The matter Lagrangian and the energy-momentum tensor in modified gravity with non-minimal coupling between matter and geometry

TL;DR

This paper demonstrates that in modified gravity models with non-minimal matter-geometry coupling, the matter Lagrangian and energy-momentum tensor are uniquely determined by the coupling form, leading to a more general energy-momentum tensor with additional terms.

Contribution

It provides a variational derivation showing the unique determination of matter Lagrangian and energy-momentum tensor in such models, highlighting their differences from standard general relativity.

Findings

The energy-momentum tensor includes an extra, equation-of-state dependent term.

The extra-force from matter-geometry coupling is always non-zero, even for dust.

The matter Lagrangian is uniquely fixed by the coupling form.

Abstract

We show that in modified $f(R)$ type gravity models with non-minimal coupling between matter and geometry, both the matter Lagrangian, and the energy-momentum tensor, are completely and uniquely determined by the form of the coupling. This result is obtained by using the variational formulation for the derivation of the equations of motion in the modified gravity models with geometry-matter coupling, and the Newtonian limit for a fluid obeying a barotropic equation of state. The corresponding energy-momentum tensor of the matter in modified gravity models with non-minimal coupling is more general than the usual general-relativistic energy-momentum tensor for perfect fluids, and it contains a supplementary, equation of state dependent term, which could be related to the elastic stresses in the body, or to other forms of internal energy. Therefore, the extra-force induced by the coupling between matter and geometry never vanishes as a consequence of the thermodynamic properties of the system, or for a specific choice of the matter Lagrangian, and it is non-zero in the case of a fluid of dust particles.