The Entropy Function for the Black Holes of Nariai Class

TL;DR

This paper extends the entropy function formalism to Nariai class black holes, showing exact agreement with Bekenstein-Hawking entropy and revealing unique temperature properties for these extremal black holes.

Contribution

The paper introduces a novel entropy function formalism for Nariai class black holes, generalizing the extremal case to include de Sitter near-horizon geometries.

Findings

Exact match with Bekenstein-Hawking entropy for Nariai black holes.

Higher order corrections are significant only in dimensions greater than four.

The temperature of Nariai black holes should be non-zero despite extremality.

Abstract

Based on the fact that the near horizon geometry of the extremal Schwarzschild-de Sitter black holes is Nariai geometry, we define the black holes of Nariai class as the configuration whose near-horizon geometry is factorized as two dimensional de Sitter space-time and some compact topology, that is Nariai geometry. We extend the entropy function formalism to the case of the black holes of Nariai class. The conventional entropy function (for the extremal black holes) is defined as Legendre transformation of Lagrangian density, thus the `Routhian density', over two dimensional anti-de Sitter. As for the black holes of Nariai class, it is defined as {\em minus} `Routhian density' over two dimensional de Sitter space-time. We found an exact agreement of the result with Bekenstein-Hawking entropy. The higher order corrections are nontrivial only when the space-time dimension is over four, that is, $d>4$. There is a subtlety as regards the temperature of the black holes of Nariai class. We show that in order to be consistent with the near horizon geometry, the temperature should be non-vanishing despite the extremality of the black holes.