Entanglement Entropy and Thermodynamics of Dynamical Black Holes (2509.05700v1)

Abstract: We explore the thermodynamic and entanglement properties of dynamical black holes based on the recently proposed dynamical black hole entropy by Hollands-Wald-Zhang. We first provide a direct proof that, under first order perturbations, the dynamical black hole entropy in any $f(R)$ theory equals the Wald entropy evaluated on the generalized apparent horizon. Then, we compute the gravitational entanglement entropy explicitly using both the event horizon and the apparent horizon as the entangling surfaces, and show that only the apparent horizon prescription reproduces the correct dynamical black hole entropy satisfying the physical process first law. Furthermore, we reinterpret the generalized second law by identifying the modified von Neumann entropy as the matter entanglement across the (generalized) apparent horizon. This allows us to express the total entropy as the renormalized generalized entropy evaluated on this surface for both Einstein's gravity and $f(R)$ gravity.

• The paper demonstrates that the correct identification of black hole entropy in dynamic settings is achieved by using the apparent horizon rather than the event horizon.
• Methodologically, Gaussian null coordinates and perturbed Raychaudhuri equations are employed to derive a first law of dynamical black hole entropy in both Einstein and f(R) gravity.
• The study establishes that entanglement entropy, calculated via the replica trick, aligns with the dynamical entropy on the apparent horizon, ensuring compliance with the generalized second law.

Entanglement Entropy and Thermodynamics of Dynamical Black Holes

Introduction and Motivation

This work addresses the thermodynamic and entanglement properties of dynamical black holes, focusing on the correct identification of black hole entropy in non-stationary settings and its relation to entanglement entropy. The analysis is performed in both Einstein gravity and $f(R)$ gravity, leveraging the recently proposed dynamical black hole entropy by Hollands, Wald, and Zhang. The central question is whether the event horizon or the apparent horizon provides the appropriate surface for defining black hole entropy and entanglement entropy in dynamical (non-stationary) spacetimes.

Dynamical Black Hole Entropy: Geometric and Thermodynamic Aspects

Geometric Setup and Gauge Fixing

The analysis employs Gaussian null coordinates (GNC) to describe the geometry near the horizon of a perturbed black hole. The event horizon is taken as a bifurcate Killing horizon in the stationary background, and perturbations are introduced via infalling matter. The apparent horizon is defined as the locus where the outgoing null expansion vanishes, and its location is perturbed away from the event horizon under non-stationary dynamics.

Raychaudhuri Equation and the Physical Process First Law

The Raychaudhuri equation is used to relate the change in the expansion of null geodesics to the stress-energy flux across the horizon. For Einstein gravity, integrating the perturbed Raychaudhuri equation yields a "physical process" first law for arbitrary cross-sections of the horizon: $$\frac{\kappa}{2\pi} \Delta \delta S_{\text{dyn}} = \Delta \delta M - \Omega_{\mathcal{H}} \Delta \delta J$$ where $S_{\text{dyn}}$ is the dynamical black hole entropy, $\kappa$ is the surface gravity, and the right-hand side encodes the matter energy and angular momentum fluxes.

Identification of Entropy with Apparent Horizon Area

A key result is that, to first order in perturbations, the dynamical black hole entropy in Einstein gravity is given by the area of the apparent horizon: $S_{\text{dyn}} = \frac{A[\mathcal{T}(v)]}{4G}$ where $\mathcal{T}(v)$ is a cross-section of the apparent horizon at affine parameter $v$. This result generalizes to $f(R)$ gravity, where the entropy is given by the Wald entropy evaluated on the generalized apparent horizon: $$S_{\text{dyn}} = S_{\text{Wald}}[\tilde{\mathcal{T}}(v)] = \frac{1}{4G} \int_{\tilde{\mathcal{T}}(v)} f'(R)$$ This equivalence is established via explicit calculation in GNC, showing that the dynamical correction to the Wald entropy matches the entropy of the generalized apparent horizon.

Gravitational Entanglement Entropy: Event Horizon vs. Apparent Horizon

Replica Trick and Path Integral Approach

The entanglement entropy is computed using the replica trick in the Euclidean path integral formalism, both for matter fields and for the gravitational sector. The calculation is performed semiclassically, focusing on the leading area law divergence.

Event Horizon as Entangling Surface

When the future event horizon is used as the entangling surface, the entanglement entropy reproduces the area law: $$S_{\text{ent}}[\mathcal{C}(v)] = \frac{A[\mathcal{C}(v)]}{4G}$$ for Einstein gravity, and

$$S_{\text{ent}}[\mathcal{C}(v)] = S_{\text{Wald}}[\mathcal{C}(v)]$$

for $f(R)$ gravity. However, this entropy does not match the dynamical black hole entropy $S_{\text{dyn}}$ in the presence of non-stationary perturbations, nor does it satisfy the physical process first law.

Apparent Horizon as Entangling Surface

When the apparent horizon (or its generalization in $f(R)$ gravity) is used as the entangling surface, the entanglement entropy matches the dynamical black hole entropy: $S_{\text{ent}}[\mathcal{T}(v)] = S_{\text{dyn}}$ This result holds to first order in perturbations and is valid for both Einstein and $f(R)$ gravity. The calculation demonstrates that the apparent horizon is the correct surface for defining entanglement entropy in dynamical black holes, as it yields an entropy that satisfies the physical process first law and the generalized second law.

Generalized Second Law and Modified von Neumann Entropy

Quantum Null Energy Condition and GSL

The quantum null energy condition (QNEC) is used to establish the generalized second law (GSL) for dynamical black holes. The analysis shows that the sum of the dynamical black hole entropy and a modified von Neumann entropy is non-decreasing: $$\frac{d}{dv} \left( \delta S_{\text{dyn}} + \delta \tilde{S}_{\text{vN}} \right) \geq 0$$ where the modified von Neumann entropy is defined as

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

This quantity is interpreted as the matter entanglement entropy across the apparent horizon.

Renormalized Generalized Entropy

The total entropy, including both gravitational and matter contributions, is identified with the renormalized generalized entropy evaluated on the apparent horizon: $$S_{\text{gen}}[\mathcal{T}(v)] = \frac{1}{4G_{\text{ren}}} A[\mathcal{T}(v)] + \tilde{S}_{\text{vN}}$$ This formulation ensures the finiteness of the total entropy and provides a consistent thermodynamic interpretation in both Einstein and $f(R)$ gravity.

Implications and Future Directions

Theoretical Implications

• The analysis demonstrates that the apparent horizon, not the event horizon, is the correct surface for defining both thermodynamic and entanglement entropy in dynamical black holes.
• The results provide a concrete prescription for the entropy entering the generalized second law in non-stationary settings, with clear identification of the relevant geometric surface.
• The equivalence between dynamical black hole entropy and entanglement entropy across the apparent horizon supports the interpretation of black hole entropy as entanglement entropy in a broad class of gravitational theories.

Extensions and Open Problems

• The extension to higher curvature theories beyond $f(R)$, such as $f(\text{Riemann})$ gravity, is a natural next step. This would require a careful treatment of extrinsic curvature corrections and potentially new definitions of generalized expansions and horizons.
• The analysis is restricted to first order perturbations; a fully non-perturbative treatment of dynamical black holes remains an open challenge, likely requiring numerical relativity and more sophisticated horizon definitions (e.g., dynamical or isolated horizons).
• The results have direct implications for holography and the AdS/CFT correspondence in dynamical spacetimes, suggesting that the apparent horizon should play a central role in the holographic entanglement entropy prescription for non-equilibrium systems.

Conclusion

This work provides a detailed and rigorous analysis of the thermodynamic and entanglement properties of dynamical black holes in both Einstein and $f(R)$ gravity. By establishing the equivalence between dynamical black hole entropy and the entanglement entropy across the apparent horizon, it clarifies the correct geometric surface for entropy calculations in non-stationary settings. The results have significant implications for the formulation of the generalized second law, the interpretation of black hole entropy as entanglement entropy, and the extension of these concepts to more general gravitational theories and holographic contexts. Future work should address higher curvature corrections, non-perturbative dynamics, and the role of apparent horizons in quantum gravity and holography.