Generalized gravitational entropy

TL;DR

This paper generalizes the black hole entropy formula to Euclidean gravity solutions with non-contractible circles, explaining the Ryu-Takayanagi minimal surface prescription for entanglement entropy in holography.

Contribution

It extends the minimal surface entropy formula to a broader class of Euclidean solutions without requiring a Killing vector, providing a theoretical foundation for the Ryu-Takayanagi conjecture.

Findings

Entropy given by minimal surface area in generalized solutions

Provides theoretical explanation for Ryu-Takayanagi formula

Connects Euclidean gravity solutions to quantum entanglement measures

Abstract

We consider classical Euclidean gravity solutions with a boundary. The boundary contains a non-contractible circle. These solutions can be interpreted as computing the trace of a density matrix in the full quantum gravity theory, in the classical approximation. When the circle is contractible in the bulk, we argue that the entropy of this density matrix is given by the area of a minimal surface. This is a generalization of the usual black hole entropy formula to euclidean solutions without a Killing vector. A particular example of this set up appears in the computation of the entanglement entropy of a subregion of a field theory with a gravity dual. In this context, the minimal area prescription was proposed by Ryu and Takayanagi. Our arguments explain their conjecture.