Coarse-graining black holes out of equilibrium with boundary observables on time slice

In black hole thermodynamics, defining coarse-grained entropy for dynamical black holes has long been a challenge, and various proposals, such as generalized entropy, have been explored. Guided by the AdS/CFT, we introduce a new definition of coarse-grained entropy for a dynamical black hole in Lorentzian Einstein gravity. On each time slice, this entropy is defined as the horizon area of an auxiliary Euclidean black hole that shares the same mass, (angular) momenta, and asymptotic normalizable matter modes with the original Lorentzian solution. The entropy is shown to satisfy a generalized first law within Einstein theory and, through holography, the second law as well. This second law corresponds to the positivity of the relative entropy in the CFT. Furthermore, by applying this thermodynamics to several Vaidya models in AdS and flat spacetime, we discover a connection between the second law and the null energy condition.