Classical and Quantum Descriptions of the Geometrodynamics of Black Holes

Abstract

Analytical aspects of the classical and quantum geometrodynamics of charged black hole are considered. We start with a reduced action for a spherically symmetric of Maxwell-Einstein system written in characteristic variables. The feature of these configurations is that they admit two motion integrals, the total mass and charge. Momenta and constraint are introduced. Using the conservation laws and the Hamiltonian constraint, the momenta as functions of configuration variables are found. It turns out that the system of equations relating momenta and functional derivatives of an action on a configuration space (CS) is integrable. This allows us to obtain the action functional, as a solution of the Einstein-Hamilton-Jacobi equation in functional derivatives. Variations of the action functional with respect to mass and charge of the configuration lead to the motion trajectories in the CS. Further, the space-temporal action is transformed into an action in the configuration space similar to the Jacobi action of classical mechanics. This induces a metric in the CS. Thus, the metric on the CS is introduced and its geometry is studied. The field variables transformation is obtained which brings the CS metric to the "quasi-Lorentzian" form. On this basis, quantization is considered. Taking into account the structure of the CS, the momentum operators, DeWitt equations, mass and charge operators are constructed. Further, for comparison, consider the reduced CBH model defined in the T-region. In this simplified formulation, the T-model equations are integrated and lead to CBH with a continuous spectrum of mass and charge.