Quantum Structure of Space Near a Black Hole Horizon

Abstract

We describe a midi-superspace quantization scheme for generic single horizon black holes in which only the spatial diffeomorphisms are fixed. The remaining Hamiltonian constraint yields an infinite set of decoupled eigenvalue equations: one at each spatial point. The corresponding operator at each point is the product of the outgoing and ingoing null convergences, and describes the scale invariant quantum mechanics of a particle moving in an attractive $1/X^2$ potential. The variable $X$ that is analoguous to particle position is the square root of the conformal mode of the metric. We quantize the theory via Bohr quantization, which by construction turns the Hamiltonian constraint eigenvalue equation into a finite difference equation. The resulting spectrum gives rise to a discrete spatial topology exterior to the horizon. The spectrum approaches the continuum in the asymptotic region.

arXiv commentReferences added and typos corrected. 21 pages, 1 figure

References in corpus (22)

• Evanescent Black Holes
• Quantum Nature of the Big Bang
• The Two-Nucleon Potential from Chiral Lagrangians
• Dilaton Gravity in Two Dimensions
• Black Hole Entropy from Conformal Field Theory in Any Dimension
• Quantum gravity, shadow states, and quantum mechanics
• Conformal description of horizon's states
• Exact Dirac Quantization of All 2-D Dilaton Gravity Theories
• Observables for Two-Dimensional Black Holes
• Renormalization of the Inverse Square Potential
• Near-Horizon Conformal Symmetry and Black Hole Entropy
• Anomaly in conformal quantum mechanics: From molecular physics to black holes
• On Singularity Resolution in Quantum Gravity
• Quantum resolution of black hole singularities
• Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics. I. General Theory
• Black Hole Thermodynamics from Near-Horizon Conformal Quantum Mechanics
• Origin of the anomalies: the modified Heisenberg equation
• Flat slice Hamiltonian formalism for dynamical black holes
• Quantum black holes from null expansion operators
• Anomalous Commutator Algebra for Conformal Quantum Mechanics
• Semiclassical Methods in Curved Spacetime and Black Hole Thermodynamics
• Two-dimensional electron gas in a uniform magnetic field in the presence of a delta-impurity

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