Eur. Phys. J. C (2017) 77:878 https://doi.org/10.1140/epjc/s10052-017-5452-y Regular Article - Theoretical Physics 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 Riemann tensor 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 general relativity 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 curvature tensor linearly, and thus the Weyl tensor (F) tensor for matter, and Tijkl the energy-momentum tensor for as well. We show that this equation, written in n dimensions, 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 cosmological constant 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 Riemann curvature tensor 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 tensors constructed from the Ricci Because of the properties of the Weyl tensor, its contraction tensor R and from the scalar curvature 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.