Black Holes Without Singularities

We study the properties of a completely integrable deformation of the CGHS dilaton gravity model in two dimensions. The solution is shown to represent a singularity free black hole that at large distances asymptoticaly joins to the CGHS solution. One of the fundamental unsolved problems in theoretical physics is the unification of quantum theory and gravity. One of the reason why this is so difficult stems from the complicated nonlinear structure of the equations of general relativity. These equations are much simpler in lower dimensions. For this reason there has recently been much activity related to the quantization of gravity in two and three dimensions. One of the most important results in 2d was the exactly solvable dilaton gravity model constructed by Callan, Giddings, Harvey and Strominger [1]. The CGHS model has 2d black hole solutions that are remarkably similar to the Schwarzschild solution of general relativity. Of the four fundamental interactions in nature, gravity is by far the weakest. For this reason, we can hope to see quantum effects only in the vicinity of classical singularities. Penrose and Hawking have shown that these singularities are endemic in general relativity. The general belief is that quantization will rid gravitation of singularities, just as in atomic physics it got rid of the singularity of the Coulomb potential. If this is indeed the case, then there must exist a non-singular gravitational effective action whose classical equations encode the full quantum theory. This effective action must have the Planck length LPlanck in it as an input parameter. For L ≫ LPlanck the effective model must be indistinguishable from the classical gravity action. The search for such an effective model parallels Landau’s treatment of phase transitions in ferromagnets. Landau chose (the simplest) effective action (Gibbs potential in statistical mechanics parlance) that led to a qualitatively correct discription of phase transitions. 1 An important result in dilaton gravity has been the work of Louis-Martinez and Kunstatter [2], who reduced the solution of the general dilaton gravity model to the solution of two ordinary integrals, i.e. to two quadratures. In a previous paper [3] we used their proceedure to construct a dilaton model that yields a black hole without a singularity. In this paper we will review the central results of this derivation, construct the deformed CGHS model and show that it leads to a maximal curvature proportional to L Planck. The action of all dilaton gravity models can be put into the general form