From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited

TL;DR

This paper revisits the entropy calculation of Reissner-Nordstr"om and Kerr black holes, highlighting differences between extremal and non-extremal cases using Euclidean methods and topological theorems.

Contribution

It clarifies the limitations of Euclidean and topological methods in determining extremal black hole entropy, emphasizing the non-uniqueness in the extremal case.

Findings

Non-extremal black holes have a well-defined entropy via Euclidean methods.

Extremal black holes lack a unique Euclidean periodicity, leading to ambiguous entropy.

Chern-Gauss-Bonnet theorem fails to constrain extremal black hole entropy.

Abstract

In this paper, we derive the entropy of Reissner-Nordstr\"om (RN) and Kerr black holes using the Hawking-Gibbons path integral method. We determine the periodicity of the Euclidean time coordinate using two approaches: first, by analyzing the near-horizon geometry, and second, by applying the Chern-Gauss-Bonnet (CGB) theorem. For non-extremal cases, both these methods yield a consistent and unique periodicity, which in turn leads to a well-defined expression for the entropy. In contrast, the extremal case exhibits a crucial difference. The absence of a conical structure in the near-horizon geometry implies that the periodicity of the Euclidean time is no longer uniquely fixed within the Hawking-Gibbons framework. The CGB theorem also fails to constrain the periodicity, as the corresponding Euler characteristic vanishes. As a result, the entropy cannot be uniquely determined using either method.