A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices

TL;DR

This work establishes a general semiclassical proof of the generalized second law (GSL) for rapidly evolving quantum fields crossing a causal horizon by introducing a horizon-restricted observable algebra A(H) that satisfies four axioms. It grounds the proof in horizon thermality (KMS) and the monotonicity of relative entropy, linking area changes to a boost-energy generator K and showing that the generalized entropy S_out + S_H increases along any horizon slice. The paper provides an explicit construction for a free scalar on the horizon, extends the framework to spinors, photons, and gravitons, and discusses the status and limits when interactions are included, including 1+1 CFTs and higher-dimensional conformal theories. It also addresses renormalization and the impact of higher-curvature and nonminimal coupling gravities on the horizon entropy, outlining when the GSL can be expected to hold in broader settings. Overall, it offers a comprehensive, symmetry-informed approach to enforcing thermodynamic consistency in semiclassical gravity across dynamic horizons.

Abstract

The generalized second law is proven for semiclassical quantum fields falling across a causal horizon, minimally coupled to general relativity. The proof is much more general than previous proofs in that it permits the quantum fields to be rapidly changing with time, and shows that entropy increases when comparing any slice of the horizon to any earlier slice. The proof requires the existence of an algebra of observables restricted to the horizon, satisfying certain axioms (Determinism, Ultralocality, Local Lorentz Invariance, and Stability). These axioms are explicitly verified in the case of free fields of various spins, as well as 1+1 conformal field theories. The validity of the axioms for other interacting theories is discussed.

Figures (1)

• Figure 1: a) An eternal black hole spacetime is shown. The GSL says that the generalized entropy must increase from time slice $1$ to time slice $2$. However, all information outside of the horizon must either fall across the horizon $H$ or else reach future null infinity $\mathcal{I}^+$ (Determinism). Hence one can "push forward" each of the two time slices to part of $H$ and all of $\mathcal{I}^+$ without losing any information. In addition to the Killing symmetry which acts on the horizon as a dilation, there is a translation symmetry of $H$ (shown as an arrow) which is not a symmetry of the whole spacetime. b) A transverse view of $H$ in the same spacetime. Vertical lines represent horizon generators. Each horizon generator can be independently translated and dilated (Local Lorentz Symmetry); this permits any two slices on $H$ to be translated into each other, and ensures that region above each slice on $H$ is thermal with respect to dilations about that slice. In order to prove the GSL this thermal property is needed for both slice $1$ and slice $2$.