Apparent Horizons Associated with Dynamical Black Hole Entropy

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.

💡 Research Summary

The paper introduces a novel geometric construct, the Entropic Marginally Outer Trapped Surface (E‑MOTS), which generalizes the familiar notion of apparent horizons by incorporating an arbitrary entropy density. Starting from a given entropy density s associated with a diffeomorphism‑invariant gravitational action, the authors define an “entropic expansion” θₛ as the variation of s along the null generators of a hypersurface, weighted by the boost weight of the underlying tensorial quantities. This definition is formulated in affinely‑parametrized Gaussian null coordinates (GNC), ensuring covariance under the natural boost rescaling that preserves the null structure.

An E‑MOTS is then defined as a co‑dimension‑2 surface on which the entropic expansion vanishes, θₛ = 0. In the limit where the entropy density reduces to the usual area density (Einstein‑Hilbert action), the condition reproduces the standard marginally outer trapped surface (MOTS) condition, but for more general theories the surface can differ substantially. The authors prove that, for any diffeomorphism‑invariant theory, an E‑MOTS exists (at least perturbatively) in the neighborhood of a stationary black‑hole background when one considers first‑order perturbations.

The central result concerns the Hollands‑Wald‑Zhang (HWZ) dynamical black‑hole entropy, a quantity defined on an arbitrary cross‑section of a background Killing horizon using the covariant phase‑space formalism. By carefully analysing the symplectic potential, Noether charge, and the Jacobson‑Kang‑Myers (JKM) ambiguities, the authors show that the variation of the HWZ entropy under a first‑order perturbation coincides exactly with the variation of the Wall entropy—the Noether charge entropy associated with the same diffeomorphism‑invariant Lagrangian—when evaluated on an E‑MOTS that is related to the original Killing horizon by a diffeomorphism.

In practical terms, the proof proceeds as follows. The background stationary black hole is described in GNC with a Killing vector ξᵃ = κ(v kᵃ − u ℓᵃ), where kᵃ and ℓᵃ are the null generators of the horizon and its affine parameter, respectively. The boost weight analysis shows that the entropy density’s components scale with a factor e^{wσ} under a boost of parameter σ. By choosing a boost that aligns the perturbed null surface with the original Killing horizon, the authors map the HWZ entropy functional onto the Wall functional evaluated on the perturbed surface. The condition θₛ = 0 ensures that the extra boost‑dependent terms cancel, leaving the two entropies identical to linear order.

A crucial consistency check is performed: in any stationary regime (where the perturbation vanishes), the E‑MOTS coincides with the bifurcation surface of the Killing horizon, and the HWZ entropy reduces to the classic Wald entropy. Thus the HWZ proposal satisfies the first law of black‑hole thermodynamics not only on the bifurcation surface but on any cross‑section that can be associated with an E‑MOTS, and it automatically respects the second law in the linearized regime because the area (or more generally the entropy density) of the E‑MOTS cannot decrease under the null energy condition.

The authors also discuss extensions beyond Einstein gravity. For f(R) theories, they show that the entropic expansion can be expressed in terms of the derivative f′(R) and that the HWZ entropy matches the Iyer‑Wald entropy evaluated on the surface where the generalized expansion vanishes. This demonstrates that the method does not rely on the existence of an Einstein frame and is truly applicable to any diffeomorphism‑invariant action.

Technical appendices provide a systematic derivation of boost weights for arbitrary tensor components, the construction of Noether charges in the covariant phase‑space, and the handling of Lie derivatives of scalar densities. These tools make the analysis readily adaptable to higher‑curvature theories, theories with non‑minimal couplings, or even effective field theory corrections.

In summary, the paper establishes a robust bridge between the HWZ dynamical entropy and the Wall (Noether‑charge) entropy by introducing the E‑MOTS as the natural geometric locus where the entropic expansion vanishes. This bridge guarantees that the HWZ entropy reduces to Wald’s entropy in every stationary phase, validates the first law on arbitrary cross‑sections, and provides a clear, covariant prescription for defining a local entropy current in dynamical black‑hole spacetimes across a wide class of gravitational theories.