Evaporation of Three Dimensional Black Hole in Quantum Gravity

We discuss an evaporation of (2+1)-dimensional black hole by using quantum gravity holding in the vicinity of the black hole horizon. It is shown that the black hole evaporates at a definite rate by emitting matters through the quantum tunneling effect. A relation of the present formalism to the black hole entropy is briefly commented. † E-mail address: [email protected] The canonical formalism provides us with a useful starting point for quantization by Dirac [1]. In particular when applied for general relativity, the dynamics is entirely controlled by constraints [1, 2]. It is well known that their imposition as operator equations on the physical states produces the Wheeler-DeWitt equation [3] although its physical interpretation is sometimes unclear and difficult. Our aim in the present study is to make use of the canonical quantization formalism for a system having a black hole in order to understand the black hole evaporation [4] in three dimensions in the framework of quantum gravity. Since an announcement by Hawking [4] that black holes are not, after all, completely black, but emit a thermal radiation due to quantum effects, there have been many efforts to extend his semiclassical analysis where the background metric is treated classically, but the matter fields quantum mechanically, to a completely quantum-mechanical analysis. At the moment, it seems to be fair to say that we do not yet have a fully satisfactory and consistent theory of quantum gravity. Recently, however, there appeared one interesting and fully quantum-mechanical approach that is based on both the black-hole minisuperspace model and the canonical quantization formalism, and was applied for understanding of the black hole radiation [5, 6] and the mass inflation in four dimensions [7] † . The key observation is that near the black hole horizon the Hamiltonian constraint becomes proportional to the supermomentum constraint (we will later clarify the reason) and, at the same time, has a tractable form by which we can solve the Wheeler-DeWitt equation analytically. Even if not completely satisfactory, it seems to be very reasonable at least for the present author to consider a theory of quantum gravity in the vicinity of the black hole horizon by the following reasons: it is nowadays thought that some important properties of quantum black holes such as the black hole thermodynamics [8] and the Hawking radiation [4] have an origin of the existence of the horizon [9, † P.Moniz has independently considered a similar model from a different motivation (private communication).