[Paper Review] Quantum Singularities

D. A. Konkowski, T. M. Helliwell+1 more|ArXiv.org|Aug 12, 2004

Geophysics and Sensor Technology1 references5 citations

TL;DR

This paper investigates quantum singularities in spherically and cylindrically symmetric spacetimes using Horowitz and Marolf's criterion: a spacetime is quantum mechanically nonsingular if the spatial Klein-Gordon operator is essentially self-adjoint. The key finding is that a sufficiently strong repulsive potential near a classical singularity suppresses quantum wave packet penetration, rendering the spacetime quantum nonsingular—demonstrated in infinite line mass and cosmic string spacetimes where quantum singularities vanish due to repulsive barriers.

ABSTRACT

The definitions of classical and quantum singularities in general relativity are reviewed. The occurence of quantum mechanical singularities in certain spherically symmetric and cylindrically symmetric (including infinite line mass)spacetimes is considered. A strong repulsive ``potential'' near the classical singularity is shown to turn a classically singular spacetime into a quantum mechanically nonsingular spacetime.

Motivation & Objective

• To determine whether classical singularities in general relativity remain singular under quantum mechanical probing using wave packets.
• To apply Horowitz and Marolf's criterion for quantum non-singularity based on essential self-adjointness of the spatial Klein-Gordon operator.
• To investigate how the strength and nature of the spacetime potential near singularities affect quantum behavior.
• To classify quantum singularities in specific spacetimes, including infinite line mass and cosmic string geometries.

Proposed method

• Use the Horowitz and Marolf criterion: a spacetime is quantum mechanically nonsingular if the spatial operator in the Klein-Gordon equation is essentially self-adjoint.
• Apply the test equation $(\nabla^2 \pm i)\Phi = 0$ to determine the number of $L^2$-normalizable solutions near the singularity.
• Transform the radial Klein-Gordon equation into Schrödinger form via appropriate variable substitution to analyze the effective potential.
• Analyze the behavior of solutions near $r=0$ to determine $L^2$ integrability and assess the existence of multiple solutions.
• Focus on static, spherically and cylindrically symmetric spacetimes with timelike singularities.
• Use separation of variables and mode decomposition (e.g., $e^{im\phi}$) to reduce the PDE to a radial ODE for analysis.

Experimental results

Research questions

• RQ1 Under what conditions does a classical singularity in a spherically symmetric spacetime become quantum mechanically nonsingular?
• RQ2 How does the effective potential near a singularity influence the essential self-adjointness of the Klein-Gordon operator?
• RQ3 Are infinite line mass spacetimes quantum mechanically singular, and can repulsive potentials shield the singularity?
• RQ4 Does the type of quantum particle (scalar, spinor, etc.) affect the presence of quantum singularities in these spacetimes?
• RQ5 Can boundary conditions at the singularity be avoided if the potential is sufficiently repulsive?

Key findings

• Spherical spacetimes with $p < 3/2$ (except $p=1$) are quantum mechanically singular due to multiple $L^2$ solutions near $r=0$.
• Spherical spacetimes with $p \geq 3/2$ are quantum mechanically nonsingular when the effective potential is repulsive.
• Cylindrical spacetimes with $a + b < 3$ (excluding Minkowski) are quantum mechanically singular; those with $a + b \geq 3$ are nonsingular.
• The Levi-Civita spacetime for infinite line mass ($0 < \sigma < 1/2$) is quantum mechanically singular, except in the Minkowski limit ($\sigma = 0, c=1$).
• Cosmic string spacetimes ($\sigma = 0, c \neq 1$) are generically quantum mechanically singular for arbitrary modes, but become nonsingular if $|m|c \geq 1$.
• A strong repulsive potential at the classical singularity suppresses wave packet penetration, rendering the spacetime quantum nonsingular, even for quasiregular singularities.