QUANTIZATION OF THE SPHERICALLY SYMMETRIC CONFIGURATION OF THE GRAVITATIONAL AND ELECTROMAGNETIC FIELDS

Анотація

To construct a quantum model of a charged black hole (CBH), we introduce a modified description of the configuration of the electromagnetic and gravitational fields in a spherically symmetric space-time, which consists of T- and R- regions. We choose such coordinate system that the desired metric functions depend only on the time coordinate in the T-region, and on the space coordinate in the R-region. Then, the initial action for the configuration decays into terms which describe the fields in the T- and R-regions with the time and the space evolutionary coordinate respectively. We define new coordinates in the R- and T- regions, what allows us to unify the form of the Lagrangians, in each of them and carry out their uniform analysis. Then we construct the canonical formalism for obtained degenerate system according to the method of D.M.Gitman and I.V.Tyutin. It appears that system contains non-physical degrees of freedom. For their expicit separation we carry out the canonical transformation to new canonical variables. In these variables the constraints are reduced to the canonical form and physical part of the Hamilton function of the system is identically equal to zero. This leads to the fact that the desired wave function is determined only by the eigenvalue equations for the operators of observable physical quantities. According that considered system has only two observables – charge and mass of black hole – for further construction of quantum model of this system we introduce its mass and charge functions and find their expression in the new canonical variables. The solution of eigenvalue equations for corresponding operators leads to continuous spectra of charge and mass in considered model of CBH