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 $R_N$ 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.