Non-minimal fluid Lagrangian couplings

Abstract

Gravitational models with non-minimal couplings involving functions of the matter Lagrangian and curvature have become popular in recent decades. By coupling the matter Lagrangian directly to the gravitational Lagrangian, one hopes to construct theories that can explain dark energy or dark matter without introducing additional sources. When this matter Lagrangian describes a perfect fluid, some technicalities are involved in its variational formulation. We present a careful derivation of the gravitational field equations together with the complete set of fluid equations using two approaches: Schutz's velocity potential and Brown's Lagrange multipliers. We find that with non-minimal couplings, the energy-momentum conservation equations give rise to a non-vanishing contribution which acts in a different direction depending on the approach. This also leads to the emergence of different effective thermodynamic quantities such as the number density, chemical potential, free energy, and temperature. We demonstrate the non-equivalence of the Lagrangian formulations of Schutz and Brown for these types of models and provide a detailed interpretation of our results.