A ug 1 99 6 Exact Solutions of 1 + 1 Dimensional Dilaton Gravity Coupled to Matter

A class of integrable models of 1+1 dimensional dilaton gravity coupled to scalar and electromagnetic fields is obtained and explicitly solved. More general models are reduced to 0+1 dimensional Hamiltonian systems, for which two integrable classes (called s-integrable) are found and explicitly solved. As a special case, static spherical solutions of the Einstein gravity coupled to electromagnetic and scalar fields in any real space-time dimension are derived. A generalization of the 'no-hair' theorem is pointed out and the Hamiltonian formulation that enables quantizing s-integrable systems is outlined. 1. The general 1+1 dimensional dilaton gravity [1] recently attracted a good deal of attention in connection with the black hole physics (for a review and references see e.g. [2]-[4]). An important special case of this theory – the 1+3 dimensional Schwarzschild black hole (SBH) – was recently attempted to quantize by using different approaches [5]-[7]. The key property of SBH, which makes the quantization possible, is integrability 1 [5], [7]. In fact, the most general 1+1 dilaton gravity (DG) also has this nice property. Due to the generalized Birkhoff theorem [4], it actually reduces to a finite dimensional constrained system (FDC) [1], and thus can be explicitly quantized following the approach of Ref.[7]. Though quantizing SBH might seem to be most important, in view of their special role in the Einstein gravity theory, the more general models have to be considered. An explicitly integrable system related to string models and not re-ducible to FDC had been proposed in [10] (CGHS) and was analyzed in detail for the last three years (see e.g. [11], [12] and references therein). A more complex integrable model, related to the Kerr black hole, was recently studied with employing the full machinery of modern methods for quantizing integrable systems (see [13] where references to other papers of the authors may be found). 1 For a relevant review of integrable dynamical systems see e.g. [8], [9]. Integrable systems considered here are explicitly integrable, meaning that their classical solutions can be found in terms of elementary functions or reduced to quadratures.