Palatini formulation of modified gravity with a nonminimal curvature-matter coupling

TL;DR

This paper develops the Palatini formalism for modified gravity theories with nonminimal curvature-matter coupling, deriving field equations, equations of motion, and analyzing the non-geodesic motion of test particles.

Contribution

It provides a novel derivation of field equations and particle motion in Palatini formalism for theories with arbitrary matter-geometry coupling, highlighting non-conservation of energy-momentum.

Findings

Explicit form of equations of motion for perfect fluids

Expression of extra-force in terms of coupling functions

Motion generally non-geodesic with orthogonal extra-force

Abstract

We derive the field equations and the equations of motion for massive test particles in modified theories of gravity with an arbitrary coupling between geometry and matter by using the Palatini formalism. We show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, matter Lagrangian dependent metric, which is related with the physical metric by means of a conformal transformation. Similarly to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We derive the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra-force is obtained in terms of the matter-geometry coupling functions and of their derivatives. Generally, the motion is non-geodesic, and the extra force is orthogonal to the four-velocity.