Generalized gravity theory with curvature, torsion and nonmetricity
K. Yesmakhanova, N. Myrzakulov, S. Myrzakul, G. Yergaliyeva, K. Myrzakulov, K. Yerzhanov, R. Myrzakulov
aa r X i v : . [ g r- q c ] J a n Generalized gravity theory with curvature, torsion andnonmetricity
K. Yesmakhanova , , S. Myrzakul , , G. Yergaliyeva , , K. Myrzakulov , ,K. Yerzhanov , and R. Myrzakulov , Eurasian National University, Nur-Sultan 010008, Kazakhstan Ratbay Myrzakulov Eurasian International Centre for TheoreticalPhysics, Nur-Sultan 010009, Kazakhstan
January 15, 2021
Abstract
In this article, the generalized gravity theory with the curvature, torsion and nonmetricywas studied. For the FRW spacetime case, in particular, the Lagrangian, Hamilatonian andgravitational equations are obtained. The particular case F ( R, T ) = αR + βT + µQ + ν T is investigated in detail. In quantum case, the corresponding Wheeler-DeWitt equation isobtained. Finally, some gravity theories with the curvature, torsion and nonmetricity arepresented. At present, General Relativity (GR) is considered the best accepted fundamental theory describinggravity. GR is described in terms of the Levi-Civita connection, which is the basis of Riemanniangeometry with the Ricci curvature scalar R . But GR can be described in terms of differentgeometries from the Riemannian one, for example, F ( R ) gravity. There are several other alternativegravity theories. For example, one of the alternative gravity theory is the so-called teleparallelgravity with the torsion scalar T or its generalization F ( T ) gravity. Another possible alternativeis symmetric teleparallel gravity with the nonmetricity scalar Q or its generalization F ( Q ) gravity.In this paper, we will consider the more general gravity theory, the so-called MG-VIII with theaction S = Z √− gd x [ F ( R, T, Q, T ) + L m ] . (1.1)This paper is organized as follows. In Sec. 2, we briefly review the geometry of the underlyingspacetime. In Sec. 3, we present a main information on the MG-VIII. FRW cosmology of theMG-VIII is studied in Sec.4. The specific model F ( R, T ) = αR + βT + µQ + ν T is analized inSec.5. The cosmological power-law solution is sdudied in Sec. 6. In Sec. 7, the Wheeler - DeWittequation is derived. Some other gravity theories related with the MG-VIII are presented in Sec.8. Final conclusions and remarks are provided in Sec. 9. Consider a general spacetime with curvature, torsion and nonmetricity. The corresponding con-nection is given by Γ ρ µν = ˘Γ ρ µν + K ρµν + L ρ µν , (2.1)1here ˘Γ ρ µν is the Levi–Civita connection, K ρµν is the contorsion tensor and L ρ µν is the disfor-mation tensor. These three tensors have the following forms˘Γ l jk = g lr ( ∂ k g rj + ∂ j g rk − ∂ r g jk ) , (2.2) K ρµν = 12 g ρλ (cid:0) T µλν + T νλµ + T λµν (cid:1) = − K ρνµ , (2.3) L ρ µν = 12 g ρλ (cid:0) − Q µνλ − Q νµλ + Q λµν (cid:1) = L ρ νµ , (2.4)where T αµν = 2Γ α [ µν ] , Q ρµν ≡ ∇ ρ g µν = 0 (2.5)are the torsion tensor and the non-metricity tensor, respectively. In this generalized spacetimewith the curvature, torsion and nonmetricity, let us introduce three scalars as R = g µν R µν , (2.6) T = S ρµν T ρµν , (2.7) Q = − g µν ( L αβµ L βνα − L αβα L βµν ) , (2.8)where R is the curvature scalar, T is the torsion scalar and Q is the nonmetricity scalar. Here R jk = ∂ i Γ ijk − ∂ j Γ iik + Γ iip Γ pjk − Γ ijp Γ pik , (2.9) S pµν = K µνp − g pν T σµσ + g pµ T σνσ , (2.10) K νpµ = 12 ( T νp µ + T νµ p − T νpµ ) . (2.11)are the Ricci tensor, the potential and the contorsion tensor, respectively. The key moment of ourconstruction is following: as in our previous paper [12], here we assume that these three scalarshave the following forms R = u + R s , (2.12) T = v + T s , (2.13) Q = w + Q s , (2.14)where u = u ( x i ; g ij , ˙ g ij , ¨ g ij , ... ; f j ), v = v ( x i ; g ij , ˙ g ij , ¨ g ij , ... ; g j ) and w = w ( x i ; g ij , ˙ g ij , ¨ g ij , ... ; h j ) aresome functions to be defined. Here: i) R s = R ( LC ) is the curvature scalar corresponding to theLevi-Civita connection with the vanishing torsion and nonmetricity ( T = Q = 0); ii) T s = T ( W C ) is the torsion scalar for the purely Weitzenbock connection with the vanishing curvature andnonmetricity ( R = Q = 0); iii) Q s = Q ( NM ) is the nonmetricity scalar with the vanishing torsionand curvature ( R = T = 0).Consider the Friedmann-Robertson-Walker (FRW) spacetime. The flat FRW spacetime is de-scribed by the metric ds = − N ( t ) dt + a ( t )( dx + dy + dz ) , (2.15)where a = a ( t ) is the scale factor, N ( t ) is the lapse function. The orthonormal tetrad components e i ( x µ ) are related to the metric through g µν = η ij e iµ e jν , (2.16)where the latin indices i , j run over 0...3 for the tangent space of the manifold, while the greekletters µ , ν are the coordinate indices on the manifold, also running over 0...3. With the FRWmetric ansatz the three variables R s , T s , Q s look like (we assume that N = 1) R s = R LC = 6( ˙ H + 2 H ) , (2.17) T s = T W C = − H , (2.18) Q s = Q NM = 6 H , (2.19)2here H = (ln a ) t is the Hubble parameter. Therefore, the three scalars ( R, T, Q ) of the metric -affine spacetime take the following forms R = u + 6( ˙ H + 2 H ) , (2.20) T = v − H , (2.21) Q = w + 6 H , (2.22)where u, v, w are some real functions of t, a, ˙ a, ¨ a, R s , T s , Q s ... and so on. F ( R, T , Q, T ) gravity Let us consider a general spacetime with the curvature, the torsion and the nonmetricity. In thisspacetime, the action of the Myrzakulov F ( R, T, Q, T ) gravity (GMG) (or shortly MG-VIII) isgiven by [12] S = Z √− g d x [ F ( R, T, Q, T ) + L m ] , (3.1)where R stands for the Ricci scalar (curvature scalar), T is the torsion scalar, Q is the nonmetricityscalar, T is trace of the energy-momentum tensor of matter Lagrangian L m . These fourth scalarsare given by R = g µν R µν (3.2) T = S ρµν T ρµν , (3.3) Q = − g µν ( L αβµ L βνα − L αβα L βµν ) , (3.4) T = g µν T µν , (3.5)where R µν = R αµνα , (3.6) T µν = − √− g δ ( √− gL m ) δg µν (3.7)are R βµνα is the Riemann curvature tensor and T µν is the energy-momentum tensor, respectively.Now we want rewrite the action of the MG-VIII with the lagrangian multipliers as S = Z √− gd x [ F − λ ( R − R s − u ) − λ ( T − T s − v ) − λ ( Q − Q s − w ) − λ ( T − T s ) + L m ] . (3.8)The variations with respect to R, T, Q, T of the action give λ = F R , λ = F T , λ = F Q , λ = F T respectively. Thus the action of the M-VIII takes the form S = Z √− gd x [ F − F R ( R − R s − u ) − F T ( T − T s − v ) − F Q ( Q − Q s − w ) − F T ( T − T s ) + L m ] . (3.9) For a simplicity, we consider the flat FLRW metric in the following form: ds = − dt + a ( t ) δ ij dx i dx j , (4.1)where a ( t ) stands for the scale factor. If we write down Lagrangian of F ( R, T, Q, T ) for this metricand if we assumed that the Universe is filled with matter fields with effective pressure p and energydensity ρ , we obtain T = 3 p − ρ . The point like Lagrangian of the M-VIII has the form S = Z L dt, (4.2)where L = L + ¯ L m . (4.3)3fter an integration part by part is written as the following L == L + ¯ L m = a (cid:16) F − RF R − T F T − QF Q − T F T (cid:17) (4.4) − a ˙ a (cid:16) F R + F T − F Q ) − F Rt a ˙ a + a ( T s F T + ¯ L m ) , where we suppose that L m = − p ( a ) and L = a (cid:16) F − RF R − T F T − QF Q − T F T (cid:17) − a ˙ a (cid:16) F R + F T − F Q ) − F Rt a ˙ a, (4.5)¯ L m = a ( uF R + vF T + wF Q + T s F T + L m ) . (4.6)Introduce R = 6( ¨ aa + ˙ a a ) + u, (4.7) T = − a a + υ, (4.8) Q = 6 ˙ a a + w, (4.9) T = T s = 3 p − ρ. (4.10)The associated Euler-Lagrange equations are given by ddt ( ∂ L ∂ ˙ q ) − ∂ L ∂q = 0 , (4.11)where q ≡ { a, R, T, Q, T } . One more equation we get from the Hamiltonian constraint H = ˙ a ∂ L ∂ ˙ a + ˙ R ∂ L ∂ ˙ R + ˙ T ∂ L ∂ ˙ T + ˙ Q ∂ L ∂ ˙ Q + ˙ T ∂ L ∂ ˙ T − L = 0 . (4.12)This constraint gives 6 a ˙ a (2) + 6 a ˙ aF Rt + a (cid:2) (1) − B + ¯ L m (cid:3) = 0 , (4.13)were B = ∂ ¯ L m ∂t . Finally we have the following system of 6 equations3 a (1) + 6 (cid:0) ˙ a + 2 a ¨ a (cid:1) (2) + 12 a ˙ a (2) t + 6 F Rtt a + ∂ (cid:0) a ¯ L m (cid:1) a − (cid:18) a ∂ ¯ L m ∂ ˙ a (cid:19) t = 0 , (4.14) a (cid:18) uF RR − vF RT + wF QR + T F R T − ∂ ¯ L m ∂R (cid:19) + (cid:18) a ∂ ¯ L m ∂ ˙ R (cid:19) t = 0 , (4.15) a (cid:18) uF RT + vF T T + wF QT + T F T T − ∂ ¯ L m ∂T (cid:19) + (cid:18) a ∂ ¯ L m ∂ ˙ T (cid:19) t = 0 , (4.16) a (cid:18) uF RQ + vF T Q + wF QQ + T F T Q − ∂ ¯ L m ∂Q (cid:19) + (cid:18) a ∂ ¯ L m ∂ ˙ Q (cid:19) t = 0 , (4.17) a (cid:18) uF R T + vF T T + wF Q T + T F T T − ∂ ¯ L m ∂ T (cid:19) + (cid:18) a ∂ ¯ L m ∂ ˙ T (cid:19) t = 0 , (4.18)6 a ˙ a (2) + 6 a ˙ aF R,t + a (cid:2) (1) − B + ¯ L m (cid:3) = 0 , (4.19)where B = ˙ a ∂ ¯ L m ∂ ˙ a + ˙ R ∂ ¯ L m ∂ ˙ R + ˙ T ∂ ¯ L m ∂ ˙ T + ˙ Q ∂ ¯ L m ∂ ˙ Q + ˙ T ∂ ¯ L m ∂ ˙ T , (4.20)(1) = F − RF R − − T F T − QF Q − T F T , (4.21)(2) = F R + F T − F Q . (4.22)4 FRW cosmology of F = αR + βT + µQ + ν T To understand the physical and mathematical nature of the GMG that is the MG-VIII, in thissection, we consider the following particular model F ( R, T, Q ) = αR + βT + µQ + ν T , (5.1)where α, β, µ, ν are some real constants. Then the Lagrangian (3.16) takes the form L = − µa ˙ a + a ¯ L m , (5.2)where ¯ L m = αu + βv + λw + γ T s + L m . (5.3)As result we get the following two equations6 µ (cid:0) ˙ a + 2 a ¨ a (cid:1) + (cid:2) a ( αu + βv + λw + γ T s ) (cid:3) a ++ ∂ (cid:0) a L m (cid:1) ∂a − (cid:2) a ( αu ˙ a + βv ˙ a + λw ˙ a + γ T s ˙ a ) (cid:3) t − (cid:2) a L m (cid:3) t = 0 , (5.4)6 µa ˙ a + a (cid:16) αu + βv + λw + γ T s + L m − B (cid:17) = 0 , (5.5)where B = ˙ a (cid:18) αu ˙ a + βv ˙ a + λw ˙ a + γ T s ˙ a + ∂L m ∂ ˙ a (cid:19) . (5.6)We can rewrite these two equations in the following standard forms3 H = ρ, (5.7)2 ˙ H = − ( ρ + p ) . (5.8)Here the matter density ρ and the pressure p have the form ρ = 12 µ (cid:16) B − αu + βv − λw − γ T s − L m (cid:17) , (5.9) p = 16 µa Z, (5.10)where Z = [ a ( αu + βv + λw + γ T s )] a + ∂∂a ( a L m ) − [ a ( αu ˙ a + βv ˙ a + λw ˙ a + γ T s ˙ a )] t − [ a L m ] t . (5.11)The EoS have the form ω = pρ = Z a (cid:16) Bαu + βv + λw + γ T s − L m (cid:17) . (5.12)For simplicity, we assume the u, v, w are functions only of a and L m = ρ a . (5.13)Then Z = [ a ( αu + βv + λw + γ T s )] a , B = 0 . (5.14)5 Cosmological solutions
As example of the cosmological solutions, let us consider the power-law solution a = a t n , (6.1)where a , n are some constants. Then ρ = 3 n t , p = n (2 + 3 n ) t . (6.2)We have two equations for the functions u, v, w : αu + βv + λw + γ T s + L m = − µn t , (6.3)( a ( αu + βv + λw + γ T s )) a = 6 µn (2 + 3 n ) a t n − (6.4)or αu + βv + λw + γ T s + L m = − µn a n a − n , (6.5)( a ( αu + βv + λw + γ T s )) a = 6 µn (2 + 3 n ) a n a n − n . (6.6)Therefore we obtain αu + βv + λw + γ T s + L m = − µn t , (6.7) a ( αu + βv + λw + γ T s ) = 6 µn (2 + 3 n ) a n − t n − + C. (6.8)Hence we get αu + βv + λw + γ T s = Ca t n + 6 µn (2 − n )(3 n − t = Ca t n − µn t . (6.9)These equations give L m = − µn t − Ca t n + 6 µn t . (6.10)On the other hand, we assume that L m = ρ a = ρ a t n . (6.11)Thus we obtain C = − ρ . (6.12)Finally we get the following expression for the function w : w = 1 λ [ − ρ a t n − µn t − αu − βv − γ T s ] = 1 λ " − ρ a − µn a n a n − αu − βv − γ T s . (6.13) In the Hamiltonian formulation of ordinary classical mechanics the key concept is the Poissonbracket (PB). In this formalism, the canonical coordinate system consists of canonical position q i and momentum p i variables which satisfy the following fundamental canonical PB relations { q i , p j } = δ ij . (7.1)Here the PB reads as { f, g } = N X i =1 (cid:18) ∂f∂q i ∂g∂p i − ∂f∂p i ∂g∂q i (cid:19) , (7.2)6here f, g are the phase space functions. Correspondingly, the Hamilton equations have the fol-lowing forms ˙ q i = { q i , H } , (7.3)˙ p i = { p i , H } , (7.4)which can be interpreted as the flow or orbit in phase space generated by H . In quantum casethe q, p are promoted to quantum operators ˆ q, ˆ p on a Hilbert space with the following canonicalcommutation [ˆ q, ˆ p ] = i ~ . (7.5)These operators satisfy the following equationsˆ qψ ( q ) = qψ ( q ) , (7.6)ˆ pψ ( q ) = − i ~ ddq ψ ( q ) . (7.7)Finally we get the following Schr¨odinger equation i ~ ∂∂t ψ = ˆ Hψ, (7.8)where ˆ H is the operator form of the Hamiltonian H with the usual replacements q q, p
7→ − i ~ ddq . (7.9)The momenta conjugate to variable a is given by π = ∂L∂ ˙ a = − µa ˙ a + a ∂ ¯ L m ∂ ˙ a . (7.10)Therefore the Hamiltonian takes the form H = ˙ a ∂ L ∂ ˙ a − L = − µa ˙ a + a ˙ a ∂ ¯ L m ∂ ˙ a − a ¯ L m (7.11)or ˆ H = − µa [ π − a ( ∂ ¯ L m ∂ ˙ a ) ] − a ¯ L m . (7.12)The classical dynamics is governed by the following Hamiltonian equations˙ a = { a, H} = ∂ H ∂π , (7.13)˙ π = { π , H} = − ∂ H ∂a . (7.14)Therefore, we have ˙ a = − π µa , (7.15)˙ π = π µa + { a ¯ L m − a µ ( ∂ ¯ L m ∂ ˙ a ) } a . (7.16)According to the Dirac quantization approach, the quantum states of the universe should beannihilated by the operator version of the Hamiltonian, that isˆ H Ψ = (cid:20) − a ( α + β − µ ) [ π − a ( ∂ ¯ L m ∂ ˙ a ) ] − a ¯ L m (cid:21) Ψ = 0 , (7.17)where Ψ = Ψ( a ) is the wave function of the universe. We now use the standard representation π → − i∂ a . Then we obtain the Wheeler - DeWitt equation ((WDWE) [18]-[19] (cid:20) − a ( α + β − µ ) [ − ∂ ∂ a − a ( ∂ ¯ L m ∂ ˙ a ) ] − a ¯ L m (cid:21) Ψ = 0 , (7.18)or (cid:20) ∂ ∂ a + 2 a ( ∂ ¯ L m ∂ ˙ a ) + 24 µa ¯ L m (cid:21) Ψ = 0 . (7.19)7 Relation with the soliton theory
Let us rewrite the WDWE as L Ψ = − (cid:2) ∂ a − U (cid:3) Ψ = κ Ψ , (8.1)where U = − [2 a ( ∂ ¯ L m ∂ ˙ a ) + 24 µa ¯ L m ] + κ. (8.2)Introduce the operator A as A = 4 ∂ a − U ∂ a + ∂ a U ] . (8.3)Then the Lax equation L Λ = [ L, A ] (8.4)gives the famous Korteweg-de Vries equation U Λ + 6 U U a + U aaa = 0 . (8.5) We consider some generalized metric - affine spacetime with the curvature, torsion and nonmetric-ity. In the previous sections, we have considered the MG-VIII theory. In this section, we want topresent some other MG theories.
The action of the Myrzakulov F ( R, T ) gravity or the MG-I has the following form [12] S = Z √− gd x [ F ( R, T ) + L m ] , (9.1)where R is the curvature scalar, T is the torsion scalar and L m is the matter Lagrangian. ThisMG is some kind generalizations of the well-known F ( R ) and F ( T ) gravity theories. If exactly,the MG is the unification of the F ( R ) and F ( T ) theories. Note that there are some other gravitytheories of this kind. Let us we present some of them. The action of the Myrzakulov F ( R, Q ) gravity or MG-II reads as [12] S = Z √− gd x [ F ( R, Q ) + L m ] , (9.2)where R is the curvature scalar and Q is the nonmetricity scalar. F ( R, T ) gravity The action of the F ( R, T ) gravity reads as [9] S = Z √− gd x [ F ( R, T ) + L m ] , (9.3)where R is the curvature scalar and T is the trace of the energy-momentum tensor.8 .4 F ( G, T ) gravity The action of the F ( G, T ) gravity reads as [17] S = Z √− gd x [ F ( G, T ) + L m ] , (9.4)where G is the Gauss-Bonnet term and T is the trace of the energy-momentum tensor. F ( Q, T ) gravity The action of the F ( Q, T ) gravity reads as [5] S = Z √− gd x [ F ( Q, T ) + L m ] , (9.5)where Q is the nonmetricity scalar and T is the trace of the energy-momentum tensor. F ( T, T ) gravity The action of the F ( T, T ) gravity reads as [11] S = Z √− gd x [ F ( T, T ) + L m ] , (9.6)where T is the torsion scalar and T is the trace of the energy-momentum tensor. The action of the MG-III reads as [12] S = Z √− gd x [ F ( T, Q ) + L m ] , (9.7)where T is the torsion scalar and Q is the nonmetricity scalar. The action of the MG-IV has the following form [12] S = Z √− gd x [ F ( R, T, T ) + L m ] , (9.8)where R is the curvature scalar, T is the torsion scalar and T is the trace of the energy-momentumtensor. The action of the MG-V is given by [12] S = Z √− gd x [ F ( R, T, Q ) + L m ] , (9.9)where R is the curvature scalar, T is the torsion scalar and Q is the nonmetricity scalar. The action of the MG-VII reads as [12] S = Z √− gd x [ F ( R, Q, T ) + L m ] , (9.10)where R is the curvature scalar, Q is the nonmetricity scalar and T is the trace of the energy-momentum tensor. 9 .11 MG-VII The action of the MG-VII reads as [12] S = Z √− gd x [ F ( T, Q, T ) + L m ] , (9.11)where T is the torsion scalar, Q is the nonmetricity scalar and T is the trace of the energy-momentum tensor. The action of the MG-VIII reads as [12] S = Z √− gd x [ F ( R, T, Q, T ) + L m ] , (9.12)where R is the curvature scalar, T is the torsion scalar, Q is the nonmetricity scalar and T is thetrace of the energy-momentum tensor (the trace of the stress-energy tensor).
10 Conclusion
As we mentioned in the introduction, GR has several generalizations like F ( R ) , F ( T ) and soon. Among these generalizations of GR, the metric-affine gravity theories have a nice featureby extending to admit not only curvature but both torsion and nonmetricity. This means theMAG is described by a pseudo - Riemannian geometry. The geometrical structure of the MAGcan be studied once a metric tensor and a connection are given. In this way, we can calculatethe affine connection for the underlying theory. In this paper, we have considered the so-calledgeneralized Myrzakulov gravity or MG-VIII which can be considered as the particular case of theMAG. To simplify the problem, we consider the FRW spacetime case in detail. For this case thepoint-like Lagrangian and Hamiltonian of the theory is derived. Using this Lagrangian and theEuler-Lagrangian equation, the gravitational equations of the GMG is presented. For simplicity,the particular case of the GMG when F = αR + βT + µQ + ν T is investigated. For this particularcase, the gravitational equations is considered in detail. For the quantum case, the correspondingWheeler - DeWitt equation is presented. The relation with the soliton theory is shortly discussed.These results show that altogether one can say that some ingredients of the GMG are present andwork as expected, but some other aspects remain to be properly understood. These aspects of theGMG certainly worth further investigation (see e.i. refs. [4]-[8]). Acknowledgments
The work was supported by the Ministry of Education and Science of the Republic of Kazakhstan,Grant AP08856912.
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