Abstract
A “time”-covariant Schrödinger equation is defined for the minisuperspace model of the Reissner–Nordström (RN) black hole, as a “hybrid” between the “intrinsic time” Schrödinger 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→f(r˜). As a result, the evolution of states with respect to the parameter r and the 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 Dirac’s delta and Gaussian initial states. A geometrical interpretation of the wavefunctions is presented via Bohm analysis. Alongside this, a criterion is presented to adjudicate which, between two singular spacetimes, is “more” or “less” singular. Two ways to adjudicate 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 those of a 3D electromagnetic pp-wave spacetime is revealed.
Highlights
The quantization of gravity possesses a fundamental place in the realm of theoretical physics
The purpose of this work is to provide some sort of a Schrödinger-type equation for the minisuperspace Reissner–Nordström black hole, without imposing any gauge conditions and without introducing additional functional χA for the purpose of choosing “time” variables
If we focus on a specific distance x, we observe that as β grows, meaning stronger quantum effects, the region of accepted values of α is closer to α = 0, which corresponds to a Schwarzschild black hole
Summary
The quantization of gravity possesses a fundamental place in the realm of theoretical physics. If a coherent quantum theory exists that describes those fundamental interactions, gravity should be included as well. The second motivation is related to the appearance of singularities in the classical theory of gravity, at is described by general relativity. The main reason that such a problem arises is that in the conventional quantum theory, time is an external parameter, while in the general theory of relativity, there is no preferred or fundamental notion of time. This is due to the diffeomorphism invariance of the theory. The relation between constraint equations and diffeomorphisms can be found in [8], as well as in [9]
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