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 $p_t$, 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.

Figures (3)

• Figure 1: The free energy of first-order phase transition and interior structure of the scalar field with $\lambda=-0.3$, $P=0.03~(L=1.994)$. The top panel is free energy of the first-order phase transition. The dashed line represents the normal solution. The blue and green solid lines correspond to scalarized black hole solutions 1 and 2, respectively. The red solid line indicates the unstable black hole solution. The bottom panel is the behavior of the scalar field inside the black hole. Curves of different colors correspond to selected points on the respective solutions in the top panel.
• Figure 2: The dependence of the kasner exponents $p_t$ and a function of temperature $T$. The top panel shows $p_t(T)$ in the first-order phase transition region, and the bottom panel shows $p_t(T)$ in the supercritical region.
• Figure 3: The phase diagram of the Kasner exponent $p_t$ as a function of temperature and pressure. The red line indicates the critical point from the normal phase to the scalarized phase. The black solid line denotes the first-order phase transition points, while the black dashed lines represent the spinodal lines of the first-order phase transition. The orange dashed line mark the points where the slope of $p_t$ changes abruptly, and the green dashed line corresponds to the Widom line, defined as $(\partial^2 G/\partial T^2)/(\partial G/\partial T)$.