Minisuperspace Canonical Quantization of the Reissner-Nordstrom 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 $t$, $\theta$ and $\varphi$ 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 $r$-reparametrization invariance since one dependent variable remains unfixed, and the two quadratic relations satisfied by the integration constants. We then quantize the model using 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.

LaTeX2e source file, 21 pages, no figures

References in corpus (6)

• Spherically symmetric solutions in f(R)-gravity via Noether Symmetry Approach
• Noether symmetry in $f(R)$ cosmology
• Bianchi spacetimes in noncommutative phase-space
• Conditional Symmetries and the Canonical Quantization of Constrained Minisuperspace Actions: the Schwarzschild case
• Singularity Avoidance of Charged Black Holes in Loop Quantum Gravity
• Higher dimensional black holes as constrained systems

Cited by in corpus (23)

• Cartan symmetries and global dynamical systems analysis in a higher-order modified teleparallel theory
• Noether analysis of Scalar-Tensor Cosmology
• Conditional symmetries in axisymmetric quantum cosmologies with scalar fields and the fate of the classical singularities
• A late time accelerated FRW model with scalar and vector fields via Noether symmetry
• Quasinormal Modes and Thermodynamics of Regular Black Holes
• Canonical Quantization of the BTZ Black Hole using Noether Symmetries
• Decoupling of the re-parametrization degree of freedom and a generalized probability in quantum cosmology
• Variational Contact Symmetries of Constraint Lagrangians
• Schrödinger Symmetry in Gravitational Mini-Superspaces
• Reduced Lagrangians and analytic solutions in Einstein-æther Cosmology
• Infinite dimensional symmetry groups of the Friedmann equations
• Quantization of the Szekeres System
• Wheeler-DeWitt equation and Lie symmetries in Bianchi scalar-field cosmology
• Quantum cosmology of a Bianchi III LRS geometry coupled to a source free electromagnetic field
• Dynamical symmetries of homogeneous minisuperspace models
• Discrete spectrum of the quantum Reissner - Nordström geometry
• Exact solution of the Einstein-Skyrme model in a Kantowski-Sachs spacetime
• A new symmetry of the spatially flat Einstein-Friedmann equations
• "Time"-covariant Schrödinger equation and the canonical quantization of the Reissner-Nordström black hole
• Closed-form solutions of the Wheeler-DeWitt equation in a scalar-vector field cosmological model by Lie symmetries
• Classical and quantum (2+1)-dimensional spatially homogeneous string cosmology
• Quantum fate of timelike naked singularity with scalar hair
• Time-covariant Schrödinger equation and invariant decay probability: The $\Lambda$-Kantowski-Sachs universe