"Time"-covariant Schr\"odinger equation and the canonical quantization of the Reissner-Nordstr\"om black hole

A “time”-covariant Schr"{o}dinger equation is defined for the minisuperspace model of the Reissner-Nordstr"{o}m (RN) black hole, as a “hybrid” between the “intrinsic time” Schr"{o}dinger and Wheeler-DeWitt(WDW) equations. To do so, a reduced, regular and “time®”-dependent Hamiltonian density was constructed, without “breaking” the re-parametrization covariance $r\rightarrow f(\tilde{r})$\mathrm{.}Asaresult,evolutionofstateswithrespecttotheparameter$r$ and probabilistic interpretation of the resulting quantum description is possible, while quantum schemes for different gauge choices are equivalent by construction. The solutions are found for a Dirac’s delta and a Gaussian initial states. A geometrical interpretation of the wavefunctions is presented via Bohm analysis. Alongside, a criterion is presented to adjudicate which, between two singular spacetimes is “more” or “less” singular. Two ways to adjudicate about the existence of singularities are compared (vanishing of the probability density at the classical singularity and semi-classical spacetime singularity). Finally, an equivalence of the reduced equations with these of a 3D electromagnetic pp-wave spacetime is revealed.