Edge states and entropy of two-dimensional black holes

Abstract

In several recent publications Carlip, as well as Balachandran, Chandar, and Momen, has proposed a statistical-mechanical interpretation for black hole entropy in terms of ``would-be gauge'' degrees of freedom that become dynamical on the boundary to spacetime. After critically discussing several routes for deriving a boundary action, we examine their hypothesis in the context of generic 2D dilaton gravity. We first calculate the corresponding statistical-mechanical entropy of black holes in $1+1$ de Sitter gravity, which has a gauge theory formulation as a BF theory. Then we generalize the method to dilaton gravity theories that do not have a (standard) gauge theory formulation. This is facilitated greatly by the Poisson \ensuremath{\sigma}-model formulation of these theories. It turns out that the phase space of the boundary particles coincides precisely with a symplectic leaf of the Poisson manifold that enters as target space of the \ensuremath{\sigma} model. Despite this qualitatively appealing picture, the quantitative results are discouraging: In most of the cases, the symplectic leaves are noncompact and the number of microstates yields a meaningless infinity. In those cases where the particle phase space is compact---such as, e.g., in the Euclidean de Sitter theory---the edge state degeneracy is finite, but generically it is far too small to account for the semiclassical Bekenstein-Hawking entropy.