We use the conditional symmetry approach to study the r evolution of a minisuperspace spherically symmetric model both at the classical and the quantum level. After integration of the coordinates t, θ and ϕ in the gravitational plus electromagnetic action the configuration space dependent dynamical variables turn out to correspond to the r-dependent metric functions and the electrostatic field. In the context of the formalism for constrained systems (Dirac-Bergmann, Arnowitt-Deser-Misner) with respect to the radial coordinate r, we set up a pointlike reparametrization invariant Lagrangian. It is seen that, in the constant potential parametrization of the lapse, the corresponding minisuperspace is a Lorentzian three-dimensional flat manifold which obviously admits six Killing vector fields plus a homothetic one. The weakly vanishing r Hamiltonian guarantees that the phase space quantities associated to the six Killing fields are linear holonomic integrals of motion. The homothetic field provides one more rheonomic integral of motion. These seven integrals are shown to comprise the entire classical solution space, i.e. the space-time of a Reissner-Nordström black hole, the r-reparametrization invariance since one dependent variable remains unfixed, and the two quadratic relations satisfied by the integration constants. We then quantize the model using as supplementary conditions acting on the wave function, the quantum analogues of the various subalgebras of the classical conditional symmetries. We find that, as a semiclassical analysis shows, in all but one allowed case the ensuing solutions to the Wheeler-DeWitt equation exhibit a good correlation with the classical regime. In the remaining case, the emerging semiclassical geometry is a four-dimensional homogeneous space-time, thus exhibiting no curvature singularity.
Christodoulakis, T., et al. “Minisuperspace Canonical Quantization of the Reissner-Nordstrom Black Hole via Conditional Symmetries.” Physical Review D, vol. 89, no. 4, American Physical Society, Feb. 2014, p. 044031, doi:10.1103/PhysRevD.89.044031.
Physical Review D