4-Index theory of gravity and its relation with the violation of the energy–momentum conservation law

Abstract

Recently, a 4-index generalization of the Einstein theory has been proposed by Moulin [F. Moulin, Eur. Phys. J. C 77, 878 (2017)]. 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.