f(R,L_m) gravity

TL;DR

This paper extends $f(R)$ gravity models by incorporating an arbitrary function of the Ricci scalar and matter Lagrangian, deriving new field equations and particle motion equations, including non-geodesic trajectories and Newtonian limits.

Contribution

It introduces a generalized $f(R,L_m)$ gravity framework with novel field equations and explores test particle dynamics with non-geodesic motion and specific exponential models.

Findings

Derived gravitational field equations for $f(R,L_m)$ gravity.

Established equations of motion showing non-geodesic trajectories.

Analyzed Newtonian limit and specific exponential models.

Abstract

We generalize the $f(R)$ type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar $R$ and of the matter Lagrangian $L_m$. We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the energy-momentum tensor. The equations of motion for test particles can also be derived from a variational principle in the particular case in which the Lagrangian density of the matter is an arbitrary function of the energy-density of the matter only. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equation of motion is also considered, and a procedure for obtaining the energy-momentum tensor of the matter is presented. The gravitational field equations and the equations of motion for a particular model in which the action of the gravitational field has an exponential dependence on the standard general relativistic Hilbert--Einstein Lagrange density are also derived.