Black Hole Entropy Beyond the Wald Term in Nonminimally Coupled Gravity: A Covariant Phase Space Decomposition

TL;DR

This paper investigates black hole entropy in theories with nonminimal matter-curvature couplings, decomposing the entropy into Wald and non-Wald parts, and applies the method to various black hole solutions to assess the completeness of Wald entropy.

Contribution

It introduces a decomposition of horizon surface charge variation into Wald and non-Wald parts, providing a criterion to test the sufficiency of Wald entropy in nonminimally coupled gravity theories.

Findings

Kalb--Ramond black holes have entropy equal to Wald entropy.

Bumblebee black holes show either additional non-Wald contributions or cancellations.

Extended Gauss--Bonnet black holes require both Wald and non-Wald corrections.

Abstract

We study the entropy of static, spherically symmetric black holes in diffeomorphism-invariant theories with nonminimal matter--curvature couplings, using the covariant phase space formalism. For regular bifurcate Killing horizons, the Iyer--Wald construction gives the standard Wald entropy. If a matter field cannot be smoothly extended to the regular bifurcation surface, however, the horizon surface charge variation can contain finite contributions that are not included in the Wald entropy density. In the representative obtained by directly varying the action, and after ordinary work terms are subtracted, we decompose the entropy entering the first law of black hole thermodynamics as \(S_{\mathrm H}=S_{\mathrm W}+S_1+\Delta S\). Here \(S_{\mathrm W}\) is the Wald entropy, \(S_1\) is the non-Wald part of the Noether charge, and \(\Delta S\) is the remaining integrable part of the horizon surface charge variation. Applying this criterion to Kalb--Ramond, bumblebee, and extended Gauss--Bonnet black holes, we find that the regular Kalb--Ramond branch has \(S_{\mathrm H}=S_{\mathrm W}\), the bumblebee branches yield either \(S_1=0\) with \(\Delta S\neq0\) or a cancellation between \(S_1\) and \(\Delta S\), and the Weyl-vector extended Gauss--Bonnet examples require both corrections. This gives a direct test of whether the Wald entropy density is sufficient, or whether the full horizon surface charge variation has to be used.