Unifying topological, geometric, and complex classifications of black hole thermodynamics

TL;DR

This paper unifies three different classification schemes of black hole thermodynamics—geometric, topological, and complex analysis—by linking their core concepts through the critical point structure of temperature functions.

Contribution

It establishes the equivalence of these schemes via two dictionaries, connecting stability, state count, and foliation number, simplifying analysis and enabling exploration of complex black holes.

Findings

Number of temperature extrema determines classification across schemes.

Counting extrema yields topological invariants and phase transition insights.

Unification simplifies thermodynamic analysis and aids in studying complex black holes.

Abstract

Black hole thermodynamics has recently witnessed three distinct classification schemes: based on local geometric properties of the temperature function, global topological invariants, and Riemann surface foliations in the complex plane. We show that these schemes are equivalent in the real domain via two dictionaries: one linking thermal stability to the monotonicity of the temperature curve, and the other connecting the number of black hole states to the foliation number of a Riemann surface. The number of extremal points of the temperature curve determines the classification in all three frameworks, tracing this unification to the critical point structure of the black hole solution space. As an illustration, several black holes demonstrate how counting extrema yields topological invariants and phase transition information. This unified framework simplifies black hole thermodynamic analysis and provides a foundation for exploring more complex black holes.