We study a two-dimensional dilaton gravity model related by a conformal transformation of the metric to the Callan-Giddings-Harvey-Strominger model. We find that most of the features and problems of the latter can be simply understood in terms of the class

In the past few years,a great deal of work has been dedicated to the study of a two-dimensional model of dilaton gravityfirst proposed by Callan et al.(CGHS)[1].The reason of this interest is due to the fact that such model provides a good approximation of low-energy scattering from a nearly extremal black hole of four-dimensional string gravity. Such approximation is much more tractable than the original model and in fact permits the investigation of the process of formation and evaporation of a black hole in a semiclassical approximation.

The model is described by the two-dimensional action:

S= d2x 2 (ˆ∇f i)2 ,(1)

whereφis the dilaton and f i are a set of scalarfields.

At the classical level,this model admits an exact solution describing the formation of a black hole caused by a shock wave of incident matter.It is then possible to discuss the Hawking radiation of the black hole by means of the standard semiclassical calculation performed by quantizing the matterfields f i in the background constituted by the classical solution.As is well known,in this approximation the contribution of the scalars to the energy-momentum tensor is proportional to the one-loop anomaly,which in the conformal gauge is in turn proportional to the curvature of the background metric.

Thefinal result of the calculation is that a constantflux of radiation is emitted,which is independent of the mass of the black hole[1].This surprising result can be improved by making a better approximation,taking into account the backreaction of the gravitational field to the radiation.This topic has been widely investigated[2].

A further problem arises when one considers conformal transformations of the original metric.As is well known,in fact,a conformal transformation in two dimensions consists essentially in a redefinition of thefields and therefore the physical content of the theory should not depend on it.In particular,if one defines a rescaled metric

gµν=e−2φˆgµν,(2) the action(1)becomes

S= d2x√2 (∇f i)2 .(3)

However,the action(3)admits onlyflat solutions(in Rindler coordinates)with non-trivial dilaton.It has therefore been argued that the theory defined by(1)should be trivial,since it is equivalent toflat space under conformal transformations and in particular does not admit Hawking radiation,since the conformal anomaly is of course zero forflat space[3], at variance with the results of[1].In this letter,we wish to clarify this point,by observing that in order to solve the puzzle,one has to take into due account the role of the dilaton field,which should be considered on the same footing as the metric in the discussion of the structure of the spacetime.In particular,the requirement of reality of the dilaton field implies that evenflat solutions have a non-trivial structure,since one is forced to

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