# Decoding the Apparent Horizon: A Coarse-Grained Holographic Entropy

## TL;DR

This paper establishes that the apparent horizon's area in a black hole corresponds to the maximum coarse-grained holographic entropy consistent with external measurements, linking geometric and informational aspects in holography.

## Contribution

It proves the apparent horizon's area equals the maximum holographic entropy compatible with exterior data, clarifying its thermodynamic and informational significance.

## Key findings

- Apparent horizon area equals maximum coarse-grained entropy.
- Identifies the boundary dual to this entropy.
- Shows the entropy obeys a Second Law of Thermodynamics.

## 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.