Equivalence of modified gravity equation to the Clausius relation

TL;DR

This paper demonstrates that the equations of motion for various modified gravity theories can be derived from the thermodynamic Clausius relation, linking gravity and thermodynamics.

Contribution

It explicitly establishes the equivalence between modified gravity equations and thermodynamic relations, extending the thermodynamic perspective to multiple gravity models.

Findings

Modified gravity equations are equivalent to the Clausius relation.

The entropy expression includes contributions from both matter and modified gravity.

The energy flux definition incorporates effects from the modified gravity terms.

Abstract

We explicitly show that the equations of motion for modified gravity theories of $F(R)$-gravity, the scalar-Gauss-Bonnet gravity, $F(\mathcal{G})$-gravity and the non-local gravity are equivalent to the Clausius relation in thermodynamics. In addition, we discuss the relation between the expression of the entropy and the contribution from the modified gravity as well as the matter to the definition of the energy flux (heat).