Long time scales and eternal black holes

TL;DR

This work analyzes how eternal black holes in AdS/CFT influence the long-time behavior of correlation functions. It argues that semiclassical gravity violates a unitarity-based bound and shows that incorporating a subleading master field and topology-changing Euclidean saddles can recover part of the required structure, but fully capturing Poincaré recurrences requires nonperturbative, likely stringy, effects. The authors compare the continuous spectrum of AdS black holes with the discrete spectrum of vacuum AdS to derive a Heisenberg time $t_H \sim \beta e^{S(beta)}$ and quantify instanton corrections that scale as $e^{-2\Delta I} \sim e^{-N^2}$. They conclude that while semiclassical descriptions reproduce coarse-grained entropy, the detailed unitary time evolution demands microscopic dynamics beyond semiclassical gravity, such as a stretched horizon or other stringy physics.

Abstract

We discuss the various scales determining the temporal behaviour of correlation functions in the presence of eternal black holes. We point out the origins of the failure of the semiclassical gravity approximation to respect a unitarity-based bound suggested by Maldacena. We find that the presence of a subleading (in the large-N approximation involved) master field does restore the compliance with one bound but additional configurations are needed to explain the more detailed expected time dependence of the Poincare recurrences and their magnitude.

Figures (6)

• Figure 1: The energy spectrum of a CFT representing ${\rm AdS}_{d+1}$ quantum gravity. The spectrum is discrete on a sphere of radius $R$, with gap of order $1/R$. The asymptotic energy band of very dense "black hole" states sets in beyond energies of order $N^2 /R$. The corresponding density of states is that of a conformal fixed point in $d$ spacetime dimensions.
• Figure 2: Schematic representation of the very long time behaviour of the normalized time correlator $L(t)$ in bounded systems. The initial decay with lifetime of order $\Gamma^{-1}$ is followed by O(1) "resurgences" after the Heisenberg time $t_H \sim \beta\,\exp(S)$ has elapsed. Poincaré recurrence times can be defined by demanding the resurgences to approach unity with a given a priori accuracy, and scale like a double exponential of the entropy.
• Figure 3: The effective potential determining the semiclassical normal frequency modes in a large AdS black hole background (left). In Regge--Wheeler coordinates the horizon is at $r_* = -\infty$, whereas the boundary of AdS is at $r_* = \pi R/2$ (only the region exterior to the horizon appears). There is a universal exponential behaviour in the near-horizon (Rindler) region. The effective one-dimensional Schrödinger problem represents a semi-infinite barrier with a continuous energy spectrum. This contrasts with the analogous effective potential in vacuum AdS with global coordinates (right). The domain of $r_*$ is compact and the spectrum of normal modes is discrete with gap of order $1/R$.
• Figure 4: Summing over large-scale fluctuations of the thermal ensemble in which a black hole spontaneously turns into radiation (and viceversa) is represented in the Euclidean formalism as the coherent sum of thermal saddle points of different topology. The "cigar-like" geometry $X$ represents the black-hole master field (in the CFT language) and the cylindrical topology $Y$ represents the thermal gas of particles.
• Figure 5: The instanton approximation to the correlator $L(t)_{\rm inst}$ features the expected exponential decay $\exp(-\Gamma \,t)$ induced by the contribution of the $X$-manifold, whereas the resurgences are entirely due to the interference with the $Y$-manifold, leading to small bumps of order $\exp(-2\Delta I) \sim \exp(-N^2)$, separated a time $t_H (Y) \sim N^0$. These bumps are noticeable against the background of the $X$-manifold after a time $t_c \sim \Delta I /\Gamma$.