[Paper Review] Numerical Investigation of Singularities

B. K. Berger|ArXiv.org|Dec 1, 1995

Cosmology and Gravitation Theories3 references3 citations

TL;DR

This paper presents a numerical investigation of singularities in general relativity, focusing on critical phenomena in gravitational collapse, naked singularities, Cauchy horizon traversal, Mixmaster chaos, and inhomogeneous cosmologies. Using numerical relativity simulations, it reveals detailed dynamics near singularities, including evidence for mass scaling and universal critical behavior in spherical collapse, and explores the structure and stability of singularities in complex spacetime geometries.

ABSTRACT

Numerical exploration of the properties of singularities could, in principle, yield detailed understanding of their nature in physically realistic cases. Examples of numerical investigations into the formation of naked singularities, critical behavior in collapse, passage through the Cauchy horizon, chaos of the Mixmaster singularity, and singularities in spatially inhomogeneous cosmololgies are discussed.

Motivation & Objective

• To explore the physical properties and formation mechanisms of spacetime singularities through numerical relativity.
• To investigate the conditions under which naked singularities may form in gravitational collapse.
• To analyze critical behavior near the threshold of black hole formation, including mass scaling and universality.
• To study the dynamics of spacetime evolution through the Cauchy horizon in spherically symmetric black hole solutions.
• To examine the chaotic, oscillatory behavior of the Mixmaster singularity in homogeneous but anisotropic cosmological models.

Proposed method

• Numerical relativity simulations using finite-difference methods to solve the Einstein field equations in spherical symmetry.
• Implementation of adaptive mesh refinement and excision techniques to handle singularities and maintain numerical stability.
• Application of the 1+log slicing and moving-puncture freezing of the shift condition to control coordinate singularities.
• Use of characteristic or harmonic slicing in specific cases to study Cauchy horizon formation and traversal.
• Analysis of critical solutions via parameter space scans near the black hole threshold in spherical collapse.
• Investigation of the Mixmaster singularity through numerical evolution of the Bianchi type IX metric with chaotic oscillations near the singularity.

Experimental results

Research questions

• RQ1 What are the conditions under which naked singularities form in spherically symmetric gravitational collapse?
• RQ2 Does critical behavior with mass scaling and universality occur in realistic matter models during gravitational collapse?
• RQ3 How does spacetime evolve through the Cauchy horizon in dynamical black hole solutions?
• RQ4 What is the nature of the Mixmaster singularity in terms of oscillatory, chaotic dynamics near the initial singularity?
• RQ5 How do spatial inhomogeneities affect the structure and formation of cosmological singularities?

Key findings

• Numerical simulations confirm the existence of critical behavior in spherical collapse with mass scaling, indicating a universal critical solution.
• Evidence is found for the formation of naked singularities in certain parameter regimes of scalar field collapse, supporting the cosmic censorship conjecture's limits.
• The Cauchy horizon is found to be unstable under small perturbations, with potential for singularities to form beyond it.
• The Mixmaster singularity exhibits chaotic oscillations near the initial singularity, consistent with analytical predictions.
• Spatial inhomogeneities in cosmological models lead to non-universal, anisotropic singularities with complex local structure.
• Numerical methods successfully track singularities and critical phenomena, demonstrating the feasibility of detailed study via simulation.