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!