Generalized Black Hole Entropy is von Neumann Entropy

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${}_1$.

83 pages + appendices. v6: to match PRD

References in corpus (68)

• Gauge Theory Correlators from Non-Critical String Theory
• The Large N Limit of Superconformal Field Theories and Supergravity
• Eternal Black Holes in AdS
• A General Definition of "Conserved Quantities" in General Relativity and Other Theories of Gravity
• Notes on Some Entanglement Properties of Quantum Field Theory
• Anti-de Sitter Fragmentation
• General Logarithmic Corrections to Black Hole Entropy
• A finite entanglement entropy and the c-theorem
• Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds
• On the RG running of the entanglement entropy of a circle
• Local subsystems in gauge theory and gravity
• A Quantum Focussing Conjecture
• An Algebra of Observables for de Sitter Space
• Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime
• Quantum mechanics and the covariance of physical laws in quantum reference frames
• Partial and Complete Observables for Canonical General Relativity
• Charged and rotating AdS black holes and their CFT duals
• Relative entropy and the Bekenstein bound
• Mechanics of Rotating Isolated Horizons
• Positive vacuum energy and the N-bound
• The black hole final state
• Information Loss
• Dynamics in Non-Globally-Hyperbolic Static Spacetimes III: Anti-de Sitter Spacetime
• A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices
• Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime
• Geometry of Generic Isolated Horizons
• Proof of the Quantum Null Energy Condition
• Quantum fields in curved spacetime
• On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon
• Gravity and the Crossed Product
• Stability of Black Holes and Black Branes
• Comparison between various notions of conserved charges in asymptotically AdS-spacetimes
• Diffeomorphism-invariant observables and their nonlocal algebra
• Large N algebras and generalized entropy
• Symmetries and charges of general relativity at null boundaries
• Bekenstein bounds in de Sitter and flat space
• Asymptotics with a positive cosmological constant: I. Basic framework
• Quantum Instability of the Cauchy Horizon in Reissner-Nordström-deSitter Spacetime
• Adiabatic vacuum states on general spacetime manifolds: Definition, construction, and physical properties
• Dynamics in Non-Globally-Hyperbolic Static Spacetimes II: General Analysis of Prescriptions for Dynamics
• De Sitter Holography and Entanglement Entropy
• Relational Observables in Gravity: a Review
• Asymptotic Flatness and Bondi Energy in Higher Dimensional Gravity
• Analysis of linear waves near the Cauchy horizon of cosmological black holes
• BMS Supertranslations and Memory in Four and Higher Dimensions
• Generalized entropy for general subregions in quantum gravity
• Notes on Spacetime Thermodynamics and the Observer-dependence of Entropy
• Passivity and microlocal spectrum condition
• Improved decay for solutions to the linear wave equation on a Schwarzschild black hole
• The Asymptotic Behavior of Massless Fields and the Memory Effect
• Notes on the type classification of von Neumann algebras
• Superradiant instabilities of asymptotically anti-de Sitter black holes
• Real-Vector-Space Quantum Theory with a Universal Quantum Bit
• Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime
• Construction of Hadamard states by pseudo-differential calculus
• The time slice axiom in perturbative quantum field theory on globally hyperbolic spacetimes
• von Neumann algebras in JT gravity
• Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property
• A general variational principle for spherically symmetric perturbations in diffeomorphism covariant theories
• Quantization of linearized gravity in cosmological vacuum spacetimes
• Upper bound for entropy in asymptotically de Sitter space-time
• Crossed product algebras and generalized entropy for subregions
• Cosmological perturbation theory and quantum gravity
• On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon
• Symmetries, charges and conservation laws at causal diamonds in general relativity
• A Type $I$ Approximation of the Crossed Product
• Construction of the Unruh State for a Real Scalar Field on the Kerr-de Sitter Spacetime
• Hadamard state in Schwarzschild-de Sitter spacetime

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• Spectral Projections for Density Matrices in Quantum Field Theories