Research ArticleA Covariant Canonical Quantization of General Relativity

Stuart Marongwe

Department of Physics and Astronomy Botswana International University of Science and Technology P Bag 16 Palapye Botswana

Correspondence should be addressed to Stuart Marongwe stuartmarongwegmailcom

Received 13 June 2018 Accepted 8 November 2018 Published 19 December 2018

Guest Editor Farook Rahaman

Copyright copy 2018 Stuart Marongwe is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited epublication of this article was funded by SCOAP3

A Hamiltonian formulation of General Relativity within the context of the Nexus Paradigm of quantum gravity is presented Weshow that the Ricci flow in a compact matter free manifold serves as the Hamiltonian density of the vacuum as well as a timeevolution operator for the vacuum energy density e metric tensor of GR is expressed in terms of the Bloch energy eigenstatefunctions of the quantum vacuum allowing an interpretation of GR in terms of the fundamental concepts of quantum mechanics

1 Introduction

e gravitational field which is elegantly described by Ein-steinrsquos field equations has so far eluded a quantum descrip-tion Much effort has been placed into formulating GeneralRelativity (GR) in terms of Hamiltonrsquos equations since aHamiltonian formulation of a classical field theory leadsnaturally to its quantization e earliest such attempt is theADM formalism [1] named for its authors Richard ArnowittStanley Deser and Charles W Misner first published in1959 is formalism starts from the assumption that spaceis foliated into a family of time slices Σ119905 labeled by theirtime coordinate t and with space coordinates on each slicegiven by 119909119896 e dynamic variables of this theory are thentaken to be the metric tensor of three-dimensional spatialslices 120574119894119895(119905119909119896) and their conjugate momenta 120587119894119895(119905119909119896) Usingthese variables it is possible to define a Hamiltonian andthereby write the equations of motion for GR in Hamiltonsform e time slices are then welded together using fourLagrange multipliers and components of a shi13 vector fieldAn extensive review of this formalism can be found in theliterature notably in [2ndash5]

eADM formalismwas first applied by BryceDeWitt in1967 [6] to quantize gravitywhich resulted in theWheelerndashDeWitt equation of quantum gravity It is a functional differen-tial equation in which the three-dimensional spatial metricshave the form of an operator acting on a wave functionis wave function contains all of the information about

the geometry and matter content of the universe of eachtime slice However the Hamiltonian no longer determinesthe evolution of the system and leads to the problem oftimelessness [7 8] Hawking rightly points out that the veryact of splitting space-time into space and time destroys thespirit of GR (general covariance) and therefore not muchcan be gained from this approach to quantization of gravityPerturbative covariant approaches to the problemof quantumgravity have an inherent weakness in that they depend ona fine classical background It is therefore difficult to obtaina self-consistent quantum theory of gravity with a classicalbackground space-time Steven Carlip [9 10] and ClausKiefer [11 12] have made excellent and extensive reviews ofthe problems faced by current approaches to the problemof quantum gravity Carlip [9] in particular singles out thelack of a firm conceptual understanding of the foundationalconcepts of quantum gravity as the source of much of thedifficulty in understanding quantum gravity ese deepconceptual issues result in technical problems in the attemptto develop a consistent quantum theory of gravity

In this paper we report a successful covariant canonicalquantization of the gravitational field which preserves thesuccess of GR while simultaneously explaining Dark Energy(DE) and Dark Matter (DM) is approach to quantizationtakes place in 4-space of metric signature (-1111) in whichthe quanta are excitations of the quantum vacuum calledNexus gravitons ough the Nexus Paradigm has beenintroduced in the following papers [13ndash15] the aim of this

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 4537058 7 pageshttpsdoiorg10115520184537058

2 Advances in High Energy Physics

study is to explicitly express the Hamiltonian formulationof the theory using the Bloch eigenstate functions of thequantum vacuumese wave functions contain informationabout the energy state of the quantum vacuum which in turndictates the geometry of space-time

2 Methods

Our first step towards a covariant canonical quantizationbegins with defining a quantized space-time and its quantaWe thenmodify Einsteinrsquos vacuum equations to be consistentwith the quantized space-time followed by the defining ofHamiltonrsquos equations of the quantized space-time is stepis then followed by the Poisson brackets which provide thebridge between classical and quantum mechanics (QM) ecovariant canonical quantization procedure is carried outwithin the context of theNexus Paradigmof quantumgravity

21 Quantization of 4-Space and the Nexus Graviton epri-mary objective of physics is the study of functional relation-ships amongst measurable physical quantities In particulara unifying paradigm of physical phenomena should revealthe functional relationship between the fundamental physicalquantities of 4-space and 4-momentum Currently GR andQM offer the best predictions of the results of measurementof physical phenomena in their respective domains usingdifferent languages GR describes gravitation in the languageof geometry and thus far it has been difficult to applythe the language of wave functions used in QM to giveit a QM description e problem of quantum gravity istherefore to interpret GR in terms of the wavefunctions ofQMTranslating the language ofmeasurement ofGR into thatof QM becomes the primary objective of this present attemptto resolve the problem of quantum gravity

Measurements in GR take place in a local patch of aReimannian manifold is local patch can be consideredas a flat Minkowski space e line element in Minkowskispace which is the subject of measurement can be computedthrough the inner product of the local coordinates as

Δ119909120583Δ119909120583 = Δ1199092 + Δ1199102 + Δ1199112 minus 1198882Δ1199052= (119860Δ119909 + 119861Δ119910 + 119862Δ119911 + 119894119863119888Δ119905)sdot (119860Δ119909 + 119861Δ119910 + 119862Δ119911 + 119894119863119888Δ119905)

(1)

On multiplying the right hand side we see that to get all thecross terms such as Δ119909Δ119910 to cancel out we must assume

119860119861 + 119861119860 = 01198602 = 1198612 = sdot sdot sdot = 1 (2)

e above conditions therefore imply that the coefficients(119860 119861 119862119863) generate a Clifford algebra and therefore mustbe matrices We rewrite these coefficients in the 4-tupleform as (1205741 1205742 1205743 1205740) which may be summarized using theMinkowski metric on space-time as follows

120574120583 120574] = 2120578120583] (3)e gammas are of course the Dirac matrices us in orderto satisfy (1) we can express a displacement 4-vector as

Δ119909120583 = 119886120574120583 (4)

where 119886 is the amplitude of a displacement vector inMinkowski space If we consider the Hubble diameter asthe maximum dimension of the local patch of space then119886 = 119903119867119878 where 119903119867119878 is the Hubble radius is choice of amaximum amplitude is justified from the fact that we cannotphysically interact with objects beyond the Hubble 4-radiusIt is important to note that the line element is the squareof the amplitude of the displacement 4-vector us if weare to express GR in terms of the language of QM we mustmake the radical assumption that the displacement vectorsin Minkowski space are pulses of 4-space which can beexpressed in terms of Fourier functions as follows

Δ119909120583119899 = 2119903119867119878119899120587 120574120583 intinfinminusinfin

sinc (119896119909) 119890119894119896119909119889119896= 120574120583 intinfin

minusinfin119886119899119896120593(119899119896119909)119889119896

(5)

Where2119903119867119878119899120587 = 119896=+infinsum

119896=minusinfin

119886119899119896 (6)

Here 120593(119899119896119909) = sinc(119896119909)119890119894119896119909 are Bloch energy eigenstatefunctions e Bloch functions can only allow the four wavevector to assume the following quantized values

119896120583 = 119899120587119903120583119867119878 119899 = plusmn1 plusmn2 1060 (7)

e minimum 4-radius in Minkowski space is the Planck 4-length since it is impossible to measure this length withoutforming a black hole e 1060 states arise from the ratio ofHubble 4-radius to the Planck 4-length e displacement 4-vectors in each eigenstate of space-time generate an infiniteBravais 4-lattice Also condition (7) transforms (5) to

Δ119909120583119899 = 120574120583 int1198991198961minus1198991198961

119886119899119896120593(119899119896119909)119889119896 (8)

e second assumption wemake is that each displacement 4-vector is associatedwith a conjugate pulse of four-momentumwhich can also be expressed as a Fourier integral

Δ119901(119899)120583 = 21198991199011120587 120574120583 int1198991198961minus1198991198961

120593(119899119896119909)119889119896= 120574120583 int1198991198961

minus1198991198961

119888119899119896120593(119899119896119909)119889119896(9)

where 119901(1)120583 is the four-momentum of the ground stateA displacement 4-vector and its conjugate 4-momentum

satisfy the Heisenberg uncertainty relation

Δ119909119899Δ119901119899 ge ℎ2 (10)

e Uncertainty Principle plays the important role of gener-ating a vector bundle out of the total uncertainty space E oftrivial displacement 4-vectors from which a closed compactmanifold X is formed ie (120587 119864 997888rarr 119883) Each point on themanifold is associated with a vector which is along a normalto the manifold

Advances in High Energy Physics 3

ewave packet described by (8) is essentially a particle offour-space e spin of this particle can be determined fromthe fact that each component of the four-displacement vectortransforms according to the law

Δ1199091015840120583119899 = exp(18120596120583120592 [120574120583 120574120592])Δ119909120583119899 (11)

where120596120583120592 is an antisymmetric 4x4matrix parameterizing thetransformation

erefore each component of the 4-vector has a spin halfA summation of all the four half spins yields a total spin of2 We give the name the Nexus graviton to this particle of 4-space since the primary objective of quantumgravity is to findthe nexus between the concepts of GR and QM

From (7) the norm squared of the 4-momentum of then-th state graviton is

(ℎ)2 119896120583119896120583 = 11986421198991198882 minus3 (119899ℎ1198670)21198882 = 0 (12)

where H0 is the Hubble constant (22 x 10minus18 sminus1) and can be

expressed in terms of the cosmological constant Λ asΛ 119899 = 1198642119899(ℎ119888)2 =

31198962119899(2120587)2 = 1198992Λ (13)

We infer from (13) that the Nexus graviton (or displacement4-vector) in the n-th quantum state forms a trivial vectorbundle via the Uncertainty Principle which generates acompact Riemannian manifold of positive Ricci curvaturethat can be expressed in the form

119866(119899119896)120583120592 = 1198992Λ119892(119899119896)120583120592 (14)

where 119866(119899119896)120583120592 is the Einstein tensor of space-time in then-th state Equation (14) depicts a contracting geodesicball and as explained in [13ndash15] this is DM which is anintrinsic compactification of the elements space-time in then-th quantum state is compactification is a result of thesuperposition of several plane waves as described by (8) toform an increasingly localized wave packet as more waves areadded Similary the converse is also truee loss of harmonicwaves expands the elements of space-time which gives rise toDE us the DE arises from the emission of a ground stategraviton such that (14) becomes

119866(119899119896)120583120592 = (1198992 minus 1)Λ119892(119899119896)120583120592 (15)

ese are Einsteinrsquos vacuum field equations in the quantizedspace-time If the graviton field is perturbed by the presenceof baryonic matter then (15) becomes

119866(119899119896)120583120592 = 119896119879120583] + (1198992 minus 1)Λ119892(119899119896)120583120592= 119896119879120583] + (1198992 minus 1) 119896120588119863119864119892(119899119896)120583120592 (16)

where 120588119863119864 is the density of DEFrom [14] the static solution to (14) for a spherically

symmetric Nexus graviton is computed as

1198891199042 = minus(1 minus ( 21198992 )) 11988821198891199052 + (1 minus ( 21198992 ))minus1 1198891199032

+ 1199032 (1198891205792 + sin2 1205791198891205932) (17)

ere are no singularities in (17) At high energies character-ized by microcosmic scale wavelengths of the Nexus gravitonand high values of n space-time is flat and highly compactis implies a continuous length contraction of the localcoordinates via the addition of more waves resulting in anincrease in localization Also from aworld line perspective ofa test particle it implies that the deviation from a rectilineartrajectory due to uncertainities in its location in 4-space issmall us at microcosmic scales in the realm of subatomicparticles space-time is extremely flat e world line beginsto substantially deviate from a well defined rectilinear path atlow energies as the uncertainities in its location are increasedat is at low values of n the world line becomes degenerateallowing amultiplicity of trajectorieswhich inGRare averagedby the Ricci curvature tensor us gravity is a low energyphenomenonwhich vanishes asymptotically at high energies

e gravitational effects of the Nexus graviton manifestat large scales and for galaxies these effects begin to manifestwhen the density of DE is equal to the density of baryonicmatter as described by (16) or when the acceleration due tobaryonic matter is equal and opposite to the acceleration dueto the emission of the ground state graviton as described in[14] A solution to eqn (16) from [14] in the weak field atgalactic scale radii is

11988921199031198891199052 = 119866119872(119903)1199032 + 1198670V119899 minus 1198670119888 (18)

Here c is the speed of lighte first term on the right is the Newtonian gravitational

acceleration the second term is a radial acceleration due tospace-time in the n-th quantum state and the final term isacceleration due to DEe dynamics becomes strongly non-Newtonian when

119866119872(119903)1199032 = 1198670119888 = V2119899119903 (19)

ese are conditions in which the acceleration due to bary-onic matter is annulled by that due to the DE Under suchconditions

119903 = V21198991198670119888 (20)

Substituting for r in (19) yieldsV4119899 = 119866119872(119903)1198670119888 (21)

is is the Baryonic TullyndashFisher relation e conditionspermitting the DE to cancel out the acceleration due tobaryonic matter leave quantum gravity as the unique sourceof gravity us condition (19) reduces (18) to

11988921199031198891199052 = 119889V119899119889119905 = 1198670V119899 (22)

from which we obtain the following equations of galactic andcosmic evolution

119903119899 = 11198670 119890(1198670119905) (119866119872 (119903)1198670119888)14 (23)

4 Advances in High Energy Physics

V119899 = 119890(1198670119905) (119866119872 (119903)1198670119888)14 (24)

119886119899 = 1198670119890(1198670119905) (119866119872 (119903)1198670119888)14 (25)

Here 119903119899 is the radius of curvature of space-time in the n-th quantum state (which is also the radius of the n-th statenexus graviton) V119899 the radial velocity of objects embedded inthat space-time and 119886119899 their radial acceleration within iteamplification of the radius of curvature with time explainsthe existence of ultra-diffuse galaxies and the spiral shapesof most galaxies e increase in radial velocity with timeexplains why early type galaxies composed of population IIstars are fast rotators Equation (25) explains late time cosmicacceleration which began once condition (19) was satisfied orequivalently from (16) when the density of baryonic matterwas at the same value as that of DE us condition (19) alsoexplains the Coincidence Problem

22 Canonical Transformations in the Nexus Paradigm Inclassical mechanics a system is described by n indepen-dent coordinates (1199021 1199022 119902119899) together with their conjugatemomenta (1199011 1199012 119901119899) In the Nexus Paradigm the labeling119902119899 refers to a creation of a Nexus graviton in the n-thquantum state associated with a conjugate momentum 119901119899e Hamiltonian equation

119902119899 = 120597119867120597119901119899 (26)

refers to the rate of expansion or contraction of space-timegenerated by the addition or emission of harmonic waves tothe Nexus graviton and

119899 = minus120597119867120597119902119899 (27)

refers to the force field associated with the addition oremission of harmonic waves to the Nexus graviton It isimportant to note that this force field generates an isotropicexpansion or contraction of space-timewithin the spatiotem-poral dimensions of the graviton

We can also rewrite the Hamiltonian equations in termsof Poisson brackets which are invariant under canonicaltransformations as 119902119899 = 119902119899 119867 119899 = 119901119899 119867 (28)

ePoisson brackets provide the bridge between classical andQM and in QM these brackets are written as119902119899 = [119902119899 ] 119901119899 = [119901119899 ] (29)

and obey the following commutation rules[119902119899 119902119904] = 0[119901119899 119901119904] = 0[119902119899 119901119904] = 120575119899119904

(30)

23 e Hamiltonian Formulation for the Quantum Vacuume Nexus graviton is a pulse of 4-space which can only

expand or contract and does not execute translationalmotionimplying that the Hamiltonian density of the system is equalto the Lagrangian density 119867 = 119871 (31)

GR is a metric field in which the energy density in four-space determines its value Since the Bloch energy eigenstatefunctions determine the energy of space-time we mustseek to express the metric in terms of the Bloch wavefunctions To this end we shall express the eigenstate four-space components of the Nexus graviton in the k-th band as

Δ119909120583119899119896

= 119911120583119899119896

= 119886119899119896120574120583sinc (119896119909) 119890119894119896119909 (32)

such that an infintesimal four-radius within the k-th band iscomputed as

119889119903119899119896 = 120597119911120583119899119896120597119896120583 119889119896120583 = 119894119909119886119899119896120574120583sinc (119896119909) 119890119894119896119909119889119896 (33)

In (33) the first-order derivative of the periodic sinc functionis equal to zero for all integral values of n e interval withinthe band is then computed as

1198891199042 = 1198891199032119899119896 = 120597119911120583119899119896120597119896120583120597119911120583119899119896120597119896] 119889119896120583119889119896]= 120572120573120574120583120574]120593(119899119896119909)120593(119899119896119909)119889119896120583119889119896]

(34)

Here the interval is described in terms of the reciprocal latticewith 120572 = 119894119909120583119886119899119896 and 120573 = 119894119909]119886119899119896 e metric tensor offour-space in the k-th band is therefore associated with theBloch energy eigenstate functions of the quantum vacuum asfollows 119892120583]

(119899119896)= 120574120583120574]120593(119899119896119909)120593(119899119896119909) = 120578120583]120593(119899119896119909)120593(119899119896119909) (35)

Equation (35) translates the geometric language of GR intothe wave function language of QM

We initiate the translation procedure of GR into QM byfirst finding the Lagrange density for the quantized vacuumfrom (15) which following Einstein and Hilbert is found to be

119871119864119867 = 119896 (119877 minus 2 (1198992 minus 1)Λ) (36)

Given that the Einstein tensor in a compact manifold is equalto the Ricci flow

minus120597119905119892120583] = Δ119892120583] = 119877120583] minus 12119877119892120583] = 119866120583] (37)

therefore the equations of motion of the quantum vacuumobtained from (36) yield the following quantized field equa-tions minus 120597119905 (120574120583120593(119899119896119909)120574]120593(119899119896119909))

= (1198992 minus 1)Λ120574120583120593(119899119896119909)120574]120593(119899119896119909) (38)

which can be written as120597119905 (120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909))= minus1198942(2120587)2nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)

= 141205872nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)(39)

where

120593(119899minus1119896119909) = sinc ((119899 minus 1) 1198961119909) 119890119894(119899minus1)1198961119909 (40)

Advances in High Energy Physics 5

120593(119899+1119896119909) = sinc ((119899 + 1) 1198961119909) 119890119894(119899+1)1198961119909 (41)

311989621(2120587)2 = Λ (42)

For large values of n the Bloch functions statisfy the condition120593(119899minus1119896119909) asymp 120593(119899119896119909) asymp 120593(119899+1119896119909) (43)

equantumvacuumcan therefore be interpreted as a systemin which there are a constant annihilation and creation ofquanta as implied by (40) and (41) which causes the Nexusgraviton to either expand or contract

24 e Hamiltonian Formulation in the Presence of MatterFields We now seek to introduce matter fields into thequantum vacuum If we compare the quantized metric of(17) describing the gravity within a Nexus graviton withthe Schwarzschild metric describing the gravitational fieldaround baryonic matter we notice that we can describe thegravitational field around baryonic matter in terms of thequantum state of space-time through the relation

21198992 = 21198661198721198882119903 (44)

is yields a relationship between the quantum state of space-time and the amount of baryonic matter embedded within itas follows

1198992 = 1198882119903119866119872 = 1198882V2

(45)

Equation (45) reveals a family of concentric stable circularorbits 119903119899 = 11989921198661198721198882 with corresponding orbital speeds ofV119899 = 119888119899 us in the Nexus Paradigm unlike in GR theinnermost stable circular orbit ocurrs at 119899 = 1 or at halfthe Schwarzchild radius which implies that the event horizonpredicted by the Nexus Paradigm is half the size predictedin GR Also (45) reveals how the Nexus graviton in the n-th quantum state imitates DM if M is considered as theapparent mass of the DMrough this comparision we canalso deduce that the deflection of light through gravitationallensing by space-time in the n-th quantum state is

120572 = 41198992 (46)

us gravitational lensing can be used to constrain the valueof the quantum state n of space-time within a lensing system

Having obtained the relationship between the quantumstate of space-time and the amount of baryonic matterembedded within it the result of (45) is then added to (39)to yield the time evolution of the quantum vacuum in thepresence of baryonic matter as

120597119905 (120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= 141205872nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)minus 1198992Λ120574120583120593(119899119896119909)120574]120593(119899119896119909)

(47)

us the time evolution of the quantum vacuum in thepresence of matter resembles thermal flow in the presence of

a heat sinke second term on the RHS of (47) is an 8-cellor 4-cube that operates as a sinc filter with a four-wave cut-offof

119896119888 = 1198991198961 = 2120587radicΛ3 ∙ 1198882119903119866119872 (48)

e filtration of high frequencies from the vacuum lowersthe quantum vacuum state and generates a gravitational fieldin much the same way as the Casimir Effect is generatede physical process of filtration occurs as follows A 4-cubewith baryonic matter embedded with in it acquires inertiae inertia gives the cell inductive impedance and becomesless reponsive to high frequency vibrations of the quantumvacuum us the heavier the cell the lower the cut-off fre-quency Equation (48) shows that when the radius of the cellis equal to half the Schwarzschild radius the only permitedfrequency is that of the ground stateus the inside of a blackhole is in the ground state having a negative metric signature

We now introduce a test particle of mass m into thequantum vacuumperturbed bymatter fieldse particle willflow along with the Ricci flow and the Hamiltonian of thesystem becomes

(120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= P120583P]2119898 (120574120583120593(119899minus1119896119909)120574]120593( 119899+1119896119909))minus 119881 (120574120583120593(119899119896119909)120574]120593(119899119896119909))

(49)

Here

119881 = 1198992 ℎ22119898Λ = 11990311988821198661198723ℎ2119896212119898 (50)

120583 = minus119894ℎnabla120583 (51)

= minus119894ℎ120597119905 (52)

Equation (49) is equivalent to (31) in which the Hamiltonianis equal to the Lagrangian

(120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= L (120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909)) (53)

eLagrangian is not relativistic because the energy referenceframes or space-time states involved in the transition dynam-ics 119899 minus 1 119899 and 119899 + 1 are separated by a small energy gap119864 = radic3 ∙ ℎ1198670 e exchanged quantum of energy betweenstates is responsible for ldquoweldingrdquo them together

Equation (50) is the gravitational interaction in reciprocalspace-time e weakness of the gravitational interaction isdue to the small value of the cosmological constant

3 Line Elements and Information in K-Space

A close inspection of (34) reveals the relationship between aline element in 4-space and its conjugate line in K-space

1198891199042 = 1205721205731198891205812 (54)

6 Advances in High Energy Physics

52

39

26

13

13

n =1

n =2

0

0

13

13

26

2626

39

3939 5252

horizon the quantum state ofspace-time is n =1

arcseconds At the eventShadow diameter = 26 micro Shadow of Sagittarius Alowast

arcseconds

rn =52n2GM

2c2

2c2r1 =

52GM

r2 =104GM

c2

Figure 1 Shadow of Sagittarius Alowast

where

1198891205812 = 120574120583120593(119899119896119909)120574]120593(119899119896119909)119889119896120583119889119896] (55)

e term 120574120583120593(119899119896119909)120574]120593(119899119896119909) refers to the normalized infor-mation flux density passing through an elementary surface119889119896120583119889119896] in K-spacee gradient of the flux density yields thebaseband bandwidth 119896(119899)120583

120597120583120574120583120593(119899119896119909)120574]120593(119899119896119909) = 119896(119899)120583120574120583120593(119899119896119909)120574]120593(119899119896119909) (56)

is gradient is also a measure of the acutance or sharpnessof the information A large bandwidth provides detailedinformation while a small bandwidth provides diffuse infor-mation Equation (49) describes the diffusion of informationas it flows into a gravitational well Unitarity is preserved ifthe flux is summed over the total diffusion surface

∬Σ0120574120583120593(119899119896119909)120574^120593(119899119896119909)119889119896120583119889119896] = 1 (57)

Here the integral implies that one bit of information is foundon a surface Σ = 1198962119899 in K-space which corresponds to anarea 119860119899 = 412058721198962119899 in 4-space e 1198961 bandwidth permittedwithin a black hole generates the smallest elementary surfaceor pixel in K-space of 11989621 suggesting that information insidea blackhole is diluted to one bit per area equivalent to thesquare of the Hubble 4-radius in 4-space or from (48) theinformation surface density is 11989621 = 41205872Λ3 us thecosmological constant can also be interpreted as a unit ofinformation surface densitye ratio of the Planck surface inK-space to the smallest elementary surface in K-space yields10120 is huge number represents the maximum number ofpixels that can fit on the largest surface in K-space Hencethe expression 119882 = 1198962p1198962n represents the number of empty

pixel slots available in the n-th quantum state or the numberof degenerate energy levels

1003816100381610038161003816120593⟩119899 = 1198881 10038161003816100381610038161205951⟩119899 + 1198882 10038161003816100381610038161205952⟩119899 119888119882 1003816100381610038161003816120595119882⟩119899 (58)

e entropy of the n-state therefore becomes

119878 = 119896119861 ln119882 = 119896119861 ln 1198962p1198962n = 119896119861 ln 1198601198991198972p = 119896119861 ln 1198883119860119899Gℎ (59)

Here we observe that objects fall into a gravitational fieldbecause it leads to an increase in entropy In strong fields theinformation is delocalized A black hole does not annihilateinformation it simply diffuses or dilutes it by increasing theentropy It can also be interpreted as a low pass filter thatpermits one bit of information to pass per Hubble timeSince information is delocalized close to a black hole thenwe do not expect the Event Horizon Telescope to observe asilhouette surrounded by a well defined accretion disc buta circular silhouette surrounded by a cloud of degenerate(delocalized) matter (grey area) as depicted in Figure 1

e radii are calculated from (45) and the magnificationfactor 522 arises from gravitational lensing e silhouetteis half the size predicted in GR

4 Discussion

A successful covariant canonical quantization of the gravita-tional field has been presented in which we find that gravityis akin to a thermal flow of space-time and that space-timecan also be described in terms of a reciprocal lattice epresence of matter creates an impure lattice and a potentialwell arises in the region of perturbation through filtration ofhigh frequencies from the quantum vacuum A test particleflows along with the quantum vacuum and the presence of

Advances in High Energy Physics 7

a potential well will cause it to flow towards the sink isformulation will find important applications in high energylattice gauge field theories where it may help expand thestandard model of particle physics by eliminating divergentterms in the current theory and predicting hitherto unknownphenomena

Data Availability

e results of this theoretical research will be confirmed bydata from the Event Horizon Collaboration on the size andshape of the shadow of Sgr AlowastConflicts of Interest

e author declares no conflicts of interest

Acknowledgments

e author gratefully appreciates the funding and supportfrom the Department of Physics and Astronomy at theBotswana International University of Science and Technol-ogy (BIUST)

References

[1] R Arnowitt S Deser and C W Misner ldquoDynamical structureand definition of energy in general relativityrdquo Physical ReviewAAtomic Molecular and Optical Physics vol 116 no 5 pp 1322ndash1330 1959

[2] Y Sendouda N Deruelle M Sasaki and D Yamauchi ldquoHighercurvature theories of gravity in the adm canonical formalismrdquoInternational Journal of Modern Physics Conference Series vol01 pp 297ndash302 2011

[3] E Gourgoulhon ldquo3+1 formalism and bases for numericalrelativityrdquo 2007 httpsarxivorgabsgr-qc0703035

[4] N Kiriushcheva and S Kuzmin ldquoeHamiltonian formulationof general relativity myths and realityrdquoOpen Physics vol 9 no3 pp 576ndash615 2011

[5] E Anderson ldquoProblem of time in quantum gravityrdquo Annalender Physik vol 524 no 12 pp 757ndash786 2012

[6] B S DeWitt ldquoQuantum theory of gravity I e canonicaltheoryrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 160 article 1113 1967

[7] M Bojowald and P A Hohn ldquoAn Effective approach to theproblem of timerdquo Classical and Quantum Gravity vol 28 no3 Article ID 035006 2011

[8] C Kiefer ldquoDoes time exist in Quantum Gravityrdquo in Towardsa eory of Spacetime eories Einstein Studies D LehmkuhlG Schiemamm and E Scholz Eds vol 13 pp 287ndash295Birkhauser New York NY USA 2017

[9] S Carlip ldquoQuantum gravity a progress reportrdquo Reports onProgress in Physics vol 64 no 8 pp 885ndash942 2001

[10] S Carlip D-W Chiou W-T Ni and R Woodard ldquoQuantumgravity a brief history of ideas and some prospectsrdquo Interna-tional Journal of Modern Physics D Gravitation AstrophysicsCosmology vol 24 no 11 Article ID 1530028 23 pages 2015

[11] C Kiefer ldquoQuantum gravity general introduction and recentdevelopmentsrdquo Annalen der Physik vol 15 no 1-2 pp 129ndash1482006

[12] C Kiefer ldquoConceptual problems in quantum gravity and quan-tum cosmologyrdquo ISRN Mathematical Physics vol 2013 ArticleID 509316 17 pages 2013

[13] S Marongwe ldquoe Nexus graviton A quantum of Dark EnergyandDarkMatterrdquo International Journal of GeometricMethods inModern Physics vol 11 no 06 Article ID 1450059 2014

[14] S Marongwe ldquoe Schwarzschild solution to the Nexus gravi-ton fieldrdquo International Journal of GeometricMethods inModernPhysics vol 12 no 4 Article ID 1550042 10 pages 2015

[15] S Marongwe ldquoe electromagnetic signature of gravitationalwave interactionwith the quantumvacuumrdquo International Jour-nal of Modern Physics D Gravitation Astrophysics Cosmologyvol 26 no 3 Article ID 1750020 15 pages 2017

Advances in High Energy Physics 3

ewave packet described by (8) is essentially a particle offour-space e spin of this particle can be determined fromthe fact that each component of the four-displacement vectortransforms according to the law

Δ1199091015840120583119899 = exp(18120596120583120592 [120574120583 120574120592])Δ119909120583119899 (11)

where120596120583120592 is an antisymmetric 4x4matrix parameterizing thetransformation

erefore each component of the 4-vector has a spin halfA summation of all the four half spins yields a total spin of2 We give the name the Nexus graviton to this particle of 4-space since the primary objective of quantumgravity is to findthe nexus between the concepts of GR and QM

From (7) the norm squared of the 4-momentum of then-th state graviton is

(ℎ)2 119896120583119896120583 = 11986421198991198882 minus3 (119899ℎ1198670)21198882 = 0 (12)

where H0 is the Hubble constant (22 x 10minus18 sminus1) and can be

expressed in terms of the cosmological constant Λ asΛ 119899 = 1198642119899(ℎ119888)2 =

31198962119899(2120587)2 = 1198992Λ (13)

We infer from (13) that the Nexus graviton (or displacement4-vector) in the n-th quantum state forms a trivial vectorbundle via the Uncertainty Principle which generates acompact Riemannian manifold of positive Ricci curvaturethat can be expressed in the form

119866(119899119896)120583120592 = 1198992Λ119892(119899119896)120583120592 (14)

where 119866(119899119896)120583120592 is the Einstein tensor of space-time in then-th state Equation (14) depicts a contracting geodesicball and as explained in [13ndash15] this is DM which is anintrinsic compactification of the elements space-time in then-th quantum state is compactification is a result of thesuperposition of several plane waves as described by (8) toform an increasingly localized wave packet as more waves areadded Similary the converse is also truee loss of harmonicwaves expands the elements of space-time which gives rise toDE us the DE arises from the emission of a ground stategraviton such that (14) becomes

119866(119899119896)120583120592 = (1198992 minus 1)Λ119892(119899119896)120583120592 (15)

ese are Einsteinrsquos vacuum field equations in the quantizedspace-time If the graviton field is perturbed by the presenceof baryonic matter then (15) becomes

119866(119899119896)120583120592 = 119896119879120583] + (1198992 minus 1)Λ119892(119899119896)120583120592= 119896119879120583] + (1198992 minus 1) 119896120588119863119864119892(119899119896)120583120592 (16)

where 120588119863119864 is the density of DEFrom [14] the static solution to (14) for a spherically

symmetric Nexus graviton is computed as

1198891199042 = minus(1 minus ( 21198992 )) 11988821198891199052 + (1 minus ( 21198992 ))minus1 1198891199032

+ 1199032 (1198891205792 + sin2 1205791198891205932) (17)

ere are no singularities in (17) At high energies character-ized by microcosmic scale wavelengths of the Nexus gravitonand high values of n space-time is flat and highly compactis implies a continuous length contraction of the localcoordinates via the addition of more waves resulting in anincrease in localization Also from aworld line perspective ofa test particle it implies that the deviation from a rectilineartrajectory due to uncertainities in its location in 4-space issmall us at microcosmic scales in the realm of subatomicparticles space-time is extremely flat e world line beginsto substantially deviate from a well defined rectilinear path atlow energies as the uncertainities in its location are increasedat is at low values of n the world line becomes degenerateallowing amultiplicity of trajectorieswhich inGRare averagedby the Ricci curvature tensor us gravity is a low energyphenomenonwhich vanishes asymptotically at high energies

e gravitational effects of the Nexus graviton manifestat large scales and for galaxies these effects begin to manifestwhen the density of DE is equal to the density of baryonicmatter as described by (16) or when the acceleration due tobaryonic matter is equal and opposite to the acceleration dueto the emission of the ground state graviton as described in[14] A solution to eqn (16) from [14] in the weak field atgalactic scale radii is

11988921199031198891199052 = 119866119872(119903)1199032 + 1198670V119899 minus 1198670119888 (18)

Here c is the speed of lighte first term on the right is the Newtonian gravitational

acceleration the second term is a radial acceleration due tospace-time in the n-th quantum state and the final term isacceleration due to DEe dynamics becomes strongly non-Newtonian when

119866119872(119903)1199032 = 1198670119888 = V2119899119903 (19)

ese are conditions in which the acceleration due to bary-onic matter is annulled by that due to the DE Under suchconditions

119903 = V21198991198670119888 (20)

Substituting for r in (19) yieldsV4119899 = 119866119872(119903)1198670119888 (21)

is is the Baryonic TullyndashFisher relation e conditionspermitting the DE to cancel out the acceleration due tobaryonic matter leave quantum gravity as the unique sourceof gravity us condition (19) reduces (18) to

11988921199031198891199052 = 119889V119899119889119905 = 1198670V119899 (22)

from which we obtain the following equations of galactic andcosmic evolution

119903119899 = 11198670 119890(1198670119905) (119866119872 (119903)1198670119888)14 (23)

4 Advances in High Energy Physics

V119899 = 119890(1198670119905) (119866119872 (119903)1198670119888)14 (24)

119886119899 = 1198670119890(1198670119905) (119866119872 (119903)1198670119888)14 (25)

Here 119903119899 is the radius of curvature of space-time in the n-th quantum state (which is also the radius of the n-th statenexus graviton) V119899 the radial velocity of objects embedded inthat space-time and 119886119899 their radial acceleration within iteamplification of the radius of curvature with time explainsthe existence of ultra-diffuse galaxies and the spiral shapesof most galaxies e increase in radial velocity with timeexplains why early type galaxies composed of population IIstars are fast rotators Equation (25) explains late time cosmicacceleration which began once condition (19) was satisfied orequivalently from (16) when the density of baryonic matterwas at the same value as that of DE us condition (19) alsoexplains the Coincidence Problem

22 Canonical Transformations in the Nexus Paradigm Inclassical mechanics a system is described by n indepen-dent coordinates (1199021 1199022 119902119899) together with their conjugatemomenta (1199011 1199012 119901119899) In the Nexus Paradigm the labeling119902119899 refers to a creation of a Nexus graviton in the n-thquantum state associated with a conjugate momentum 119901119899e Hamiltonian equation

119902119899 = 120597119867120597119901119899 (26)

refers to the rate of expansion or contraction of space-timegenerated by the addition or emission of harmonic waves tothe Nexus graviton and

119899 = minus120597119867120597119902119899 (27)

refers to the force field associated with the addition oremission of harmonic waves to the Nexus graviton It isimportant to note that this force field generates an isotropicexpansion or contraction of space-timewithin the spatiotem-poral dimensions of the graviton

We can also rewrite the Hamiltonian equations in termsof Poisson brackets which are invariant under canonicaltransformations as 119902119899 = 119902119899 119867 119899 = 119901119899 119867 (28)

ePoisson brackets provide the bridge between classical andQM and in QM these brackets are written as119902119899 = [119902119899 ] 119901119899 = [119901119899 ] (29)

and obey the following commutation rules[119902119899 119902119904] = 0[119901119899 119901119904] = 0[119902119899 119901119904] = 120575119899119904

(30)

23 e Hamiltonian Formulation for the Quantum Vacuume Nexus graviton is a pulse of 4-space which can only

expand or contract and does not execute translationalmotionimplying that the Hamiltonian density of the system is equalto the Lagrangian density 119867 = 119871 (31)

GR is a metric field in which the energy density in four-space determines its value Since the Bloch energy eigenstatefunctions determine the energy of space-time we mustseek to express the metric in terms of the Bloch wavefunctions To this end we shall express the eigenstate four-space components of the Nexus graviton in the k-th band as

Δ119909120583119899119896

= 119911120583119899119896

= 119886119899119896120574120583sinc (119896119909) 119890119894119896119909 (32)

such that an infintesimal four-radius within the k-th band iscomputed as

119889119903119899119896 = 120597119911120583119899119896120597119896120583 119889119896120583 = 119894119909119886119899119896120574120583sinc (119896119909) 119890119894119896119909119889119896 (33)

In (33) the first-order derivative of the periodic sinc functionis equal to zero for all integral values of n e interval withinthe band is then computed as

1198891199042 = 1198891199032119899119896 = 120597119911120583119899119896120597119896120583120597119911120583119899119896120597119896] 119889119896120583119889119896]= 120572120573120574120583120574]120593(119899119896119909)120593(119899119896119909)119889119896120583119889119896]

(34)

Here the interval is described in terms of the reciprocal latticewith 120572 = 119894119909120583119886119899119896 and 120573 = 119894119909]119886119899119896 e metric tensor offour-space in the k-th band is therefore associated with theBloch energy eigenstate functions of the quantum vacuum asfollows 119892120583]

(119899119896)= 120574120583120574]120593(119899119896119909)120593(119899119896119909) = 120578120583]120593(119899119896119909)120593(119899119896119909) (35)

Equation (35) translates the geometric language of GR intothe wave function language of QM

We initiate the translation procedure of GR into QM byfirst finding the Lagrange density for the quantized vacuumfrom (15) which following Einstein and Hilbert is found to be

119871119864119867 = 119896 (119877 minus 2 (1198992 minus 1)Λ) (36)

Given that the Einstein tensor in a compact manifold is equalto the Ricci flow

minus120597119905119892120583] = Δ119892120583] = 119877120583] minus 12119877119892120583] = 119866120583] (37)

therefore the equations of motion of the quantum vacuumobtained from (36) yield the following quantized field equa-tions minus 120597119905 (120574120583120593(119899119896119909)120574]120593(119899119896119909))

= (1198992 minus 1)Λ120574120583120593(119899119896119909)120574]120593(119899119896119909) (38)

which can be written as120597119905 (120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909))= minus1198942(2120587)2nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)

= 141205872nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)(39)

where

120593(119899minus1119896119909) = sinc ((119899 minus 1) 1198961119909) 119890119894(119899minus1)1198961119909 (40)

Advances in High Energy Physics 5

120593(119899+1119896119909) = sinc ((119899 + 1) 1198961119909) 119890119894(119899+1)1198961119909 (41)

311989621(2120587)2 = Λ (42)

For large values of n the Bloch functions statisfy the condition120593(119899minus1119896119909) asymp 120593(119899119896119909) asymp 120593(119899+1119896119909) (43)

equantumvacuumcan therefore be interpreted as a systemin which there are a constant annihilation and creation ofquanta as implied by (40) and (41) which causes the Nexusgraviton to either expand or contract

24 e Hamiltonian Formulation in the Presence of MatterFields We now seek to introduce matter fields into thequantum vacuum If we compare the quantized metric of(17) describing the gravity within a Nexus graviton withthe Schwarzschild metric describing the gravitational fieldaround baryonic matter we notice that we can describe thegravitational field around baryonic matter in terms of thequantum state of space-time through the relation

21198992 = 21198661198721198882119903 (44)

is yields a relationship between the quantum state of space-time and the amount of baryonic matter embedded within itas follows

1198992 = 1198882119903119866119872 = 1198882V2

(45)

Equation (45) reveals a family of concentric stable circularorbits 119903119899 = 11989921198661198721198882 with corresponding orbital speeds ofV119899 = 119888119899 us in the Nexus Paradigm unlike in GR theinnermost stable circular orbit ocurrs at 119899 = 1 or at halfthe Schwarzchild radius which implies that the event horizonpredicted by the Nexus Paradigm is half the size predictedin GR Also (45) reveals how the Nexus graviton in the n-th quantum state imitates DM if M is considered as theapparent mass of the DMrough this comparision we canalso deduce that the deflection of light through gravitationallensing by space-time in the n-th quantum state is

120572 = 41198992 (46)

us gravitational lensing can be used to constrain the valueof the quantum state n of space-time within a lensing system

Having obtained the relationship between the quantumstate of space-time and the amount of baryonic matterembedded within it the result of (45) is then added to (39)to yield the time evolution of the quantum vacuum in thepresence of baryonic matter as

120597119905 (120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= 141205872nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)minus 1198992Λ120574120583120593(119899119896119909)120574]120593(119899119896119909)

(47)

us the time evolution of the quantum vacuum in thepresence of matter resembles thermal flow in the presence of

a heat sinke second term on the RHS of (47) is an 8-cellor 4-cube that operates as a sinc filter with a four-wave cut-offof

119896119888 = 1198991198961 = 2120587radicΛ3 ∙ 1198882119903119866119872 (48)

e filtration of high frequencies from the vacuum lowersthe quantum vacuum state and generates a gravitational fieldin much the same way as the Casimir Effect is generatede physical process of filtration occurs as follows A 4-cubewith baryonic matter embedded with in it acquires inertiae inertia gives the cell inductive impedance and becomesless reponsive to high frequency vibrations of the quantumvacuum us the heavier the cell the lower the cut-off fre-quency Equation (48) shows that when the radius of the cellis equal to half the Schwarzschild radius the only permitedfrequency is that of the ground stateus the inside of a blackhole is in the ground state having a negative metric signature

We now introduce a test particle of mass m into thequantum vacuumperturbed bymatter fieldse particle willflow along with the Ricci flow and the Hamiltonian of thesystem becomes

(120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= P120583P]2119898 (120574120583120593(119899minus1119896119909)120574]120593( 119899+1119896119909))minus 119881 (120574120583120593(119899119896119909)120574]120593(119899119896119909))

(49)

Here

119881 = 1198992 ℎ22119898Λ = 11990311988821198661198723ℎ2119896212119898 (50)

120583 = minus119894ℎnabla120583 (51)

= minus119894ℎ120597119905 (52)

Equation (49) is equivalent to (31) in which the Hamiltonianis equal to the Lagrangian

(120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= L (120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909)) (53)

eLagrangian is not relativistic because the energy referenceframes or space-time states involved in the transition dynam-ics 119899 minus 1 119899 and 119899 + 1 are separated by a small energy gap119864 = radic3 ∙ ℎ1198670 e exchanged quantum of energy betweenstates is responsible for ldquoweldingrdquo them together

Equation (50) is the gravitational interaction in reciprocalspace-time e weakness of the gravitational interaction isdue to the small value of the cosmological constant

3 Line Elements and Information in K-Space

A close inspection of (34) reveals the relationship between aline element in 4-space and its conjugate line in K-space

1198891199042 = 1205721205731198891205812 (54)

6 Advances in High Energy Physics

52

39

26

13

13

n =1

n =2

0

0

13

13

26

2626

39

3939 5252

horizon the quantum state ofspace-time is n =1

arcseconds At the eventShadow diameter = 26 micro Shadow of Sagittarius Alowast

arcseconds

rn =52n2GM

2c2

2c2r1 =

52GM

r2 =104GM

c2

Figure 1 Shadow of Sagittarius Alowast

where

1198891205812 = 120574120583120593(119899119896119909)120574]120593(119899119896119909)119889119896120583119889119896] (55)

e term 120574120583120593(119899119896119909)120574]120593(119899119896119909) refers to the normalized infor-mation flux density passing through an elementary surface119889119896120583119889119896] in K-spacee gradient of the flux density yields thebaseband bandwidth 119896(119899)120583

120597120583120574120583120593(119899119896119909)120574]120593(119899119896119909) = 119896(119899)120583120574120583120593(119899119896119909)120574]120593(119899119896119909) (56)

is gradient is also a measure of the acutance or sharpnessof the information A large bandwidth provides detailedinformation while a small bandwidth provides diffuse infor-mation Equation (49) describes the diffusion of informationas it flows into a gravitational well Unitarity is preserved ifthe flux is summed over the total diffusion surface

∬Σ0120574120583120593(119899119896119909)120574^120593(119899119896119909)119889119896120583119889119896] = 1 (57)

Here the integral implies that one bit of information is foundon a surface Σ = 1198962119899 in K-space which corresponds to anarea 119860119899 = 412058721198962119899 in 4-space e 1198961 bandwidth permittedwithin a black hole generates the smallest elementary surfaceor pixel in K-space of 11989621 suggesting that information insidea blackhole is diluted to one bit per area equivalent to thesquare of the Hubble 4-radius in 4-space or from (48) theinformation surface density is 11989621 = 41205872Λ3 us thecosmological constant can also be interpreted as a unit ofinformation surface densitye ratio of the Planck surface inK-space to the smallest elementary surface in K-space yields10120 is huge number represents the maximum number ofpixels that can fit on the largest surface in K-space Hencethe expression 119882 = 1198962p1198962n represents the number of empty

pixel slots available in the n-th quantum state or the numberof degenerate energy levels

1003816100381610038161003816120593⟩119899 = 1198881 10038161003816100381610038161205951⟩119899 + 1198882 10038161003816100381610038161205952⟩119899 119888119882 1003816100381610038161003816120595119882⟩119899 (58)

e entropy of the n-state therefore becomes

119878 = 119896119861 ln119882 = 119896119861 ln 1198962p1198962n = 119896119861 ln 1198601198991198972p = 119896119861 ln 1198883119860119899Gℎ (59)

Here we observe that objects fall into a gravitational fieldbecause it leads to an increase in entropy In strong fields theinformation is delocalized A black hole does not annihilateinformation it simply diffuses or dilutes it by increasing theentropy It can also be interpreted as a low pass filter thatpermits one bit of information to pass per Hubble timeSince information is delocalized close to a black hole thenwe do not expect the Event Horizon Telescope to observe asilhouette surrounded by a well defined accretion disc buta circular silhouette surrounded by a cloud of degenerate(delocalized) matter (grey area) as depicted in Figure 1

e radii are calculated from (45) and the magnificationfactor 522 arises from gravitational lensing e silhouetteis half the size predicted in GR

4 Discussion

A successful covariant canonical quantization of the gravita-tional field has been presented in which we find that gravityis akin to a thermal flow of space-time and that space-timecan also be described in terms of a reciprocal lattice epresence of matter creates an impure lattice and a potentialwell arises in the region of perturbation through filtration ofhigh frequencies from the quantum vacuum A test particleflows along with the quantum vacuum and the presence of

Advances in High Energy Physics 7

a potential well will cause it to flow towards the sink isformulation will find important applications in high energylattice gauge field theories where it may help expand thestandard model of particle physics by eliminating divergentterms in the current theory and predicting hitherto unknownphenomena

Data Availability

e results of this theoretical research will be confirmed bydata from the Event Horizon Collaboration on the size andshape of the shadow of Sgr AlowastConflicts of Interest

e author declares no conflicts of interest

Acknowledgments

e author gratefully appreciates the funding and supportfrom the Department of Physics and Astronomy at theBotswana International University of Science and Technol-ogy (BIUST)

References

[1] R Arnowitt S Deser and C W Misner ldquoDynamical structureand definition of energy in general relativityrdquo Physical ReviewAAtomic Molecular and Optical Physics vol 116 no 5 pp 1322ndash1330 1959

[2] Y Sendouda N Deruelle M Sasaki and D Yamauchi ldquoHighercurvature theories of gravity in the adm canonical formalismrdquoInternational Journal of Modern Physics Conference Series vol01 pp 297ndash302 2011

[3] E Gourgoulhon ldquo3+1 formalism and bases for numericalrelativityrdquo 2007 httpsarxivorgabsgr-qc0703035

[4] N Kiriushcheva and S Kuzmin ldquoeHamiltonian formulationof general relativity myths and realityrdquoOpen Physics vol 9 no3 pp 576ndash615 2011

[5] E Anderson ldquoProblem of time in quantum gravityrdquo Annalender Physik vol 524 no 12 pp 757ndash786 2012

[6] B S DeWitt ldquoQuantum theory of gravity I e canonicaltheoryrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 160 article 1113 1967

[7] M Bojowald and P A Hohn ldquoAn Effective approach to theproblem of timerdquo Classical and Quantum Gravity vol 28 no3 Article ID 035006 2011

[8] C Kiefer ldquoDoes time exist in Quantum Gravityrdquo in Towardsa eory of Spacetime eories Einstein Studies D LehmkuhlG Schiemamm and E Scholz Eds vol 13 pp 287ndash295Birkhauser New York NY USA 2017

[9] S Carlip ldquoQuantum gravity a progress reportrdquo Reports onProgress in Physics vol 64 no 8 pp 885ndash942 2001

[10] S Carlip D-W Chiou W-T Ni and R Woodard ldquoQuantumgravity a brief history of ideas and some prospectsrdquo Interna-tional Journal of Modern Physics D Gravitation AstrophysicsCosmology vol 24 no 11 Article ID 1530028 23 pages 2015

[11] C Kiefer ldquoQuantum gravity general introduction and recentdevelopmentsrdquo Annalen der Physik vol 15 no 1-2 pp 129ndash1482006

[12] C Kiefer ldquoConceptual problems in quantum gravity and quan-tum cosmologyrdquo ISRN Mathematical Physics vol 2013 ArticleID 509316 17 pages 2013

[13] S Marongwe ldquoe Nexus graviton A quantum of Dark EnergyandDarkMatterrdquo International Journal of GeometricMethods inModern Physics vol 11 no 06 Article ID 1450059 2014

[14] S Marongwe ldquoe Schwarzschild solution to the Nexus gravi-ton fieldrdquo International Journal of GeometricMethods inModernPhysics vol 12 no 4 Article ID 1550042 10 pages 2015

[15] S Marongwe ldquoe electromagnetic signature of gravitationalwave interactionwith the quantumvacuumrdquo International Jour-nal of Modern Physics D Gravitation Astrophysics Cosmologyvol 26 no 3 Article ID 1750020 15 pages 2017

4 Advances in High Energy Physics

V119899 = 119890(1198670119905) (119866119872 (119903)1198670119888)14 (24)

119886119899 = 1198670119890(1198670119905) (119866119872 (119903)1198670119888)14 (25)

Here 119903119899 is the radius of curvature of space-time in the n-th quantum state (which is also the radius of the n-th statenexus graviton) V119899 the radial velocity of objects embedded inthat space-time and 119886119899 their radial acceleration within iteamplification of the radius of curvature with time explainsthe existence of ultra-diffuse galaxies and the spiral shapesof most galaxies e increase in radial velocity with timeexplains why early type galaxies composed of population IIstars are fast rotators Equation (25) explains late time cosmicacceleration which began once condition (19) was satisfied orequivalently from (16) when the density of baryonic matterwas at the same value as that of DE us condition (19) alsoexplains the Coincidence Problem

22 Canonical Transformations in the Nexus Paradigm Inclassical mechanics a system is described by n indepen-dent coordinates (1199021 1199022 119902119899) together with their conjugatemomenta (1199011 1199012 119901119899) In the Nexus Paradigm the labeling119902119899 refers to a creation of a Nexus graviton in the n-thquantum state associated with a conjugate momentum 119901119899e Hamiltonian equation

119902119899 = 120597119867120597119901119899 (26)

refers to the rate of expansion or contraction of space-timegenerated by the addition or emission of harmonic waves tothe Nexus graviton and

119899 = minus120597119867120597119902119899 (27)

refers to the force field associated with the addition oremission of harmonic waves to the Nexus graviton It isimportant to note that this force field generates an isotropicexpansion or contraction of space-timewithin the spatiotem-poral dimensions of the graviton

We can also rewrite the Hamiltonian equations in termsof Poisson brackets which are invariant under canonicaltransformations as 119902119899 = 119902119899 119867 119899 = 119901119899 119867 (28)

ePoisson brackets provide the bridge between classical andQM and in QM these brackets are written as119902119899 = [119902119899 ] 119901119899 = [119901119899 ] (29)

and obey the following commutation rules[119902119899 119902119904] = 0[119901119899 119901119904] = 0[119902119899 119901119904] = 120575119899119904

(30)

23 e Hamiltonian Formulation for the Quantum Vacuume Nexus graviton is a pulse of 4-space which can only

expand or contract and does not execute translationalmotionimplying that the Hamiltonian density of the system is equalto the Lagrangian density 119867 = 119871 (31)

GR is a metric field in which the energy density in four-space determines its value Since the Bloch energy eigenstatefunctions determine the energy of space-time we mustseek to express the metric in terms of the Bloch wavefunctions To this end we shall express the eigenstate four-space components of the Nexus graviton in the k-th band as

Δ119909120583119899119896

= 119911120583119899119896

= 119886119899119896120574120583sinc (119896119909) 119890119894119896119909 (32)

such that an infintesimal four-radius within the k-th band iscomputed as

119889119903119899119896 = 120597119911120583119899119896120597119896120583 119889119896120583 = 119894119909119886119899119896120574120583sinc (119896119909) 119890119894119896119909119889119896 (33)

In (33) the first-order derivative of the periodic sinc functionis equal to zero for all integral values of n e interval withinthe band is then computed as

1198891199042 = 1198891199032119899119896 = 120597119911120583119899119896120597119896120583120597119911120583119899119896120597119896] 119889119896120583119889119896]= 120572120573120574120583120574]120593(119899119896119909)120593(119899119896119909)119889119896120583119889119896]

(34)

Here the interval is described in terms of the reciprocal latticewith 120572 = 119894119909120583119886119899119896 and 120573 = 119894119909]119886119899119896 e metric tensor offour-space in the k-th band is therefore associated with theBloch energy eigenstate functions of the quantum vacuum asfollows 119892120583]

(119899119896)= 120574120583120574]120593(119899119896119909)120593(119899119896119909) = 120578120583]120593(119899119896119909)120593(119899119896119909) (35)

Equation (35) translates the geometric language of GR intothe wave function language of QM

We initiate the translation procedure of GR into QM byfirst finding the Lagrange density for the quantized vacuumfrom (15) which following Einstein and Hilbert is found to be

119871119864119867 = 119896 (119877 minus 2 (1198992 minus 1)Λ) (36)

Given that the Einstein tensor in a compact manifold is equalto the Ricci flow

minus120597119905119892120583] = Δ119892120583] = 119877120583] minus 12119877119892120583] = 119866120583] (37)

therefore the equations of motion of the quantum vacuumobtained from (36) yield the following quantized field equa-tions minus 120597119905 (120574120583120593(119899119896119909)120574]120593(119899119896119909))

= (1198992 minus 1)Λ120574120583120593(119899119896119909)120574]120593(119899119896119909) (38)

which can be written as120597119905 (120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909))= minus1198942(2120587)2nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)

= 141205872nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)(39)

where

120593(119899minus1119896119909) = sinc ((119899 minus 1) 1198961119909) 119890119894(119899minus1)1198961119909 (40)

Advances in High Energy Physics 5

120593(119899+1119896119909) = sinc ((119899 + 1) 1198961119909) 119890119894(119899+1)1198961119909 (41)

311989621(2120587)2 = Λ (42)

For large values of n the Bloch functions statisfy the condition120593(119899minus1119896119909) asymp 120593(119899119896119909) asymp 120593(119899+1119896119909) (43)

equantumvacuumcan therefore be interpreted as a systemin which there are a constant annihilation and creation ofquanta as implied by (40) and (41) which causes the Nexusgraviton to either expand or contract

24 e Hamiltonian Formulation in the Presence of MatterFields We now seek to introduce matter fields into thequantum vacuum If we compare the quantized metric of(17) describing the gravity within a Nexus graviton withthe Schwarzschild metric describing the gravitational fieldaround baryonic matter we notice that we can describe thegravitational field around baryonic matter in terms of thequantum state of space-time through the relation

21198992 = 21198661198721198882119903 (44)

is yields a relationship between the quantum state of space-time and the amount of baryonic matter embedded within itas follows

1198992 = 1198882119903119866119872 = 1198882V2

(45)

Equation (45) reveals a family of concentric stable circularorbits 119903119899 = 11989921198661198721198882 with corresponding orbital speeds ofV119899 = 119888119899 us in the Nexus Paradigm unlike in GR theinnermost stable circular orbit ocurrs at 119899 = 1 or at halfthe Schwarzchild radius which implies that the event horizonpredicted by the Nexus Paradigm is half the size predictedin GR Also (45) reveals how the Nexus graviton in the n-th quantum state imitates DM if M is considered as theapparent mass of the DMrough this comparision we canalso deduce that the deflection of light through gravitationallensing by space-time in the n-th quantum state is

120572 = 41198992 (46)

us gravitational lensing can be used to constrain the valueof the quantum state n of space-time within a lensing system

Having obtained the relationship between the quantumstate of space-time and the amount of baryonic matterembedded within it the result of (45) is then added to (39)to yield the time evolution of the quantum vacuum in thepresence of baryonic matter as

120597119905 (120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= 141205872nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)minus 1198992Λ120574120583120593(119899119896119909)120574]120593(119899119896119909)

(47)

us the time evolution of the quantum vacuum in thepresence of matter resembles thermal flow in the presence of

a heat sinke second term on the RHS of (47) is an 8-cellor 4-cube that operates as a sinc filter with a four-wave cut-offof

119896119888 = 1198991198961 = 2120587radicΛ3 ∙ 1198882119903119866119872 (48)

e filtration of high frequencies from the vacuum lowersthe quantum vacuum state and generates a gravitational fieldin much the same way as the Casimir Effect is generatede physical process of filtration occurs as follows A 4-cubewith baryonic matter embedded with in it acquires inertiae inertia gives the cell inductive impedance and becomesless reponsive to high frequency vibrations of the quantumvacuum us the heavier the cell the lower the cut-off fre-quency Equation (48) shows that when the radius of the cellis equal to half the Schwarzschild radius the only permitedfrequency is that of the ground stateus the inside of a blackhole is in the ground state having a negative metric signature

We now introduce a test particle of mass m into thequantum vacuumperturbed bymatter fieldse particle willflow along with the Ricci flow and the Hamiltonian of thesystem becomes

(120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= P120583P]2119898 (120574120583120593(119899minus1119896119909)120574]120593( 119899+1119896119909))minus 119881 (120574120583120593(119899119896119909)120574]120593(119899119896119909))

(49)

Here

119881 = 1198992 ℎ22119898Λ = 11990311988821198661198723ℎ2119896212119898 (50)

120583 = minus119894ℎnabla120583 (51)

= minus119894ℎ120597119905 (52)

Equation (49) is equivalent to (31) in which the Hamiltonianis equal to the Lagrangian

(120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= L (120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909)) (53)

eLagrangian is not relativistic because the energy referenceframes or space-time states involved in the transition dynam-ics 119899 minus 1 119899 and 119899 + 1 are separated by a small energy gap119864 = radic3 ∙ ℎ1198670 e exchanged quantum of energy betweenstates is responsible for ldquoweldingrdquo them together

Equation (50) is the gravitational interaction in reciprocalspace-time e weakness of the gravitational interaction isdue to the small value of the cosmological constant

3 Line Elements and Information in K-Space

A close inspection of (34) reveals the relationship between aline element in 4-space and its conjugate line in K-space

1198891199042 = 1205721205731198891205812 (54)

6 Advances in High Energy Physics

52

39

26

13

13

n =1

n =2

0

0

13

13

26

2626

39

3939 5252

horizon the quantum state ofspace-time is n =1

arcseconds At the eventShadow diameter = 26 micro Shadow of Sagittarius Alowast

arcseconds

rn =52n2GM

2c2

2c2r1 =

52GM

r2 =104GM

c2

Figure 1 Shadow of Sagittarius Alowast

where

1198891205812 = 120574120583120593(119899119896119909)120574]120593(119899119896119909)119889119896120583119889119896] (55)

e term 120574120583120593(119899119896119909)120574]120593(119899119896119909) refers to the normalized infor-mation flux density passing through an elementary surface119889119896120583119889119896] in K-spacee gradient of the flux density yields thebaseband bandwidth 119896(119899)120583

120597120583120574120583120593(119899119896119909)120574]120593(119899119896119909) = 119896(119899)120583120574120583120593(119899119896119909)120574]120593(119899119896119909) (56)

is gradient is also a measure of the acutance or sharpnessof the information A large bandwidth provides detailedinformation while a small bandwidth provides diffuse infor-mation Equation (49) describes the diffusion of informationas it flows into a gravitational well Unitarity is preserved ifthe flux is summed over the total diffusion surface

∬Σ0120574120583120593(119899119896119909)120574^120593(119899119896119909)119889119896120583119889119896] = 1 (57)

Here the integral implies that one bit of information is foundon a surface Σ = 1198962119899 in K-space which corresponds to anarea 119860119899 = 412058721198962119899 in 4-space e 1198961 bandwidth permittedwithin a black hole generates the smallest elementary surfaceor pixel in K-space of 11989621 suggesting that information insidea blackhole is diluted to one bit per area equivalent to thesquare of the Hubble 4-radius in 4-space or from (48) theinformation surface density is 11989621 = 41205872Λ3 us thecosmological constant can also be interpreted as a unit ofinformation surface densitye ratio of the Planck surface inK-space to the smallest elementary surface in K-space yields10120 is huge number represents the maximum number ofpixels that can fit on the largest surface in K-space Hencethe expression 119882 = 1198962p1198962n represents the number of empty

pixel slots available in the n-th quantum state or the numberof degenerate energy levels

1003816100381610038161003816120593⟩119899 = 1198881 10038161003816100381610038161205951⟩119899 + 1198882 10038161003816100381610038161205952⟩119899 119888119882 1003816100381610038161003816120595119882⟩119899 (58)

e entropy of the n-state therefore becomes

119878 = 119896119861 ln119882 = 119896119861 ln 1198962p1198962n = 119896119861 ln 1198601198991198972p = 119896119861 ln 1198883119860119899Gℎ (59)

Here we observe that objects fall into a gravitational fieldbecause it leads to an increase in entropy In strong fields theinformation is delocalized A black hole does not annihilateinformation it simply diffuses or dilutes it by increasing theentropy It can also be interpreted as a low pass filter thatpermits one bit of information to pass per Hubble timeSince information is delocalized close to a black hole thenwe do not expect the Event Horizon Telescope to observe asilhouette surrounded by a well defined accretion disc buta circular silhouette surrounded by a cloud of degenerate(delocalized) matter (grey area) as depicted in Figure 1

e radii are calculated from (45) and the magnificationfactor 522 arises from gravitational lensing e silhouetteis half the size predicted in GR

4 Discussion

A successful covariant canonical quantization of the gravita-tional field has been presented in which we find that gravityis akin to a thermal flow of space-time and that space-timecan also be described in terms of a reciprocal lattice epresence of matter creates an impure lattice and a potentialwell arises in the region of perturbation through filtration ofhigh frequencies from the quantum vacuum A test particleflows along with the quantum vacuum and the presence of

Advances in High Energy Physics 7

a potential well will cause it to flow towards the sink isformulation will find important applications in high energylattice gauge field theories where it may help expand thestandard model of particle physics by eliminating divergentterms in the current theory and predicting hitherto unknownphenomena

Data Availability

e results of this theoretical research will be confirmed bydata from the Event Horizon Collaboration on the size andshape of the shadow of Sgr AlowastConflicts of Interest

e author declares no conflicts of interest

Acknowledgments

e author gratefully appreciates the funding and supportfrom the Department of Physics and Astronomy at theBotswana International University of Science and Technol-ogy (BIUST)

References

[1] R Arnowitt S Deser and C W Misner ldquoDynamical structureand definition of energy in general relativityrdquo Physical ReviewAAtomic Molecular and Optical Physics vol 116 no 5 pp 1322ndash1330 1959

[2] Y Sendouda N Deruelle M Sasaki and D Yamauchi ldquoHighercurvature theories of gravity in the adm canonical formalismrdquoInternational Journal of Modern Physics Conference Series vol01 pp 297ndash302 2011

[3] E Gourgoulhon ldquo3+1 formalism and bases for numericalrelativityrdquo 2007 httpsarxivorgabsgr-qc0703035

[4] N Kiriushcheva and S Kuzmin ldquoeHamiltonian formulationof general relativity myths and realityrdquoOpen Physics vol 9 no3 pp 576ndash615 2011

[5] E Anderson ldquoProblem of time in quantum gravityrdquo Annalender Physik vol 524 no 12 pp 757ndash786 2012

[6] B S DeWitt ldquoQuantum theory of gravity I e canonicaltheoryrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 160 article 1113 1967

[7] M Bojowald and P A Hohn ldquoAn Effective approach to theproblem of timerdquo Classical and Quantum Gravity vol 28 no3 Article ID 035006 2011

[8] C Kiefer ldquoDoes time exist in Quantum Gravityrdquo in Towardsa eory of Spacetime eories Einstein Studies D LehmkuhlG Schiemamm and E Scholz Eds vol 13 pp 287ndash295Birkhauser New York NY USA 2017

[9] S Carlip ldquoQuantum gravity a progress reportrdquo Reports onProgress in Physics vol 64 no 8 pp 885ndash942 2001

[10] S Carlip D-W Chiou W-T Ni and R Woodard ldquoQuantumgravity a brief history of ideas and some prospectsrdquo Interna-tional Journal of Modern Physics D Gravitation AstrophysicsCosmology vol 24 no 11 Article ID 1530028 23 pages 2015

[11] C Kiefer ldquoQuantum gravity general introduction and recentdevelopmentsrdquo Annalen der Physik vol 15 no 1-2 pp 129ndash1482006

[12] C Kiefer ldquoConceptual problems in quantum gravity and quan-tum cosmologyrdquo ISRN Mathematical Physics vol 2013 ArticleID 509316 17 pages 2013

[13] S Marongwe ldquoe Nexus graviton A quantum of Dark EnergyandDarkMatterrdquo International Journal of GeometricMethods inModern Physics vol 11 no 06 Article ID 1450059 2014

[14] S Marongwe ldquoe Schwarzschild solution to the Nexus gravi-ton fieldrdquo International Journal of GeometricMethods inModernPhysics vol 12 no 4 Article ID 1550042 10 pages 2015

[15] S Marongwe ldquoe electromagnetic signature of gravitationalwave interactionwith the quantumvacuumrdquo International Jour-nal of Modern Physics D Gravitation Astrophysics Cosmologyvol 26 no 3 Article ID 1750020 15 pages 2017

Advances in High Energy Physics 5

120593(119899+1119896119909) = sinc ((119899 + 1) 1198961119909) 119890119894(119899+1)1198961119909 (41)

311989621(2120587)2 = Λ (42)

For large values of n the Bloch functions statisfy the condition120593(119899minus1119896119909) asymp 120593(119899119896119909) asymp 120593(119899+1119896119909) (43)

equantumvacuumcan therefore be interpreted as a systemin which there are a constant annihilation and creation ofquanta as implied by (40) and (41) which causes the Nexusgraviton to either expand or contract

24 e Hamiltonian Formulation in the Presence of MatterFields We now seek to introduce matter fields into thequantum vacuum If we compare the quantized metric of(17) describing the gravity within a Nexus graviton withthe Schwarzschild metric describing the gravitational fieldaround baryonic matter we notice that we can describe thegravitational field around baryonic matter in terms of thequantum state of space-time through the relation

21198992 = 21198661198721198882119903 (44)

is yields a relationship between the quantum state of space-time and the amount of baryonic matter embedded within itas follows

1198992 = 1198882119903119866119872 = 1198882V2

(45)

Equation (45) reveals a family of concentric stable circularorbits 119903119899 = 11989921198661198721198882 with corresponding orbital speeds ofV119899 = 119888119899 us in the Nexus Paradigm unlike in GR theinnermost stable circular orbit ocurrs at 119899 = 1 or at halfthe Schwarzchild radius which implies that the event horizonpredicted by the Nexus Paradigm is half the size predictedin GR Also (45) reveals how the Nexus graviton in the n-th quantum state imitates DM if M is considered as theapparent mass of the DMrough this comparision we canalso deduce that the deflection of light through gravitationallensing by space-time in the n-th quantum state is

120572 = 41198992 (46)

us gravitational lensing can be used to constrain the valueof the quantum state n of space-time within a lensing system

Having obtained the relationship between the quantumstate of space-time and the amount of baryonic matterembedded within it the result of (45) is then added to (39)to yield the time evolution of the quantum vacuum in thepresence of baryonic matter as

120597119905 (120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= 141205872nabla120583nabla]120574120583120593(119899minus1119896119909)120574]120593(119899+1119896119909)minus 1198992Λ120574120583120593(119899119896119909)120574]120593(119899119896119909)

(47)

us the time evolution of the quantum vacuum in thepresence of matter resembles thermal flow in the presence of

a heat sinke second term on the RHS of (47) is an 8-cellor 4-cube that operates as a sinc filter with a four-wave cut-offof

119896119888 = 1198991198961 = 2120587radicΛ3 ∙ 1198882119903119866119872 (48)

e filtration of high frequencies from the vacuum lowersthe quantum vacuum state and generates a gravitational fieldin much the same way as the Casimir Effect is generatede physical process of filtration occurs as follows A 4-cubewith baryonic matter embedded with in it acquires inertiae inertia gives the cell inductive impedance and becomesless reponsive to high frequency vibrations of the quantumvacuum us the heavier the cell the lower the cut-off fre-quency Equation (48) shows that when the radius of the cellis equal to half the Schwarzschild radius the only permitedfrequency is that of the ground stateus the inside of a blackhole is in the ground state having a negative metric signature

We now introduce a test particle of mass m into thequantum vacuumperturbed bymatter fieldse particle willflow along with the Ricci flow and the Hamiltonian of thesystem becomes

(120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= P120583P]2119898 (120574120583120593(119899minus1119896119909)120574]120593( 119899+1119896119909))minus 119881 (120574120583120593(119899119896119909)120574]120593(119899119896119909))

(49)

Here

119881 = 1198992 ℎ22119898Λ = 11990311988821198661198723ℎ2119896212119898 (50)

120583 = minus119894ℎnabla120583 (51)

= minus119894ℎ120597119905 (52)

Equation (49) is equivalent to (31) in which the Hamiltonianis equal to the Lagrangian

(120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909))= L (120574120583120593(119899minus1119899119896119909)120574]120593(119899+1119899119896119909)) (53)

eLagrangian is not relativistic because the energy referenceframes or space-time states involved in the transition dynam-ics 119899 minus 1 119899 and 119899 + 1 are separated by a small energy gap119864 = radic3 ∙ ℎ1198670 e exchanged quantum of energy betweenstates is responsible for ldquoweldingrdquo them together

Equation (50) is the gravitational interaction in reciprocalspace-time e weakness of the gravitational interaction isdue to the small value of the cosmological constant

3 Line Elements and Information in K-Space

A close inspection of (34) reveals the relationship between aline element in 4-space and its conjugate line in K-space

1198891199042 = 1205721205731198891205812 (54)

6 Advances in High Energy Physics

52

39

26

13

13

n =1

n =2

0

0

13

13

26

2626

39

3939 5252

horizon the quantum state ofspace-time is n =1

arcseconds At the eventShadow diameter = 26 micro Shadow of Sagittarius Alowast

arcseconds

rn =52n2GM

2c2

2c2r1 =

52GM

r2 =104GM

c2

Figure 1 Shadow of Sagittarius Alowast

where

1198891205812 = 120574120583120593(119899119896119909)120574]120593(119899119896119909)119889119896120583119889119896] (55)

e term 120574120583120593(119899119896119909)120574]120593(119899119896119909) refers to the normalized infor-mation flux density passing through an elementary surface119889119896120583119889119896] in K-spacee gradient of the flux density yields thebaseband bandwidth 119896(119899)120583

120597120583120574120583120593(119899119896119909)120574]120593(119899119896119909) = 119896(119899)120583120574120583120593(119899119896119909)120574]120593(119899119896119909) (56)

is gradient is also a measure of the acutance or sharpnessof the information A large bandwidth provides detailedinformation while a small bandwidth provides diffuse infor-mation Equation (49) describes the diffusion of informationas it flows into a gravitational well Unitarity is preserved ifthe flux is summed over the total diffusion surface

∬Σ0120574120583120593(119899119896119909)120574^120593(119899119896119909)119889119896120583119889119896] = 1 (57)

Here the integral implies that one bit of information is foundon a surface Σ = 1198962119899 in K-space which corresponds to anarea 119860119899 = 412058721198962119899 in 4-space e 1198961 bandwidth permittedwithin a black hole generates the smallest elementary surfaceor pixel in K-space of 11989621 suggesting that information insidea blackhole is diluted to one bit per area equivalent to thesquare of the Hubble 4-radius in 4-space or from (48) theinformation surface density is 11989621 = 41205872Λ3 us thecosmological constant can also be interpreted as a unit ofinformation surface densitye ratio of the Planck surface inK-space to the smallest elementary surface in K-space yields10120 is huge number represents the maximum number ofpixels that can fit on the largest surface in K-space Hencethe expression 119882 = 1198962p1198962n represents the number of empty

pixel slots available in the n-th quantum state or the numberof degenerate energy levels

1003816100381610038161003816120593⟩119899 = 1198881 10038161003816100381610038161205951⟩119899 + 1198882 10038161003816100381610038161205952⟩119899 119888119882 1003816100381610038161003816120595119882⟩119899 (58)

e entropy of the n-state therefore becomes

119878 = 119896119861 ln119882 = 119896119861 ln 1198962p1198962n = 119896119861 ln 1198601198991198972p = 119896119861 ln 1198883119860119899Gℎ (59)

Here we observe that objects fall into a gravitational fieldbecause it leads to an increase in entropy In strong fields theinformation is delocalized A black hole does not annihilateinformation it simply diffuses or dilutes it by increasing theentropy It can also be interpreted as a low pass filter thatpermits one bit of information to pass per Hubble timeSince information is delocalized close to a black hole thenwe do not expect the Event Horizon Telescope to observe asilhouette surrounded by a well defined accretion disc buta circular silhouette surrounded by a cloud of degenerate(delocalized) matter (grey area) as depicted in Figure 1

e radii are calculated from (45) and the magnificationfactor 522 arises from gravitational lensing e silhouetteis half the size predicted in GR

4 Discussion

A successful covariant canonical quantization of the gravita-tional field has been presented in which we find that gravityis akin to a thermal flow of space-time and that space-timecan also be described in terms of a reciprocal lattice epresence of matter creates an impure lattice and a potentialwell arises in the region of perturbation through filtration ofhigh frequencies from the quantum vacuum A test particleflows along with the quantum vacuum and the presence of

Advances in High Energy Physics 7

a potential well will cause it to flow towards the sink isformulation will find important applications in high energylattice gauge field theories where it may help expand thestandard model of particle physics by eliminating divergentterms in the current theory and predicting hitherto unknownphenomena

Data Availability

e results of this theoretical research will be confirmed bydata from the Event Horizon Collaboration on the size andshape of the shadow of Sgr AlowastConflicts of Interest

e author declares no conflicts of interest

Acknowledgments

e author gratefully appreciates the funding and supportfrom the Department of Physics and Astronomy at theBotswana International University of Science and Technol-ogy (BIUST)

References

[1] R Arnowitt S Deser and C W Misner ldquoDynamical structureand definition of energy in general relativityrdquo Physical ReviewAAtomic Molecular and Optical Physics vol 116 no 5 pp 1322ndash1330 1959

[2] Y Sendouda N Deruelle M Sasaki and D Yamauchi ldquoHighercurvature theories of gravity in the adm canonical formalismrdquoInternational Journal of Modern Physics Conference Series vol01 pp 297ndash302 2011

[3] E Gourgoulhon ldquo3+1 formalism and bases for numericalrelativityrdquo 2007 httpsarxivorgabsgr-qc0703035

[4] N Kiriushcheva and S Kuzmin ldquoeHamiltonian formulationof general relativity myths and realityrdquoOpen Physics vol 9 no3 pp 576ndash615 2011

[5] E Anderson ldquoProblem of time in quantum gravityrdquo Annalender Physik vol 524 no 12 pp 757ndash786 2012

[6] B S DeWitt ldquoQuantum theory of gravity I e canonicaltheoryrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 160 article 1113 1967

[7] M Bojowald and P A Hohn ldquoAn Effective approach to theproblem of timerdquo Classical and Quantum Gravity vol 28 no3 Article ID 035006 2011

[8] C Kiefer ldquoDoes time exist in Quantum Gravityrdquo in Towardsa eory of Spacetime eories Einstein Studies D LehmkuhlG Schiemamm and E Scholz Eds vol 13 pp 287ndash295Birkhauser New York NY USA 2017

[9] S Carlip ldquoQuantum gravity a progress reportrdquo Reports onProgress in Physics vol 64 no 8 pp 885ndash942 2001

[10] S Carlip D-W Chiou W-T Ni and R Woodard ldquoQuantumgravity a brief history of ideas and some prospectsrdquo Interna-tional Journal of Modern Physics D Gravitation AstrophysicsCosmology vol 24 no 11 Article ID 1530028 23 pages 2015

[11] C Kiefer ldquoQuantum gravity general introduction and recentdevelopmentsrdquo Annalen der Physik vol 15 no 1-2 pp 129ndash1482006

[12] C Kiefer ldquoConceptual problems in quantum gravity and quan-tum cosmologyrdquo ISRN Mathematical Physics vol 2013 ArticleID 509316 17 pages 2013

[13] S Marongwe ldquoe Nexus graviton A quantum of Dark EnergyandDarkMatterrdquo International Journal of GeometricMethods inModern Physics vol 11 no 06 Article ID 1450059 2014

[14] S Marongwe ldquoe Schwarzschild solution to the Nexus gravi-ton fieldrdquo International Journal of GeometricMethods inModernPhysics vol 12 no 4 Article ID 1550042 10 pages 2015

[15] S Marongwe ldquoe electromagnetic signature of gravitationalwave interactionwith the quantumvacuumrdquo International Jour-nal of Modern Physics D Gravitation Astrophysics Cosmologyvol 26 no 3 Article ID 1750020 15 pages 2017

6 Advances in High Energy Physics

52

39

26

13

13

n =1

n =2

0

0

13

13

26

2626

39

3939 5252

horizon the quantum state ofspace-time is n =1

arcseconds At the eventShadow diameter = 26 micro Shadow of Sagittarius Alowast

arcseconds

rn =52n2GM

2c2

2c2r1 =

52GM

r2 =104GM

c2

Figure 1 Shadow of Sagittarius Alowast

where

1198891205812 = 120574120583120593(119899119896119909)120574]120593(119899119896119909)119889119896120583119889119896] (55)

e term 120574120583120593(119899119896119909)120574]120593(119899119896119909) refers to the normalized infor-mation flux density passing through an elementary surface119889119896120583119889119896] in K-spacee gradient of the flux density yields thebaseband bandwidth 119896(119899)120583

120597120583120574120583120593(119899119896119909)120574]120593(119899119896119909) = 119896(119899)120583120574120583120593(119899119896119909)120574]120593(119899119896119909) (56)

is gradient is also a measure of the acutance or sharpnessof the information A large bandwidth provides detailedinformation while a small bandwidth provides diffuse infor-mation Equation (49) describes the diffusion of informationas it flows into a gravitational well Unitarity is preserved ifthe flux is summed over the total diffusion surface

∬Σ0120574120583120593(119899119896119909)120574^120593(119899119896119909)119889119896120583119889119896] = 1 (57)

Here the integral implies that one bit of information is foundon a surface Σ = 1198962119899 in K-space which corresponds to anarea 119860119899 = 412058721198962119899 in 4-space e 1198961 bandwidth permittedwithin a black hole generates the smallest elementary surfaceor pixel in K-space of 11989621 suggesting that information insidea blackhole is diluted to one bit per area equivalent to thesquare of the Hubble 4-radius in 4-space or from (48) theinformation surface density is 11989621 = 41205872Λ3 us thecosmological constant can also be interpreted as a unit ofinformation surface densitye ratio of the Planck surface inK-space to the smallest elementary surface in K-space yields10120 is huge number represents the maximum number ofpixels that can fit on the largest surface in K-space Hencethe expression 119882 = 1198962p1198962n represents the number of empty

pixel slots available in the n-th quantum state or the numberof degenerate energy levels

1003816100381610038161003816120593⟩119899 = 1198881 10038161003816100381610038161205951⟩119899 + 1198882 10038161003816100381610038161205952⟩119899 119888119882 1003816100381610038161003816120595119882⟩119899 (58)

e entropy of the n-state therefore becomes

119878 = 119896119861 ln119882 = 119896119861 ln 1198962p1198962n = 119896119861 ln 1198601198991198972p = 119896119861 ln 1198883119860119899Gℎ (59)

Here we observe that objects fall into a gravitational fieldbecause it leads to an increase in entropy In strong fields theinformation is delocalized A black hole does not annihilateinformation it simply diffuses or dilutes it by increasing theentropy It can also be interpreted as a low pass filter thatpermits one bit of information to pass per Hubble timeSince information is delocalized close to a black hole thenwe do not expect the Event Horizon Telescope to observe asilhouette surrounded by a well defined accretion disc buta circular silhouette surrounded by a cloud of degenerate(delocalized) matter (grey area) as depicted in Figure 1

e radii are calculated from (45) and the magnificationfactor 522 arises from gravitational lensing e silhouetteis half the size predicted in GR

4 Discussion

A successful covariant canonical quantization of the gravita-tional field has been presented in which we find that gravityis akin to a thermal flow of space-time and that space-timecan also be described in terms of a reciprocal lattice epresence of matter creates an impure lattice and a potentialwell arises in the region of perturbation through filtration ofhigh frequencies from the quantum vacuum A test particleflows along with the quantum vacuum and the presence of

Advances in High Energy Physics 7

a potential well will cause it to flow towards the sink isformulation will find important applications in high energylattice gauge field theories where it may help expand thestandard model of particle physics by eliminating divergentterms in the current theory and predicting hitherto unknownphenomena

Data Availability

e results of this theoretical research will be confirmed bydata from the Event Horizon Collaboration on the size andshape of the shadow of Sgr AlowastConflicts of Interest

e author declares no conflicts of interest

Acknowledgments

e author gratefully appreciates the funding and supportfrom the Department of Physics and Astronomy at theBotswana International University of Science and Technol-ogy (BIUST)

References

[1] R Arnowitt S Deser and C W Misner ldquoDynamical structureand definition of energy in general relativityrdquo Physical ReviewAAtomic Molecular and Optical Physics vol 116 no 5 pp 1322ndash1330 1959

[2] Y Sendouda N Deruelle M Sasaki and D Yamauchi ldquoHighercurvature theories of gravity in the adm canonical formalismrdquoInternational Journal of Modern Physics Conference Series vol01 pp 297ndash302 2011

[3] E Gourgoulhon ldquo3+1 formalism and bases for numericalrelativityrdquo 2007 httpsarxivorgabsgr-qc0703035

[4] N Kiriushcheva and S Kuzmin ldquoeHamiltonian formulationof general relativity myths and realityrdquoOpen Physics vol 9 no3 pp 576ndash615 2011

[5] E Anderson ldquoProblem of time in quantum gravityrdquo Annalender Physik vol 524 no 12 pp 757ndash786 2012

[6] B S DeWitt ldquoQuantum theory of gravity I e canonicaltheoryrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 160 article 1113 1967

[7] M Bojowald and P A Hohn ldquoAn Effective approach to theproblem of timerdquo Classical and Quantum Gravity vol 28 no3 Article ID 035006 2011

[8] C Kiefer ldquoDoes time exist in Quantum Gravityrdquo in Towardsa eory of Spacetime eories Einstein Studies D LehmkuhlG Schiemamm and E Scholz Eds vol 13 pp 287ndash295Birkhauser New York NY USA 2017

[9] S Carlip ldquoQuantum gravity a progress reportrdquo Reports onProgress in Physics vol 64 no 8 pp 885ndash942 2001

[10] S Carlip D-W Chiou W-T Ni and R Woodard ldquoQuantumgravity a brief history of ideas and some prospectsrdquo Interna-tional Journal of Modern Physics D Gravitation AstrophysicsCosmology vol 24 no 11 Article ID 1530028 23 pages 2015

[11] C Kiefer ldquoQuantum gravity general introduction and recentdevelopmentsrdquo Annalen der Physik vol 15 no 1-2 pp 129ndash1482006

[12] C Kiefer ldquoConceptual problems in quantum gravity and quan-tum cosmologyrdquo ISRN Mathematical Physics vol 2013 ArticleID 509316 17 pages 2013

[13] S Marongwe ldquoe Nexus graviton A quantum of Dark EnergyandDarkMatterrdquo International Journal of GeometricMethods inModern Physics vol 11 no 06 Article ID 1450059 2014

[14] S Marongwe ldquoe Schwarzschild solution to the Nexus gravi-ton fieldrdquo International Journal of GeometricMethods inModernPhysics vol 12 no 4 Article ID 1550042 10 pages 2015

[15] S Marongwe ldquoe electromagnetic signature of gravitationalwave interactionwith the quantumvacuumrdquo International Jour-nal of Modern Physics D Gravitation Astrophysics Cosmologyvol 26 no 3 Article ID 1750020 15 pages 2017

Advances in High Energy Physics 7

a potential well will cause it to flow towards the sink isformulation will find important applications in high energylattice gauge field theories where it may help expand thestandard model of particle physics by eliminating divergentterms in the current theory and predicting hitherto unknownphenomena

Data Availability

e results of this theoretical research will be confirmed bydata from the Event Horizon Collaboration on the size andshape of the shadow of Sgr AlowastConflicts of Interest

e author declares no conflicts of interest

Acknowledgments

e author gratefully appreciates the funding and supportfrom the Department of Physics and Astronomy at theBotswana International University of Science and Technol-ogy (BIUST)

References

[1] R Arnowitt S Deser and C W Misner ldquoDynamical structureand definition of energy in general relativityrdquo Physical ReviewAAtomic Molecular and Optical Physics vol 116 no 5 pp 1322ndash1330 1959

[2] Y Sendouda N Deruelle M Sasaki and D Yamauchi ldquoHighercurvature theories of gravity in the adm canonical formalismrdquoInternational Journal of Modern Physics Conference Series vol01 pp 297ndash302 2011

[3] E Gourgoulhon ldquo3+1 formalism and bases for numericalrelativityrdquo 2007 httpsarxivorgabsgr-qc0703035

[4] N Kiriushcheva and S Kuzmin ldquoeHamiltonian formulationof general relativity myths and realityrdquoOpen Physics vol 9 no3 pp 576ndash615 2011

[5] E Anderson ldquoProblem of time in quantum gravityrdquo Annalender Physik vol 524 no 12 pp 757ndash786 2012

[6] B S DeWitt ldquoQuantum theory of gravity I e canonicaltheoryrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 160 article 1113 1967

[7] M Bojowald and P A Hohn ldquoAn Effective approach to theproblem of timerdquo Classical and Quantum Gravity vol 28 no3 Article ID 035006 2011

[8] C Kiefer ldquoDoes time exist in Quantum Gravityrdquo in Towardsa eory of Spacetime eories Einstein Studies D LehmkuhlG Schiemamm and E Scholz Eds vol 13 pp 287ndash295Birkhauser New York NY USA 2017

[9] S Carlip ldquoQuantum gravity a progress reportrdquo Reports onProgress in Physics vol 64 no 8 pp 885ndash942 2001

[10] S Carlip D-W Chiou W-T Ni and R Woodard ldquoQuantumgravity a brief history of ideas and some prospectsrdquo Interna-tional Journal of Modern Physics D Gravitation AstrophysicsCosmology vol 24 no 11 Article ID 1530028 23 pages 2015

[11] C Kiefer ldquoQuantum gravity general introduction and recentdevelopmentsrdquo Annalen der Physik vol 15 no 1-2 pp 129ndash1482006

[12] C Kiefer ldquoConceptual problems in quantum gravity and quan-tum cosmologyrdquo ISRN Mathematical Physics vol 2013 ArticleID 509316 17 pages 2013

[13] S Marongwe ldquoe Nexus graviton A quantum of Dark EnergyandDarkMatterrdquo International Journal of GeometricMethods inModern Physics vol 11 no 06 Article ID 1450059 2014

[14] S Marongwe ldquoe Schwarzschild solution to the Nexus gravi-ton fieldrdquo International Journal of GeometricMethods inModernPhysics vol 12 no 4 Article ID 1550042 10 pages 2015

[15] S Marongwe ldquoe electromagnetic signature of gravitationalwave interactionwith the quantumvacuumrdquo International Jour-nal of Modern Physics D Gravitation Astrophysics Cosmologyvol 26 no 3 Article ID 1750020 15 pages 2017

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