Dynamical Wald Entropy Explained

• Dynamical Wald Entropy is a generalized concept that extends the Noether charge definition to time-dependent, non-equilibrium black hole horizons.
• It incorporates explicit dynamical corrections using covariant phase space methods, local expansion, and shear to ensure compatibility with the laws of thermodynamics.
• The framework adapts to higher-curvature and quantum corrections, providing a robust tool for verifying the first and second laws in various gravitational theories.

Dynamical Wald Entropy is a generalization of black hole entropy that extends the Noether charge framework of Wald to incorporate time-dependent, non-equilibrium processes in gravitational theories with arbitrary diffeomorphism invariant Lagrangians. Unlike the stationary Wald entropy, which associates entropy with the Noether charge evaluated on a cross-section of a Killing horizon, the dynamical extension defines entropy on arbitrary (possibly non-stationary) horizon cross-sections, includes explicit dynamical correction terms, and is constructed to satisfy physically sensible first and second-laws of black hole mechanics even in generic dynamical regimes. This concept has deep connections to quasi-local definitions of horizons, effective field theory corrections, energy conditions (including quantum modifications), and plays a central role in efforts to unify black hole thermodynamics with semiclassical gravity and statistical mechanics.

1. From Wald Entropy to Dynamical Generalizations

Wald entropy, in its classic formulation, is the Noether charge associated with diffeomorphism invariance evaluated on a bifurcation surface of a stationary black hole's Killing horizon. For a Lagrangian $L[g_{ab}, R_{abcd}]$ (with possible dependence on higher curvature invariants), the entropy on a spatial section $\mathcal{C}$ of the Killing horizon is given by

$$S_{\text{Wald}} = 2\pi \int_{\mathcal{C}} \tilde{Q}_\xi/\kappa$$

where $\tilde{Q}_\xi$ is the Noether charge $(n-2)$-form associated with the horizon-generating Killing field $\xi^a$ and $\kappa$ is the surface gravity (Das, 2010).

To extend this to non-stationary cases, recent works propose an "improved" entropy expression, the Hollands-Wald-Zhang (HWZ) entropy, valid for dynamical black hole horizons, including systems with higher curvature corrections (Hollands et al., 1 Feb 2024, Visser et al., 11 Mar 2024, Kong et al., 1 Dec 2024, Furugori et al., 18 Jul 2025). For a cross-section $\mathcal{C}(v)$ of the (weakly) dynamical event horizon parametrized by an affine parameter $v$, the entropy is given, schematically, by

$$S_{\text{dyn}}(\mathcal{C}) = (1 - v\partial_v) S_{\text{Wald}}(\mathcal{C})\,.$$

For Einstein gravity, this reduces to

$$S_{\text{dyn}} = \frac{A(\mathcal{C})}{4G} - \frac{1}{4G} \int_{\mathcal{C}} v\,\theta\,dA$$

where $\theta$ is the expansion of the null generators, and the second term is the dynamical correction (Hollands et al., 1 Feb 2024, Visser et al., 11 Mar 2024, Kong et al., 1 Dec 2024). For generic diffeomorphism invariant theories, $S_{\text{Wald}}$ is replaced by the appropriate functional—to first order, often the Wall entropy (Visser et al., 11 Mar 2024).

This construction ensures covariance, reduces to the standard result in stationary regimes, and incorporates the true non-equilibrium geometry of the horizon by explicitly encoding local expansion and shear of horizon generators.

2. Covariant Phase Space, Noether Charge, and Entropy Currents

The dynamical entropy formula arises in the covariant phase space framework, where the symplectic potential $\theta$, the Noether current $J_\xi$, the Noether charge $Q_\xi$, and an additional boundary term $B_H^+$ are utilized to define an "improved" Noether charge on any cross-section $\mathcal{C}$ of the (possibly nonstationary) null horizon. The structural relation

$\underline{\theta} = \delta B_H^+$

on the horizon allows for this prescription (Hollands et al., 1 Feb 2024, Visser et al., 11 Mar 2024), resolving the ambiguities (JKM ambiguities) that appear in nonstationary contexts, and allowing extension to non-minimally coupled matter fields.

The dynamical correction naturally emerges in Gaussian null coordinates, with the operator $(1 - v\partial_v)$ mapping the canonical stationary entropy to its dynamical generalization on each cross-section (Visser et al., 11 Mar 2024). The explicit use of boost weight (Lie algebra) arguments and horizon-adapted coordinates ensures gauge covariance and practical computability, as further elucidated in higher curvature and effective field theory frameworks (Kar et al., 7 Mar 2024, Davies et al., 2023).

Additionally, the construction of a local entropy current $s^\mu$ on the horizon and the analysis of its divergence (entropy production) constitute dynamical versions of the second law for nonstationary black holes in higher derivative and anomalous gravity theories (Chapman et al., 2012, Kar et al., 7 Mar 2024).

3. Apparent Horizons, E-MOTSs, and the Geometric Substrate

In the dynamical context, the notion of entropy must be connected to geometric objects beyond the teleological event horizon. The concept of entropic marginally outer trapped surfaces (E-MOTSs) generalizes the standard apparent horizon: an E-MOTS is a codimension-2 surface where the "entropic expansion" associated with a chosen entropy density vanishes,

$$\Theta_+ = 0,\quad \Theta_w = \frac{1}{s}\partial_\mu (w^\mu s) = w^\mu \partial_\mu \log s + \partial_\mu w^\mu,$$

with $s$ as the relevant entropy density (Furugori et al., 18 Jul 2025).

The HWZ dynamical black hole entropy, evaluated on an arbitrary cross-section of the background Killing horizon, can—in first-order perturbation theory—be equivalently expressed as the Wall entropy on the associated E-MOTS. Thus, to leading order in perturbations, the dynamical entropy can be interpreted as the equilibrium entropy (Wald/Wall) evaluated not on the background cross-section, but on a dynamically shifted apparent horizon surface fixing the entropic expansion, providing a quasi-local, physically meaningful boundary for entropy assignment in dynamical gravity (Furugori et al., 18 Jul 2025). This connection guarantees that all entropy definitions coincide in the stationary limit.

4. Laws of Black Hole Mechanics in the Dynamical Regime

The HWZ entropy satisfies both physical process and comparison versions of the first law of black hole mechanics: $$\delta M - \Omega_H \delta J = T \delta S_{\text{dyn}}$$ on arbitrary cross-sections (Hollands et al., 1 Feb 2024, Visser et al., 11 Mar 2024, Kong et al., 1 Dec 2024, Furugori et al., 18 Jul 2025). For source-driven (non-vacuum) perturbations, the change in entropy is directly tied to the energy flux across the horizon; for vacuum perturbations, first-order corrections vanish, and the leading change is quadratic in the metric perturbation, captured by the canonical energy flux quadratic in expansion and shear. This ensures that, under the null energy condition, the entropy does not decrease—thus satisfying a dynamical, local second law to leading (and, for many theories, to second) order (Hollands et al., 1 Feb 2024, Kong et al., 1 Dec 2024).

The method is robust under higher derivative corrections, nonminimally coupled matter, Kaluza-Klein reduction, and is field-redefinition invariant when analyzed via the Einstein frame for theories such as $f(R)$ and scalar-tensor gravity (Kong et al., 1 Dec 2024, Gomez-Fayren et al., 2023).

5. Ambiguity, Covariance, and Comparison with Other Entropy Proposals

Dynamical generalizations of Wald entropy feature inherent ambiguities associated with field redefinitions, boundary terms, and coordinate choices (Iyer-Wald and JKM ambiguities). The HWZ procedure fixes these at first order by requiring that the physical process first law hold, by demanding gauge covariance under horizon-adapted reparametrizations, and by employing boost weight analysis in horizon-coordinates (Kar et al., 7 Mar 2024, Visser et al., 11 Mar 2024). The explicit transformation properties of the entropy current components under horizon-friendly coordinate changes have been worked out for large classes of higher derivative and nonminimally coupled theories, with implications for horizon symmetry enhancements and possible relations to "soft hair" (Kar et al., 7 Mar 2024).

Comparison with other dynamical entropy functionals—most notably, the Dong-Wall entropy—reveals that the HWZ entropy is strictly local-in-time: its first-order change is nonzero only when energy crosses the horizon, whereas alternatives retain some "teleological" anticipation inherited from the event horizon's global definition. Both approaches reduce to the same result in stationary regimes; their differences are most pronounced in non-equilibrium settings and can be linked to subtle distinctions in the way they handle matter entropy and the quantum null energy condition (QNEC) in semiclassical gravity (Hollands et al., 1 Feb 2024).

6. Higher-Curvature, Nonlocal, and Quantum Corrections

Dynamical Wald entropy formulas are well-defined for all diffeomorphism-invariant Lagrangians, including those with higher curvature, nonlocal, or quantum corrections. In these contexts, the entropy density functional used in the HWZ or Wall entropy must be constructed from all available geometric and matter data, including extrinsic curvature corrections (e.g., in the Wall formula),

$$S_{\text{Wall}} = -8\pi \int_{\mathcal{C}} dA \left[\frac{\partial L}{\partial R_{uvuv}} - 4 \frac{\partial L}{\partial R_{uiuj}} R_{vkv l} \, \bar{K}_{ij} K_{kl} \right]$$

(Visser et al., 11 Mar 2024). Quantum field theory effects (trace anomaly, logarithmic corrections) enter as universal corrections via the Noether charge construction for the effective action, and can be related to entanglement entropy and the QNEC in semiclassical dynamical scenarios (Aros et al., 2013, Hollands et al., 1 Feb 2024).

For nonlocal or auxiliary field-reduced gravities, the localization approach allows the application of the same general Noether charge-based recipe (with covariant functional derivatives taken with respect to the Riemann tensor or its substitute, as appropriate) (Teimouri, 2017).

7. Impact and Interpretational Consequences

Dynamical Wald entropy constitutes a robust, covariant thermodynamic quantity for dynamical black holes, enjoying universality across a wide class of gravitational theories. It (i) guarantees compatibility with the equilibrium laws in stationary backgrounds, (ii) incorporates dynamical corrections reflecting the true causal horizon structure, (iii) connects local geometric data (expansion, shear, entropy current divergence) with entropy production and the generalized second law, (iv) can be interpreted quasi-locally as the entropy evaluated on E-MOTSs, and (v) is compatible with field-redefinition invariance.

Its connection with local, gauge-covariant entropy currents, with anomaly-induced corrections, and with quasi-local horizons highlights its relevance for gravitational effective field theory, semiclassical gravity, holography, and quantum information-theoretic approaches to spacetime. Violations or marginal satisfaction of the dynamical second law in certain higher curvature theories (notably, critical gravity) serve as highly sensitive probes of physical viability and can expose pathologies in candidate semiclassical or quantum gravity models (Maeda et al., 2018).

The dynamical Wald entropy framework unifies diverse threads in gravitational thermodynamics, establishing a clear standard against which entropy functionals for black hole horizons should be compared in both classical and quantum gravity.