On energy-momentum conservation in non-minimal geometry-matter coupling theories

TL;DR

This work analyzes non-minimal geometry–matter couplings in $f(R,T)$ gravity, focusing on the class $f(R,T)=R+\lambda h(T)$. It shows that with a single matter source described by nonlinear electrodynamics or a scalar field, the modified gravity can be recast as GR with a nonlinear, effectively modified matter sector, preserving a conserved energy–momentum tensor. When multiple matter sources are present and $h(T)$ is nonlinear, cross terms couple scalar and vector sectors, preventing a simple GR reinterpretation. The results clarify when energy exchange is purely gravitational versus when it manifests as nonlinear matter self-interactions, and they identify invariant NEDs where GR-like dynamics are recovered despite the extra coupling.

Abstract

In this work, we discuss the conditions that allow the establishment of an equivalence between $f(R,T)=R+λh(T)$ gravity models and General Relativity (GR) coupled to a modified matter sector. We do so by considering a $D$-dimensional spacetime and the matter sector to be described by nonlinear electrodynamics and/or a scalar field. We find that, for this particular family of models, the action and field equations can indeed be written in terms of a modified matter source within GR. However, when several matter sources are combined, this interpretation is no longer possible if $h(T)$ is a nonlinear function, due to the emergence of crossed terms that mix together the scalar and vector sectors.