Dynamical Entropy Is a Noether Charge

Abstract

Black hole thermodynamics for generic dynamical, non-equilibrium regimes remains a fundamental challenge. We establish dynamical entropy as the Noether charge associated with a generic evolving null surface subject to Dirichlet boundary conditions. We specify the symmetry generator associated with the dynamical entropy, which is a null vector on the null surface, upon requiring physically motivated geometric conditions that yield a notion of ``dynamical zeroth law.'' We prove that this Noether charge density satisfies the second law of thermodynamics strictly at each instant in time, bypassing the teleological final conditions traditionally required by event horizons. Thus, we extend and generalize the notion of dynamical entropy introduced in \cite{Hollands:2024vbe}, in some different ways: We do not impose background stationarity; our dynamical entropy and the associated second law are local in time and work for generic dynamical gravitational systems.