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 nondegenerate critical points in the temperature function $$T(r_+),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where $$r_+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> is the horizon radius, which fold the parameter space into a three-sheeted covering structure. As a direct application, we propose that for the small/large black hole first-order phase transition in the canonical ensemble, a black hole undergoes such a transition if and only if its $$T(r_+)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> curve has two extrema. Accordingly, we establish a classification scheme, denoted A 1, A 2, 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.