Restoring unitarity in the Banados-Teitelboim-Zanelli black hole

Abstract

Whether or not a system is unitary can be seen from the way it, if perturbed, relaxes back to equilibrium. The relaxation of a semiclassical black hole can be described in terms of a correlation function which exponentially decays with time. In the momentum space it is represented by an infinite set of complex poles to be identified with the quasinormal modes. This behavior is in sharp contrast to the relaxation in unitary theory in finite volume: the correlation function of the perturbation in this case is a quasiperiodic function of time and, in general, is expected to show the Poincar\'e recurrences. In this paper I demonstrate how restore unitarity in the Banados-Teitelboim-Zanelli (BTZ) black hole, the simplest example of an eternal black hole in finite volume. I start with reviewing the relaxation in the semiclassical BTZ black hole and how this relaxation is mirrored in the boundary conformal field theory as suggested by the anti-de Sitter/conformal field theory correspondence. I analyze the sum over $SL(2,\mathbf{Z})$ images of the BTZ space-time and suggest that it does not produce a quasiperiodic relaxation, as one might have hoped, but results in a correlation function which decays by power law. I develop an earlier suggestion and consider a nonsemiclassical deformation of the BTZ space-time that has the structure of a wormhole connecting two asymptotic regions semiclassically separated by a horizon. The small deformation parameter $\ensuremath{\lambda}$ is supposed to have a nonperturbative origin to capture the finite $N$ behavior of the boundary theory. The discrete spectrum of perturbation in the modified space-time is computed and is shown to determine the expected unitary behavior: the corresponding time evolution is quasiperiodic with a hierarchy of large time scales $\mathrm{ln}1/\ensuremath{\lambda}$ and $1/\ensuremath{\lambda}$ interpreted, respectively, as the Heisenberg and Poincar\'e time scales in the system.