Superspace approach to the quantization of charged black holes with allowance for the cosmological constant
In the present paper, we study the geometry of a mini-superspace and its relation to the corresponding space-time geometry of a spherically
symmetric conﬁguration of electromagnetic and gravitational ﬁelds, taking into account the cosmological constant, and the construction of the wave function of a quantum system. By the generalized Birkhoﬀ
theorem, for this conﬁguration we can introduce the R- and T-regions, which simpliﬁes the description of the dynamical system. Proceeding from the standard classical Einstein-Hilbert action, a Lagrangian of the
ﬁelds conﬁguration is constructed for a spherically symmetric space-time. The Lagrangian of the system is degenerate and contains a nondynamic degree of freedom, which leads to a constraint. After eliminating the constraints, we proceed to the description of the dynamic system in the conﬁguration space (minisuperspace). We consider additional conserved physical quantities: the total mass and the charge of the system. We note that the geometry of the minisuperspace turns out to be conformally ﬂat. In addition to the standard horizons inherent in a charged black hole, space-time has an additional cosmological horizon.
In the conﬁguration space the simplest invariants of the curvature tensor: the scalar curvature, the square of the Ricci tensor, the Kretschmann invariant, are vanish, while the components of the Ricci tensor and the curvature tensor diverge on the minisuperspace
analogue of the cosmological horizon. Within the framework of canonical quantum gravity with material sources, physical states are found by
solving the Hamiltonian constraint in the operator form for the wave function of the system deﬁned on the minisuperspace, taking into account conserved additional quantities. Formal quantization in the
R-region can be regarded as an analytic continuation of solutions from the T-region. In this approach, taking into account the mass and charge operators leads to a continuous spectra of mass and charge.
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