Thermodynamic geometry as the missing link: toward a unified framework for black hole first-order phase transitions
Abstract
Black hole first-order phase transitions have been described by several seemingly independent frameworks, including local geometry, global topology, complex analysis, and thermodynamic geometry. While the first three have been unified, thermodynamic geometry has remained outside. We prove that the divergence points of the normalized Ruppeiner curvature scalar RN coincide exactly with the solutions of T'(rh)=0, where rh is the horizon radius. These solutions include extremal points (spinodal points) and stationary inflection points (thermodynamic critical points). Thus, the divergence of RN is a necessary but not sufficient condition for a first-order phase transition. This clarifies the mathematical origin of curvature divergence and why thermodynamic geometry can reliably indicate but not alone confirm phase transitions. Using the local geometric framework as a central framework, we incorporate Ruppeiner geometry into this unified picture; a similar analysis also applies to Weinhold geometry. Consequently, the four frameworks are unified within a single structure based on the local folding of the temperature function. This advances our understanding of the mathematical structure of black hole first-order phase transitions and provides clues for possible extensions to other types of phase transitions.
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BibTeX
@misc{zhang2026thermodynamic,
title = {{Thermodynamic geometry as the missing link: toward a unified framework for black hole first-order phase transitions}},
author = {Shi-Hao Zhang and Jing-Fei Zhang and Xin Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31144},
eprint = {2609.31144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31144},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Thermodynamic geometry as the missing link: toward a unified framework for black hole first-order phase transitions
AU - Shi-Hao Zhang
AU - Jing-Fei Zhang
AU - Xin Zhang
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31144
UR - https://arxiv.org/abs/2609.31144
AB - Black hole first-order phase transitions have been described by several seemingly independent frameworks, including local geometry, global topology, complex analysis, and thermodynamic geometry. While the first three have been unified, thermodynamic geometry has remained outside. We prove that the divergence points of the normalized Ruppeiner curvature scalar R_N coincide exactly with the solutions of T'(r_h)=0, where r_h is the horizon radius. These solutions include extremal points (spinodal points) and stationary inflection points (thermodynamic critical points). Thus, the divergence of R_N is a necessary but not sufficient condition for a first-order phase transition. This clarifies the mathematical origin of curvature divergence and why thermodynamic geometry can reliably indicate but not alone confirm phase transitions. Using the local geometric framework as a central framework, we incorporate Ruppeiner geometry into this unified picture; a similar analysis also applies to Weinhold geometry. Consequently, the four frameworks are unified within a single structure based on the local folding of the temperature function. This advances our understanding of the mathematical structure of black hole first-order phase transitions and provides clues for possible extensions to other types of phase transitions.
ER -