Gravitational algebras and the generalized second law

TL;DR

This paper derives the generalized second law (GSL) for Killing horizons using crossed-product gravitational algebras, connecting quantum field theory, algebraic structures, and gravitational entropy in both free and interacting theories.

Contribution

It introduces a novel algebraic framework for the GSL on Killing horizons, including a generalization for interacting theories in asymptotically flat spacetimes.

Findings

Crossed product entropy matches the generalized entropy in semiclassical limit.

Reproduces Wall's result linking GSL to relative entropy monotonicity.

Proposes a new algebraic approach for interacting theories at infinity.

Abstract

We derive the generalized second law (GSL) for arbitrary cuts of Killing horizons from the perspective of crossed-product gravitational algebras, making use of a recent proposal by one of us for the construction of local gravitational algebras. This construction relies on the existence of a state whose modular flow is geometric on the horizon. In both free and interacting quantum field theories, such states are guaranteed to exist by the properties of half-sided translations on the horizon. Using geometric identities derived from the canonical analysis of general relativity on null surfaces, we show that the crossed product entropy agrees with the generalized entropy of the horizon cut in a semiclassical limit, and further reproduce Wall's result relating the GSL to monotonicity of relative entropy of the quantum field algebras. We also give a novel generalization of the GSL for interacting theories in asymptotically flat spacetimes involving the concept of an algebra at infinity for a half-sided translation, which accounts for triviality of the algebra of fields smeared only on the horizon. Going beyond the semiclassical limit, we compute subleading corrections to the crossed product entropy, but are unable to determine if the GSL continues to hold after accounting for these. We speculate that an improved GSL could follow from a hidden subalgebra structure of the crossed products, assuming the existence of an operator-valued weight between horizon cut algebras.