A Proof of the Generalized Second Law for Two-Dimensional Black Holes

We investigate the generalized second law for two-dimensional black holes in equilibrium states (Hartle-Hawking) and in non-equilibrium states (Unruh) with Hawking radiation surrounding the black holes {\it without} assuming the quasi-stationarity to an evolution of the black holes. Using the dynamic first law proposed by Hayward and the covariant formulation of black hole thermodynamics, we obtain a very simple expression for change of total entropy in terms of covariant thermodynamic variables, which is valid for non-equilibrium states as well as for equilibrium states up to leading order. Evaluating the change of total entropy, we show that the generalized second law holds for any two-dimensional static black holes and for non-static two-dimensional black holes of the S-wave sector in four-dimensional general relativity