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

TL;DR

This paper introduces a novel time-covariant Schr"odinger equation for the Reissner-Nordstr"om black hole, enabling a consistent quantum description that preserves covariance and provides insights into singularity analysis.

Contribution

It develops a re-parametrization covariant quantum framework for RN black holes, combining intrinsic time and Wheeler-DeWitt approaches, with solutions and geometrical interpretations.

Findings

Solutions for initial states are obtained.

A criterion for spacetime singularity comparison is proposed.

An equivalence with 3D electromagnetic pp-wave spacetime is established.

Abstract

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(r)"-dependent Hamiltonian density was constructed, without "breaking" the re-parametrization covariance $r\rightarrow f(\tilde{r})$. As a result, evolution of states with respect to the parameter $r$ 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.