Deriving the First Law of Black Hole Thermodynamics without Entanglement

TL;DR

This paper explores how the bulk first law of black hole thermodynamics can be understood without entanglement, using coarse-graining in the boundary theory and analyzing geons to relate Wald entropy to a new coarse-grained entropy.

Contribution

It demonstrates that the bulk first law can be interpreted through a specific coarse-graining of the boundary theory, extending the understanding beyond entanglement-based interpretations.

Findings

Wald entropy matches the one-point entropy for small perturbations in geon spacetimes.

The one-point entropy is shown to be equal to the causal holographic information in a broader regime.

The approach provides a new perspective on the bulk first law without relying on entanglement.

Abstract

In AdS/CFT, how is the bulk first law realized in the boundary CFT? Recently, Faulkner et al. showed that in certain holographic contexts, the bulk first law has a precise microscopic interpretation as a first law of entanglement entropy in the boundary theory. However, the bulk can also satisfy a first law when the boundary density matrix is pure, i.e. in the absence of entanglement with other degrees of freedom. In this note we argue that the bulk first law should generally be understood in terms of a particular coarse-graining of the boundary theory. We use geons, or single-exterior black holes, as a testing ground for this idea. Our main result is that for a class of small perturbations to these spacetimes the Wald entropy agrees to first order with the one-point entropy, a coarse-grained entropy recently proposed by Kelly and Wall. This result also extends the regime over which the one-point entropy is known to be equal to the causal holographic information of Hubeny and Rangamani.