aa r X i v : . [ phy s i c s . g e n - ph ] S e p Interaction of Fermionic Matter and ECSK BlackHole with Torsion
Emre DilFaculty of Engineering, Beykent University,34398, Sariyer, Istanbul-TURKEYe-mail: [email protected] 15, 2020
Abstract
The interactions between the spin of fermionic matter and torsion inthe Einstein–Cartan–Sciama–Kibble (ECSK) theory of gravity providesa repulsive gravitational potential at the very dense states of fermionicmatter, which prevents the formation of black hole singularities inside thedeeper horizon. While the fermionic matter in the black hole is attractedby the black hole at the beginning, after a critical point it is repelled tobounce at a critical high density and then expand into other side of thehorizon as a newly created space, which may be considered as a nonsin-gular, closed universe. We constructed the action of these fermions in ablack hole with torsion in the framework of ECSK theory of gravity fromwhich the free Dirac action is inferred to obtain the interaction potential.The creation of a bouncing universe with the extremely repulsive poten-tial may be related to the running vacuum and the modified Friedmannequations yield the consistent cosmological parameters with present FRW niverse. Finally, this scenario naturally solves the flatness and horizonproblems of cosmology without introducing finely tuned scalar fields, ormore complicated functions of the Ricci Scalar R in the gravitational ac-tion. Keywords: bouncing universe – spinor field – Einstein-Cartan field equa-tions – black hole physics – inflation.
In curved space-time, the conservation law for the total angular momentum offermions represented by the Dirac equation requires an asymmetric affine con-nection. Because metric general relativity has constraints on the symmetry ofthe connection, the ECSK theory of gravity correspondingly fixes the constraintof general relativity. ECKS relates the spin density of fermionic matter and theantisymmetric part of the affine connection as the torsion tensor, and assigns itas a dynamical variable like the metric itself. The intrinsic spin of fermions gen-erates a torsion for the space-time by coupling to the connection of the curvedspace-time. On the other hand, The ECSK theory and general relativity givesame predictions for the high densities as in the nuclear scales due to the neg-ligibly small contribution from torsion to the Einstein equations, therefore theECKS is validated for all cases as is general relativity [1-19].If the densities are extremely high as in the black holes and as in the veryearly universe, this situation leads to a spinor–torsion interaction behaves likea gravitational repulsion, and it prevents the creation of singularities in theblack holes due to the spin of the fermionic matter [20-31]. It leads to that thesingular big bang is turned out to be a nonsingular big bounce, before whichthe universe was contracting [32-35]. This model also provides a solution forthe flatness-horizon problem of cosmology other than terminating the initial2ingularity of the universe [29-31], [34-37]. Instead of the cosmic inflation whichis verified by CMB radiation inhomogeneities and requires additional scalar fieldmatter components for solving the flatness-horizon problem [38-40], torsion canbe considered as the simplest and most natural mechanism to provide solutionfor these major problems of the standard big bang cosmology [17,41,42].There can be found noteworthy studies which consider the nonsingular andbouncing cosmologies led by a universe inside a black hole whose event horizonopens to another new universe as a white hole [43-46]. The contraction of auniverse in a black hole could correspond to gravitational collapse of matterinside the black hole existing in another universe. Therefore, the collapsingmatter inside the black hole should have a bounce at a finite density, after thatshould expand into a new region on the other side of the event horizon, which canbe considered as a nonsingular, new born closed universe [17,26,32,35], [47-54].This is why the ECSK theory is expected to give the evolution of each spatialpoint in the universe toward a state of an extremely high but finite densitydue to the spinor-torsion interaction of fermions in the black hole. For sucha scenario, the local contraction of universe finished, and the matter evolvestoward a bounce, and the expansion of universe starts.In this paper we will be concerned with the behavior of fermion fields near thehorizons of a spherically symmetric space-time with torsion in the generalizedEinstein-Cartan-Kibble-Sciama theory of gravity by considering higher orderterms to investigate the big bounce behavior of the universe. We will study theproperties of a spinor field equation in the background of an ECSK black holeby considering the spin-torsion interactions of the conventional fermion field andthe space-time. 3
Spinor Field in ECSK formalism
In order to study the dynamics of spinor fields in curved space-times we shoulduse the formalism of ECSK theory, since it holds for the sources with spin.In the curved space-times the theory of Dirac spinors is a complex example ofthe quantum field theories for obtaining the energy-momentum tensor of thefield from the variation of spinor Lagrangian. Although the energy-momentumtensor of scalar fields describes the reaction of Lagrangian to the variations ofthe metric, as the field is itself held constant during the variation of the metric,the same does not apply for the spinor field energy-momentum tensor from thevariation of metric tensor only, where the spinor fields are the sections of aspinor bundle. This spinor bundle is obtained as an associated vector bundledue to the bundle of tetrad spin frames. The bundle of spin frames is a doublecovering of the bundle of oriented and time-oriented orthonormal frames. Forspinor fields, when one varies the metric, the components of the spinor fieldsalso varies and they cannot be held fixed with respect to some fixed holonomicframe as in the scalar field case [55]. Therefore, we need to give the algebraicstructure of the ECSK formalism and the spinor field structures.The metric-affine formulation of the gravity has the dynamical variables ofthe tetrad frame field e ia and the spin connection ω abk = e aj e jb ; k = e aj ( e jb,k +Γ jik e ib ),where the comma denotes the ordinary partial derivative with respect to the x k coordinate, and the semicolon represents the covariant derivative with respectto the affine connection Γ ijk whose antisymmetric lower indices give the torsiontensor S ijk = Γ i [ jk ] . Moreover, the tetrad frames link the space-time coordinateswith indices i, j, ... and the local Lorentz coordinates with indices a, b, ... , suchthat V a = V i e ai for a Lorentz vector V a and a standard vector V i . While theLorentz vectors have a covariant derivative denoted by a bar and defined interms of the spin connection: V a | i = V a,i + ω abi V b and V a | i = V a,i − ω bai V b , the4tandard vectors have a semicolon covariant derivative defined in terms of theaffine connection, V k ; i = V k,i + Γ kli V l and V k ; i = V k,i − Γ lki V l . The Local Lorentzcoordinates are lowered-raised by the Minkowski metric η ab , and the space-timecoordinates are lowered-raised by the metric tensor g ik . Also, the metricitycondition g i j ; k = 0 yields the affine connection defined to be Γ ki j = { ki j } + C ki j in terms of the Christoffel symbols { ki j } = (1 / g km ( g mi,j + g m j,i − g i j,m ) andthe contortion tensor C ijk = S ijk + 2 S i ( jk ) . While the symmetrization is used by A ( jk ) = (1 / A jk + A k j ), the antisymmetrization is used by A [ jk ] = (1 / A jk − A k j )throughout the paper. In ECSK gravity, the metric g ik = η ab e ai e bk and thetorsion S jik = ω j [ ik ] + e a [ i,k ] e ja can also be taken as the dynamical variables morethan the tetrad frames and the spin connection [1-19].From the definition, a tensor density T D is related to the correspondingtensor T by T D = eT , where e = det e ai = √− det g ik , then the spin densityand the energy-momentum densities are given as σ i jk = e s i jk and T ik = eT ik .These tensors are called as metric spin tensor and metric energy-momentumtensor since they are specified by the space-time coordinate indices, and theyare obtained from the variation of the Lagrangian with respect to the torsion (orcontortion) tensor C i jk and the metric tensor g i j , respectively. Accordingly, themetric spin tensor is given as s ki j = (2 /e )( δℓ m /δC i jk ) = (2 /e )( ∂ℓ m /∂C i jk ), whilethe metric energy-momentum tensor is T i j = (2 /e )( δℓ m /δg i j ) = (2 /e )[ ∂ℓ m /∂g i j − ∂ k ( ∂ℓ m /∂ ( g i j,k ))]. The Lagrangian density of the source matter field is here ℓ m = eL m . If the local Lorentz coordinates are used to specify these tensorsas σ iab = e s iab and T ai = eT ai , then s iab and T ai are called as the dynamical spin tensor and dynamical energy-momentum tensor, respectively, and they areobtained from the variation of the Lagrangian with respect to the tetrad e ia andthe spin connection ω abi , such that s iab = (2 /e )( δℓ m /δω abi ) = (2 /e )( ∂ℓ m /∂ω abi ),and T ai = (1 /e )( δℓ m /δe ia ) = (1 /e )[ ∂ℓ m /∂e ia − ∂ j ( ∂ℓ m /∂ ( e ia,j ))] [1-19].5otal action of the gravitational field with spinor field dark matter in metric-affine ECSK theory is given in the same form with the classical Einstein-Hilbertaction, such as S = κ R ( ℓ g + ℓ ψ ) d x , where κ = 8 π G and ℓ g = − (1 / κ ) eR ,and ℓ ψ are the gravitational, and fermionic matter Lagrangian densities. TheRicci scalar is R = R bj e jb where R bj = R bcjk e kc is the Ricci tensor obtained fromthe curvature tensor R bcjk . Also, the curvature tensor is related to the spinconnection, such that R abi j = ω abj,i − ω abi,j + ω aci ω cbj − ω acj ω cbi . The variationof the action with respect to the contortion tensor gives the Cartan equations S ji k − S i δ jk + S k δ ji = − ( κ/ e ) σ jik , and the variation with respect to the metrictensor yields the Einstein equations G ik = κ ( T ψik + U ψik ), where G ik = P ji jk − (1 / P lmlm g ik is the Einstein tensor. Here P ji jk is also the Riemann curvaturetensor given by the relation R iklm = P iklm + C ikm : l − C ikl : m + C jkm C ijl − C jkl C ijm ,where colon represents the Riemannian covariant derivative with respect to theLevi-Civita connection { kli } , such as V k : i = V k,i + { kli } V l and V k : i = V k,i − { lki } V l .Also, the curvature tensor transforms into the Riemann tensor for torsion-freegeneral relativity theory. The U ik term in the Einstein equations is the spincontribution U ik = κ ( − s ij [ l s klj ] − (1 / s i jl s kjl +(1 / s jli s kjl +(1 / g ik ( − s lj [ m s jml ] + s jlm s jlm ), and the total energy-momentum tensor of the spinor field is given byΘ ψik = T ψik + U ψik [1-19].Combining the above algebraic relations for the metric-affine ECSK formu-lation of gravity for (+ , − , − , − ) metric signature, spinor field fermionic matteris described by the Lagrangian densities of the form: ℓ ψ = e ( i/ ψγ k ψ ; k − ¯ ψ ; k γ k ψ ) − em ψ ¯ ψψ, (1)where ψ and ¯ ψ = ψ + γ are the spinor and the adjoint spinor fields, respectively.The semicolon covariant derivative of the spinor and the adjoint spinor fields6re given by ψ ; k = ψ , k − Γ k ψ, (2)¯ ψ ; k = ¯ ψ , k − Γ k ¯ ψ, (3)where Γ k = − (1 / ω abk γ a γ b is the Fock-Ivanenko spin connection, γ k and γ a are the metric and dynamical Dirac gamma matrices as; γ k = e ka γ a , γ ( k γ m ) = g km I and γ ( a γ b ) = η ab I . One can decompose the semicolon covariant deriva-tive of the spinor field into a colon Riemannian covariant derivative with thecontortion tensor C i jk term as ψ ; k = ψ : k + (1 / C i jk γ [ i γ j ] ψ, (4)¯ ψ ; k = ¯ ψ : k − (1 / C i jk ¯ ψγ [ i γ j ] . (5)The colon Riemannian covariant derivative is also defined to be ψ : k = ψ , k + (1 / g ik { ijm } γ j γ m ψ, (6)¯ ψ : k = ¯ ψ , k − (1 / g ik { ijm } γ j γ m ¯ ψ. (7)Although the spinor field Lagrange density contains covariant derivatives in-cluding the contortion tensor C i jk , the explicit form of the contortion tensoris obtained from the Cartan equations whose right hand side involves the spintensor density. Then, the spin tensor is led by the variation of the spinor La-grangian with respect to the contortion tensor, such as s i jk = (1 /e ) σ i jk = − (1 /e ) ε i jkl s l , (8)7here ε i jkl is the Levi-Civita symbol, and s i = (1 /
2) ¯ ψγ i γ ψ (9)is the spin pseudo-vector, and γ = iγ γ γ γ . Substituting the spin tensor ofspinor field in the Cartan equations leads to the torsion tensor as S i jk = C i jk = (1 / κε i jkl s l , (10)which will be found in the spinor field Lagrange density [1-19].Then, the variation of the spinor fermionic matter Lagrangian density withrespect to the adjoint spinor ( ∂ℓ ψ /∂ ¯ ψ ) − ( ∂ℓ ψ /∂ ¯ ψ : k ) : k = 0 gives the ECSKDirac equation iγ k ψ : k − m ψ ψ + 38 κ ( ¯ ψγ k γ ψ ) γ k γ ψ = 0 , (11)while the variation with respect to the spinor itself ( ∂ℓ ψ /∂ψ ) − ( ∂ℓ ψ /∂ψ : k ) : k = 0gives adjoint ECSK Dirac equation as i ¯ ψ : k γ k + m ψ ¯ ψ − κ ( ¯ ψγ k γ ψ ) ¯ ψγ k γ = 0 . (12) A closed universe with quantum effects of fermionic spinor field in a curvedspace-time of a black hole provides an oscillatory universe as the big bounce.After the big bounce a first accelerated expansion phase of the universe becomesin a torsion-dominated era. The fermions composing the spin fluid prevents to8orm a singularity in the black hole due to the spin-torsion coupling. Moreover,the spin–torsion term triggers a big bounce from the other side of the eventhorizon after the universe collapse into minimum scale in the black hole [17,41].Therefore, it is necessary to define a black hole with torsion in the ECSKformalism which corresponds an action [56-60] S BH = Z e (cid:20) − κ ( L + a L ) (cid:21) d x, (13)where a is a coupling constant, L is ECSK Lagrangian and it is given as L = R + 14 S i jk S i jk + 12 S i jk S ji k + S i ji S kjk + 2 S i ji ; j , (14)and L is L = RS i jk S i jk + 14 S i jk S i jk (cid:2) S l mn (2 S m ln + S l mn ) + 8 S l ml ; m (cid:3) + S i ji S kjk S l mn S l mn . (15)The action of an ECSK black hole yields a static spherically symmetric space-time with the metric d s = H ( r ) dt − dr F ( r ) − r (cid:0) dθ + sin θ dφ (cid:1) , (16)where [60] F ( r ) = 1 − c c √ r + 6 c c − c ln(3 c − c √ r )6 c r ,H ( r ) = (cid:18) − c c √ r (cid:19) . (17)Here c , c and c are the integration constants related to the torsion. Thereexists a constraint on these constants due to the logarithmic term in F ( r ), suchthat, the term 3 c − c √ r must be positive at the physical region. If c > c < c − c √ r , but c > c − c √ r with the constraint √ r < c / c yielding adivergent F ( r ) function. Because of the divergent nature of F ( r ), c < c > a in13 is taken as a = 2 / c in order for the metric 16 to be asymptotically flat atthe spatial infinity for c < c < c = − F ( r ) = 0 case istaken into account and the critical values for constant c are obtained from thenumerical analysis because of the complexity of the logarithmic term: c crit = c ln(3 c ) , c > c [3 ln( − c ) − , c < . (18)For the case c > c crit , no horizon exists in the space-time and the geometryis a naked singularity. However, for the case c < c crit the black hole has onehorizon in c > c < Bouncing universe is proposed to be led by the existence of spinor matter in anECSK black hole with torsion. In order to investigate the validity of our as-sumption, we now consider the spinor field 1 in the ECSK black hole background16 with the action S = Z e (cid:18) i ψγ k ψ ; k − ¯ ψ ; k γ k ψ ) − m ψ ¯ ψψ (cid:19) d x, (19)10here e = s H ( r ) F ( r ) r sin θ, (20)for the metric of ECSK black hole. We expand the semicolon covariant deriva-tives in the action 19 by using the equations 2-10, such that ψ ; k = ψ , k + 14 (cid:20) g ik { ijm } γ j γ m + 14 κε ijkl ¯ ψγ l γ ψ γ [ i γ j ] (cid:21) ψ, (21)¯ ψ ; k = ¯ ψ , k − ¯ ψ (cid:20) g ik { ijm } γ j γ m + 14 κε ijkl ¯ ψγ l γ ψ γ [ i γ j ] (cid:21) . (22)We then take the derivatives for k = 0 , , , { ijm } Christoffel connections are obtained as { } = { } = 12 H ′ ( r ) H ( r ) , { } = 12 H ′ ( r ) F ( r ) , { } = − F ′ ( r ) F ( r ) , { } = − r F ( r ) , { } = − r F ( r ) sin θ, { } = { } = { } = { } = 1 r , { } = − sin θ cos θ, { } = { } = cot θ. (23)By applying the identities about the components of γ matrices given belowequation 3 and 9, we find the components of semicolon covariant derivatives ofspinor field such that ψ ;0 = ψ , , (24)¯ ψ ;0 = ¯ ψ , , (25) ψ ;1 = ψ , − (cid:20) H ′ ( r )8 H ( r ) + F ′ ( r )8 F ( r ) + 12 r (cid:21) ψ, (26)11 ψ ;1 = ¯ ψ , + ¯ ψ (cid:20) H ′ ( r )8 H ( r ) + F ′ ( r )8 F ( r ) + 12 r (cid:21) , (27) ψ ;2 = ψ , + 14 cot θr ψ, (28)¯ ψ ;2 = ¯ ψ , −
14 ¯ ψ cot θr , (29) ψ ;3 = ψ , , (30)¯ ψ ;3 = ¯ ψ , . (31)We now substitute these semicolon covariant derivative components into theaction 19, then S = R (cid:16) i ( ¯ ψγ ψ , − ¯ ψ , γ ψ + ¯ ψγ ψ , − ¯ ψ , γ ψ − h H ′ ( r )4 H ( r ) + F ′ ( r )4 F ( r ) + r i ¯ ψγ ψ + ¯ ψγ ψ , − ¯ ψ , γ ψ +
12 cot θr ¯ ψγ ψ + ¯ ψγ ψ , − ¯ ψ , γ ψ ) − m ψ ¯ ψψ (cid:1) d x , (32)and S = Z (cid:18) i ψγ k ψ ,k − ¯ ψ ,k γ k ψ ) − m ψ ¯ ψψ + V k ¯ ψγ k ψ (cid:19) d x. (33)where the free action is inferred from 32 as S free = Z (cid:18) i ψγ k ψ ,k − ¯ ψ ,k γ k ψ ) − m ψ ¯ ψψ (cid:19) d x. (34)Therefore, the remaining terms in 32 gives the vector potential of the spinormatter due to an ECSK black hole geometry with the components V r = − (cid:20) H ′ ( r )4 H ( r ) + F ′ ( r )4 F ( r ) + 1 r (cid:21) , V θ = 12 cot θr . (35)12or the case c = − c > F ( r ), H ( r ) and derivatives, as F ( r ) = 1 + c √ r + 9 c ln(3 c ) − c ln(3 c + 2 √ r )6 r ,H ( r ) = (cid:18) c √ r (cid:19) , (36)and F ′ ( r ) = − c r / − c c r / + 4 r / − c ln(3 c )2 r / + 3 c ln(3 c + √ r )2 r / ,H ′ ( r ) = − c r / − c r / . (37)Using 36 and 37 there can be found the components of the vector potential interms of the torsion related constants. For the radial component of the potential35 the behavior of the potential with respect to radius is illustrated in Figure 1for various positive c values. As can be obtained from 18 these constants lead tothe Schwarzschild radius of the ECSK black hole of 0 . c , and the potential hasminimum values at approximately for the 0 . c , which are higher density statesin deeper horizon. These minimum values occur for c = 2 GM at r = 0 . GM ,for c = 2 . GM at r = 0 . GM , for c = 3 . GM at r = 0 . GM and for c = 3 . GM at r = 0 . GM . The potential values before r = 0 . c is extremelyrepulsive. This implies a gravitational repulsion inside of the horizon of theECSK black hole after a critical value for the Schwarzschild radius. On theother hand, the tangential component of the potential gives also the similarbehavior for r ≈ . c but with the polar angle θ ≈ . π .This behavior expresses for the extremely high densities of fermions in thetorsion dominated black holes as in the very early universe there leads to aspinor–torsion interaction behaves like a very strong gravitational repulsion13igure 1: Repulsive and Attractive Behavior of Radial Potential V r s = 18 (¯ hcn f ) , (38)where n f is the number density of fermions and the effective energy density,and also pressure of the spinor fluid read˜ ε = ε − α n f , ˜ p = p − α n f , (39)where α = κ (¯ hc ) /
32 and α n f is the spin-torsion coupling term. The spin-torsion term decreases faster than ε during the expansion phase. The universeenters the phase of acceleration to expand infinity with the repulsive potential V r which may be thought as it is turning out to be a running vacuum as thevarying cosmological constant Λ ( r ) which has been previously proposed by usand other studies in the literature [67-70]. Then, the other side of the horizoncreates a bouncing FRW universe with an accelerated expansion obeying theusual Friedmann equation with the running vacuum Λ ( r ),˙ a c + 1 = Da + 13 Λ ( r ) a , (40)15here D = 13 κ h ∗ T eq (cid:18) ˜ a i a i (cid:19) ( a i T i ) (41)comes from the spinor matter energy density. In equation 41, h ∗ = ( π / g ∗ k B / (¯ hc ) with g ∗ = (7 / g f , where g f = P i g i is summed over fermions and g i is thespin states for each particle species i . Moreover, ˜ a i is the scale factor at tem-perature T i in the expanding phase. For the expanding phase ˜ a i is greaterthan the scale factor a i before the expansion begins. Then, ˜ a i > a i representsthe expansion of the universe with the fermion creation in the other side ofthe ECSK black hole horizon. The consistent resolutions of the cosmologicalparameters from 41 is found in the references [17,26,34] as T eq = 8820 K andΛ /κ = 5 . × − P a which gives the ratio ˜ a i /a i > representing the ex-pansion of the universe, and the deviation of the density parameter from theunity ˜Ω min − min − a i /a i ) < − solving both flatness and horizonproblems of cosmology. In this paper, we considered the ECKS theory of gravity with torsion andfermion field in a black hole. Using spin density of fermionic matter as thesource of torsion leads to a natural physical interpretation which does not in-troduce additional fields or coupling constants to form a repulsive force causingaccelerated expansion of universe. Therefore, we first give the dynamics ofspinor field in ECSK theory of gravity as constructing action to give the matterLagrangian and modified Dirac equation. After that, we explore the geometryof an ECSK black hole in which the spinor field couples to the torsion of thespace time in order to produce a repulsive potential which will drive the newlycreated exhausted particles from the other side of the black hole horizon.16ith the completed tools in sections 2 and 3, we have investigated behaviorof spinor field in the ECSK black hole which forms a bouncing universe with aradial potential V r whose behavior is illustrated in Figure 1. When the radialcoordinate is less than the Schwarzschild radius by a factor of 0.1, the potentialbecomes extremely repulsive as the event horizon is approximately 0 . c . Forsome particular values of c = 2 GM , c = 2 . GM , c = 3 . GM , and c =3 . GM the repulsive potential forms at coordinates less than r = 0 . GM , r = 0 . GM , r = 0 . GM and r = 0 . GM , respectively. At these highdensity regions in the black hole horizon strong repulsive potential behaves likea running vacuum of Λ ( r ) launches the accelerated expansion of the bouncinguniverse from the back side of the black hole horizon.The modified Friedmann equations of this newly created bouncing universeprovides the consistent cosmological parameters as scale factor and fixed flat-ness horizon problems with a values of ˜ a i /a i > and ˜Ω min − < − ,respectively.At extremely high densities as in the deeper horizon of the black holes withthe fermionic matter, a significant gravitational repulsion is generated by spinand torsion interactions. This repulsive potential prevents to form the singu-larities in black holes or big bang of the universe. With the contribution of thegravitational repulsion exhausting fermions from the other side of the horizoncreate particles, and all black holes may construct a new bouncing universe onthe other side of its event horizon [26-28]. In addition, it simultaneously solvesthe flatness and horizon problems without requiring finely tuned scalar fields,or modified version of the Ricci Scalar R in the action of gravitational field bymore complex functions. 17 eferenceseferences