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

TL;DR

This paper shows that the interior geometry near black hole singularities can serve as a new probe for phase transitions, revealing changes in spacetime structure independent of traditional thermodynamic methods.

Contribution

It introduces the Kasner exponent as a novel interior diagnostic for black hole phase transitions, including first-order and supercritical crossover regimes.

Findings

Kasner exponent $p_t$ oscillates with temperature across the transition

$p_t$ becomes smooth and monotonic away from the transition

A Kasner crossover line is identified in the supercritical region

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 $p_t$, which characterizes the approach to the singularity, undergoes a dramatic transformation. On one side of the transition, $p_t$ 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, $p_t(T)$ 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 black hole singularity 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.