Covariant conservation of energy momentum in modified gravities

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.

8 pages. v2: more discussion

References in corpus (11)

• The Cosmology of Generalized Modified Gravity Models
• Dark Energy Dominance and Cosmic Acceleration in First Order Formalism
• Consistent modified gravity: dark energy, acceleration and the absence of cosmic doomsday
• Cosmological perturbations in the Palatini formulation of modified gravity
• The matter power spectrum in f(R) gravity
• Post-Newtonian Parameters from Alternative Theories of Gravity
• Conformal aspects of Palatini approach in Extended Theories of Gravity
• Acceleration of the universe in the Einstein frame of a metric-affine f(R) gravity
• The One-Loop Effective Action in phi^4 Theory Coupled Non-Linearly with Curvature Power and Dynamical Origin of Cosmological Constant
• A Note on Energy-Momentum Conservation in Palatini Formulation of L(R) Gravity
• On the Dirac field in the Palatini form of 1/R gravity

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