Covariant conservation of energy momentum in modified gravities

TL;DR

This paper proves that energy-momentum conservation is inherently maintained in various modified gravity theories, regardless of the formalism used, ensuring consistency with fundamental physical principles.

Contribution

It provides a general proof of covariant energy-momentum conservation in modified gravities with nonlinear curvature dependence, including metric and Palatini formalisms.

Findings

Energy-momentum conservation is a built-in feature of the field equations.

The covariant divergence of the energy-momentum tensor vanishes in these theories.

In the Einstein frame, a bi-metric structure emerges.

Abstract

An explicit proof of the vanishing of the covariant divergence of the energy-momentum tensor in modified theories of gravity is presented. The gravitational action is written in arbitrary dimensions and allowed to depend nonlinearly on the curvature scalar and its couplings with a scalar field. Also the case of a function of the curvature scalar multiplying a matter Lagrangian is considered. The proof is given both in the metric and in the first-order formalism, i.e. under the Palatini variational principle. It is found that the covariant conservation of energy-momentum is built-in to the field equations. This crucial result, called the generalized Bianchi identity, can also be deduced directly from the covariance of the extended gravitational action. Furthermore, we demonstrate that in all of these cases, the freely falling world lines are determined by the field equations alone and turn out to be the geodesics associated with the metric compatible connection. The independent connection in the Palatini formulation of these generalized theories does not have a similar direct physical interpretation. However, in the conformal Einstein frame a certain bi-metricity emerges into the structure of these theories. In the light of our interpretation of the independent connection as an auxiliary variable we can also reconsider some criticisms of the Palatini formulation originally raised by Buchdahl.