Decoding the Apparent Horizon: A Coarse-Grained Holographic Entropy

TL;DR

The paper proposes that the area of the apparent horizon provides a holographic, coarse-grained entropy for black holes formed from collapse, when exterior geometry is fixed. It proves that the outer entropy of the apparent horizon equals $Area[\mu]/(4G\hbar)$ by constructing a bulk dual with the same outer wedge and a matching extremal surface, and it identifies a boundary dual, the simple entropy, via maximization over states constrained by simple, causal experiments. Together with an argument that both bulk and boundary entropies obey a Second Law, the work extends the holographic dictionary beyond the usual HRT prescription to encompass apparent horizons and their interior information. The results offer a concrete geometric and holographic mechanism for black hole entropy increases and contribute to the broader understanding of holographic encoding of interior degrees of freedom.

Abstract

When a black hole forms from collapse in a holographic theory, the information in the black hole interior remains encoded in the boundary. We prove that the area of the black hole's apparent horizon is precisely the entropy associated to coarse graining over the information in its interior, subject to knowing the exterior geometry. This is the maximum holographic entanglement entropy that is compatible with all classical measurements conducted outside of the apparent horizon. We identify the boundary dual to this entropy and explain why it obeys a Second Law of Thermodynamics.

Figures (2)

• Figure 1: The coarse-grained spacetime dual to the state $\rho'$ with maximal $S[\rho']$ and fixed $O_{W}[\mu]$ (shaded gray). The null congruence $N_{-k}$ (red) is fired from $\mu$ towards the $-k$ direction and is stationary. The congruence $N_{-l}$, the past boundary of $O_{W}[\mu]$, is fired in the $-\ell$ direction from $\mu$. $X$ is the HRT surface of the coarse-grained spacetime. Tilded quantities represent the CPT mirror reverse.
• Figure 2: (a) We fire a null congruence $N_{-\ell}$ into the bulk from time $t=t_{i}$. The surface $\mu$ is the first cross-section of $N_{-\ell}$ with vanishing $k$ expansion. We can recover all the data in $O_{W}[\mu]$, at least when the black hole is near equilibrium, by means of a "simple experiment" performed after time $t_{i}$. (b) A spacelike holographic screen (purple) has increasing area in a spacelike direction, going from 1 to 3. The corresponding outer wedges are nested, implying that the outer entropy must increase outwards. Similarly, the simple entropy must increase with $t$ from $t_{1}$ to $t_{3}$.