Action and entropy of black holes in spacetimes with a cosmological constant

TL;DR

This paper calculates the actions and entropies of Reissner-Nordström-de Sitter black holes in Euclidean spacetime, revealing how entropy varies with horizon configurations and relating it to topological Euler numbers.

Contribution

It provides explicit formulas for black hole and cosmological horizon entropies and explores their topological connections in spacetimes with a cosmological constant.

Findings

Entropy equals quarter the sum of horizon areas when temperatures are equal.

Entropy reduces to quarter the cosmological horizon area when horizons coincide.

Entropy vanishes when all three horizons coincide.

Abstract

In the Euclidean path integral approach, we calculate the actions and the entropies for the Reissner-Nordstr\"om-de Sitter solutions. When the temperatures of black hole and cosmological horizons are equal, the entropy is the sum of one-quarter areas of black hole and cosmological horizons; when the inner and outer black hole horizons coincide, the entropy is only one-quarter area of cosmological horizon; and the entropy vanishes when the two black hole horizons and cosmological horizon coincide. We also calculate the Euler numbers of the corresponding Euclidean manifolds, and discuss the relationship between the entropy of instanton and the Euler number.