Gravitational algebras and the generalized second law

TL;DR

This work develops a crossed-product von Neumann algebra framework for subregions bounded by horizon cuts in semiclassical gravity, showing that the associated renormalized entropy reproduces the generalized entropy up to a state-independent constant and that Wall-style monotonicity of relative entropy follows from half-sided modular inclusions. By incorporating the gravitational constraints and an algebra at infinity, the authors extend the GSL to asymptotically flat and de Sitter spacetimes and demonstrate a global GSL in semifinite gravitational algebras, with explicit perturbative corrections analyzed. They further explore the potential existence of an operator-valued weight between horizon-cut algebras, which could yield subalgebra monotonicity and possibly an improved data-processing inequality beyond leading order. The paper also addresses subtleties in rotating black holes and Unruh-type vacua, and discusses connections to dynamical horizon entropy and quantum focusing, outlining promising directions for applying crossed-product entropy to QFC/QNEC questions. Collectively, the work provides a regulator-free, algebraic route to horizon entropy and GSL, with implications for semiclassical entropy relations and quantum gravity phenomenology.

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.

Figures (4)

• Figure 1: The two-sided AdS black hole is shown in the Penrose diagram above. The bifurcation surface is labeled as the point $b$, and $\mathscr{H}_b^+$ denotes the half of the event horizon to the future of $b$. $\mathscr{H}_b^+$ serves as a Cauchy surface for the right exterior region $R_b$, shown in blue. The causal complement $R_b'$ involving the left exterior of the black hole is shown in green. The red dot labeled $c_\lambda$ denotes a cut of the future horizon, and the portion of the horizon to the future of this cut, $\mathscr{H}_\lambda^+$, is a Cauchy surface for the subregion $R_\lambda$ in the right exterior, depicted by the red lines. The causal complement of this region is $R_\lambda'$, labeled by the orange lines, and consists of the entire left exterior as well as a portion of the black hole interior.
• Figure 2: The extended Schwarzschild-de Sitter spacetime provides an example where one finds an algebra at infinity for the horizon translation. This spacetime is described as an infinite chain of black hole and cosmological horizons, indicated by the ellipses "$\cdots$" in the Penrose diagram above. The algebra for a cut $c_\lambda$ is defined as all quantum fields supported in the region spacelike to the right of the cut, shown by the red lines above. The algebra at infinity is associated with the region $R_\infty$ shown in purple, consisting of fields localized beyond the cosmological horizon.
• Figure 3: The Penrose diagram of the conformally extended Schwarzschild solution is shown above. The standard maximally extended Schwarzschild solution is the region shown in yellow, and the triangular regions in the corners are the conformal extension of the spacetime beyond $\mathscr{I}^+$ and $\mathscr{I}^-$. The algebra of massless or conformal fields in this spacetime naturally allows for smearing of the fields in the extended regions. This motivates defining the algebra at infinity for the Schwarzschild black hole to be associated with the region $R_\infty$ shown in purple, and this region captures the notion of outgoing quantum field modes that never enter the black hole horizon.
• Figure 4: Shown above are the conformal extensions of the Schwarzschild and Minkowski spacetimes which are asymptotically conformal to the Einstein static universe near spatial infinity $i^0$, which shrinks to a point in this conformal compactification. The algebra at infinity for the Schwarzschild solution lives in the purple region $R_\infty$. Note that the spacetime can have pathologies such as naked singularities in the region $R_\infty$ and singular behavior near $i^+$, but the figure still motivates defining a nontrivial algebra for these regions. In the conformal extension of Minkowski space, the maroon region indicates the Rindler wedge, and we see that the region at infinity for this patch is the single generator of $\mathscr{I}^+$ shown in purple. Because this is not an open region in the Einstein static universe, we do not expect an algebra at infinity for half-sided translations associated with Rindler horizons in higher than two spacetime dimensions.