Topology Change and Unitarity in Quantum Black Hole Dynamics

TL;DR

The paper investigates whether semiclassical topology change can restore unitarity in black hole dynamics within the AdS/CFT framework. By analyzing very long-time correlations in a finite-volume CFT, it shows that recurrences are expected at a Heisenberg time $t_H$ and that the infinite-time average of the normalized correlator $\overline L$ scales as $\overline L \sim e^{-S(\beta)}$, reflecting coarse-grained unitarity. The authors then examine a topology-change proposal in which the Euclidean saddles $X$ (black-hole) and $Y$ (thermal gas) are summed, yielding an instanton-corrected correlator $L(t)_{\rm inst} = L(t)_{X} + C e^{-2\Delta I} L(t)_{Y}$ with $\Delta I = I_Y - I_X$ and $I = -\log Z(\beta)$. They find that the resulting infinite-time average $\overline L_{\rm inst} \sim e^{-2\Delta I}$ reproduces the correct order of magnitude (scaling like $e^{-N^2}$ at large $N$) but fails to generate the detailed recurrences seen in the full CFT, suggesting that semiclassical topology fluctuations provide only coarse-grained unitarity. The work thus points toward the necessity of microscopically stringy dynamics or a stretched horizon to achieve full unitarity and highlights the limits of semiclassical topological corrections in resolving the information-loss problem.

Abstract

We discuss to what extent semiclassical topology change is capable of restoring unitarity in the relaxation of perturbations of eternal black holes in thermal equilibrium. The Poincare recurrences required by unitarity are not correctly reproduced in detail, but their effect on infinite time-averages can be mimicked by these semiclassical topological fluctuations. We also discuss the possible implications of these facts to the question of unitarity of the black hole S-matrix.

Figures (11)

• Figure 1: The energy spectrum of a CFT representing ${\rm AdS}_d$ 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 $d-1$ spacetime dimensions.
• Figure 2: A detailed analysis of dissipation of fluctuations in a finite thermal system can reveal the effect of large quantum fluctuations in which a black hole turns into thermal radiation and viceversa. In the semiclassical approximation to quantum gravity, these processes are represented by a coherent sum over saddle points of different topology. In the case at hand we can use AdS space as an effective finite-volume box.
• Figure 3: 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 $t_H \sim \Gamma^{-1}\,\exp(S)$ 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 4: 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 5: The Euclidean black hole manifold $X$ is simply connected, unlike standard thermal manifolds in quantum field theory.