Semiclassical extremal black holes.

Extremal black holes are studied in a two dimensional model motivated by a dimensional reduction from four dimensions. Their quantum corrected geometry is calculated semiclassically and a mild singularity is shown to appear at the horizon. Extensions of the geometry past the horizon are not unique but there are continuations free from malevolent singularities. A few comments are made about the relevance of these results to four dimensions and to the study of black hole entropy and information loss. ⋆ Research supported in part by the U.S. Department of Energy under contract no. DEAC-03-81ER4050. Introduction: Hawking’s discovery of blackhole radiance [18,19] raises several intriguing questions. It shows that black holes have an entropy which can be elegantly expressed in terms of their geometry but which remains mysterious in terms of any underlying microstates. It also suggests that because of the thermal nature of the outgoing radiation, a loss of quantum coherence might occur in processes involving black holes. Extremal black holes provide a convenient setting in which to address both these issues. Their zero temperature suggests that their entropy should be explained in terms of a degeneracy of ground states. It also makes them convenient toy laboratories in which to study scattering and a potential loss of quantum coherence. In this paper, we study a model consisting of dimensionally reduced gravity and electromagnetism coupled to two dimensional scalar fields. Classically, this model has Reissner Nordstrom black hole solutions, obtained from dimensionally reducing the usual four dimensional charged black hole solutions. In this paper, we concentrate for the most part on the extremal black holes which in Planck units have a mass equal to their charge and have zero Hawking temperature. We show that contrary to expectations, the vacuum polarisation of a scalar field in the background of such an extremal black hole blows up at the horizon. This raises the possibility of their geometry being drastically modified in the vicinity of the horizon and their entropy being very different from what classical considerations would suggest. A semiclassical calculation of their quantum corrected geometry shows however, that this is not true. For large black holes the value of the dilaton at the horizon and hence their entropy † , stays large. A singularity does appear at the horizon but it is very mild. For example, tidal forces and the curvature stay finite at the horizon. This suggests that there should be a continuation of the geometry past the horizon. In fact, there is more than one such continuation even when we restrict ourselves to static solutions. In one of these the causal structure † The entropy of these black holes depends on the dilaton and is large if the value of the dilaton at the horizon is large.