Two-dimensional quantum-corrected eternal black hole

The one-loop quantum corrections to the geometry and thermodynamics of a black hole are studied for the two-dimensional RST model. We choose boundary conditions corresponding to the eternal black hole being in thermal equilibrium with Hawking radiation. The equations of motion are exactly integrated. One of the solutions obtained is the constant curvature space-time with the dilaton being a constant function. Such a solution is absent in classical theory. One the other hand, we derive a quantum-corrected metric written in Schwarzschild-like form which is a deformation of the classical black hole solution. The space-time singularity is milder than in classical theory and the solution admits two asymptotically flat black hole space-times lying at ``different sides'' of the singularity. The thermodynamics of the classical black hole and its quantum counterpart is formulated. The thermodynamical quantities (energy, temperature, entropy) are calculated and are the same for both the classical and quantum-corrected black holes. So, no quantum corrections to thermodynamics are observed. The possible relevance of the results obtained in the four-dimensional case is discussed.