No Ensemble Averaging Below the Black Hole Threshold

Abstract

In the AdS/CFT correspondence, amplitudes associated to connected bulk manifolds with disconnected boundaries have presented a longstanding mystery. A possible interpretation is that they reflect the effects of averaging over an ensemble of boundary theories. But in examples in dimension $D\ge 3$, an appropriate ensemble of boundary theories does not exist. Here we sharpen the puzzle by identifying a class of "sub-threshold" observables that we claim do not show effects of ensemble averaging. These are amplitudes that do not involve black hole states. To support our claim, we explore the example of $D=3$, and show that connected solutions of Einstein's equations with disconnected boundary never contribute to sub-threshold observables. To demonstrate this requires some novel results about the renormalized volume of a hyperbolic three-manifold, which we prove using modern methods in hyperbolic geometry. Why then do any observables show apparent ensemble averaging? We propose that this reflects the chaotic nature of black hole physics and the fact that the Hilbert space describing a black hole does not have a large $N$ limit.

43pp, clarifications and corrections in v. 2 and some more precise results about volumes, minor corrections in v. 3

References in corpus (5)

• Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information
• AdS${}_3$ gravity and random CFT
• The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral
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• Synthetic Fuzzballs: A Linear Ramp from Black Hole Normal Modes
• A precision test of averaging in AdS/CFT
• Combinatoric topological string theories and group theory algorithms
• Vacuum Transitions in Two-Dimensions and their Holographic Interpretation
• Holographic QFTs on AdS${}_d$, wormholes and holographic interfaces
• Quantum evolution of the Hawking state for black holes
• Higher spin JT gravity and a matrix model dual
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• Replicas, averaging and factorization in the IIB matrix model
• A universal sum over topologies in 3d gravity