A note on covariant conservation of energy–momentum in modified gravities

Abstract

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