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Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Background-Independence
Gordon Belot
University of Pittsburgh & CASBS
March 14, 2007
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Something for Everyone
For Physicists: Examples.
For Philosophers: Counter-Examples.
For Everyone Else: Examples and Counter-Examples!
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Einstein: The Usual Conception of Space and Time
“Ask an intelligent man who is not a scholar what space andtime are, and he will perhaps answer as follows. If we imagineall physical things, all stars, all light taken out of the universe,what then remains is something like a giant vessel withoutwalls called ‘space.’ With respect to what is happening in theworld, it plays the same role as the stage in a theaterperformance. In this space, in this vessel without walls, there isan eternally uniformly occurring tick-tock . . . that is ‘time.’Most natural scientists, up to the present, had this conceptionabout the essence of space and time . . . .”
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Einstein’s Globes
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Einstein on the Globes
“In classical mechanics, and no less in the special theory ofrelativity, there is an inherent epistemological defect . . . .”
“What is the reason for this difference in the two bodies?No answer can be admitted as epistemologicallysatisfactory, unless the reason given is an observable factof experience.”
“Newtonian mechanics does not give a satisfactory answerto this question. It pronounces as follows:—The laws ofmechanics apply to the space R1, in respect to which thebody S1 is at rest, but not to the space R2, in respect towhich the body S2 is at rest. But the privileged space R1
of Galileo, thus introduced, is a merely factitious cause[bloß fingierte Ursache], and not a thing that can beobserved.”
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Factitious?
factitious adj.
1. (Obsolete) Made by or resulting from art; artificial.Example: Beer, Ale, or other factitious drinks.
2. Got up, made up for a particular occasion orpurpose; arising from custom, habit, or design; notnatural or spontaneous; artificial, conventional.Example: The momentary and factitious joy which hadgreeted the day of William’s crowning died utterly away.
3. (Medical) Of a disorder, symptom, or sign: feignedor self-induced by a patient.Example: Factitious purulent ophthalmia produced bythe liquorice liana, or jequirity.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Upshot
Einstein set out to create a theory in which space and timewere among the actors rather than providing a fixed stage.
These days, one says that general relativity isbackground-independent (i.e., the theory does not featurea spacetime geometry given a priori).
This notion plays some role in polemics about the futureof physics.
How can one make the intuitive notion ofbackground-independence precise?
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Master-Builders and Philosophers
“The commonwealth of learning is not at this time withoutmaster-builders, whose mighty designs, in advancing thesciences, will leave lasting monuments to the admiration ofposterity: but every one must not hope to be a Boyle or aSydenham; and in an age that produces such masters as thegreat Huygenius and the incomparable Mr. Newton, with someothers of that strain, it is ambition enough to be employed asan under-labourer in clearing the ground a little, and removingsome of the rubbish that lies in the way to knowledge ... .”
—Locke, An Essay Concerning Human Understanding
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Preliminary Topics
A. Symmetries and Patterns.
B. Relativistic Geometries.
C. Relativistic Field Theories.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Example: Symmetry
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Example: Pattern-Preserving Maps
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
More Formally
Suppose we have configurations C1,C2, . . . . Each configurationinvolves an assignment of properties and relations to some fixedset of objects D. Consider d : x 7→ x ′, a means of matching upobjects.
d is a symmetry of configuration Ci if for each x ∈ D, xand x ′ play the same role in Ci .
d is a pattern-preserving map for Ci and Cj if for eachx ∈ D, x plays the same role in Ci that x ′ plays in Cj . Wesay that Ci and Cj instantiate the same pattern if there issuch a pattern-preserving map. We write Ci ∼ Cj .
(Sometimes D has internal structure; this will be preserved byany pattern-preserving map.)
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Relativistic Geometries
Ingredients: a manifold V and a metric g of Lorentz signature(special case: flat metrics—Minkowski spacetime).
V provides a very stretchy canvass (topology but nogeometry).
g provides V with a geometry—including notions ofstraightness, distance, timelikeness, etc.—by assigninggeometrical properties to each point x ∈ V .
Any given g has relatively few symmetries—so few mapsd : x 7→ x ′ preserve the structure of (V , g).
But many maps d : x 7→ x ′ preserve the intrinsic structureof V . So for any g , there will be many g ′ such that g ∼ g ′.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Relativistic Field Theories
Ingredients:
Spacetime V .
A set Θ of fixed fields on V .
A dynamical fields, φ1, . . . , φk . We denote a configurationof these by Φ.
Technical conditions, determining the space K ofkinematically possible Φ.
Differential equations determining the space S ⊂ K ofsolutions of the theory.
A relativistic metric g must be among the fields of the theory.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Pattern-Preserving Maps for Field Theories
A pattern-preserving map relating Φ,Φ′ ∈ K is a meansd : x 7→ x ′ of matching up points of V such that:
(i) d leaves invariant the fixed fields of the theory and theintrinsic structure of V ;
(ii) for each x ∈ V , Φ assigns x the same properties that Φ′
assigns x ′.
NB. If Φ ∼ Φ′ then both are solutions or neither are—ourequations care only about structure.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Example: Theory with Fixed Fields
: Pattern-Preserving Maps are Rigid and Scarce
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Example: Theory without Fixed Fields
: Pattern-Preserving Maps are Floppy and Common
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Paragons
Ordinary Wave Equation Let V = R4 with a fixed Minkowskimetric η. A real-valued dynamical field φ subjectto �ηφ = 0.Fully background-dependent: the field propagatesagainst Minkowski metric in each solution.
Cosmological General Relativity V is spatially compact. Nofixed fields. A single dynamical field—a metric gsubject to Ricci[g ] = 0.Fully background-independent: maximal variationof geometry from solution to solution.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Background-Independence = No Fixed Fields?
Any theory with fixed fields is background-dependent. Anidea of Einstein: perhaps the converse is also true?
No. Consider the following near-relative of our paragon ofbackground-dependence:
V = R4.No fixed fields.Two dynamic fields—a metric g subject to Riem[g ] = 0,and a real-valued φ subject to �gφ = 0.
In this new theory, as in our paragon, each solutionconsists of a field obeying the wave equation living inMinkowski spacetime.
Moral: A theory lacking fixed fields can be fullybackground-dependent.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Background-Independence = No Absolute Objects?
Another idea of Einstein: in pre-1915 physics, geometryacts on matter but not vice versa; it is a virtue of generalrelativity to abandon this.Following Anderson, Friedman, et al. we say that a fieldtheory has an absolute object if one of its fieldsinstantiates the same pattern in every solution.A natural idea: a theory is background-independent iff itfeatures no absolute objects.No. Consider general relativity with asymptotic boundaryconditions: one takes V = R4 and includes in K onlythose g that are asymptotically flat at spatial infinity.Consensus view: such a theory enjoys a large degree ofbackground-independence—but involves the introductionof geometrical background at infinity.Moral: Absoluteness test fails to detect some forms ofbackground-dependence.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Background-Independence a Matter of Degree
We have seen examples of full background-dependenceand full background-independence.
We have also seen a theory that falls just short of fullbackground-independence.
It is also possible to cook up theories that fall just short offull background-dependence.
Example: V = R× S3 with two dynamic fields: a metric gsubject to Weyl[g ] = 0 andRicci[g ]− 1
4gR[g ] = 0; and a real-valued φsubject to �gφ = 0. In each solution, g is deSitter—so spacetime has constant positivecurvature k—but the value of k can vary fromsolution to solution.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
A Geometrically Ambiguous Example
A variant on Nordstrom’s scalar theory of gravity.
V = R4; no fixed fields.
Dynamical fields: scalar field φ; metrics, η and g .
Field Equations:
Riem[η] = 0 �ηφ = −4Gφ3T
Weyl[g ] = 0 R[g ] = 24πGT
g = φ2η φ = (−detg)18 η = g(−detg)
−14
Background dependent if η encodes the geometry;background-independent if g encodes the geometry.
Moral: Background-independence is not a formal feature.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Desiderata
Theories lacking fixed fields can be background-dependent.
Background-(in)dependence is a matter of degree.
Background-(in)dependence is not a formal matter.
Asymptotic boundary conditions can lead to a degree ofbackground-dependence.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
A Proposal Sketched: Background-Independence asFine Dependence of Geometry on Fields
Basic Idea: A theory is background-independent to the extentthat the geometry of a solution depends on thefields of the theory.
At one extreme we have theories in which thisdependence is as fine as possible: two solutionshave the same geometry iff they represent thesame physical possibility.
At the other extreme we have theories in whichthere is no such dependence: every solution hasthe same geometry.
Required Ingredients: Appropriate notion of geometry;Schemes for counting possibilities.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Geometry and Content
When do we think of a field theory as physics rather thanmathematics?
Plausible answer: when we understand its solutions arerepresenting spatiotemporal processes.
This is a substantive step: we assign a geometry to eachsolution; lay down a notion of sameness of geometry;require covariance.
We restrict attention to cases where this is almostautomatic: one of the fields of the theory encodes theLorentzian geometry of spacetime; two solutions encodethe same geometry iff their metrics are related by apattern preserving map.
But: In more general settings, things can become moreinteresting.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Counting Possibilities: Simple Cases
When possible, the standard physicist assumes that thespace of solutions of a theory parameterizes the space ofpossibilities allowed by a theory.
In even the simplest cases, philosophers object:
Haecceitist: Too few!Anti-Haecceitist: Too many!
Claim: The disagreement between the standard physicistand the haecceitist is largely terminological.
In what follows: focus on the strategy of the standardphysicist; other strategies can be plugged into finalproposal.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Counting Possibilities: Theories without FixedFields
Rough and ready notion of determinism: if theinstantaneous states are the same, then the global statesare the same. In a well-behaved classical theory,indeterminism should be rare and illuminating (givenappropriate boundary conditions).
Consider a theory without fixed fields.
It is easy to find distinct Φ,Φ′ ∈ S that induce the sameinitial data on some hypersurface Σ ⊂ V—look forsolutions related by a pattern preserving map that is theidentity on a neighbourhood of Σ.Indiscriminate and uninteresting indeterminism threatensunless we:(a) Deny that Φ |Σ and Φ′ |Σ correspond to the same
instantaneous state. Bad idea.(b) Deny that Φ and Φ′ correspond to distinct physical
possibilities. Good idea.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Determinism Threatened
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Gauge Equivalence
Star: We write Φ ∗ Φ′ if Φ and Φ′ instantiate the samepattern and induce the same initial data on someΣ ⊂ V .
Gauge Equivalence is the equivalence relation on S generatedby ∗: Φ and Φ′ are gauge equivalent iff thereexist solutions Φ1, . . . ,Φk such that Φ = Φ1,Φ′ = Φk , and Φi ∗Φi+1 for each i = 1, . . . , k − 1.
Everyone agrees that gauge equivalent solutions alwaysrepresent the same possibility.
The standard physicist’s approach is to take solutions torepresent the same possibility iff they are gaugeequivalent. Under this approach, the space of equivalenceclasses of gauge equivalence parameterizes the space ofpossibilities of a theory.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Features of Gauge Equivalence
Gauge equivalent solutions instantiate the same pattern.
In a theory with g a fixed field, each solution is gaugeequivalent only to itself.
In a theory without fixed fields:
If there are no asymptotic boundary conditions thensolutions are gauge equivalent iff they instantiate the samepattern.If there are asymptotic boundary conditions then(typically) solutions are gauge equivalent iff related by apattern-preserving map asymptotic to the identity atinfinity.
NB some further technical conditions required . . .
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Asymptotically Flat GR: Pattern-Preserving Maps
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Asymptotically Flat GR: Gauge Equivalence
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Proposal in Full
A theory is fully background-dependent if every pair ofsolutions correspond to the same abstract geometry.
A theory is nearly background-dependent if the family ofabstract geometries instantiated is small(finite-dimensional, say).
A theory is fully background-independent if two solutionscorrespond to the same abstract geometry iff theyrepresent the same physical possibility.
A theory is nearly background-independent if for eachabstract geometry, the corresponding family of physicalpossibilities is small (finite-dimensional, say).
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Asymptotically Flat General Relativity
Under the standard scheme for counting possibilities thetheory is nearly (but not fully) background-independent:the space of possibilities corresponding to any instantiatedabstract geometry is ten-dimensional.
There is an alternative scheme under which solutionscorefer iff related by a pattern-preserving map. Under thisscheme the theory is fully background-independent.
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Desiderata Revisited
Lack of fixed fields is consistent with fullbackground-dependence. (Klein–Gordon field on flatspacetime.)
Background-(in)dependence is non-formal, depending onchoice of geometrization and on the choice of scheme forcounting possibilities. (Nordstrom’s theory; asymptoticallyflat general relativity.)
Background-(in)dependence is a matter of degree.(Klein–Gordon field on de Sitter spacetime; asymptoticallyflat general relativity.)
Full background-independence can be spoiled byasymptotic boundary conditions. (Asymptotically flatgeneral relativity.)
Background-Independence
G. Belot
Motivation
Preliminaries
Examples
Background-Independence
Geometry
CountingPossibilities
Proposal
Worries
Worries
What about trivial field theories?
What about non-relativistic theories?
Is string theory background-independent?
What about non-spatiotemporal symmetries?
What if some geometry is solution-independent?
What about weirder asymptotic boundary conditions?
What happens if matter is included in general relativity?