A Universal Geometric Framework for Black Hole Phase Transitions: From Multivaluedness to Classification

Abstract

Recent studies have revealed synchronized multivalued behavior in thermodynamic, dynamical, and geometric quantities during the black hole first-order phase transition, which enables a diagnosis from different perspectives, yet its fundamental origin has remained poorly understood. By constructing a unified geometric framework integrating real analysis and covering space theory, we reveal the universal mathematical mechanism behind this phenomenon. We prove that this multivaluedness originates from two non-degenerate critical points in the temperature function $T(r_+)$, where $r_+$ is the horizon radius, which fold the parameter space into a three-sheeted covering structure. As a direct application, we propose that a black hole undergoes a first-order phase transition if and only if its $T(r_+)$ curve has two extrema. Accordingly, we establish a classification scheme, denoted $A1$, $A2$, and $B$ for black holes. This scheme offers a complementary perspective to classifications based on global topological invariants. Our work provides a theoretical foundation for diagnosing phase transitions via multivaluedness and establishes a unified geometric perspective on black hole thermodynamics, chaotic dynamics, and spacetime structure during first-order phase transitions.

Figures (2)

• Figure 1: Thermodynamic and geometric signatures of the first-order phase transition in RN-AdS black holes. Upper panel: Free energy $\tilde{F}$ versus temperature $\tilde{T}$ for RN-AdS black holes. Lower panel: Gaussian curvature $K$ of unstable null orbits versus temperature $\tilde{T}$ for RN-AdS black holes. $\tilde{Q}=\frac{1}{8.66},~\tilde{Q}_c=\frac{1}{6},~\tilde{Q}<\tilde{Q}_c$. The synchronized multivalued behavior of $K$ in the spinodal region $\tilde{T} \in (\tilde{T}_{1},\, \tilde{T}_{2})$ corresponds to the swallowtail structure in the free energy, with the phase transition occurring at $\tilde{T}_p$.
• Figure 2: Temperature curves $T(r_+)$ for the RN‚ÄìAdS black hole with and without a first‚Äëorder phase transition. Upper panel: For $Q=(\ell /8.66)<Q_c$, a first‚Äëorder phase transition occurs, and the $T(r_+)$ curve exhibits two local extrema ($T_1$ and $T_2$), signaling multivaluedness. Lower panel: For $Q>Q_c$ ($Q_1=\ell /5.9,~Q_2=\ell /5,~Q_3=\ell /4.5,~Q_4=\ell /4$ with $\ell=8.66$), the curve shows no local extremum, indicating the absence of a first‚Äëorder phase transition.