How to Prove a Differential Form of the Generalized Second Law

Abstract

A new method is given for proving the semiclassical generalized second law (GSL) of horizon thermodynamics. Unlike previous methods, this method can be used to prove that entropy increases for arbitrary slices of causal horizons, even when the matter fields falling across the horizon are rapidly changing with time. Chapter I discusses how to define the GSL, and critically reviews previous proofs in the literature. Chapter II describes the proof method in the special case of flat planar slices of Rindler horizons, assuming the existence of a valid renormalization scheme. Chapter III generalizes the proof method to arbitrary slices of semiclassical causal horizons, by the technique of restricting the fields to the horizon itself. In the case of free fields it is clear that this restriction is possible, but for interacting fields the situation is murkier. Each of the three parts has been, or will be, separately published elsewhere.