Minisuperspace canonical quantization of the Reissner-Nordström black hole via conditional symmetries

Abstract

We use the conditional symmetry approach to study the r-evolution of a minisuperspace spherically symmetric model both at the classical and 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, ADM) with respect to the radial coordinate r, we set up a point-like reparameterization 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 rreparametrization invariance since one dependent variable remains unfixed, and the two quadratic relations satisfied by the integration constants. We then quantize the model using the quantum analogues of the classical conditional symmetries, and show that the existence of such symmetries yields solutions to the Wheeler-DeWitt equation which, as a semiclassical analysis shows, exhibit a good correlation with the classical regime. We use the resulting wave functions to investigate the possibility of removing the classical singularities.

Exact solution of the Einstein–Skyrme model in a Kantowski–Sachs spacetime

“…Because the solution of the WdW equation is known we study the quantum effects in the classical behaviour which follow from the quantum potential in the semiclassical approach of Bohmian mechanics [38,39]. A method which has been applied recently in various models [27,[40][41][42]. …”

Conditional symmetries in axisymmetric quantum cosmologies with scalar fields and the fate of the classical singularities

“…This allows for the presence of extra symmetries in the configuration space called conditional symmetries (see e.g. [1,[21][22][23][24][25]). In [26], the relation between the Lie point symmetries and the conditional symmetries of the minisuperspace was established which in the constant potential lapse parametrization coincide with the conditional symmetries in the phase space.…”

“…Therefore, the quantum potential will vanish rendering the solution for this case same as the classical metric. Indeed, if we solve the semiclassical solutions 25) with phase function S =…”

FLRW metric f(R) cosmology with a perfect fluid by generating integrals of motion

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