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Eur. Phys. J. C (2017) 77:878 https://doi.org/10.1140/epjc/s10052-017-5452-y

Regular Article - Theoretical

Generalization of Einstein’s gravitational field equations

Frédéric Moulina Département de Physique, Ecole Normale Supérieure Paris-Saclay, 61 av du président Wilson, 94235 Cachan, France

Received: 2 November 2017 / Accepted: 5 December 2017 / Published online: 16 December 2017 © The Author(s) 2017. This article is an open access publication

Abstract The is the cornerstone of general In Sect. 2 we generalize the Einstein–Hilbert Lagrangian relativity, but as is well known it does not appear explicitly in of by introducing new scalars constructed Einstein’s equation of gravitation. This suggests that the lat- from Rijkl and R jl. ter may not be the most general equation. We propose here for In Sects. 3 and 4 we apply the principle of least action to the first time, following a rigorous mathematical treatment this Lagrangian and obtain the generalized 4-index Einstein based on the variational principle, that there exists a gen- equation written with the total energy-momentum tensor, = (M) + (F) (M) eralized 4-index gravitational field equation containing the Tijkl Tijkl Tijkl, where Tijkl is the energy-momentum Riemann tensor linearly, and thus the (F) tensor for matter, and Tijkl the energy-momentum tensor for as well. We show that this equation, written in n , the gravitational field itself. We show that the tensor contrac- contains the energy-momentum tensor for matter and that tion of this new generalized version of the Einstein equation of the gravitational field itself. This new 4-index equation restores the usual 2-index general relativity equation. remains completely within the framework of general relativ- In Sect. 5 we impose total energy-momentum conserva- ity and emerges as a natural generalization of the familiar tion, and show that the generalized equation can be written 2-index Einstein equation. Due to the presence of the Weyl ∗ ∗ with the double-dual Riemann tensor Rijkl. In the last part, tensor, we show that this equation contains much more infor- the is also introduced. mation, which fully justifies the use of a fourth-order theory.

2 Lagrangian formulation 1 Introduction We want that the new equation of general relativity neces- sarily contains the Riemann tensor linearly, and thus be a The of general relativity R can ijkl fourth-order tensor equation with the same index symme- be split into the Weyl conformal tensor C , and parts which ijkl tries as R . To be as general as possible, this equation must involve only the Ricci tensor R and the curvature scalar R. ijkl jl also contain fourth-order constructed from the Ricci Because of the properties of the Weyl tensor, its contraction tensor R and from the R. To apply the vanishes, gikC = 0, and thus the information it contains jl ijkl principle of least action, we must first define a very general (namely the information as regards the gravitational field in gravitational action, S( ), written with the Lagrangian, L( ), vacuum) is not present in the famous Einstein equation. G G including all these various terms linearly: The aim of this paper is to find a generalized gravitational  field equation explicitly containing the Riemann curvature 1 √ S(G) =− L(G)(Rijkl, R jl, R) −g dΩ (1) tensor linearly. For this purpose, we have implemented a 2χc rigorous mathematical treatment with a classical variational where χ = 8πG/c4 is the Einstein constant. principle using a generalized Lagrangian containing Rijkl, R R jl and . 2.1 Riemann tensor symmetries The paper is organized as follows: The Riemann tensor symmetries are well known in the con- text of general relativity; antisymmetry on the first two a e-mail: [email protected] indices, antisymmetry on the last two indices and symme- 123 878 Page 2 of 8 Eur. Phys. J. C (2017) 77 :878 try obtained by exchanging the first pair with the second pair where a1, a2, a3 are three arbitrary parameters that can be [1,2]: determined by contraction and identification with Eq. (6):

R =−R =−R = R . (2) (1 − a1) (1 − a1) ijkl jikl ijlk klij a = , a =− . (8) 2 (n − 2) 3 (n − 1)(n − 2) The Ricci tensor, which comes from the contraction of the ik Riemann tensor R jl = g Rijkl, is therefore symmetric on Here, we have used the following mathematical relationships jl its two indices, and the scalar curvature R = g R jl,obvi- for the tensorial contractions: ously does not have any symmetry. To construct two fourth- jl order tensors from R jl and R, and having exactly the same g Rijkl = Rik, (9) symmetries as the Riemann tensor, we shall naturally take a g jl(g R p + g R p ) = (n − 2)R + g R, (10) combination involving metric tensors. It is easy to check that ijkp l ijpl k ik ik jl( ) = ( − ) , the combinations (gik R jl − g jk Ril + g jl Rik − gil R jk) and g gijklR n 1 gik R (11) (g g R − g g R) are unique and obey exactly the same ik jl ik jl il jk g g Rijkl = R, (12) symmetries as in (2). For convenience in what follows, we ik jl p p g g (gijkpR l + gijplR k) = 2(n − 1)R, (13) define the tensor gijkl = gikg jl − gilg jk, and we note that ik jl g g (gijklR) = n(n − 1)R, (14) p p gik R jl − g jk Ril + g jl Rik − gil R jk = gijkpR l + gijplR k, (3) jl = δ j = with g g jl j n in a n-dimensional space. gikg jl R − gilg jk R = gijklR. (4) The two relations (8) therefore, allow us to write a gener- alized Lagrangian in a form that involves only one parameter a (with a = a ): These combinations of metric tensors are often found in many 1 well-known references [1–6], and it is therefore these specific  ( − )   fourth-order tensors we will use in our calculations. ik jl 1 a p p L( ) = g g aR + g R + g R G ijkl ( − ) ijkp l ijpl k n 2 ( − ) 2.2 Lagrangian terms 1 a − gijklR . (15) (n − 1)(n − 2) The Einstein–Hilbert Lagrangian of the general relativity is purely gravitational and is defined by the scalar curvature This new Lagrangian is a natural generalization, and we can [1,2]: verify that the contraction with gikg jl naturally restores the Einstein–Hilbert Lagrangian, L(G) = R, whatever the values L(G) = R. (5) of a and n. This Lagrangian is therefore physically compati- ble with the general theory of relativity and a 4-index Einstein However, it is well known that to obtain Einstein’s familiar equation will be obtained in the next chapter by a variational 2-index equation by a least action principle, we have to per- principle using Eq. (15). form the calculations using an action including a Lagrangian ik written with the Ricci tensor, L(G) = g Rik:

 3 Principle of least action 1 √ S( ) =− R −g dΩ G χc 2  It is well known that the total action S is the sum of a purely √ 1 ik =− g Rik −g dΩ. (6) gravitational Einstein–Hilbert action S(G), and a matter-field 2χc action S(MF) which describes any matter and fields living on the space-time [1]: In this paper, we remain within the familiar framework of  general relativity, and so it is also physically equivalent to per- 1 √ S = S( ) + S( ) =− L( ) −g Ω form the calculations using a generalized action constructed G MF χ G d 2 c from the three interesting tensors, (2), (3) and (4): √ + 1 − Ω.  L(MF) g d (16)   c 1 ik jl p S( ) =− g g a Rijkl + a gijkpR l G 2χc 1 2   √ In order to obtain the gravitation field equation, we should + p + − Ω gijplR k a3 gijklR g d (7) vary the total action with respect to the metric: 123 Eur. Phys. J. C (2017) 77 :878 Page 3 of 8 878  1 √ jlδ( p + p ) δS = δS( ) + δS( ) =− δ L( ) −g Ω g gijkpR l gijplR k using (10): G MF χ G d 2 c √ + 1 δ − Ω.   L(MF) g d (17) jl p p jl p p c g δ(gijkpR l + gijplR k) = δ g (gijkpR l + gijplR k) jl p p − δg (gijkpR l + gijplR k) The variation of S(MF) is given by = ( − )δ + δ( )  n 2 Rik gik R √ 1 − δg jl(g R p + g R p ), δS(MF) = δ L(MF) −g dΩ ijkp l ijpl k c  (23) √ 1 jl = Tjl δg −g dΩ 2c  √ jlδ( ) 1 ik jl g gijklR using (11): = Tijkl g δg −g dΩ (18) 2c   jl jl jl where Tjl is the familiar total energy-momentum tensor g δ(gijklR) = δ g (gijklR) − δg (gijklR) defined in [1], jl = (n − 1)δ(gik R) − δg (gijklR), (24) √ 2 δ(L(MF) −g ) Tjl = √ , (19) √ √ − δ jl jl ik g g δ −g =−1/2 g jl δg −g [1], and using gijkl g = (n− 1)g jl we can write and Tijkl a generalized 4-index total energy-momentum ten- sor such that √ 1 √ δ −g =− g gikδg jl −g. (25) ik ( − ) ijkl Tjl = g Tijkl. (20) 2 n 1

The variation of S(G) is obtained by using the lagrangian (15): Equations (22), (23), (24), (25) allow one to rewrite δS(G)  (21)intheform 1 √ δSG =− δ L(G) −g dΩ 2χc     ( − )   1 (1 − a) 1 1 a p p =− δ ik jl + p δS(G) =− aRijkl+ gijkpR l +gijplR k g g aRijkl gijkpR l 2χc (n−2) 2χc (n − 2)   ( − ) √  (1 − a) √ n 2a ik jl + p − − Ω − gijklR g δg −g dΩ gijplR k gijklR g d 2(n − 1)(n − 2) (n − 1)(n − 2)     √ ( − )  + ikδ − Ω. 1 ik jl 1 a p g Rik g d (26) =− g g a δRijkl + δ gijkpR l 2χc (n − 2)   ( − ) √ p 1 a + gijplR k − δ(gijklR) −g δ( ) (n − 1)(n − 2) We note that the terms containing gik R have been sim-  plified by themselves. The terms containing gikδR result ( − )   ik 1 a p p + aRijkl + gijkpR l + gijplR k in an integral of a covariant and hence by the (n − 2)  Stokes theorem are equal to a boundary contribution at infin- ( − ) √ 1 a ik jl ity, which we can set to zero√ by making the variation vanish − gijklR 2 g δg −g ik (n − 1)(n − 2) at infinity [5,6]: g δRik −g dΩ = 0.  √ Using Eqs. (18) and (26), the variation of the total action, + gikg jlδ −g dΩ. (21) δS, thus becomes

It is not so easy to directly calculate the variations of the δS = δS(G) + δS(MF) different terms in the last line, and so we first demonstrate   ( − )  some useful mathematical relations: 1 1 a p p =− aRijkl + gijkpR l + gijplR k g jlδR 2χc (n − 2) ijkl using (9):  ( − ) √   n 2a ik jl jlδ = δ jl − δ jl − gijklR − χTijkl g δg −g dΩ. g Rijkl g Rijkl g Rijkl 2(n − 1)(n − 2) jl = δRik − δg Rijkl, (22) (27)

123 878 Page 4 of 8 Eur. Phys. J. C (2017) 77 :878

4 Generalized 4-index Einstein equation where we introduced here a new fourth-order tensor: ( − )   1 a p p 4.1 Fourth-order equation Gijkl = aRijkl + gijkpR l + gijplR k (n − 2) ( − ) Since the expression for the variation of the total action δS n 2a − gijklR. (32) (27) should hold for any variation δg jl,wemusthave 2(n − 1)(n − 2)  ( − )  We can verify that ik 1 a p p g aRijkl + gijkpR l + gijplR k (n − 2) ik 1  g Gijkl = R jl − g jl R = G jl (33) (n − 2a) 2 − gijklR − χTijkl = 0. (28) 2(n − 1)(n − 2) where we recognize here the famous 2-index G jl. The fourth-order tensor Gijkl, containing the Riemann In general, for arbitrary Tijkl, the expression in the paren- tensor, the Ricci tensor and the scalar curvature, can therefore thesis is not equal to zero, and we cannot simply exclude be considered as a generalization of G jl. the contraction factor gik. However, it is always mathemati- cally possible to choose a particular energy-momentum ten- 4.3 Weyl tensor sor Tijkl which has the same number of components and the same symmetries as Rijkl, and which acts as a source term, The fourth-order Einstein tensor, Gijkl, depends partly on allowing us to write the sought-after generalized 4-index Ein- the mathematical parameter a and can therefore be split in stein equation: two distinct parts of which only one depends on a: ( − )  G = aC + B (34) 1 a p p ijkl ijkl ijkl aRijkl + gijkpR l + gijplR k (n − 2) and we can write by identification with (32) (n − 2a) − gijklR = χ Tijkl. (29)   2(n − 1)(n − 2) 1 p p Cijkl = Rijkl − gijkpR l + gijplR k (n − 2) We can easily check that the result of the tensorial contrac- 1 + gijklR, (35) tion of this equation gives the famous Einstein equation of (n − 1)(n − 2) general relativity, whatever the values of a and n (surprising as it may seem): 1   B = g R p + g R p 1 ijkl ( − ) ijkp l ijpl k R jl − g jl R = χ Tjl. (30) n 2 2 n − gijklR. (36) This is of course an important result for checking our calcula- 2(n − 1)(n − 2) tions, where we point out that we did not use Einstein’s equa- We recognize here the Weyl tensor C written in n dimen- tion in any intermediate calculations to obtain the generalized ijkl sions [3–7], and a new, hitherto unknown tensor B .Itis equation (29). This first result also shows us unambiguously ijkl easy to check that the tensorial contractions of (35) and (36) that the 2-index Einstein equation (30) has the same form in give any space-time of n, which is also an important result. ik g Cijkl = 0, (37) To the best of our knowledge, this new 4-index gravita- ik 1 tional field equation, obtained by means of a rigorous math- g Bijkl = R jl − g jl R. (38) ematical treatment using the variational principle, is written 2 here for the first time. Due to the presence of the Riemann This last result clearly show us that some of the information tensor, and therefore the Weyl tensor, we will show later that contained in the generalized equation (31) is not recovered Eq. (29) contains, as expected, more information than Eq. in the 2-index Einstein equation (30). This is the part of the (30). Weyl tensor.

4.2 Fourth-order Einstein tensor 4.4 Two part decomposition of total energy-momentum tensor Tijkl Equation (29) can also be written in a more compact form: It is well known that Einstein himself, in agreement with Gijkl = χ Tijkl (31) major consensus among famous physicists (see [1,3,8,9]for 123 Eur. Phys. J. C (2017) 77 :878 Page 5 of 8 878

(M) example), has emphasized that the gravitational field must The energy-momentum tensor for matter Tijkl actsasa also have an energy-momentum tensor, as do all other phys- source term for Bijkl, and it is not difficult to check that ical fields. In this paper, the 4-index tensor Tijkl, which describes any matter present plus the gravitational field con- ik (M) = , g Tijkl Tjl tained in space-time, can be simply and naturally divided into jl (M) = , two parts: g Tijkl Tik (M) =− (M) =− (M) = (M). = (M) + (F) Tijkl Tjikl Tijlk Tklij (46) Tijkl Tijkl Tijkl (39)

( ) In addition, one can easily check that the tensorial contraction where T M represents the generalized energy-momentum ijkl of Eq. (45) gives Einstein’s equation (30). (F) tensor of the matter content, and Tijkl the energy-momentum tensor of the gravitational field itself [10]. With (34) and (39) 4.6 Energy-momentum tensor of the gravitational field the generalized equation (31) becomes (F) Tijkl

+ = χ (F) + (M) . With Eq. (40), simplified by (45), we find a simple aCijkl Bijkl Tijkl Tijkl (40) between the energy-momentum tensor for the free gravita- (F) tional field T , and the Weyl tensor Cijkl: (M) ijkl 4.5 Energy-momentum tensor of the matter Tijkl = χ (F). aCijkl Tijkl (47) By rearranging the Einstein equation (30), it is possible to (F) rewrite the Ricci tensor R jl and the scalar curvature R in the Tijkl acts as a source term for the Weyl tensor, and the prop- form erties of Cijkl imply that

( ) 1 gikT F = 0, R jl = χ Tjl − g jlT (41) ijkl (n − 2) ( ) g jlT F = 0, 2 ijkl R =− χ T, ( ) ( ) ( ) ( ) ( − ) (42) F =− F =− F = F . n 2 Tijkl Tjikl Tijlk Tklij (48) which we now substitute into (36), obtaining The tensorial contraction of Eq. (47) is equal to zero. The ( ) two tensors, C and T F , are logically linked because they  ijkl ijkl   only concern the free gravitational field in vacuum, and it is 1 p p Bijkl = χ gijkpT l + gijplT k (n − 2) precisely this part which is not present in the standard theory  of general relativity. The information contained in Eq. (47) 1 − g T . is not contained in the 2-index Einstein equation, and for this ( − )( − ) ijkl (43) n 1 n 2 reason a fourth-order theory is fully justified. For a vanishing Ricci tensor, R jl = 0, the Riemann cur- Equations (38) and (43) together indicate that the tensor Bijkl vature tensor is equal to the Weyl tensor, Rijkl = Cijkl, and is directly linked to the energy-momentum tensor of the mat- it is well known that the latter describes the free gravitational ter content present in the standard theory of general relativity field in vacuum and provides curvature to space-time [6,7]. A by means of the tensor T . The right-hand side of (43) rep- (M) = jl flat space-time implies that there is no matter with Tijkl 0, resents a 4-index energy-momentum tensor for matter, and (F) no field with T = 0, and likewise Rijkl = 0, as can be in our theory there is only one tensor that can play this role, ijkl ( ) seen from the generalized Einstein equation (29). M ( ) ( ) namely Tijkl : = M + F The total energy-momentum tensor, Tijkl Tijkl Tijkl, gives the most complete description of the medium with   (M) 1 p p respect to its effect on the geometry. It describes both the T = gijkpT l + gijplT k ijkl (n − 2) state of matter and the state of the gravitational field in vac- 1 uum, but it appears to be very non-trivial to find the form of − gijklT, (44) (F) (n − 1)(n − 2) the tensor Tijkl. The determination of this tensor, however, and therefore the form of Eq. (47), would allow us to find which allows us to write important solutions concerning the contribution of the grav- (F) itational field itself. The tensor Tijkl naturally takes its place = χ (M). Bijkl Tijkl (45) in the new generalized 4-index equation, and we shall show 123 878 Page 6 of 8 Eur. Phys. J. C (2017) 77 :878   in the next chapter that Tijkl obeys total energy-momentum 1 p p 1 − Rijkl + gijkpR l + gijplR k − gijklR conservation. (n − 3) 2 = χTijkl (52) 4.7 Energy-momentum conservation where we recognize here the double (Hodge) dual Riemann tensor [2,13]: In the preceding chapters the parameter, a, has been treated as a mathematical parameter with no particular meaning. We ∗ ∗ 1 pqrs will now show that a takes on a particular physical value R = eijpq R eklrs ijkl 4 in the context of covariant conservation of the total 4-index p p 1 energy-momentum tensor: =−Rijkl + gijkpR l + gijplR k − gijklR (53) 2 i (F)i (M)i ∇i T jkl =∇i T jkl + T jkl = 0 (49) with eijkl the Levi-Civita tensor. where ∇i is the operator. The general equation (52) can thus be written simply as With the definition of the generalized Enstein tensor (32), ∗ ∗ = χ( − ) . Rijkl n 3 Tijkl (54) and using the usual metric compatibility ∇i g jl = 0, we can write It is not difficult to check that the tensorial contraction of  this equation again yields the Einstein equation (30), with ( − ) ∗ ∗ i i 1 a i i i gik R = (n−3)(R − 1 g R), and that (54), as expected, ∇i G jkl =∇i aR jkl + δ R jl − g jk R l + g jl R k ijkl jl 2 jl (n − 2) k  also satisfies total energy-momentum conservation:  ( − ) i n 2a i i − δ R jk − (δ g jl − δ g jk)R ∗ ∗i l 2(n − 1)(n − 2) k l ∇i R jkl = 0, (55) ( ) ( ) (1 − a)  ∇ T i =∇(T F i + T M i ) = 0. (56) = a∇ Ri + ∇ R − g ∇ Ri i jkl i jkl jkl i jkl ( − ) k jl jk i l n 2  i + g jl∇i R k −∇l R jk According to (45) and (47), it is also possible to write Eq. (n − 2a)   (54) in terms of two parts: − g jl∇k R − g jk∇l R . 2(n − 1)(n − 2) =−χ( − ) (F), Cijkl n 3 Tijkl (57) ( ) We make use of the contracted Bianchi identities, [3,5]: B = χ T M . i i ijkl ijkl (58) ∇i R jkl =∇k R jl −∇l R jk and ∇i R k = 1/2 ∇k R to deduce  + ( − ) As we have already seen, these two equations are not coupled; i 1 a n 3 ∇i G jkl =− ∇l R jk −∇k R jl they are independent, and their solutions found individually (n − 2)  must be added to satisfy total energy-momentum conserva- 1   tion. As an analogy in classical mechanics, for example, it is + g jl∇k R − g jk∇l R 2(n − 1) akin to the kinetic energy and the potential energy that derive 1 + a(n − 3) from different relations, but which must finally be added to =− C jkl (50) (n − 2) have conservation of the total mechanical energy.

= χ (M) where we identify here the C jkl [11,12]. 5.2 Particular solution of the equation Bijkl Tijkl Imposing energy conservation thus gives us the physical value of a: In the very important case of a centrally symmetric field in vacuum, that is, outside of the masses producing the field, i i 1 ( ) ∇i G jkl = 0 ⇒∇i T jkl = 0 ⇒ a =− . (51) M = = (n − 3) Tijkl 0, it is possible to solve Eq. (58), for n 4:

1 p p 1 Bijkl(n=4) = (gijkpR l + gijplR k) − gijklR = 0. 5 Generalized equation with a =−1/(n − 3) 2 3 (59) ∗ ∗ 5.1 Double-dual Riemann tensor Rijkl If we use spherical space coordinates (r,θ,φ), then the gen- eral expression for ds2 is In the case where energy is conserved, taking a =−1/(n − 2 = ( ) 2 2 − ( ) 2 − 2 θ 2 − 2 2θ φ2. 3), the generalized 4-index Einstein equation of general rel- ds A r c dt B r dr r d r sin d ativity (29) becomes (60) 123 Eur. Phys. J. C (2017) 77 :878 Page 7 of 8 878

To get the differential equations of gravitation with the func- 5.3 Cosmological constant tions A(r) and B(r), we must first calculate the components of the tensor Bijkl(n=4). Using a familiar method, it is not It is interesting to verify that Eq. (54) can be deduced from difficult (but it involves a long computation) to obtain the a principle of least action using the Lagrangian (15) written following independent relations: with a =−1/(n − 3):       = 1 ik jl − + p + p 1  A 2 A B AB L(G) g g Rijkl gijkpR l gijplR k =− − − + (n − 3) B0101 A  6 2A 2B rB 1   − g R . 4AB A 4A ( − ) ijkl (67) + − − , (61) n 1 r 2 r r 2      We will work again within the framework of general rel- r 2  A 2 A B 5AB B0202 = A − − − ativity, where the cosmological constant Λ is introduced 12B 2A 2B rB − Λ   into the Lagrangian by an additional term [5]: 2 .In 2AB A 2A − − + , (62) the same way as in (14), this term can be rewritten as 2 2 ik jl r r r −2Λ =−2/n(n − 1) g g (gijkl Λ). The principle of least      r 2  A 2 A B AB action applied to this term modifies Eq. (54), giving us B =− A − − + 1212 ( − ) 12A 2A 2B rB ∗ ∗ + n 3 Λ = χ( − ) .   Rijkl gijkl n 3 Tijkl (68) 2AB 5A 2A (n − 1) − + + , (63) r 2 r r 2 Contracting the latter equation indeed restores Einstein’s      r 4 sin2 θ  A 2 A B AB standard equations with the cosmological constant [5,6]: B2323 =− A − − − 3AB 2A 2B 2rB − 1 + Λ = χ .   R jl g jl R g jl Tjl (69) AB A A 2 + + − . (64) r 2 2r r 2 6 Conclusion By combining these differents equations, we finally obtain a well-known solution, the standard for In this paper we have rigorously demonstrated, by using the the space-time [1,2]: principle of least action, that there exists a much more general equation than that of Einstein. This new 4-index equation explicitly and linearly includes the Riemann tensor and hence 1 rg A(r) = = 1 + , (65) the Weyl tensor as well as the energy of the gravitational B(r) r (F) − field Tijkl. The contraction of this equation restores the usual rg rg 1 ds2 = 1 + c2dt2 − 1 + dr 2 2-index Einstein equation. The paper highlights two main r r results: − r 2dθ 2 − r 2 sin2 θdφ2. (66) = χ (M) – the first result is given by the equation, Bijkl Tijkl , This is not surprising because Eq. (59) contains only the Ricci which could allow us to find the exact solutions for any tensor and the scalar curvature, and therefore we expect to n dimension space; find the same solutions as those obtained with the Einstein – the second important result is given by the equation =−χ( − ) (F) equation (a favorable omen). Equation (58), however, goes Cijkl n 3 Tijkl, connecting the Weyl tensor and much further because it may allow us to find the exact solu- the energy-momentum tensor of the gravitational field in tions for any dimension n of the space-time. vacuum. Because the generalized Eq. (54) is contracted to obtain the original Einstein equation, only standard solutions are The next step in our research will be to determine pre- (F) present in the familiar 2-index general theory of relativity, cisely the composition of the energy-momentum tensor Tijkl but lacking are the solutions of Eq. (57) giving the contri- in order to obtain more general solutions including the con- bution of the gravitational field itself in vacuum. The next tribution of the gravitational field energy itself. step in our future research will be to determine precisely the mathematical composition of the energy-momentum tensor Open Access This article is distributed under the terms of the Creative (F) Commons Attribution 4.0 International License (http://creativecomm Tijkl in order to obtain solutions to Eq. (57), but this appears ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, to be a very difficult task. and reproduction in any medium, provided you give appropriate credit 123 878 Page 8 of 8 Eur. Phys. J. C (2017) 77 :878 to the original author(s) and the source, provide a link to the Creative 6. M. Blau, Lecture Notes on General Relativity ( Cen- Commons license, and indicate if changes were made. ter for Fundamental Physics, Bern, 2014) Funded by SCOAP3. 7. G. Øyvind, H. Sigbjørn, Einstein’s General Theory of Relativity (Springer, Berlin, 2007) 8. E. Schrödinger, Space-time structure (Cambridge University Press, References Cambridge, 1950) 9. P.A.M. Dirac, General Theory of Relativity (Princeton University 1. L. Landau, E. Lifchitz, The classical theory of fields (Pergamon Press, Princeton, 1975) Oxford Press, New York, 1975) 10. Z. Zakir, Four-index energy-momentum tensors for gravitation and 2. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (Freeman matter. TPAC Theor. Phys. Astrophys. Cosmol. 5, 1 (2010) W.H, New York, 1973) 11. E. Cotton, Sur les variétés á trois dimensions. Ann. Fac. Sc. 3. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972) Toulouse II, 1 (1899) 4. L.P. Eisenhart, (Princeton University Press, 12. A. Garcia, F.W. Hehl, C. Heinicke, A. Macias, The Cotton tensor in Princeton, 1926) Riemannian . Class. Quantum 21, 1099 (2004) 5. S. Carroll, and geometry: an introduction to general 13. E.B. Gliner, equations of the general theory of relativity. Sov. relativity (Addison-Wesley, San Francisco, 2004) Phys. JETP 28, 1238 (1969)

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