Von-Neumann Stability and Singularity Resolution in Loop Quantized Schwarzschild Black Hole

Abstract

Though loop quantization of several spacetimes has exhibited existence of a bounce via an explicit evolution of states using numerical simulations, the question about the way central singularity is resolved in the black hole interior has remained open. The quantum Hamiltonian constraint in loop quantization turns out to be a finite difference equation whose stability is important to understand to gain insights on the viability of the underlying quantization and resulting physical implications. We take first steps towards addressing these issues for a loop quantization of the Schwarzschild interior recently given by Corichi and Singh. Von-Neumann stability analysis is performed using separability of solutions as well as a full two dimensional quantum difference equation. This results in a stability condition for black holes which have a very large mass compared to the Planck mass. For black holes of smaller masses evidence of numerical instability is found. In addition, stability analysis for macroscopic black holes leads to a constraint on the choice of the allowed states in numerical evolution. With the caveat of using kinematical norm, sharply peaked Gaussian states are evolved using the quantum difference equation and singularity resolution is obtained. A bounce is found for one of the triad variables, but for the other triad variable singularity resolution amounts to a non-singular passage through the zero volume. States are found to be peaked at the classical trajectory for a long time before and after the singularity resolution, and retain their semi-classical character across the zero volume. Our main result is that quantum bounce occurs in loop quantized Schwarzschild interior at least for macroscopic black holes.

arXiv comment10 pages, 10 figures; Discussion of results expanded. To appear in CQG

References in corpus (29)

• Loop Quantum Cosmology: A Status Report
• Quantum Nature of the Big Bang: Improved dynamics
• Robustness of key features of loop quantum cosmology
• Quantum geometry and the Schwarzschild singularity
• Loop quantum cosmology of k=1 FRW models
• Loop quantum black hole
• Loop Quantum Dynamics of the Schwarzschild Interior
• Loop quantization of the Schwarzschild black hole
• From black holes to white holes: a quantum gravitational, symmetric bounce
• Is loop quantization in cosmology unique?
• Dynamical Initial Conditions in Quantum Cosmology
• Lattice refining loop quantum cosmology, anisotropic models and stability
• Semi-classical States, Effective Dynamics and Classical Emergence in Loop Quantum Cosmology
• Physical evolution in Loop Quantum Cosmology: The example of vacuum Bianchi I
• Curvature invariants, geodesics and the strength of singularities in Bianchi-I loop quantum cosmology
• Loop quantization of spherically symmetric midi-superspaces : the interior problem
• Numerical simulations of loop quantum Bianchi-I spacetimes
• Geodesic completeness and the lack of strong singularities in effective loop quantum Kantowski-Sachs spacetime
• Numerical loop quantum cosmology: an overview
• Resolution of strong singularities and geodesic completeness in loop quantum Bianchi-II spacetimes
• Consistency Conditions for Fundamentally Discrete Theories
• Difficulties with Recollapsing models in Closed Isotropic Loop Quantum Cosmology
• Wave functions for the Schwarschild black hole interior
• Coordinate time dependence in Quantum Gravity
• Generating function techniques for loop quantum cosmology
• Phenomenology with fluctuating quantum geometries in loop quantum cosmology
• Numerical Techniques in Loop Quantum Cosmology
• Numerical solutions to lattice-refined models in loop quantum cosmology
• Unique factor ordering in the continuum limit of LQC

Cited by in corpus (25)

• Quantum Transfiguration of Kruskal Black Holes
• Quantum Extension of the Kruskal Space-time
• White Holes as Remnants: A Surprising Scenario for the End of a Black Hole
• Spherically symmetric loop quantum gravity: analysis of improved dynamics
• Properties of a recent quantum extension of the Kruskal geometry
• Generic absence of strong singularities in loop quantum Bianchi-IX spacetimes
• Spherical symmetric gravitational collapse of a dust cloud: polymerized dynamics in reduced phase space
• How Information Crosses Schwarzschild's Central Singularity
• Glimpses of Space-Time Beyond the Singularities Using Supercomputers
• Polar gravitational perturbations and quasinormal modes of a loop quantum gravity black hole
• Hamiltonian formulation and loop quantization of a recent extension of the Kruskal spacetime
• Von Neumann stability of modified loop quantum cosmologies
• Quasinormal modes of a massive scalar field nonminimally coupled to gravity in the spacetime of Self-Dual Black Hole
• Exploring alternatives to the Hamiltonian calculation of the Ashtekar-Olmedo-Singh black hole solution
• On consistent gauge fixing conditions in polymerized gravitational systems
• Noncommutativity and the Weak Cosmic Censorship
• Symmetric black-to-white hole solutions with a cosmological constant
• Quantum space and quantum completeness
• Loop Quantum Cosmology: Physics of Singularity Resolution and its Implications
• Quantum geometry and black holes
• Lessons from gauge fixing and polymerization of loop quantum black holes with a cosmological constant
• Dynamically implementing the $\overline{\mu }$-scheme in cosmological and spherically symmetric models in an extended phase space model
• Basis Function Method for Numerical Loop Quantum Cosmology: The Schwarzschild Black Hole Interior
• Directional Quantum Singularities in Curzon Spacetime
• Space-Time Collocation Method: Loop Quantum Hamiltonian Constraints