黑洞相变的统一几何框架：从多值性到分类
Universal geometric framework for black hole phase transitions: from multivaluedness to classification
AI总结：
通过构建结合实分析与覆盖空间理论的几何框架，揭示了黑洞一级相变中热力学、动力学和几何量同步多值行为的普遍数学机制，并基于温度曲线的极值点提出黑洞分类方案。
摘要：
近期研究揭示了黑洞一级相变过程中热力学、动力学和几何量存在同步的多值行为，使得从不同角度进行诊断成为可能，但其根本起源尚不清楚。通过构建一个整合实分析与覆盖空间理论的统一几何框架，我们揭示了这一现象背后的普遍数学机制。我们证明，这种多值性源于温度函数 $T(r_+)$ 中的两个非退化临界点，其中 $r_+$ 是视界半径，这些临界点将参数空间折叠成一个三叶覆盖结构。作为直接应用，我们提出黑洞经历一级相变当且仅当其 $T(r_+)$ 曲线具有两个极值点。据此，我们建立了黑洞的分类方案，记为 $A1$、$A2$ 和 $B$。该方案为基于全局拓扑不变量的分类提供了补充视角。我们的工作为通过多值性诊断相变提供了理论基础，并建立了黑洞热力学、混沌动力学和一级相变过程中时空结构的统一几何视角。
英文摘要：
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.
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