On the correspondence between energy conservation and energy-momentum tensor conservation in cosmology

TL;DR

This paper explores the relationship between energy conservation and energy-momentum tensor conservation in cosmology, extending the concepts to systems with horizons, heat sources, and non-minimal coupling, ensuring thermodynamic and gravitational consistency.

Contribution

It generalizes the correspondence between energy and energy-momentum conservation to systems with horizons and non-minimal coupling, introducing surface energy and effective pressure concepts.

Findings

Energy conservation can be maintained with a defined surface energy on horizons.

Effective pressure at the horizon ensures thermodynamic and tensor conservation consistency.

The framework extends to gravity theories with non-minimal coupling, satisfying the generalized second law.

Abstract

The correspondence between the thermodynamic energy equation satisfied by a closed co-moving volume and the conservation equation satisfied by the energy-momentum tensor of the matter inside the co-moving volume is extended to a more general system with an arbitrary cosmological horizon and a heat source. The energy of the system consisting of a cosmological horizon and its internal matter could be conserved by defining a surface energy on the horizons. Therefore, energy conservation and energy-momentum tensor conservation can always be consistent for such a system. On the other hand, from the perspective of classical thermodynamics, one can define an effective pressure at the cosmological horizon to guarantee that the thermodynamic energy equation inside the horizon is consistent with the energy-momentum tensor conservation equation of the matter inside the horizon. These systems can satisfy the generalized second law of thermodynamics under appropriate conditions. The definitions of the surface energy and the effective pressure are extended to the gravity theory with non-minimal coupling between geometry and matter, in which geometry could be regarded as a heat source.