Apparent Horizons Associated with Dynamical Black Hole Entropy

TL;DR

This work introduces entropic marginally outer trapped surfaces (E-MOTS) as a local, entropy-density–driven generalization of apparent horizons and proves that, to first order around a stationary black hole, the HWZ dynamical entropy evaluated on a background Killing horizon coincides with the Wall entropy evaluated on the corresponding E-MOTS. Using the covariant phase space formalism and a boost-weight analysis, the authors show that HWZ entropy can be decomposed into IW and non-stationary corrections and relate it to Wall entropy via a shift in the null parameter $v$, with stationarity recovering the Wald entropy. The key result is a geometric characterization: S_HWZ on a horizon cross-section equals S_Wall on the entropic cross-section $$'s EMOTS, generalized to arbitrary diffeomorphism-invariant actions and reducing to the standard GR areal entropy in the Einstein–Hilbert limit. This furnishes a robust framework for dynamical black hole entropy, connecting horizon-based quantities to quasi-local, entropic surfaces and enabling applications to broader gravity theories and dynamical settings.

Abstract

We define entropic marginally outer trapped surfaces (E-MOTSs) as a generalization of apparent horizons. We then show that, under first-order perturbations around a stationary black hole, the dynamical black hole entropy proposed by Hollands, Wald, and Zhang, defined on a background Killing horizon, can be expressed as the Wall entropy evaluated on an E-MOTS associated with it. Our result ensures that the Hollands-Wald-Zhang entropy reduces to the standard Wald entropy in each stationary regime of a dynamical black hole, thereby reinforcing the robustness of the dynamical entropy formulation.