Validity of the semiclassical approximation and back reaction.

Studying two-dimensional evaporating dilatonic black holes, we show that the semiclassical approximation, based on the background field approach, is valid everywhere in regions of weak curvature (including the horizon), as long as one takes into account the effects of back-reaction of the Hawking radiation on the background geometry. Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected] 1 The suggestion that quantum gravitational effect may play a role in the evaporation of black holes [1] received some attention recently [2]. Since we do not have a full theory of quantum gravity it is not clear how to define a criterion according to which we can check the validity of the semiclassical approximation. ’t Hooft suggests that the resolution of the information puzzle (in a unitary framework) is one of the conditions that a quantum theory of gravity should satisfy, and if the present semiclassical theory implies a non-unitary evolution, it cannot be consistent with the full theory of quantum gravity. This may very well be the case, but some try to go even further and show that the standard semiclassical approximation (based on the background field approach) breaks down in regions of weak curvature, just outside the horizon of a black hole [3,4]. Though in Ref. [3] quantum matter fields forming a black hole (and consequently emitting Hawking radiation) was considered, the back-reaction of the quantum radiation on the geometry was ignored. We show in this work that when we take into account the back-reaction, we avoid the conclusions of [3]. Namely, we show in the framework of two-dimensional (2D) dilaton Gravity (in which we can solve the semiclassical equations including the back-reaction) that the semiclassical approximation is valid as long as one includes the back-reaction. We therefore conclude that, at least in 2D dilaton Gravity, the breakdown of the background field method in regions of weak curvature is an artifact of the neglection of the back-reaction. In this work we use the same criterion as in [3] according to which we check the validity of the semiclassical approximation. One considers two almost identical classical space-times with a classical dilaton field, which may differ only at the Planck scale. Then the evolution of a quantum scalar (matter) field on these two classical backgrounds is considered. By taking identical space-like hypersurfaces (i.e., hypersurfaces with the same internal geometry) in both space-times, one can compare the evolution of the two quantum states in the two difFor the hypersurfaces that we consider the shift vector does not necessarily vanish at spatial infinity, and one can have different ADM masses for the same intrinsic geometry.