Consistent energy-momentum trace couplings of fluids

TL;DR

The paper addresses how to correctly implement trace-based nonminimal couplings in $f(R,T)$ gravity for relativistic perfect fluids, showing that common on-shell Lagrangians miss essential physics. By developing Brown's Lagrange-m multiplier and Schutz's velocity-potential formalisms, the authors derive full off-shell variations, obtain consistent field equations with an effective energy–momentum tensor, and reveal temperature and chemical-potential corrections induced by the trace coupling. They demonstrate that linear-in-$T$ models offer no new physics beyond $f(R)$ gravity and that nontrivial dynamics require $f_{TR} eq 0$, with entropy perturbations potentially impacting cosmology. The work provides a rigorous framework to reassess much of the $f(R,T)$ literature and clarifies how to interpret trace couplings in the fluid/gravity interplay, including possible dark-sector interactions. Overall, the results unify two variational pictures, highlight the importance of off-shell fluid Lagrangians, and set precise criteria for when trace couplings yield genuinely new physics.

Abstract

Gravitational models with non-minimal couplings involving the trace of the energy-momentum tensor have become increasingly popular. The idea of coupling the trace of the matter tensor to the geometry can be applied to various matter models, including relativistic perfect fluids. However, it is well-known that the variational formulation of perfect fluids involves some technicalities. We carefully derive the field equations including the trace coupling of a perfect fluid using two different approaches, namely, that given by Brown using Lagrange multipliers, and that given by Schutz using velocity potentials. We show that previous results involving such trace couplings do not match the results presented here. We demonstrate that our fluid's equations of motion are consistent with the gravitational field equations. Moreover, we present a simple on-shell argument which further supports the correctness of our results. Our work implies that a vast amount of the $f(R,T)$ literature using perfect fluids needs to be revisited.