Dynamical Black Hole Entropy

• Dynamical black hole entropy is defined on locally accessible horizon cross-sections, extending the stationary Wald approach to evolving, non-equilibrium settings.
• It satisfies local versions of the first and second laws, ensuring that entropy increases with matter flux and perturbations under prescribed energy conditions.
• The formulation integrates geometric constructs, effective field theory, and quantum corrections, addressing gauge invariance and higher-derivative modifications.

Dynamical black hole entropy generalizes the concept of black hole entropy, traditionally tied to stationary (equilibrium) horizons, to settings where horizons evolve dynamically due to time-dependent processes or perturbations. This notion has undergone substantial refinement over the last decade, producing a robust theoretical framework that unifies thermodynamic laws, geometric constructions, effective field theory, and quantum/statistical interpretations across classical and semiclassical gravity.

1. Fundamental Notion and Formulation

The entropy of a dynamical black hole is a functional assigned to spacetime cross-sections of an evolving (non-stationary) event or apparent horizon. The canonical prescription for stationary black holes is given by the Noether–charge (Wald) entropy, essentially integrating a local density (constructed from the action’s diffeomorphism-invariant Lagrangian) over the bifurcation surface. For dynamical (non-stationary) contexts, recent developments (notably by Hollands, Wald, and Zhang) have established that the correct dynamical entropy must be

$$S_{\text{dyn}}[\mathcal{C}] = \frac{A[\mathcal{C}]}{4} - \frac{1}{4} \int_{\mathcal{C}} V\,\theta$$

for general relativity (GR), where $\mathcal{C}$ is a horizon cross-section, $A[\mathcal{C}]$ its area, $V$ an affine parameter along horizon generators (with $V=0$ at the bifurcation surface), and $\theta$ the future-directed expansion scalar of the null generators (Hollands et al., 1 Feb 2024). This formula generalizes as

$S_{\text{dyn}} = (1 - v\partial_v) S_{\text{BH}}$

in GR, or

$S_{\text{dyn}} = (1 - v\partial_v) S_{\text{Wall}}$

for higher-derivative or $f(\text{Riemann})$ theories, with $S_{\text{Wall}}$ the relevant (Iyer–Wald or Wall) entropy evaluated on each cross-section (Visser et al., 11 Mar 2024, Kong et al., 1 Dec 2024).

When evaluated on the generalized apparent horizon (a marginally outer trapped surface), this dynamical entropy reduces—at first order in perturbations—to the standard Wald entropy on that surface (Furugori et al., 18 Jul 2025, Jia et al., 6 Sep 2025).

2. Geometric and Thermodynamic Properties

Dynamical entropy is distinguished by several critical properties:

• Locally Evaluated: Whereas the event horizon is teleologically defined, $S_{\text{dyn}}$ can be evaluated on quasi-local, marginally trapped surfaces (apparent horizons, E-MOTS), which depend only on local geometry and matter flux.
• Physical Process First Law: For non-stationary, but near-equilibrium, evolution, $S_{\text{dyn}}$ satisfies a local form of the first law:

$$\frac{\kappa}{2\pi} \Delta \delta S_{\text{dyn}} = \Delta \delta E$$

where $\kappa$ is the surface gravity, and $\Delta \delta E$ is the matter energy flux across the horizon (Hollands et al., 1 Feb 2024, Visser et al., 11 Mar 2024, Kong et al., 1 Dec 2024).

• Second Law in Dynamical Regimes: For first-order perturbations driven by matter obeying the null energy condition, $S_{\text{dyn}}$ is non-decreasing. For vacuum perturbations, the second law is realized at second order and is controlled by a modified canonical energy (which is positive in GR and certain higher-curvature theories) (Hollands et al., 1 Feb 2024, Kong et al., 1 Dec 2024).
• Matching to Wald Entropy: In stationary regimes or on the generalized apparent horizon, $S_{\text{dyn}}$ coincides with the traditional (Iyer–Wald/Wall) entropy (Furugori et al., 18 Jul 2025, Jia et al., 6 Sep 2025).
• Gauge Invariance and Effective Field Theory: In effective field theory (EFT) up to six derivatives in four spacetime dimensions, the definition of $S_{\text{dyn}}$ is gauge invariant under rescalings of the null generator's affine parameter, but this can break down at eight-derivative order unless further improvements are introduced (Davies et al., 2022).

3. Horizon Localization and Entropic Marginally Trapped Surfaces

A crucial insight is that the relevant cross-section for dynamical entropy, especially for first-law and second-law statements, is not the event horizon but rather an "entropic marginally outer trapped surface" (E-MOTS). The E-MOTS is the unique codimension-2 surface where the entropic expansion,

$$\Theta_+ = \frac{1}{s}\,\partial_\mu\left( n_+^\mu s \right),$$

vanishes, with $s$ the entropy density (Wald entropy density for higher-curvature theories) and $n_+^\mu$ the outer future-directed null normal (Furugori et al., 18 Jul 2025).

For Einstein gravity, the entropic expansion reduces to the usual area expansion; for generalized theories, terms from $f(R)$ (or other higher-curvature invariants) modify this expansion accordingly. Calculations show that $S_{\text{dyn}}$ defined on the E-MOTS coincides perturbatively with the Wall entropy on that surface. Thus, quasi-local entropy balance is naturally realized on E-MOTS, not teleological event horizons (Furugori et al., 18 Jul 2025, Jia et al., 6 Sep 2025).

4. Entanglement, Quantum Corrections, and the Generalized Second Law

A major development is the precise connection between dynamical black hole entropy and gravitational (entanglement) entropy:

• Replica Trick and Apparent Horizon: Direct computation using the gravitational replica trick shows that entanglement entropy computed with the apparent (generalized) horizon as entangling surface reproduces the correct $S_{\text{dyn}}$ and satisfies the physical process version of the first law. By contrast, the event horizon surface gives the area law but misses crucial time-dependent corrections (Jia et al., 6 Sep 2025).
• Relative Entropy and Conservation Laws: Algebraic quantum field theory calculations relate the relative entropy of quantum fields (compared to vacuum) outside the horizon to the increase in geometric entropy, with the sum obeying a conservation law of the form

$$\frac{d}{dv}\left[ S_{\text{rel}}(\omega | \omega_\psi) + \frac{1}{4}A(v) \right] = \text{matter flux}$$

for both static and dynamical cases; in the dynamical case, an extra work term appears reflecting the energy flow into the system (D'Angelo, 2023, D'Angelo, 2021).

• Generalized Second Law (GSL) and Modified von Neumann Entropy: To ensure monotonicity of total entropy (geometric + matter) in dynamical situations, a "modified von Neumann entropy" is introduced,

$\tilde{S}_{vN} = S_{vN} - v\,\frac{d S_{vN}}{dv}$

with $v$ the affine parameter along the horizon. The generalized second law becomes

$$\frac{d}{dv}(S_{\text{dyn}} + \tilde{S}_{vN}) \geq 0$$

provided the quantum null energy condition holds (Hollands et al., 1 Feb 2024, Jia et al., 6 Sep 2025).

5. Effective Field Theory, Higher Derivatives, and Gauge Structure

A systematic method for constructing $S_{\text{dyn}}$ in gravitational EFTs—with Lagrangians organized by increasing numbers of derivatives—was developed by Hollands, Kovács, and Reall and further refined to guarantee nonperturbative second-law behavior within the EFT’s regime of applicability (Davies et al., 2023, Davies et al., 2022). The procedures involve:

• Order-by-Order Construction: At each derivative order, the entropy density includes terms built from local invariants with prescribed "boost weight" behavior (i.e., homogeneous transformation under affine parameter rescaling) to maintain gauge-invariance and ensure physical consistency to quadratic order in perturbations.
• Unambiguous Definitions up to Six Derivatives: For 4d vacuum gravity, explicit formulas involve adding "improvement" terms to the Iyer–Wald–Wall entropy so that the resulting entropy is gauge invariant and obeys the (quadratic) second law (Davies et al., 2022).
• EGB and f(R) Theories: In theories with only four-derivative (Gauss–Bonnet-type) corrections, the improved entropy reduces to the standard Iyer–Wald result, and retains gauge invariance (Davies et al., 2022). For $f(R)$ and scalar-tensor gravity, conformal transformation to Einstein frame and back yields explicit formulas for $S_{\text{dyn}}$ that match those obtained from the Noether charge method, and both the physical process and comparison first laws are satisfied (Kong et al., 1 Dec 2024).

6. Examples Beyond Einstein Gravity and Pathological Scenarios

• BHT Massive Gravity and Critical Gravity: In 2+1 dimensions, BHT massive gravity provides Vaidya-type dynamical black holes where area can decrease under positive-energy injection, but the Wald–Kodama entropy always increases, indicating the entropy law’s greater robustness than the area law (Maeda, 2010).
• Critical Gravity Pathologies: In critical higher-curvature theories in 4d, the Wald–Kodama dynamical entropy can become negative and decrease when matter satisfying the null energy condition is injected, exposing a non-perturbative pathology absent in general relativity (Maeda et al., 2018).
• Shape Dynamics: When diffeomorphism invariance is replaced by spatial diffeomorphism and spatial Weyl invariance, as in shape dynamics, the area law is restored as entropy (after restricting to area-preserving Weyl transformations at the horizon) (Herczeg et al., 2014).

7. Synthesis and Outlook

The contemporary framework for dynamical black hole entropy, codified in the Hollands–Wald–Zhang prescription and its extensions, is characterized by the following features:

• Locality: $S_{\text{dyn}}$ is defined on locally accessible horizons (apparent or entropic trapped surfaces).
• Thermodynamic Consistency: It satisfies the physical process and first/second laws, both for matter and vacuum perturbations, within the regime of validity (linear/quadratic order, and for EFT truncations with sufficiently low derivative order).
• Gauge Structure: For up to 6-derivative EFT corrections the entropy is gauge-invariant; at higher order, more refined corrections may be needed (Davies et al., 2022, Davies et al., 2023).
• Quantum Compatibility: The inclusion of quantum field relative entropy and work terms, and the introduction of the modified von Neumann entropy, enables the formulation of a local version of the generalized second law and its deep connection with the quantum null energy condition (Hollands et al., 1 Feb 2024, Jia et al., 6 Sep 2025, D'Angelo, 2023).
• Holographic and Entanglement Interpretations: Via the replica trick, gravitational entropy across the apparent horizon matches $S_{\text{dyn}}$, further supporting the area/entanglement connection and ratifying the identification of the apparent horizon as the correct thermodynamic/entanglement boundary in dynamical spacetimes (Jia et al., 6 Sep 2025).

Research continues to address remaining challenges: higher-order corrections, uniqueness, field redefinition ambiguities, and the non-perturbative extension of the second law. Ongoing work investigates entropy currents, local versions of the second law, and semiclassical/quantum effects in cosmological horizons and black holes far from equilibrium.

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Summary Table: Key Dynamical Black Hole Entropy Prescriptions and Principles

Setting / Theory	Entropy Formula (schematic)	Satisfies Second Law
Einstein Gravity	$S_{\rm dyn} = (1 - v\partial_v)\frac{A}{4}$	Yes, to second order
$f(R)$ Gravity	$S_{\rm dyn} = (1 - v\partial_v) S_{\rm Wald}$	Yes, to second order
EFT with up to 6 derivatives	$$S_{\rm iwwhkr} = S_{\rm IWW} + \varsigma(\text{allowed})$$	Yes, up to quadratic order
Critical Gravity ($\beta=0$ branch)	$S_{\rm WK} = (1+8aA)A_h/(4G)$	Can be violated, negative
BHT massive gravity (2+1d)	$S_{\rm WK} = f(\text{null dust})$	Robust, even as area decreases

This encyclopedic synthesis captures the principal structure and results underlying state-of-the-art formulations and applications of dynamical black hole entropy.