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

Recently, a $4$-index generalization of the Einstein theory is proposed by 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.