Phase Transition and Critical Behavior in Gravitational Collapse

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 $\kappa_{hk}$ in terms of the geometric variables. We derive the specific heat capacities and identify a critical condition $\dot{\kappa}_{hk} = 0$ 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.