Geometrothermodynamics for Black holes and de Sitter Space

TL;DR

This paper develops a geometric method to derive thermodynamic quantities from Einstein solutions, applying it to black holes and de Sitter space, confirming the area theorem and unifying thermodynamics with general relativity.

Contribution

A new geometric approach to extract black hole and de Sitter thermodynamics from Einstein solutions, aligning with Wald's method and confirming key theorems.

Findings

Method successfully applied to Schwarzschild, Kerr, and Kerr-Newman black holes.

Results consistent with previous independent methods.

Supports the validity of the black hole area theorem.

Abstract

In this report, a general method to extract thermodynamic quantities from solutions of the Einstein equation is developed. In 1994, Wald established that the entropy of a black hole could be identified as a Noether charge associated with a Killing vector of a global space-time (pseudo-Riemann) manifold. We reconstruct Wald's method using geometrical language, e.g$.$, via differential forms defined on the local space-time (Minkowski) manifold. Concurrently, the abstract thermodynamics are also reconstructed using geometrical terminology, which is parallel to general relativity. The correspondence between the thermodynamics and general relativity can be seen clearly by comparing the two expressions. This comparison requires a modification of Wald's method. The new method is applied to Schwarzschild, Kerr, and Kerr--Newman black holes and de Sitter space. The results are consistent with previous results obtained using various independent methods. This strongly supports the validity of the area theorem for black holes.