Generalized Black Hole Entropy is von Neumann Entropy

TL;DR

The authors provide a universal, horizon-centered framework showing that gravitationally dressed observables on spacetimes with Killing horizons generate Type II von Neumann algebras, enabling well-defined traces and von Neumann entropies that realize the generalized entropy S_gen = A/(4G_N) + S_ext for semiclassical states. By combining a structure theorem for horizons with modular theory and a crossed-product construction, they demonstrate that dressing to horizon charges yields Type II_∞ algebras (or Type II_1 in settings with bounded charges), and that the semiclassical entropy matches the generalized entropy up to a state-independent constant. The framework is applied to exterior Kerr and Schwarzschild–AdS, Schwarzschild–de Sitter, and Kerr–AdS, as well as AdS/CFT, showing consistency with generalized entropy and offering insights into the generalized second law, boundary structures, and potential extensions to higher dimensions, interactions, and near-extremal limits. Collectively, the work provides a rigorous algebraic underpinning for black hole entropy in a broad class of spacetimes, linking gravitational charges, modular flow, and holographic perspectives to renormalized entropy measures.

Abstract

It was recently shown that the von Neumann algebras of observables dressed to the mass of a Schwarzschild-AdS black hole or an observer in de Sitter are Type II, and thus admit well-defined traces. The von Neumann entropies of "semi-classical" states were found to be generalized entropies. However, these arguments relied on the existence of an equilibrium (KMS) state and thus do not apply to, e.g., black holes formed from gravitational collapse, Kerr black holes, or black holes in asymptotically de Sitter space. In this paper, we present a general framework for obtaining the algebra of dressed observables for linear fields on any spacetime with a Killing horizon. We prove, assuming the existence of a stationary (but not necessarily KMS) state and suitable decay of solutions, a structure theorem that the algebra of dressed observables always contains a Type II factor "localized" on the horizon. These assumptions have been rigorously proven in most cases of interest. Applied to the algebra in the exterior of an asymptotically flat Kerr black hole, where the fields are dressed to the black hole mass and angular momentum, we find a product of a Type II$_{\infty}$ algebra on the horizon and a Type I$_{\infty}$ algebra at past null infinity. In Schwarzschild-de Sitter, despite the fact that we introduce an observer, the quantum field observables are dressed to the perturbed areas of the black hole and cosmological horizons and is the product of Type II$_{\infty}$ algebras on each horizon. In all cases, the von Neumann entropy for semiclassical states is given by the generalized entropy. Our results suggest that in all cases where there exists another "boundary structure" (e.g., an asymptotic boundary or another Killing horizon) the algebra of observables is Type II$_{\infty}$ and in the absence of such structures (e.g., de Sitter) the algebra is Type II$_{1}$.

Figures (7)

• Figure 1: A spacetime diagram depicting a bifurcate Killing horizon given by the union of null surfaces $\mathcal{H}^{-}\cup \mathcal{H}^{+}$ where $\mathcal{H}^{-}=\mathcal{H}^{-}_{\textrm{R}}\cup \mathcal{H}^{-}_{\textrm{L}}$ and $\mathcal{H}^{+}=\mathcal{H}^{+}_{\textrm{R}}\cup \mathcal{H}^{+}_{\textrm{L}}$. The horizons intersect at the bifurcation surface $\mathcal{B}$. The orbits of the Killing field locally around the bifurcation surface are shown. The Killing horizon divides the spacetime into four regions $\mathcal{L}, \mathcal{R},\mathcal{P}$ and $\mathcal{F}$ as shown.
• Figure 2: For an asymptotically flat spacetime, $\mathscr I^{+}$ and $\mathscr I^{-}$ denote the future and past null infinities of the spacetime, $i^{+}$ and $i^{-}$ denote the future and past timelike infinities and $i^{0}$ denotes spatial infinity. We also use the label $\textrm{L}$ and $\textrm{R}$ to denote whether these labels refer to the "left" or "right" asymptotically flat regions. The black surface represents the surface of a collapsing star and the red spacelike surface is a Cauchy surface for $\mathcal{M}_{\textrm{R}}=\mathcal{R}\cup \mathcal{F}$.
• Figure 3: Left: The Penrose diagram for de Sitter space. The green line represents an inertial observer with $\mathcal{R}$ its associated static patch. The red spacelike surface is a Cauchy surface for the spacetime. Right: Embedding of de Sitter as a hyperboloid in flat space. Spatial slices are closed and increase in (proper) volume exponentially at late times.
• Figure 4: A portion of the maximally extended Kerr geometry. The black surface represents the surface of a collapsing star and the red spacelike surface is a Cauchy surface for the globally hyperbolic spacetime $\mathcal{M}_{\textrm{R}}=\mathcal{R}\cup \mathcal{F}$.
• Figure 5: Penrose diagram of Schwarzschild-de Sitter spacetime. The maximally extended spacetime contains an infinite sequence of black holes and cosmological horizons depicted in the transparent regions. The spacetime of a single black hole can be obtained by identifying the timelike surfaces. The black surface denotes the surface of a (spherically symmetric) collapsing star. The green curve is the worldline $\gamma$ of an observer in $\mathcal{R}$ and the red line is a Cauchy slice for the physically relevant region $\mathcal{M}_{\textrm{R}}=\mathcal{F}_{1}\cup\mathcal{R}_{1}\cup \mathcal{F}_{2}$.

Theorems & Definitions (1)

• Theorem 1