Interior geometry of black holes as a probe of first-order phase transition

Abstract Traditional diagnostics of black hole phase transitions rely on thermodynamic quantities defined at the event horizon or asymptotic boundary. Here, we demonstrate that the near-singularity geometry offers a sharp, independent probe of both first-order phase transitions and supercritical crossover. For scalarized AdS black holes exhibiting a first-order phase transition, the Kasner exponent $p_t$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mip</mml:mi> mml:mit</mml:mi> </mml:msub> </mml:math> , which characterizes the approach to the singularity, undergoes a dramatic transformation. On one side of the transition, $p_t$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mip</mml:mi> mml:mit</mml:mi> </mml:msub> </mml:math> oscillates strongly with temperature, reflecting violent interior dynamics. On the other side, it becomes a smooth, monotonically varying function. These two distinct behaviors converge as the critical point is approached. Beyond the critical point, in the supercritical region, $p_t(T)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mip</mml:mi> mml:mit</mml:mi> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:miT</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> develops a distinct extremum, defining a “Kasner crossover line” that is entirely independent of traditional thermodynamic (Widom line) or dynamic (Frenkel line) criteria. Our work establishes the near-singularity geometry of scalarized black holes as a novel class of diagnostics for phase transitions, revealing that a change in the macroscopic thermodynamic state fundamentally reshapes the deepest interior structure of spacetime.