Phase Transition and Critical Behavior in Gravitational Collapse

TL;DR

This work develops a thermodynamic interpretation of spherically symmetric gravitational collapse by employing the Hayward-Kodama formalism to map horizon dynamics to thermodynamic variables. The central finding is that the apparent horizon acts as a universal critical surface, with divergences in the specific heats $C_V$ and $C_P$ occurring when the Hayward-Kodama surface gravity evolves stationarily, i.e. $\dot{\kappa}_{hk}=0$, corresponding to a stationary horizon temperature and to a stationary expansion of outgoing null geodesics per the Raychaudhuri equation. The framework is applied to three geometries (LTB collapse, conformally evolving JMN naked singularity, and conformally scaled Simpson-Visser metric), deriving a common critical condition and analyzing near-horizon and near-singularity limits, including a near-horizon solution for the critical radius via a Lambert $W$-function. The results link geometric stability and thermodynamic criticality, offering a potential bridge to observable strong-field phenomena and providing a general platform for exploring phase-transition-like behavior in gravitational collapse.

Abstract

We present a thermodynamic analysis of spherically symmetric gravitational collapse. Using the Hayward-Kodama formalism, we treat a collapsing sphere as a thermodynamic system and express the surface gravity $κ_{hk}$ in terms of the geometric variables. We derive the specific heat capacities and identify a critical condition $\dotκ_{hk} = 0$ as the locus of second order phase transition during the collapse. Through specific examples, we demonstrate that the condition is independent of singular/non-singular nature of the geometry. We also find that the critical condition of phase transition is equivalent to a stationary condition of the expansion of null congruence. This establishes a direct correspondence between geometric stability and thermodynamic criticality, allowing the identification of apparent horizon as a universal critical surface in the phase-space of gravitational collapse.