On the quantum model of a charged black hole

Abstract

We study the quantum states of the black hole model in the configuration space. To this end, we investigate the properties of the configuration space of the Einstein–Maxwell set of equations in the T-region. We limit ourselves to considering the T-region, where the fields under consideration have a dynamic meaning. Based on the standard action for the of Einstein–Maxwell set of equations, we construct the reduced action for the spherical symmetry case. Using the Hamiltonian constraint, we exclude the non-dynamic degree of freedom from the reduced action, thereby passing to the configuration space. In the new representation of the system, we study the induced dynamical system in the configuration space. It turns out that the induced supermetric is reduced to a quasi-Cartesian form. The laws of charge and mass conservation, which the system contains, together with the Hamilton constraint, completely determine the state of the black hole. They allow one to find the momenta and the action of the system in terms of field variables and conserved quantities, as well as the trajectory of motion in the configuration space. The black hole quantization is reduced to the construction of the quantum states for the system with a fixed mass and charge in a three-dimensional pseudo-Euclidean configuration space The Hamilton constraint is associated with the DeWitt equation, the latter is constructed using the Laplace–Beltrami operator, which is Hermitian with respect to the natural measure. To construct a Hermitian mass operator, it suffices to restrict ourselves to partial derivatives with their corresponding ordering. For the physical states satisfying the DeWitt equation to be also eigenfunctions of the mass operator, the compatibility conditions must be satisfied. In this case, we arrive at the corresponding ansatz. Its substitution into the DeWitt equation leads to a self-consistent solution of the quantum DeWitt equation and equation for the eigenvalue of the mass operator. The constructed model describes the quantum states of a charged black hole in the configuration space with continuous mass and charge spectra.