Unifying topological, geometric, and complex classifications of black hole thermodynamics
Shi-Hao Zhang, Shao-Wen Wei, Jing-Fei Zhang, Xin Zhang
TLDR
This paper unifies three distinct classification schemes for black hole thermodynamics using the critical point structure of temperature curves.
Key contributions
- Proves equivalence of geometric, topological, and complex classification schemes for black hole thermodynamics.
- Establishes two dictionaries linking thermal stability to temperature monotonicity and state count to Riemann surface foliation.
- Demonstrates that the number of temperature curve extrema unifies all three classification frameworks.
Why it matters
Black hole thermodynamics has been analyzed using disparate classification methods. This paper provides a unified framework, simplifying analysis and revealing the underlying critical point structure. This unification is crucial for understanding complex black holes and their phase transitions.
Original 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.
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