How non-minimal gravity-matter couplings reshape fluid thermodynamics

Non-minimal couplings between curvature and the matter Lagrangian are a popular route to explaining dark energy or dark matter without new fields. Working through the variational formulation for a perfect fluid, the authors obtain the complete gravitational and fluid equations and extract thermodynamic quantities—temperature, chemical potential, and number density—showing explicitly how the coupling distorts them. A key finding is that the Schutz and Brown Lagrangian formulations, equivalent in standard general relativity, are no longer equivalent here. The distinction matters for any physical interpretation built on either formalism.