Dynamical black hole entropy beyond general relativity from the Einstein frame

TL;DR

This paper derives and analyzes a formula for dynamical black hole entropy in modified gravity theories using conformal transformations, confirming the second law and relating it to Wald entropy.

Contribution

It provides a pedagogical derivation of dynamical black hole entropy in $f(R)$ and scalar-tensor theories via conformal transformations, extending previous work to non-vacuum and non-stationary cases.

Findings

The entropy formula satisfies the non-stationary physical process first law.

The second law holds at second order for vacuum perturbations.

The relation between dynamical entropy and Wald entropy is established.

Abstract

Recently Hollands, Wald and Zhang proposed a new formula for the entropy of a dynamical black hole for an arbitrary theory of gravity obtained from a diffeomorphism covariant Lagrangian via the Noether charge method. We present an alternative, pedagogical derivation of the dynamical black hole entropy for $f(R)$ gravity as well as canonical scalar-tensor theory by means of conformal transformations. First, in general relativity we generalize Visser and Yan's pedagogical proof of the non-stationary physical process first law to black holes that may not be in vacuum, and give a pedagogical derivation of the second-order behavior of the dynamical black hole entropy for vacuum perturbations by considering the second-order variation of the Raychaudhuri equation. Second, we apply the derivation for general relativity to theories in the Einstein frames, and then recast the conclusions in their original frames. We show that the dynamical black hole entropy formulae of these theories satisfy both the non-stationary physical process first law and the non-stationary comparison first law via the Einstein frame. We further study the second-order behavior of the dynamical black hole entropy for vacuum perturbations, and observe that the second law is obeyed at second order in those theories. Using the Einstein frame, we also determine the relationship between the dynamical black hole entropy and the Wald entropy of the generalized apparent horizon in the original frame.