A note on entropy of de Sitter black holes

TL;DR

This paper develops a formalism to compute the total entropy of de Sitter black holes with two horizons using near horizon symmetries and boundary terms, generalizing to multi-horizon spacetimes in arbitrary dimensions.

Contribution

It introduces a geometric framework for calculating total entropy in stationary axisymmetric spacetimes with multiple horizons, extending previous single-horizon approaches.

Findings

Total entropy equals the sum of horizon areas under the formalism.

The near horizon mode functions require specific restrictions for multiple horizons.

The framework applies to known solutions like Kerr-Newman-de Sitter and Plebanski-Demianski-de Sitter.

Abstract

A de Sitter black hole or a black hole spacetime endowed with a positive cosmological constant has two Killing horizons -- a black hole and a cosmological event horizon surrounding it. It is natural to expect that the total Bekenstein-Hawking entropy of such spacetimes should be the sum of the two horizons' areas. In this work we apply the recently developed formalism using the Gibbons-Hawking-York boundary term and the near horizon symmetries to derive the total entropy of such two horizon spacetimes. We construct a suitable general geometric set up for general stationary axisymmetric spacetimes with two or more than two commuting Killing vector fields in an arbitrary spacetime dimensions. This framework helps us to deal with both the horizons in an equal footing. We show that in order to obtain the total entropy of such spacetimes, the near horizon mode functions for the diffeomorphism generating vector fields have to be restricted in a certain manner, compared to the single horizon spacetimes. We next discuss specific known exact solutions belonging to the Kerr-Newman- or the Plebanski-Demianski-de Sitter families to show that they fall into the category of our general framework. We end with a sketch of further possible extensions of this work.