Equivalence of black hole thermodynamics between a generalized theory of gravity and the Einstein theory

TL;DR

This paper demonstrates that black hole thermodynamics are equivalent in a generalized gravity theory and Einstein's theory when analyzed through conformal transformations, ensuring consistent thermodynamic variables across frames.

Contribution

It shows the equivalence of black hole thermodynamics between a generalized gravity theory and Einstein's theory using conformal transformations and the Iyer--Wald formalism.

Findings

Thermodynamical variables are identical in both frames under specified conditions.

The analysis confirms the first law of thermodynamics holds in both frames.

Results may impact the understanding of the second law in generalized gravity theories.

Abstract

We analyze black hole thermodynamics in a generalized theory of gravity whose Lagrangian is an arbitrary function of the metric, the Ricci tensor and a scalar field. We can convert the theory into the Einstein frame via a "Legendre" transformation or a conformal transformation. We calculate thermodynamical variables both in the original frame and in the Einstein frame, following the Iyer--Wald definition which satisfies the first law of thermodynamics. We show that all thermodynamical variables defined in the original frame are the same as those in the Einstein frame, if the spacetimes in both frames are asymptotically flat, regular and possess event horizons with non-zero temperatures. This result may be useful to study whether the second law is still valid in the generalized theory of gravity.