Evaluating the Wald Entropy from two-derivative terms in quadratic actions

TL;DR

This paper investigates how the Wald entropy for black holes in generalized gravity theories depends solely on two-derivative terms in the Lagrangian, providing a method to evaluate entropy using linearized equations.

Contribution

It demonstrates that only two-derivative terms contribute to Wald entropy and extends the analysis to matter fields, offering a simplified evaluation method using linearized equations.

Findings

Entropy density depends only on two-derivative terms.

Higher powers of matter fields do not contribute to entropy.

Explicit examples clarify the method's application.

Abstract

We evaluate the Wald Noether charge entropy for a black hole in generalized theories of gravity. Expanding the Lagrangian to second order in gravitational perturbations, we show that contributions to the entropy density originate only from the coefficients of two-derivative terms. The same considerations are extended to include matter fields and to show that arbitrary powers of matter fields and their symmetrized covariant derivatives cannot contribute to the entropy density. We also explain how to use the linearized gravitational field equation rather than quadratic actions to obtain the same results. Several explicit examples are presented that allow us to clarify subtle points in the derivation and application of our method.