation ofthe eccentric reaches this

painfully insistentpoint,he shoutsthe word from the rooftops.

Thisisa dangerousm om ent;thetim em ustbechosen carefully.

(Hardin,1973,pp.xi{xii)

W e do not know how wellwe m ay have chosen the m om ent and place to

putforth thisunorthodox view ofwhattim e could be| the serialchaining

ofm etageom etricquantum -m echanicalthingsoutsidespacetim em irrored in

theperm anentnow ofhum an consciousness.Butwhilewedo notknow how

wellwem ay haveobserved thisrule,in thecourseofthiswork wehavewith

certainty infringed upon anothertacticsofHardin’sneverto \be forgotten

by the am bitiousstalkeroftaboos:Nevertackle m ore than one taboo ata

tim e" (p.30).For,togetherwith the assum ed nature oftim e,we have also

questioned them inortaboo on therelevanceofthePlanck scaleto quantum

gravity and,m ore signi�cantly,the taboo on the nature and relevance in

physicalscience ofthe sacred discipline ofgeom etry. W e believe that at

least the thought-taboo on tim e sim ply had to be tackled together with

theoneon geom etry| butthen again,perhapswe have been too am bitious

stalkersofthetaboosofphysics.

The deeply felt hum an dichotom y between the life-as-irretrievable- ow

and perm anent-present aspects ofconscious experience has been captured

m asterfully and with rare insight by the Argentinian writer Jorge Luis

Borges.Heconcludeshisessay \A new refutation oftim e" in an attem ptto

com e to term swith thisseem ing contradiction,thus:

Tim e isthe substance Iam m ade of.Tim e isa riverthatgrabs

m e,butIam theriver;a tigerthattearsm eapart,butIam the

tiger;a �rethatconsum esm e,butIam the�re.(Borges,1976,

p.187)

W hich ofthe two feelingsshallwe hang on to?

13.3 O verview and outlook

Physics m isrepresents tim e and fails to picture it because it views it as a

property superim posed on the existence ofthings| forphysics,thingsexist

in tim e. Buttim e isnota property ofthe existence ofthings;itisinstead

an experientialproductrelated to consciousness. Thisisto say thatthere

com m on-sense worldviewsofthem ore distantfuturem ay havein store forus,thereisno

way oftelling now.


is no actual\observation oftim e" or \perception oftim e," but only the

observation or perception ofphenom ena or events. How does tim e enter

the picture,then? These events| psychologically,com plexesofelem entary

feelings| arechained serially into thestream ofconsciousness.Itisthis ow

ofeventsin theperm anentpresentthatwe calltim e.

To give a physicalexplanation oftim e,we need to know whatplaysthe

partofthesethreekey conceptsofthings,events,and the ow ofeventsjust

m entioned in generalrelativity and in quantum m echanics,as wellas �nd

the way the theoreticalpicturesfrom both sidesare interconnected.

In generalrelativity,the role ofthings is so far vacant,and we cannot

say whether the players ofthis part should be �rst recognized in order to

progress,or whether this is inessential. Neither can we say whether these

m issinggeneral-relativisticthingsaretobem etageom etric,norwhatam eta-

geom etric form ulation ofgeneralrelativity m ay look like.

Atany rate,weintuitively seek forrelativity theory a foundation sim ilar

to the m etageom etric one we gave to quantum theory. In the latter,we

explained theappearanceofstatevectors,jaiand jbi,and thefullgeom etric

background traditionally built upon them out ofthe decom position ofthe

physically m ore essentialprobability m easurem entvalues jhbjaij2;and the

rise ofthese valueswere,in turn,explained in term sofphysically yetm ore

elem entalm etageom etric things,P(a) and P(b),associated with physical

activities ofprem easurem ents. Likewise,in relativity theory,we explained

theappearanceofthem etric�eld g�� (x),di�erentialarrow vectorsdx�,and

the fullgeom etric background builtupon them outofthe decom position of

the physically m ore essentialclock m easurem ent ds =p� g�� (x)dx

�dx�.

Butfrom whatm etageom etric,physically yetm oreelem entalfoundation do

these clock-m easurem entvaluesarise?

The role ofevents,on the otherhand,iscrystal-clear:itistaken up by

spacetim e events,understood not as spacetim e points but as phenom ena.

Betterstill,to bring forth theconnection with tim e,eventscan bepictured

by clock readingssA atthe eventA in question.Finally,the ow ofevents

also �ndsa naturalcharacterization in term softhe clock separation �s =

sB � sA ,correlating the pair ofevents A and B . Centralto this idea of

classical event correlations �s is that the focus is placed on the shared

connection between eventsinstead ofon theeventsthem selves.

In quantum m echanics,weidenti�ed thingsasm etageom etricthings,for

exam ple,P(bja). The role ofevents is now naturally �lled by the m eta-

geom etric tim e elem entsP

iP(bijai),which we can view as com plexes of

m etageom etric things. How can we then give a stillm issing characteriza-

tion ofthe ow ofthese events? W hat shallbe the m etageom etric toolof

thought that allows us to m ove forward on this ank? A toolwell-suited

to the task ofdescribing m etageom etric quantum -m echanicalcorrelations

associated with tim e elem entsiscategory theory (Table 13.1).

The basic com ponents ofa category are objects and relations between

them .Denotingobjectsas; 0,and arelation rbetween them asr

�! 0,

a category isdeterm ined asfollows:

13.3 O verview and outlook 249

Psychology Q uantum m echanics G eneralrelativity

things feelings m etageom etric

thingsP(bja)

?

events com plexes

offeelings

m etageom etric

tim e elem entsP

iP(bijai)

spacetim eeventsA;

clock readingssA

ow ofevents stream of

conscious-

ness

category-theoretical

relationsr between

groupsoftim e

elem ents

correlations �s be-

tween events

Table 13.1: Com parison between thethreekey conceptsofthings,events,and the

ow ofeventsfora physicaldescription oftim e from the pointofview ofpsychol-

ogy,quantum m echanics,and generalrelativity. The task rem ainsto provide the

m issinglinksbetween thethree�elds;in particular,tofathom am etageom etricfor-

m ulation ofgeneralrelativity and characterizem etageom etricquantum -m echanical

correlationswith the help ofcategory theory.

1. Closure: ifr

�! 0 and 0 s�! 00 are relations,their com position

s�r�! 00isalso a relation;

2. Associativity: ifr

�! 0,0 s�! 00,and 00 t

�! 000 are relations,

s�r�! 00 t

�! 000= r

�! 0 t�s�! 000;

3. Existence ofan identity:ifr isa relation,there existsrelation isuch

thatr�i�! 0=

r�! 0=

i�r�! 0;

4. Existence ofan inverse:ifr isa relation,thereexistsrelation r� such

thatr��r�! =

i�! =

r�r�

�! .11

These four properties look conspicuously fam iliar. They are precisely the

sam e ones satis�ed by a group,but with a di�erence. Now the em phasis

hasshifted from the objects to the relations between the objects. Indeed,

in a category,the inner structure ofthe objects is not a concern;on the

contrary, in a category it is the existence ofrelations that carries essen-

tialsigni�cance. W hat is m ore,a category| unlike its naturalgeom etric

counterpart,a graph| isabstractenough to be naturally free ofgeom etric

denotation:itsobjectsarenotpoints,norareitsrelationsextended arrows.

W e m entioned in passing on pages115 and 225 thata key to the prob-

lem ofthe geom etric ethercould be to look notinto the innerstructure or

physicalreality ofthe spacetim e pointsthem selvesbutinstead atthe clas-

sicalcorrelationsgiven by theintervals�s between them ;in non-geom etric

language,to look atthe classicaltem poralcorrelations �s between events

11The existence ofan inverse relation isnotan axiom ofcategory theory.


asphenom ena fora physicaldescription ofthe ow ofeventsfrom thepoint

ofview ofgeneralrelativity.

Could now category theory,with its shift offocus from objects to re-

lations, be m eaningfully used as a suitable setting for a sim ilar study of

corelations between m etageom etric tim e elem entsP

iP(bijai)? Notreally;

theconnection need notbeso straightforward.Fora physicaldescription of

m etageom etric ow from the pointofview ofquantum m echanics,instead

ofsim ply taking isolated objects (cf.tim e elem ents) or sets ofthem ,the

m ostnaturalchoiceofcategory isthecategory ofgroups:itsobjectsarethe

groupsform ed by m etageom etric tim e elem entsand itsrelationsare group

hom om orphism s.

So far,regarding a connection between \spacetim e and the quantum ,"

thisiswhatwe have achieved. Afterreducing the clock tim e ofrelativistic

physics to the absolute tim e ofclassicalphysics,we found a link between

the latterand m etageom etric quantum -m echanicaltim e:

�s= �t�! U (�t)�! Tba �!X

i

P(bijai):

The challenge that rem ains is to �nd the link between the full- edged

general-relativistic clock tim e elem ent and the quantum -m echanicalm eta-

geom etric tim e elem entfora com bined understanding oftim e.

W e believe that it is through the concept of tim e that quantum m e-

chanicsand generalrelativity are connected| thatitisthe conceptoftim e

thathasthepotentialto give physicalm eaning to thepresently disoriented

�eld ofquantum gravity. Butratherthan ask whattim e is,we should ask

why tim e is.

Epilogue

Ifyou pay a m an a salary fordoingresearch,heand you will

want to have som ething to point to at the end ofthe year

to show thatthe m oney has notbeen wasted. In prom ising

workofthehighestclass,however,resultsdo no com ein this

regular fashion,in factyearsm ay pass withoutany tangible

result being obtained, and the position of the paid worker

would be very em barrassing and he would naturally take to

work on a lower,or atany rate a di�erentplane where he

could be sure ofgetting year by year tangible results which

would justify hissalary. The position isthis:You wantthis

kind ofresearch,but,ifyou pay a m an to do it,itwilldrive

him to research ofa di�erentkind. The only thing to do is

to pay him for doing som ething else and give him enough

leisure to do research for the love ofit.

SirJoseph J.Thom son,The life ofSir J.J.Thom son

In thisthesis,webroughtto attention thetightrelationship existing be-

tween geom etric thoughtand thehum an m ind.Resem bling therem arkable

giftgiven K ing M idas by Dionysus,the m ind hasgreatly pro�ted from its

M idas touch ofturning into geom etry everything itbeholds,as a world of

understanding opened up through it;but like K ing M idas’gift,geom etry

too turned from giftto curseby enslaving us.O urexclusiveuseofgeom etry

is to blam e for the vicious perpetuation ofseveralm ysteries in our physi-

calworldviews,as wellas for restricting thought towards new and better

physics. AsK ing M idasdiscovered upon his�rstm eal,turning everything

wetouch into gold| gold though itis| isnotasdesirableaswem ighthave

thought.Butwhereshallwe�nd ourriverPactolusto divestthem ind ofits

K ing-M idastouch,ofthissuperlativegiftturned likeacurseupon ourselves?

Although farfrom �nding riverPactolus,atleastpartly through m eta-

geom etric thinking and theorizing,and based on existing observations,we

m anaged to clarify theexperim ental,m athem atical,and psychologicalfoun-

dationsofrelativity and quantum theories.W efounded theform ergeom et-

rically on clock-reading separations ds,and the latter m etageom etrically

on theexperim ent-based conceptsofprem easurem entand transition things.

M etageom etric thingswere inspired in the physically unexplored aspectof

tim e as a consciousness-related product,which allowed us to develop the

252 Epilogue

concept ofm etageom etric tim e. W e deem ed it necessary to discover the

connection between quantum -m echanicalm etageom etric tim eelem entsand

general-relativistic clock tim e elem ents ds (classicalcorrelations) in order

to attain a com bined understanding oftim e in the form ofm etageom etric

correlationsofquantum -m echanicalorigin.

In both cases, clari�cation was achieved through the identi�cation of

suitableexperim entsand m easurem entsthatM an hasaccessto and can af-

fect. Physics m ust,after all,be based noton m athem atics or professional

philosophy buton physics| a triviality thatseem sto have been largely for-

gotten. Experim ent is, in this light, both the beginning and the end of

physicaltheorizing. In the �eld ofquantum gravity,however,the physical

referent| thatwhich itisabout| hasnotbeen identi�ed.Thepresentquest

offrontierphysics,an orphan withouta m odelin theworld,isthen a search

foritsown identity.Forthisreason,theidenti�cation ofobservablesrem ains

a centralgoal.

Thekind ofm etageom etric endeavourscontem plated here,however,are

no easy task.Hum an geom etric com pulsionsare signi�cant,and ourim ag-

ination easily succum bs to its geom etric overlords. Consider the following

analogy. Im agine,aspartofthe norm aluniverse we know,an accidentally

static world populated by m otionless inhabitants,who are nonetheless en-

dowed with a good understanding oflength. W hat kind oftoiling against

m entalbarriers m ight they experience, not naturally able to conceive of

m otion (forno evolutionary advantage attached to thisability),in orderto

discover and form ulate the special-relativistic e�ect oflength contraction?

Length contraction itselfwould beperfectly com prehensibleand m easurable

forthese beings,butitsdiscovery could be im possible and itscause would

rem ain an enigm a without an understanding ofm otion. In our case,the

discovery ofquantum -m echanicale�ectsrelated to em pty spacetim em ay be

im possible,and an understandingoftheircausean enigm a,withoutprevious

com prehension ofthem etageom etric thingsfrom which they arise.

In spite ofthe evidentdi�cultiesto forgo geom etric thinking in hum an

theorizing,thesituation isnothopeless.G eom etricthinkingisadeep-rooted

hum an habit,buta habitnonetheless;an extrinsic lim itation forced on the

plastic structure ofthe hum an brain by evolutionary processes,butnotan

intrinsiconeim possibletoovercom e.A cluesupportingthisview isfound in

hum an language| itselfa m ore recentevolutionary product| which,aswe

showed with thisthesis,iscapable ofsupporting geom etric thoughtaswell

asm etageom etric rum inations.Language,asitwere,succeedsin aiding our

m etageom etricthoughtprocessesrightwhereourpicturing abilitiesleaveus

high and dry. W hether,in the space ofeons,the brain ofHom o sapiens

could evolve to naturally support m etageom etric thought in fulldepends

on whetherornot,asthe needsim posed by the environm entsin which we

livegradually change,any evolutionary advantagewould attach itselfto this

unexplored m annerofpsychologicalportrayal. Atsuch a tim e,\describing

thephysicallawswithoutreferenceto geom etry" (like Einstein said)would

be m uch m ore e�ortless and naturalthan it has been for m e to describe

253

the m etageom etric thoughts poured in these pages with these words. Itis

preem inently thepossibility ofhum an change| coupled with therecognition

that the hum an m ind is not,neither now nor ever,a universalcognitive

tool| that m akes the pronouncem ent ofultim ate anthropic principles on

physicalcognition an unwiseactivity indeed.

But leaving behind possibilities that can only be guessed at,and con-

centrating on the im m ediate here and now,whatare the psychologicalre-

quirem ents for the successfuladvancem ent in the physicalsciences ofthe

m etageom etricendeavourwehaveherebutglim psed upon or,forthatm at-

ter,ofany otherpieceofgenuinely new physics? O rto putdi�erently,what

doesittakeforhum an scientiststo becapableofhighly innovativethought?

W hatkind ofm ortalsare up to such a cyclopean task?

In sketchingan answertothisquestion (and justwhen wethoughtwehad

leftevolutionarythoughtsbehind),aparallelcan bedrawnwith evolutionary

theoryassuggested byHardin (1960,pp.258{346).Thekeytothesuccessful

perpetuation ofanyspeciesisforitsm em berstodi�erentiatethem selvesinto

races,each inhabitingareproductively isolated ecologicalniche,forthen the

species as a species enjoys high genetic diversity and is aptto spread into

and exploitvirgin nichesofopportunity(in particular,when confronted with

changesoftheirecosystem s).Speciesallofwhoseconservativem em bersare

�nely adapted only to thepeak of\M ountTory" (p.293)lack thecapacity

to stray away into virgin land: undiscovered \M ountO pportunity" isonly

available to som e ofthe less�nely adapted racesof\M ountRisky," where

each race ofthespeciesevolvesin isolation.

Thesam eisthecasein thesphereofscience.Thedevelopm entoftruly

originalideas requires unorthodox,hereticalthought. This kind ofhighly

originalthoughtneeded to reach the peak ofM ountO pportunity,however,

presupposes detachm ent from the ubiquitous in uence oforthodox ideas,

from the ebbsand ows ofscienti�c fashions. At som e point,the creative

spirit m ust draw his gaze away from his peers and look within him self,

m akingthescienti�ca�airwith thenaturalworld apersonalm atter.12 High

creativity, and the prom ise of M ount O pportunity, require thoughts and

ideastoevolvein isolation.And justliketheblind opportunism ofevolution

can,through the discarding ofun�t,wastefulproducts,create life form s

undream t-ofin the drawing board ofthe m ostintelligent designer,so can,

by the sam e m ethod,the independentconceptions ofm yriadsofgenuinely

autonom ousm en lead tothediscovery of\m orethingsin heaven and earth,"

asShakespeare’sHam letwould say totoday’steam -workingHoratios,\than

aredream tofin yourphilosophy."

A look at frontier physics reveals that the �gure ofthe intellectually

independent m an of science is regrettably far from holding court today;

creativethoughton thebasisofcollective ideasisnottheexception butthe

rule.Beforetheapprovinggazeofthescienti�ccom m unity and ofsociety as

a whole,legions ofscientists and professionalphilosopherssingle-m indedly

12CharlesD arwin,who secluded him selfwith hisfam ily in the English village ofD own

is,coincidentally,a paradigm atic exam ple ofthisway ofproceeding.

254 Epilogue

toiltoday,am ongothertrends,in thediggingofabottom lessholeargum ent,

in the exploration of hidden bends and corners of this universe and the

conjuring up ofm yriadsofothers,in theknitting oftheworld outofstrings

or loops,and| the m ost irresistible,com pelling,and widespread trend of

all| in the exploration ofthe secluded depthsofthe Planck scale. O n the

otherhand,nottofollow such trendsand goone’sown way isseverely looked

down upon:notoneoftheo�cialroadstoquantum gravity.Butifscienti�c

breakthroughs(asin K uhnian revolutions)iswhatwe are after| and so it

would seem from the grand visions ofquantum gravity| these have never

been known toarisefrom theteam work ofhundredsbutfrom theintellectof

onecreativespirit.Seeing thesam eold world in a new lightislargely a one-

m an a�air.Echoing these thoughts,and even the evolutionary parallelism ,

we hearthe reverberation ofRichard Feynm an’swords:

Theonlythingthatisdangerousisthateverybodydoesthesam e

thing![...]There are enorm ousnum bersofpossibilities...And

we have to explore. So we ought to be running around in as

m any directionsaspossible.13

Yet,as Hardin (1960) observed,this independent attitude does not com e

easy to those already \breathing an atm osphere offellowship and togeth-

erness" (p.343),i.e.to those accustom ed to whatSm olin (2001)described

as \the com fortoftravelling with a crowd oflike-m inded seekers" (p.10).

Afterall,M an isby naturea gregariousanim al.

W e recognize in the form ulaic,repetitious,and unoriginalcharacter of

frontier physics,to borrow Lem ’s (1984b,pp.173{177) words,an \age of

epigonism "14 or\repetitive ful�llm ent" ofthe geom etric physicaltradition.

Anycreativetradition| whetherartistic,scienti�c,oreven natural| isbased

on an underlying space ofpossibilities determ ined by its own m ethods. A

painterwho only usesa straight-edge willonly producestraight-line paint-

ings,justlikeaphysicistwhoonly usesgeom etricm ethodswillonly produce

geom etric theoriesofnature.Creation within an budding creative tradition

isdi�cultbecause,although the�eld ofopportunity isvast,itspossibilities

have notyeteven been m apped out,itsscholarsresem bling m ore explorers

than inventors (cf.m etageom etry). O riginalcreation is easiest when the

tradition hasbeen su�ciently explored butitspossibilitiesnoteven nearly

exhausted (cf.geom etric classicalm echanics and the geom etric di�erential

calculus). However,when a creative tradition has been nearly thoroughly

exploited,theproduction ofnew originalwork within itbecom esextrem ely

di�cult(cf.geom etricgeneralrelativity),leadssom etim estoviableyetm al-

form ed new specim ens(cf.geom etricquantum theory),and m ostofthetim e

reducesto therepetition ofolderproducts(cf.geom etricquantum gravity).

Returning once m ore to the evolutionary picture,the bestexem pli�ca-

tion ofrepetitive ful�llm ent within a creative tradition is found in nature

13In (D avies& Brown,1988,p.196).

14Epigone, from G reek epi \after" + gonos \seed": to be born after; an inferior or

undistinguished im itator,follower,orsuccessor.

255

itself. W hen a new taxonom ic class oforganism (e.g.Insecta,M am m alia)

appears,thepossiblesuccessfulspeciesallowed within thecreativeprinciple

ofthisclassisim m ense (forInsecta ofthe orderofm illions).Asecological

nichesbecom eoccupied,however,speciation stopsand naturecontentsitself

with an incessantrepetition ofitsearliersuccessfulproducts. The creative

tradition has becom e full. The geom etric physicaltradition seem s now to

have reached a sim ilarful�llm ent| itsfatefulage ofepigonism .

Ifforwardnecessarilym eansdi�erent,how com eourpresentsocialfram e-

work,in which scienceischerished sohighly,doesnotencouragethecreative

developm entofthisdiscipline? W hatisthe source ofthis state ofa�airs?

The paradoxicalpredicam ent in which science adm inistrators have placed

the scienti�c enterprise isnothing new,asitwasnoticed by J.J.Thom son

over65 yearsago (opening quote),butitiscorrectto say thatthesituation

hasonly worsened in tim e.No betteranalysisofthepresentpredicam entof

science| and no betterthoughtson which to end thisdissertation| than in

thelucid and uncom prom ising wordsofG arrettHardin:

As it becam e generally realized that an im portant fraction of

theworld’sresearch in purescience wasdoneby academ ic m en,

adm inistrators de�ned research as part of the job, and m ade

productivity in research a criterion foradvancem ent. The con-

sequencesofthism eddling havebeen aboutwhatonewould ex-

pect.Thereisnow a tendency to chooseprojectsthatarepretty

suretogivequick results,and toavoid questionson tabooed sub-

jects...As research has becom e m ore expensive,the academ ic

m an hashad to develop a talentforbegging.He getsa subsidy

from foundationsby telling com m itteeswhathehopestoaccom -

plish with theirm oney ifhegetsit.Thesuccessfulbeggaroften

givesm oreattention to thecom m itteethan hedoesto thescien-

ti�cproblem .Theresult:other-directednessisintroduced into a

realm whereithasnobusinessbeing,therealm ofinner-directed

science.O rthodoxy isencouraged.Thism ay notbetoo bad for

whatwe call\science," forits�eldsare alm ostallfreed now of

con ictwith tradition,and itsm ethodssystem atized tothepoint

whereinnum erableand im m ensely im portantdiscoveriescan be

m ade by m en who are notin the �rstrank ofthe heretics. But

thereisneed forthespiritofscienceto m oveinto �eldsnotnow

called science,into �elds where tradition stillholds court. W e

can hardly expecta com m itteeto acquiescein thedethronem ent

oftradition.O nly an individualcan do that,an individualwho

isnotresponsible to the m ob. Now thatthe truly independent

m an ofwealth has disappeared,now that the independence of

theacadem ic m an isfastdisappearing,whereareweto �nd the

conditions ofpartialalienation and irresponsibility needed for

thehighestcreativity? (Hardin,1960,pp.344{345)

M ay the last survivors ofthese endangered races ofm en conquer the

taboosnecessary to unlock the enigm asofthe cosm os!

Geom etry! You have accom panied m ankind

For longer than itcan rem em ber;

W ell-entrenched within the cham bers ofitsm ind,

You glow asthough relentlessem ber.

Anciently,your shapes enticed the Babylonians;

On clay they would record inscriptions

Reverenced in allour worldviews: from Newtonian

To quantum -world ornate descriptions.

Theories have been presented in your honour;

Correctness based upon your clarity;

Ignorance ispredicated on the goner

To do withoutyour fam iliarity.

Presently,despite your trium phs (which are ours),

Farewellwe bid you,alterego,

Fearing thatyou have run outofyour powers,

Disorienting usall,aswe go!

A ppendix A :

To picture or to m odel?

Throughoutthiswork,theverb \topicture"isused instead ofthem oreusual

\to m odel," and likewise\theoreticalpicture" isused instead of\theoretical

m odel." This is because the form er usage m akes the subjective,theorist-

ridden natureoftheoreticalportrayalin science m ore explicitby putting it

on a par with the equally subjective,artistic renderings ofa painter;the

m eaning of\to m odel," on theotherhand,isam biguous.

M oster��n (1984,pp.131{146)captured therelationship between thekey

notionssurrounding a representationalactivity in the canonicalexpression

\x paintsywhich representsz." An application toeveryday usagegives\the

painter(x)paintsthepainting(y)which representsthem odel(z)." Herethe

m odel| that which is to be subjectively m im icked,em ulated,replicated|

is the thing (e.g.a landscape) or person (e.g.a hum an m odel) which the

painter captures| according to his own biases and tools| through and in

his painting. Thus,the counterpart ofthis expression in scienti�c usage

is \the scientist (x) devises the theory (y) which represents the section of

reality (z)." Asa result,M oster��n noted thatthe use ofthe word \m odel"

m adeby naturalscientists| to denotea theoreticalconstructthatattem pts

to explain a section ofreality| isdiam etrically opposed to thatofeveryday

usage,where by \m odel" we ratherdenote thatsection ofreality which we

\copy," i.e.thatsection ofreality theorized upon by the scientistorartist.

In everyday usage\to m odel" can also referto them aking ofa m aterial

object(e.g.a m aquette ofa building ora wooden plane builtto scale). Is

thism eaning opposed to theonejustm entioned? Doesitm akethem eaning

of\m odel" as \theory"| both som ething we m ake| m ore plausible? Not

at all. W hen we m ake a m aterialm odel,we do not use it as a theoretical

toolto study som ething else,butas a scaled-down version ofthat section

ofreality we are interested in,and which we thus �nd m ore m anageable

to study. It,the m odel,is again the object ofour theoreticalstudy,not

the theoreticaltoolitself. For exam ple,we now apply the laws about the

strength ofm aterialstothem aquetteofthebuilding,orweplacethewooden

plane in a wind tunnelto study its aerodynam ics. This is what M oster��n

called \to serve asa m odel."

Incidentally,thescienti�cuseof\m odel" isalso opposed to them eaning

given itin thetheory ofm odels,which again coincideswith thatofeveryday

258 A ppendix A :To picture or to m odel?

language.Thistheory form ally studiessectionsofreality which itcallssys-

tem s;forevery system ,itconsidersatheory thatwillexplain itsfunctioning.

Ifand only ifa system worksasthetheory states,itiscalled a m odelofthe

theory.

Ultim ately,one can attribute m eaning to wordsin any way one wishes,

and the scientistisfree to continue to callhistheoriesm odelsifthatishis

inclination. But,at least from the perspective explained here,the widely

used expression \theoreticalm odel"isnaturalscientists’favourite,unwitting

oxym oron.

A ppendix B :

Preparation tim e

Isthereany quantum -m echanicalconceptoftim e| or,m oreprecisely,tim e

instant| at the geom etric level? Could such a concept be connected with

the transition function hbjai (geom etrically,the inner product)? The idea

appearsprom ising because the transition function represents the innerre-

arrangem entofphysicalsystem satpreparation,and change isattheheart

ofwhat we understand by tim e. The concept oftim e to be derived here,

leading to a prim itive form ofSchr�odinger’sequation,iscalled preparation

tim e afterthisassociation.

W e startby exam ining an in�nitesim ally sm allchange �hbjaiofhbjaiin

term softheinteraction between ketand bravectorsjaiand hbj.15 In orderto

work with �hbjai,weneed toexpressitin term sofconceptswhoseproperties

and behaviourwe know.To thisend,we introduce the �rstgeom etric tim e

postulate,

�hbjai= ihbj�Fbajai; (B.1)

which gives a geom etric representation to the in�nitesim alchange ofthe

transition function as a m ediated transition between arrow state vectors.

The m ediation is perform ed by operator �Fba,which works as the inter-

m ediary ofthe interaction between any state vector jai (in general,jaii)

associated with observableA and any statevectorjbi(in general,jbii)asso-

ciated with observable B .W e call�Fba thein�nitesim alaction operator.

Assum ingthechangeofthetransition function tobecontinuous,we�nd

that� satis�esthechain rulefora productoftransition functions:

�(hdjcihbjai)= (hdjci+ �hdjci)(hbjai+ �hbjai)� hdjcihbjai

= hdjci�hbjai+ �hdjcihbjai+ �hdjci�hbjai

= hdjci�hbjai+ �hdjcihbjai+ O [(�hbjai)2]: (B.2)

Thus,� worksm athem atically like a di�erentialinsofarasthe chain rule is

concerned,butitm ustnotbe confused with a regulardi�erential,since in

general�hbjai6= 0,wherehbjaiisacom plex num ber.Asitwillturn out,and

aswecould expect,thereisnoin�nitesim alchangeonlyforhajjaii= �ij 2 R,

i.e.when no transition actually occurs.

15See (Schwinger,1959,pp.1550{1551) form ostofthe m athem aticalform alism below

butwithoutrelation to tim e.

260 A ppendix B :P reparation tim e

Postulate (B.1), decom position relation (12.24), and chain rule (B.2)

give usenough inform ation aboutthephysicalpropertiesof�Fba.In e�ect,

because

�hbjai=X

i

h

(�hbjcii)hcijai+ hbjcii(�hcijai)

i

; (B.3)

we �nd

ihbj�Fbajai=X

i

h

ihbj�Fbcjciihcijai+ hbjciiihcij�Fcajai

i

= ihbj

�Fbc

X

i

jciihcij+ �Fca

X

i

jciihcij

!

jai; (B.4)

where we have now interpreted the quasigeom etric decom position relation

geom etrically.Thisleadsto theresult

�Fba = �Fbc+ �Fca foralla;b;c: (B.5)

This tells us that the interaction between two (geom etrically represented)

statesofa system can bem ediated by a third such statein a serialm anner.

From Eq.(B.5)weobtain two specialresults.Setting c= a,we �nd

�Fba = �Fba + �Faa; (B.6)

which im plies

�Faa = 0 foralla: (B.7)

Thism eans thatthe in�nitesim alaction operator is nullin the absence of

change;i.e.in the absence ofa transition,asin hajjaii= �ij,no instantof

preparation tim e iscreated by itsin�nitesim alchange.O n the otherhand,

thechangeofagenuinequantum -m echanicaltransition hbjaidoesdeterm ine

an instantofpreparation tim e.

Now,setting b= a in Eq.(B.5)and using Eq.(B.7),we �nd

0 = �Faa = �Fac+ �Fca; (B.8)

which im plies

�Fac = � �Fca foralla;c: (B.9)

Thistellsusthata changeofsign oftheaction operatorindicatesa reversal

ofthedirection in which itm ediatesthetransition.

Finally,usingEq.(B.9),thetransition-function postulate(12.31),and an

extension to itin thesensethatcom plex conjugation m ustreversenotonly

the direction ofthe transition butalso thatofa change in the transition|

�hbjai = (�hajbi)�| we obtain that the action operator is Herm itian. In

e�ect,because

ihbj�Fbajai= [ihaj�Fabjbi]� = � ihbj�F

y

abjai; (B.10)

we �nd

�Fba = � �Fy

ab= �F

y

ba: (B.11)

261

|a〉B〈b|

︷︸︸︷

δ|a〉︷︸︸︷

δ〈b|

|b〉

︸︷︷︸

〈b|a〉

Figure B .1: Schem aticrepresentation ofpreparation tim ein term sofa transition

box and arrow vectors.

The introduction ofa second geom etric tim e postulate,indicating the

sim plestpossibledependenceof�Fba on band a,allowsusnextto recovera

prim itiveform ofSchr�odinger’sequation linking in�nitesim alchangesofjai

and hbjwith jaiand hbjthem selves.Thesecond geom etric tim e postulate is

�Fba = G (b)� G (a): (B.12)

HeretheHerm itian operator G worksasgeneratorofin�nitesim altransfor-

m ationsofjaiand hbj.Using thispostulate,we �nd,on the one hand,

�hbjai= ihbjG (b)jai� ihbjG (a)jai: (B.13)

This is an equation for the in�nitesim alchange ofthe transition function

in term s ofket and bra vectors. O n the other hand,using the geom etric

decom position ofthe transition function16 and the chain rule,we can �nd

anotherequation forthein�nitesim alchange ofthetransition function:

�hbjai= (�hbj)jai+ hbj(�jai) (B.14)

(Figure B.1).Thetwo equationstogetherim ply

i�jai= G (a)jai (B.15)

and

� i�hbj= hbjG (b); (B.16)

wherewedism issthe alternative im plication on a physicalbasis.

Results (B.16) and (B.15) are prim itive Schr�odinger equations,respec-

tively,for the bra vector hbjat preparation and for the ket vector jai just

outside preparation. Thus,Herm itian operator G ,generator ofin�nitesi-

m alchange ofjai and hbj,can be considered a prim itive ancestor ofthe

Ham iltonian H ,itselfthe generatoroftranslationsofa system in tim e t.

In thisgeom etric analysis,then,the rise ofa quantum -m echanicaltim e

instantcom esaboutin two parts,nam ely,thein�nitesim alchange�hbjofa

bra vector atpreparation and the in�nitesim alchange �jaiofa ket vector

16Thisisperfectly reasonable because thisparticular analysis oftim e iscarried outat

the geom etric level.

262 A ppendix B :P reparation tim e

−iG(a)|a〉 = δ|a〉

|a〉

i〈b|G(b) = δ〈b|

〈b|

δ〈b|a〉

Figure B .2: Birth ofpreparation tim e,i.e.ofa quantum -m echanicalgeom etric

tim e instant,atthe borderline ofa quantum -m echanicalprem easurem entprocess

asthe com bination ofthe in�nitesim alchange �hbjofa bra vectoratpreparation

and the in�nitesim alchange�jaiofa ketvectorjustoutside preparation.

justoutside preparation. Preparation tim e isby nature inherentin prepa-

ration,and itisborn atthe preparation borderline(Figure B.2).

A quantum -m echanicalconceptoftim einspired in a sim ilaridea isthat

ofa q-tick proposed by Jaroszkiewicz (2001).Thistim earisesqualitatively

from the \irreversible acquisition ofinform ation" thatoccurrsaftera m ea-

surem ent is m ade on a quantum -m echanicalsystem . In contrast to a q-

tick,preparation tim e ariseshere from quantum -m echanicaltransitions(in

general,prem easurem ents) rather than from m easurem ents. Further,the

in�nitesim alchanges ofthe transition function giving rise to a geom etric

tim e instant are governed by the prim itive Schr�odinger equations (B.15)

and (B.16).

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