4-index theory of gravity

Recently, a 4-index generalization of the Einstein theory is proposed by Moulin [1]. Using this method, we find the most general 2-index field equations derivable from the Einstein-Hilbert action. The application of Newtonian limit, the role of gravitational coupling constant and the effects of the properties of ordinary energy-momentum tensor in obtaining a 4-index gravity theory have been studied. We also address the results of building Weyl free 4-index gravity theory. Our study displays that both the Einstein and Rastall theories can be obtained as the subclasses of a 4-index gravity theory which shows the power of 4-index method in unifying various gravitational theories. It is also obtained that the violation of the energy-momentum conservation law may be allowed in 4-index gravity theory, and moreover, the contraction of 4-index theory generally admits a non-minimal coupling between geometry and matter field in the Rastall way. This study also shows that, unlike the Einstein case, the gravitational coupling constant of 4-index Rastall theory generally differs from that of the ordinary 2-index Rastall theory.
In Riemannian geometry, the geometrical information of a manifold is encoded into the forth-order Riemann tensor, while the general relativity (GR) includes the second rank divergence-less tensors [2, 3].
Riemannian geometry, 4-index, 2-index, Einstein-Hilbert action, Rastall theories
Moradpour, H., Licata I., Corda, C., Salako, I.G. (2019). 4-index theory of gravity and its relation with the violation of the energy-momentum conservation law. Pg 1-8