---
title: Dynamical Black Hole Entropy
url: https://www.emergentmind.com/topics/dynamical-black-hole-entropy
type: topic
---

# Dynamical Black Hole Entropy

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 [2402.00818]. 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 [2403.07140, 2412.00647].

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 [2507.14105, 2509.05700].

## 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 [2402.00818, 2403.07140, 2412.00647].

- **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) [2402.00818, 2412.00647].

- **Matching to Wald Entropy**: In stationary regimes or on the generalized apparent horizon, $S_{\text{dyn}}$ coincides with the traditional (Iyer–Wald/Wall) entropy [2507.14105, 2509.05700].

- **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 [2212.09777].

## 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 [2507.14105].

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 [2507.14105, 2509.05700].

## 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 [2509.05700].

- **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 [2309.01548, 2105.04303].

- **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 [2402.00818, 2509.05700].

## 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 [2312.07659, 2212.09777]. 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 [2212.09777].

- **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 [2212.09777]. 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 [2412.00647].

## 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 [1012.5048].

- **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 [1805.00026].

- **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) [1410.4248].

## 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 [2212.09777, 2312.07659].
- 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 [2402.00818, 2509.05700, 2309.01548].
- 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 [2509.05700].

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.

Source: https://www.emergentmind.com/topics/dynamical-black-hole-entropy