A proof of the generalized second law for rapidly-evolving Rindler horizons

TL;DR

The paper proves the generalized second law for rapidly evolving Rindler horizons in semiclassical gravity for minimally coupled quantum fields. It reframes horizon entropy changes in terms of the relative entropy to a thermal vacuum, leveraging the Unruh effect and the monotonicity of relative entropy to show dS_gen/dt >= 0. The result relies on a boost symmetry of the horizon and a renormalization procedure for S_out, and is limited to flat or near flat backgrounds with small perturbations but suggests extensions to more general horizons and couplings. The work clarifies the statistical mechanics underpinning horizon thermodynamics and offers a stepping stone toward general quantum gravity insights.

Abstract

The generalized second law is proven for rapidly-evolving semiclassical Rindler horizons at each instant of time, for arbitrary interacting quantum fields minimally coupled to general relativity. The proof requires the background spacetime to have boost and null translation symmetry. Possible extensions to more general horizons and matter-gravity couplings are discussed.

Figures (2)

• Figure 1: a) The one parameter family of Rindler wedges in the $u$-$v$ coordinate system, illustrated by three particular wedges which share the same future Rindler horizon. The wedges are related by null translations in the v direction. The GSL states that each wedge should have at least as much generalized entropy as the wedges beneath it. b) The boost symmetry of a single Rindler wedge, which is used to show that the vacuum state is thermal with respect to the boost energy. The spatial slices related by the boost symmetry all have the same horizon area and the same entropy content, so the generalized entropy of each slice is constant, assuming there is no anomaly in the renormalization of the outside entropy.
• Figure 2: The wedge $W(V)$ evolves forward in time to $W(V^\prime)$. Each of the wedges contains a certain amount of boost energy $K$ all of which must either fall across the horizon $H$ or be radiated to infinity and thus contribute to $K_\mathrm{rad}$. The total amount of boost energy in each wedge is thus proportional to the area of the wedge, up to the contribution at $v = +\infty$, which is the same for both $W(V)$ and $W(V^\prime)$.