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 with 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 the second law of thermodynamics.
Highlights
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 with 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
Summary
We prove that the area of the black hole’s apparent horizon is precisely the entropy associated with coarse graining over the information in its interior, subject to knowing the exterior geometry. One natural notion of entropy is the von Neumann entropy: S1⁄2ρ 1⁄4 −trðρ ln ρÞ; ð1Þ where ρ is the density matrix of a quantum system This quantity is conserved under unitary time evolution, in apparent tension with the second law. In holographic models of quantum gravity, a black hole is dual to some boundary state ρ whose von Neumann entropy S1⁄2ρ can be computed from a compact extremal (HRT) surface in the bulk, as conjectured in Refs.
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