Abstract On the basis of the Lie derivative method in a metric-affine spacetime (L4 , g) it is shown that in the metric-affine gravitational theory the energy-momentum conservation law and therefore the equations of the matter motion are the consequence (as in the GR) of the gravitational field equations. The possibility of the detection of the space-time nonmetric properties is discussed.
1. Introduction It is well known that in the General Relativity the equations of the matter motion are the consequence of the gravitational field equations. The reason of this fact consists in the identical vanishing (as the cosequence of the Bianchi identities) of the covariant divergence of the left part of the Einstein equations, which leads to the covariant energy-momentum conservation law of matter (R)
β ∇ β Tα = 0
(here ∇ is the covariant differentiation with respect to the connection of a Riemann space-time). The equations of the matter motion are the consequence of (1). In the Einstein-Cartan theory [1, 2] the same situation takes place . The more complicated case occurs in the generalized theories of gravity with torsion in a Riemann-Cartan space-time U4 based on non-linear Lagrangians, where the some new covariant identities appear. In the theories of a such type as it was shown in [4, 5] the equations of the matter motion are also the consequence of the gravitational field equations. In this paper we generalize the results of [4, 5] to the theory of gravitation in the metric-affine space-time (L4 , g) with the curvature Rµνσλ , torsion Tµνλ and nonmetricity Qλσρ := ∇λ g σρ . The metric-affine theory of gravitation was proposed in  and now is of a great interest in connection with the problem of the relation of the gravitation and the elementary particles physics [7, 8]. 1
2. The energy-momentum conservation law as the consequence of the field equations in the metric-affine gravitational theory We shall use the Lie derivative method for deriving the differential identities in the metric-affine space (L4 , g). In a Riemann-Cartan space this method was described in detail in . Let us consider the Lie transport £ξ in the direction of an arbitrary vector field ξ σ . Then the transformation law of the gravitational √ field Lagrangian density L0 = −gL0 is ˆ σξσ , £ξ L0 = ξ σ ∇σ L0 + L0 ∇
ˆ σ := ∇σ + Tσ , and Tσ := Tσττ . Here ∇ is the covariant differentiation where ∇ with respect to the connection of (L4 , g). From the other side the result of this Lie differentiation can be calculated as the consequence of the explicit dependence of the Lagrangian density √ (3) L0 = −gL0 (g σρ , Rµνσλ , Tµνλ ) from the metric, curvature and torsion tensors of (L4 , g), the Lie differentiation of those being calculated by the corresponding rules that valid in (L4 , g): £ξ g σρ = ξ µ Qµσρ − 2 ξ (σρ) , ξµρ := ∇µ ξ ρ − Tµνρ ξ ν , £ξ Rµνσλ = ξ ρ ∇ρ Rµνσλ + Rρνσλ ξµρ + Rµρσλ ξνρ +Rµνρλ ξσρ − Rµνσρ ξρλ , £ξ Tµνλ = ξ ρ ∇ρ Tµνλ + Tρνλ ξµρ + Tµρλ ξνρ − Tµνρ ξρλ .
(4) (5) (6)
Comparing the results of the both Lie differentiation methods, we get the covariant correlation, on the base of which, taking into account that ξ σ and ∇λ ξ σ are arbitrary, we obtain the two covariant identities. Because of the arbitrariness of ∇λ ξ σ we get √
where the following notations have been used: √ √ ˆ ν (√−gJσλν [L0 ]) , −gtσλ [L0 ] := −gTσλ [L0 ] + ∇ √ √ δL0 δL0 −gTσρ [L0 ] := −2 σρ , −gJσλν [L0 ] := − . δg δΓνλσ
Because of the arbitrariness of ξ σ we get the second identity: ˆ λ − Tσλρ )√−gtρλ [L0 ] − Rσνλρ δL0 − Qσαβ δL0 = 0 . (δσρ ∇ δΓνλρ δg αβ 2
When obtaining (7) and (10) the following identities have played the essential role δL0 ˆ ρ ∂L0 + Tαβσ ∂L0 + 2 ∂L0 , = 2 ∇ δΓσνλ ∂Rσρνλ ∂Rαβνλ ∂Tσνλ ˆ σ δL0 = (Rαβσν δλρ − Rαβλρ δσν ) ∂L0 + 2∇ ˆ σ ∂L0 , ∇ δΓσνλ ∂Rαβσρ ∂Tσνλ
which take place in (L4 , g) for the the Lagrangian density (3) (, for some details see Appendix). Taking into account (8),(9), the field equations of the metric-affine gravitational theory can be introduced in the following form δg σρ : δΓνλσ :
Tσρ [L0 ] = −Tσρ , Jσλν [L0 ] = −Jσλν ,
Tσρ := Tσρ [Lm ] , Jσλν := Jσλν [Lm ] ,
where Lm is the Lagrangian density of matter, generating the gravitational field. The system (13), (14) with the help of the identity 1 1ˆ τ R[σρ] = ∇ τ Mσρ − Vσρ , 2 2
Mσρτ := Tσρτ + 2δ[σ]τ Tρ ,
where Vαβ is the tensor of homothetic curvature of (L4 , g): 1 Vαβ = Rαβττ = ∇[α Qβ] + Tαβτ Qτ , 2
Qσ := Qσττ ,
can be written in the equivalent form: tσλ [L0 ] = −tσλ ,
tσλ := tσλ [Lm ] .
Jσλν [L0 ] = −Jσλν ,
where in the right part of the gravitational field equations (17), (18) we have the canonical energy-momentum tensor and the hypermomentum tensor of all matter that generates the gravitational field. These quantities are calculated by means of the replacing the matter Lagrangian density Lm instead of L0 into (8),(9). The question of deriving the field equations (17) will be discussed in detail in the following paper . Substituting the field equations (17), (18) into (10), we obtain the quasiconservation law for the canonical energy-momentum tensor of matter in the metric-affine gravitational theory: ˆ λ (√−gtσλ ) − √−gTσλρ tρλ + √−gRσνλρ Jρλν + 1 √−gQσαβ Tαβ = 0 . ∇ 2 3
The equation (19) was derived in  (see also ) as the consequence of the matter Lagrangian invariance with respect to the infinitesimal coordinate transformations. 3. The various cases of the matter motion in the metric-affine spacetime (L4 , g) The hypermomentum tensor can be split up in the following way  1 Jσρλ = Sσρλ + J σρλ + gσρ J λ , 4
Sσρλ := J[σρ]λ ,
J σσλ = 0 .
In (L4 , g) the affine connection coefficients have the form  1 Γλνσ = g σρ ∆αβγ νλρ (∂α gβγ − Tαβγ + Qαβγ ) , 2 α β γ α β γ α β γ ∆αβγ νλρ := δν δλ δρ + δλ δρ δν − δρ δν δλ .
The quasiconservation law (19) with the help of (20), (21) yields 1√ −gVσν J ν 4 1 1 ρ [ρ ρ ˆ √ ρ] −(Tσ[λ + T λσ − Q λ]σ + Qσλ )∇ν ( −gJρλν ) 2 2 √ 1 τ λρν =0. + −g(∇[σ Qν]λρ + Tσν Qτ λρ )J 2 √
−g ∇ λ (tσλ ) +
−gRσνλρ Sρλν +
Let us consider some particular cases of the matter motion. I. The matter hypermomentum tensor is equal to the spin momentum tensor: Jσρλ = Sσρλ . In this case some terms with nonmetricity vanish in (23) exept the terms [ρ ˆ ν (√−gSρλν ) . Therefore in this case the matter motion de(Q λ]σ − 12 Qσλρ )∇ pends on both torsion and nonmetricity. II. The matter hypermomentum tensor vanishes: Jσρλ = 0 . In this case the canonical energy-momentum tensor of matter (17) reduces to the metric one: tσλ = Tσλ and the quasiconservation law (23) reduces to the matter energy-momentum conservation law in a Riemann space-time (1). Therefore we have proved the following Theorem [14, 15]. Theorem: In (L4 , g) the motion of matter without hypermomentum coinsides with the motion in the Riemann space-time, which the metric tensor coinsides with the metric tensor of (L4 , g). Thus bodies and mediums without hypermomentum are not subjected to the influence of the possible nonmetricity of the space-time (in contrast to the generally accepted opinion) and therefore can not be the tools for the detection of the deviation of the real space-time properties from the Riemann space structure. 4
Therefore for the investigation of the different manifestations of the possible space-time nonmetricity one needs to use the bodies and mediums endowed with the hypermomentum, i.e. the spin particles, the perfect spinning fluid or the perfect fluid with the intrinsic hypermomentum [16, 17, 18]. Appendix Let us consider the identities (11),(12) in more detail. The identity (11) is the direct consequence of (3) and the explicit forms of the curvature tensor Rµνσλ and the torsion tensor Tµνλ . In order to derive the identity (12) let us evaluate the covariant derivative ˆ ∇σ from the both sides of the (11): ˆ σ δL0 = 2∇[σ ∇α] ∂L0 + 2(∇σ Tα ) ∂L0 ∇ δΓσνλ ∂Rσανλ ∂Rσανλ ! ∂L0 ∂L0 ˆ σ ∂L0 . +∇σ Tαβσ + Tσ Tαβσ + 2∇ λ λ ∂Rαβν ∂Rαβν ∂Tσνλ
The first term in the right part of the (24) may be calculated with the help of Ricci identity in (L4 , g) : 2∇[α∇β] W λ = Rαβσλ W σ − Tαβσ ∇σ W λ − Vαβ W λ .
Here Vαβ is the tensor of homothetic curvature (16); W λ is a contravariant vector density, the minus appearing in the first term in the right part of the (25) in case of the covariant vector density Wλ . As a result one has ˆ σ δL0 = (2∇α Tβ + ∇σ Tαβσ + Tσ Tαβσ − 3R[αβσ]σ ) ∂L0 ∇ δΓσνλ ∂Rαβνλ ∂L0 ∂L0 ˆ σ ∂L0 . +Rαβσν − Rαβλσ + 2∇ λ σ ∂Rαβσ ∂Rαβν ∂Tσνλ
Then using in (26) the second identity for the curvature : R[αβσ]λ = ∇[α Tβσ]λ − T[αβρ Tσ]ρλ ,
we convince oneself that the terms in the parentheses in the right part of the (26) vanishes. As a result we get the identity (12).
References  T.W.B. Kibble, J. Math. Phys. 2 (1961) 212.  A. Trautman, Symp. Math. 12 (1973) 139. 5
 F.W. Hehl, Phys Letters 36A (1971) 225.  O.V. Babourova, V.N. Ponomariov, B.N. Frolov, in: Gravitation and electromagnatic field, Universitetskoe,Minsk,1988, p.6 [in Russian].  O.V. Babourova, The variational theory of perfect fluid with intrinsic degrees of freedom in generalized spaces of the modern theory of gravitation, Dr. thesis, VNICPV, Moscow, 1989 [in Rusian].  F.W. Hehl, G.D. Kerlick, P. Heyde, Z. Naturforsch 31A (1976) 111, 524, 823. ˘ cki, Ann. Phys. 120 (1979) 292.  Y. Ne´eman, Dj. Sija˘ ˘ cki, Gen. Rel. Grav. 12 (1980) 83.  F.W. Hehl, Dj. Sija˘  W. Kopczy´ nski, Ann. Phys. (NY) 203 (1990) 308.  F.W. Hehl, J.D. McCrea, E.W. Mielke, Y. Ne´eman, Found. Phys. 19 (1989) 1075.  R. Hecht et al, Phys. Lett. A172 (1992) 13.  O.V.Baburova, B.N. Frolov, On the field equations in the metric-affine gravitational theory, preprint gr-qc 9502.  F.W. Hehl, P. von der Heyde, G.D. Kerlick, J.M. Nester, Rev. Mod. Phys. 48 (1976) 393.  O.V. Babourova, in: Abstracts of contrib. papers of the Cornelius Lanczos Int. conf. (NC State Univ., USA) 1993, p. 100.  O.V.Babourova, M. Yu. Koroliov, B.N. Frolov, Izvestiya Vysshykh Uchebnykh Zavedenij. Fizika (Russian Physics Letters) N1 (1994) 76.  O.V. Babourova, in: Gravitation and fundamental interaction (theses), UDN, Moscow, 1988, p. 119 [in Russian].  O.V.Baburova, B.N. Frolov, M. Yu. Koroliov, in: 13th Int. Conf. gen. rel. grav. (Abstract of contr. papers), Cordoba (Argentina), 1992, eds. P.W. Lamberti and O.E. Ortiz, p. 131.  O.V.Babourova, B.N. Frolov, M. Yu. Koroliov, in: Nauchnye trudy MPGU (Ser,: Est. nauki) (”Prometej”, Moscow), 1994, Part 1, P. 89 [in Russian].  J.A. Schouten, Ricci-Calculus, Springer, Berlin, 1954.