Generalized Thermodynamics and Phase Transitions of Vaidya-adS Black Holes

This work investigates the thermodynamics of non-stationary anti-de Sitter (adS) black holes and their gauge-gravity duals. To model a non-equilibrium process, we employ a time-dependent Vaidya-adS metric to describe the collapse of a null fluid and the subsequent thermalization of the dual field theory. We introduce an effective version of free energy as a quasilocal probe to be used in a dynamical setup. By analyzing how this quantity evolves, we show that a first-order phase transition occurs when the final black-hole mass exceeds the limit established by the Hawking–Page temperature. Furthermore, we demonstrate that the final state only approaches the thermal behavior of the N=4 supersymmetric Yang–Mills theory for mass states much larger than the adS radius. Finally, we compute the boundary energy–momentum tensor using holographic renormalization and compare the resulting energy density with the strong-coupling limit. This demonstrates that the dual description provides a consistent framework for both the equilibrium and non-equilibrium thermodynamics of strongly coupled gauge theories.