Entropy of a Black Hole with Distinct Surface Gravities

TL;DR

This paper derives a universal formula for black hole entropy with multiple horizons using Euclidean quantum gravity, and explores its implications in Lovelock gravity and black hole creation.

Contribution

It proves a universal entropy formula for nonrotating black holes with multiple horizons in Euclidean quantum gravity.

Findings

Entropy equals one quarter the sum of Euler characteristic times horizon area.

The formula applies to Lovelock gravity and black hole quantum creation.

Entropy is related to the Euclidean action at the WKB level.

Abstract

In gravitational thermodynamics, the entropy of a black hole with distinct surface gravities can be evaluated in a microcanonical ensemble. At the $WKB$ level, the entropy becomes the negative of the Euclidean action of the constrained instanton, which is the seed for the black hole creation in the no-boundary universe. Using the Gauss-Bonnet theorem, we prove the quite universal formula in Euclidean quantum gravity that the entropy of a nonrotating black hole is one quarter the sum of the products of the Euler characteristics and the areas of the horizons. For Lovelock gravity, the entropy and quantum creation of a black hole are also studied.