Einstein-Cartan Gravity in Particle Physics and Cosmology

Nikodem J. Popławski

Department of Mathematics and PhysicsUniversity of New Haven

West Haven, CT, USA

Department of Physics SeminarIndian Institute of Technology, Hyderabad, India

22/23 November 2016

Outline

1. Einstein-Cartan-Sciama-Kibble gravity

2. Spin-torsioncouplingofDiracfields

3. Matter-antimatterasymmetryfromtorsion

4. Nonsingularfermionsfromtorsion

5. Spinfluids

6. Bigbounceandinflationfromtorsion

7. Cosmologicalconstantfromtorsion

Gravitywithtorsion

Einstein-Cartan-Sciama-Kibbletheoryofgravity

Whatistorsion?• Tensors– behaveundercoordinatetransformations likeproductsofdifferentialsandgradients.Specialcase:vectors.

• Differentiationofvectorsincurvedspacetime requiressubtractingtwoinfinitesimalvectorsattwopointsthathavedifferenttransformationproperties.

• Paralleltransportallowstobringonevectortotheoriginoftheotherone,sothattheirdifference wouldmakesense.

E = mc2

AδA

δxδAi =-Γijk Aj δxk

Affineconnection

Whatistorsion?• Curvedspacetimerequiresgeometricalstructure:affineconnection¡½¹º

• CovariantderivativerºV¹ =¶ºV¹ +¡¹½ºV½

• CurvaturetensorR½¾¹º =¶¹¡½¾º - ¶º¡½¾¹ +¡½¿¹¡¿

¾º - ¡½¿º¡¿¾¹

E = mc2

1

2

3

4

Measures thechangeofavectorparallel-transported

alongaclosedcurve:

change=curvatureXareaXvector

Whatistorsion?

• Torsiontensor – antisymmetricpartofaffineconnection

• Contortiontensor

GR– affineconnectionrestrictedtobesymmetric inlowerindices

ECSK– noconstraintonconnection:morenatural

E = mc2

Measures noncommutativity ofparalleltransports

TheoriesofspacetimeSpecialRelativity – flatspacetime (nocurvature)Dynamicalvariables:matterfields

GeneralRelativity – (curvature,notorsion)Dynamicalvariables:matterfields+metrictensor

ECSKGravity(simplesttheorywithcurvature&torsion)Dynamicalvariables:matterfields+metric+torsion

E = mc2

Moredegreesoffreedom

ECSKgravity• Riemann-Cartanspacetime– metricityr½g¹º =0→connection¡½¹º ={½¹º}+C½¹º

Christoffelsymbolscontortiontensor

• Lagrangiandensity formatter

Metricalenergy-momentum tensorSpintensor

TotalLagrangiandensity(likeinGR)

E = mc2

T.W.B.Kibble,J.Math.Phys.2,212(1961)D.W.Sciama,Rev.Mod.Phys.36,463(1964)

ECSKgravity• Curvaturetensor=Riemanntensor

+tensorquadratic intorsion+totalderivative

• Stationarityofactionunder±g¹º →Einsteinequations

R{}¹º - R{}g¹º /2=k(T¹º +U¹º)

U¹º =[C½¹½C¾º¾ - C½¹¾C¾º½ - (C½¾½C¿¾¿ -C¾½¿C¿½¾)g¹º /2]/k

• Stationarityofactionunder±C¹º½→Cartanequations

S½¹º - S¹±½º +Sº±½¹ =-ks¹º½ /2

S¹ =Sº¹º

• Cartanequationsarealgebraicandlinear: torsionα spindensity• Contributionstoenergy-momentum fromspinarequadratic

E = mc2

Samecouplingconstantk

T.W.B.Kibble,J.Math.Phys.2,212(1961)D.W.Sciama,Rev.Mod.Phys.36,463(1964)

ECSKgravity

• FieldequationswithfullRiccitensorcanbewrittenas

R¹º - Rg¹º /2=k£º¹

Tetradenergy-momentum tensor

• Belinfante-Rosenfeldrelation

£¹º =T¹º +r¤½(s¹º½ +s½º¹ +s½¹º)/2r¤

½=r½- 2S½

• Conservation lawforspin

r¤½s¹º½ =(£¹º - £º¹)

• Cyclicidentities

R¾¹º½ =-2r¹S¾º½ +4S¾¿¹S¿º½ (¹,º,½ cyclicallypermutated)

T.W.B.Kibble,J.Math.Phys.2,212(1961)D.W.Sciama,Rev.Mod.Phys.36,463(1964)

ECSKgravity

• Bianchiidentities (¹,º,½ cyclicallypermutated)

r¹R¾¿º½ =2R¾¿¼¹S¼º½

• Conservation lawforenergyandmomentum

Dº£¹º =Cº½¹£º½ +sº½¾Rº½¾¹/2Dº=r{}º

Equationsofmotionofparticles

F.W.Hehl,P.vonderHeyde,G.D.Kerlick &J.M.Nester,Rev.Mod.Phys.48,393(1976)E.A.Lord,Tensors,RelativityandCosmology(McGraw-Hill,1976)– @ IndianInstituteofScience,Bangalore,IndiaNJP,arXiv:0911.0334

T.W.B.Kibble,J.Math.Phys.2,212(1961)D.W.Sciama,Rev.Mod.Phys.36,463(1964)

ECSKgravity

• Nospinors->torsionvanishes->ECSKreducestoGR

• TorsionsignificantwhenU¹º » T¹º (atCartan density)

Forfermionic matter(quarksandleptons)½ >1045kgm-3

Nuclearmatterinneutronstars½ » 1017 kgm-3

Gravitationaleffectsoftorsionnegligibleevenforneutronstars

TorsionsignificantonlyinveryearlyUniverseand inblackholes,andforfermionsatverysmallscales

E = mc2

Spin-torsioncouplingofspinors• Diracmatrices

• SpinorrepresentationofLorentzgroup

• Spinors

• Covariantderivativeofspinor

Metricity ->->

E = mc2

Spinorconnection

Fock-Ivanenkocoefficients(1929)

Tetrad

Spinconnection

Spin-torsioncouplingofspinors• DiracLagrangiandensity

• Spindensity

Cartan equationsDiracequation

E = mc2

VariationofCó variationofω

Totallyantisymmetric

Diracspinpseudovector

;– covariantderivativewithaffineconnection

:– withChristoffelsymbols

F.W.Hehl &B.K.Datta,J.Math.Phys.12,1334(1971)– @BoseInstitute, India

Matter-antimatterasymmetry• Hehl-Dattaequation

• Chargeconjugate

SatisfiesHehl-Dattaequationwithoppositechargeanddifferentsignforthecubicterm

HDasymmetrysignificantwhentorsionis->baryogenesis ->darkmatter?

E = mc2

NJP,Phys.Rev.D83,084033(2011)

Adjoint spinor

Energylevels(effective masses)

FermionsAntifermions

Inversenormalizationforspinorwavefunction

Matter-antimatterasymmetry

E = mc2

Multipoleexpansion• Papapetrou (1951)– multipole expansion->equationsofmotion

Matterinasmallregioninspacewithcoordinatesx¹(s)Motionofanextendedbody– worldtube

Motionofthebodyasawhole– wordline X¹(s)

• ±x® =x® - X® ±x0=0

u¹ =dX¹/ds ® - spatialcoordinates

M¹º½ =-u0s±x¹ £º½(-g)1/2dVN¹º½ =u0ss¹º½(-g)1/2dV

• Dimensionsofthebodysmall->neglecthigher-order(in±x¹)integralsandomitsurfaceintegrals

E = mc2

Four-velocity

Nonsingularfermions

E = mc2

s ->Σ

K.Nomura,T.Shirafuji &K.Hayashi,Prog.Theor.Phys.86,1239(1991)

Nonsingularfermions

ForDiracfields:

Single-poleapproximationofDiracfieldcontradictsfieldequations.Diracfieldscannotrepresentpointparticles(also1Dand2Dconfigurations).Fermionsmustbe3D.

E = mc2

NJP,Phys.Lett.B690,73(2010)

NonsingularfermionsForDiracfields:

E = mc2

NJP,Phys.Lett.B690,73(2010)

Nonsingularfermions

E = mc2

Spinfluids• Conservation lawforspin->

M½¹º - M½º¹ =N¹º½ - N¹º0u½/u0

• Averagefermionic matterasacontinuum(fluid)

NeglectM½¹º ->s¹º½ =s¹ºu½ s¹ºuº =0

Macroscopicspintensorofaspinfluid

• Conservation lawforenergyandmomentum->

£¹º =c¦¹uº - p(g¹º - u¹uº)² =c¦¹u¹ s2=s¹ºs¹º /2

Four-momentumPressureEnergydensitydensity

J.Weyssenhoff &A.Raabe,Acta Phys.Pol.9,7(1947)

E = mc2

Spinfluids• Dynamicalenergy-momentum tensorforaspinfluid

EnergydensityPressure

F.W.Hehl,P.vonder Heyde &G.D.Kerlick,Phys.Rev.D10,1066(1974)

• Spinfluidoffermionswithnospinpolarization

I.S.Nurgaliev &W.N.Ponomariev,Phys.Lett.B130,378(1983)

E = mc2

forrandomspinorientation

BouncecosmologywithtorsionClosed,homogeneous& isotropicUniverse:3Dsurfaceof4Dsphere

Friedman-Lemaitre-Robertson-Walker metric(k =1)

Friedmanequationsforscalefactora(t)

Conservation law

B.Kuchowicz,Gen.Relativ.Gravit.9,511(1978)M.Gasperini,Phys.Rev.Lett.56,2873(1986)NJP,Phys.Lett.B694,181(2010)

a

Spin-torsion couplinggeneratesnegativeenergy(gravitational repulsion)

Bouncecosmologywithtorsion

Torsion and particle production

Forrelativisticmatterinthermalequilibrium,Friedmanequationscanbewrittenintermsoftemperature.

2nd Friedmanequation isrewrittenas1st lawofthermodynamicsforconstantentropy.Parker-Starobinskii-Zel’dovich particleproductionrateK,proportionaltothesquareofcurvature,producesentropyintheUniverse.Noreheatingneeded.

NP,Astrophys.J.832,96(2016) (toappearinApJ,2016)

Generating inflation with only 1 parameterNearabounce:

Toavoideternalinflation:

Duringanexpansionphase,nearcriticalvalueofparticleproductioncoefficientβ:

ExponentialexpansionlastsaboutthenT decreases.

Torsionbecomesweakandradiationdominatederabegins.Nohypothetical fieldsneeded.

E = mc2

E = mc2

• ThetemperatureatabouncedependsonthenumberofelementaryparticlesandthePlancktemperature.

• Numericalintegrationoftheequationsshowsthatthenumbersofbouncesande-foldsdependontheparticleproductioncoefficientbutarenottoosensitive totheinitialscalefactor.

DynamicsoftheearlyUniverse

β/βcr =0.9998

S.Desai&NJP,Phys.Lett.B755,183(2016)

Otherwise,theUniversecontractstoanotherbounce(withlargerscalefactor)atwhichitproducesmorematter,andexpandsagain.

Ifquantumeffectsinthegravitationalfieldnearabounceproduceenoughmatter,thentheclosedUniversecanreachasizeatwhichdarkenergybecomesdominantandexpandstoinfinity.

Otherwise,theUniversecontractstoanotherbounce(withlargerscalefactor)atwhichitproducesmorematter,andexpandsagain.

E = mc2

WasOurUniverseBorninaBlackHole?CharlesPeterson,MechanicalEngineeringMentors:Dr.NikodemPoplawski&Dr.ChrisHaynes

METHODHYPOTHESIS

RESULTSCONCLUSIONS

BACKGROUND

Black holes (regions of space from where nothing can escape)form frommassive stars that collapse because of their gravity.

The Universe is expanding, like the 3-dimensional analogue ofthe 2-dimensional surface of a growing balloon.

Problem. According to general theory of relativity, the matterin a black hole collapses to a point of infinite density(singularity). The Universe also started from a point (Big Bang).But infinities are unphysical.

Solution: Einstein-Cartan theory. Adding quantum-mechanicalangular momentum (spin) of elementary particles generates arepulsive force (torsion) at extremely high densities whichopposes gravitational attraction and prevents singularities.

We argue that the matter in a black hole collapses to an extremely high butfinite density, bounces, and expands into a new space (it cannot go back).Every black hole, because of torsion, becomes a wormhole (Einstein-Rosenbridge) to a new universe on the other side of its boundary (event horizon).

If this scenario is correct then we would expect that:• Such a universe never contracts to a point.• This universe may undergo multiple bounces between which it expands

and contracts.

Our Universe may thus have been formed in a black hole existing in anotheruniverse. The last bounce would be the Big Bang (Big Bounce). We wouldthen expect that:• The scalar spectral index (ns) obtained from mathematical analysis of our

hypothesis is consistent with the observed value ns = 0.965 ± 0.006obtained the Cosmic Microwave Background (CMB) data.

ACKNOWLEDGMENTS

To evaluate our expectations:1. We wrote a code in Fortran programming language to solvethe equations which describe the dynamics of the closeduniverse in a black hole (NP, arXiv:1410.3881) and then graphthe solutions. These equations give the size (scale factor) aand temperature T of the universe as functions of time t (seeFig. 1).

2. From the obtained graphs we found the values of the scalarspectral index ns and compared them with the observed CMBvalue (see Fig. 2).

• The dynamics of the early universe formed in a black holedepends on the quantum-gravitational particle productionrate β, but is not too sensitive to the initial scale factor a0.

• Inflation (exponential expansion) can be caused by particleproduction with torsion if β is near some critical value βcr.

• Our results for ns are consistent with the 2015 CMB data,supporting our assertion that our Universe may have beenformed in a black hole.

I would like to thank my awesome teachers and mentors, Dr.Nikodem Poplawski and Dr. Chris Haynes, Dr. Shantanu Desaifor his help, Carol Withers who organized the SummerUndergraduate Research Fellowship, and the donors who gaveme the opportunity to pursue my research.

Fig. 1. Sample scale factor a(t). Several bounces, atwhich a is minimum but always >0, may occur.

Fig. 2. The simulated values of ns in our model areconsistent with the observed CMB value ns for a smallrange of β and a wide range of a0 (m).

Itispossibletofindascalarfieldpotentialwhichgeneratesagiventimedependenceofthescalefactor,andcalculatetheparameterswhicharemeasured incosmicmicrowavebackground.

ConsistentwithPlanck2015data.

S.Desai&NP,PLB755,183(2016)

Darkenergyfromtorsion

AcknowledgmentsUNHUniversityResearchScholarProgram

UNHSummerUndergraduateResearchFellowship

Prof.RameshSharma

Dr.ShantanuDesai

q The ECSK gravity extends GR to be consistent with the Dirac equation which allowsthe orbital-spin angular momentum exchange. Spacetime must have both curvatureand torsion.

q For fermionic matter at very high densities, torsion manifests itself as gravitationalrepulsion that prevents the formation of singularities in black holes and at the bigbang. The big bang is replaced by a big bounce.

q Big-bounce cosmology with spin-torsion coupling and quantum particle productionexplains how inflation begins and ends, without hypothetical fields and with onlyone unknown parameter.

q Torsion can be the origin of the matter-antimatter asymmetry in the Universe andthe cosmological constant.

q Torsion requires fermions to be extended, which may provide a UV cutoff for fermionpropagators in QFT. Future work.

SummaryThankyou!