AdS2 and quantum stability in the CGHS model

The two-dimensional anti-de Sitter space(AdS2) is constructed in terms of the CGHS model. The geometric solutions are composed of the AdS vacuum and the AdS black hole which are locally equivalent but distinguishable by their mass. The infalling classical fields do not play any role but the quantum back reaction is crucial in the formation of the AdS vacuum and AdS black hole. In the presence of the AdS black hole, there does not exist any radiation, which is consistent with the constraint equations. Therefore the transition from the AdS black hole to the AdS vacuum is impossible and they are quantum mechanically stable . We discuss the reason why the vanishing Hawking radiation appears in the AdS2 black hole in contrast to asymptotically flat black holes. ∗electronic address:[email protected] 1 Recently, there has been much attention to the anti-de Sitter(AdS) spacetime in connection with a calculation of the statistical entropy of the Bañados, Teitelboim, and Zanelli(BTZ) black hole [1,2] which is related to the higher-dimensional black holes [3], and the supergravities on the boundary of AdS spacetime and conformal field theory correspondence [4]. It is now natural to study AdS spacetime which may be essential to resolve the quantum gravity puzzles. On the other hand, black holes as geometric solutions of gravity theory are expected to have Hawking temperatures [5] which means that one can detect some radiation from the black hole through quantum fluctuations. The calculation of Hawking radiation can be done on the generic black hole backgrounds. At first sight, this Hawking radiation seems to be applied to both asymptotically flat and nonflat geometry. In the former case, the Tolman temperature as a local temperature is coincident with the Hawking temperature in the asymptotic infinity, however, they are not compatible with each other in the latter case, especially in AdS black holes since the local temperature vanishes in the asymptotic infinity while the Hawking temperature is finite. Some years ago, black hole evaporation and the back reaction of the geometry has been studied by Callan-Giddings-Harvey-Strominger(CGHS) [6] and subsequently by RussoSusskind-Thorlacius(RST) [7] and in the many literatures [8]. In two-dimensions, the quantum back reaction of the geometry is more tractable compared to the other higher dimensional cases and may solve various quantum gravity problems [9]. In this paper, we study the quantum-mechanical generation of constant curvature spacetime of AdS2 in terms of quantum back reaction by using the CGHS(RST) model and obtain the AdS vacuum defined as a lowest energy state of geometry and AdS black hole which is regarded as a massive state. Similarly to the CGHS model, we take the large N limit where N is a number of conformal matter fields in order to maintain the validity of semiclassical approximations. The crucial difference from the CGHS solution is that we shall assume the constant dilaton background instead of the linear dilaton or spacetime-dependent dilaton background. Next, we shall calculate the Hawking radiation in this resulting AdS2 black hole 2 and infer the Hawking temperature without resort to the conventional definition of Hawking temperature. We find that the transition from the AdS black hole to the AdS vacuum is impossible and they are quantum mechanically stable. Finally some remarks and discussion will be given. Let us now consider the two-dimensional low-energy string theory given by SDG = 1 2π ∫