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arXiv · 2505.16309

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

C. Fairoos·Chiranjeeb Singha
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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.

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C. Fairoos, Chiranjeeb Singha. 2025-05-22. From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited. https://arxiv.org/abs/2505.16309

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