Entanglement Entropy and Thermodynamics of Dynamical Black Holes

TL;DR

The paper addresses how to define and compute entropy for dynamical black holes in Einstein and $f(R)$ gravity. It shows that, to first order in perturbations, the dynamical entropy $S_{\text{dyn}}$ equals the Wald entropy evaluated on the generalized apparent horizon in $f(R)$ gravity and reduces to the Bekenstein–Hawking entropy of the apparent horizon in Einstein gravity. Through replica-trick calculations, it demonstrates that entanglement entropy across the apparent horizon, not the event horizon, reproduces $S_{\text{dyn}}$ and satisfies the physical-process version of the first law, while entanglement entropy across the event horizon does not. The generalized second law emerges by interpreting the modified von Neumann entropy as matter entanglement across the apparent horizon and formulating the total generalized entropy as a renormalized quantity evaluated on the (generalized) apparent horizon, with extensions discussed for AdS/CFT and higher-curvature theories.

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.

Figures (3)

• Figure 1: (a) A bifurcate Killing horizon $\mathcal{H}$ consists of the future horizon $\mathcal{H}^+$ and the past horizon $\mathcal{H}^-$, intersecting at the bifurcation surface $\mathcal{B}$. The horizon Killing field is $\xi^{a}$, while $k^a = (\partial_v)^a$ and $l^a = (\partial_u)^a$ are future-directed tangent vectors to the affinely parameterized null geodesics of $\mathcal{H}^+$ and $\mathcal{H}^-$, respectively. $\mathcal{C}(v)$ is a cross section of $\mathcal{H}^+$ at an affine time $v$. (b) The apparent horizon $\cal A$ deviates from ${\cal H}^+$ due to the perturbation. $\mathcal{T}(v)$ is a cross section of $\cal A$ at an affine time $v$.
• Figure 2: The apparent horizon deviates from the event horizon due to the non-stationary perturbation. $\delta T_{ab}$ represents the infalling matter field that causes the perturbation near the future event horizon $\mathcal{H}^+$. The entropy change between two generic cross sections $\mathcal{C}(v_2)$ and $\mathcal{C}(v_1)$ is related to the matter Killing energy flux through the physical process first law.
• Figure 3: The conical singularity at $r= 0$ is regularized by introducing the smooth function $\beta_\delta$.