No Ensemble Averaging Below the Black Hole Threshold

TL;DR

This paper investigates the absence of ensemble averaging effects in certain boundary observables within AdS/CFT, especially those not involving black hole states, and explores geometric and physical reasons behind this phenomenon.

Contribution

It identifies a class of 'sub-threshold' observables that do not exhibit ensemble averaging effects and proves related geometric properties in hyperbolic geometry.

Findings

Connected solutions with disconnected boundaries do not contribute to sub-threshold observables.

A novel proof of the renormalized volume of hyperbolic three-manifolds is provided.

Black hole physics' chaotic nature explains the apparent ensemble averaging in some observables.

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 $D\geq 3$, 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.