Quantum gravity near the apparent horizon and two-dimensional dilaton gravity

We study Hawking radiation in a two-dimensional dilaton black hole by means of quantum gravity holding near the apparent horizon. First of all, we construct the canonical formalism of the dilaton gravity in two dimensions. Then the Vaidya metric corresponding to the dilaton black hole is established where it is shown that the dilaton field takes the form of a Linear dilaton. Based on the canonical formalism and the Vaidya metric, we proceed to analyze the quantum properties of a dynamical black hole. It is found that the mass loss rate of Hawking radiation is independent of the black hole mass and at the same time the apparent horizon recedes to the singularity as shown in other studies of two-dimensional gravity. It is interesting that one can construct quantum gravity even near the origin in the spherical coordinate and draw the same conclusion with respect to Hawking radiation as the above-mentioned picture. Unfortunately, the present formalism seems to be ignorant of the contributions from the functional measures over the gravitational field, the dilaton, and the ghosts.