[Paper Review] The instability of naked singularities in the gravitational collapse of a scalar field

Demetrios Christodoulou|arXiv (Cornell University)|Dec 31, 1998

Cosmology and Gravitation Theories11 citations

TL;DR

This paper investigates the stability of naked singularities formed during spherically symmetric gravitational collapse of a scalar field, using nonlinear hyperbolic PDEs to model the dynamics. It proves that such singularities are unstable under generic perturbations, supporting the cosmic censorship conjecture by showing that naked singularities do not form with positive probability in this model framework.

ABSTRACT

One of the fundamental unanswered questions in the general theory of relativity is whether ``naked'' singularities, that is singular events which are visible from infinity, may form with positive probability in the process of gravitational collapse. The conjecture that the answer to this question is in the negative has been called ``cosmic censorship.'' The present paper, which is a continuation previous work, addresses this question in the context of the spherical gravitational collapse of a scalar field.

Motivation & Objective

• To investigate whether naked singularities—singularities visible from infinity—can form with positive probability during gravitational collapse of a scalar field.
• To test the validity of the cosmic censorship conjecture in the context of spherically symmetric scalar field collapse.
• To analyze the stability of singularities that arise in the solutions of the Einstein-scalar field equations under small perturbations.
• To determine whether such singularities are generic or require fine-tuned initial conditions.

Proposed method

• Formulates the spherically symmetric Einstein-scalar field equations as a system of nonlinear hyperbolic partial differential equations.
• Applies a double null foliation technique to analyze the behavior of the spacetime near the singularity.
• Uses a characteristic initial value problem formulation to study the evolution from regular initial data.
• Imposes regularity and decay conditions at future null infinity and at spatial infinity to ensure physical relevance.
• Performs a detailed asymptotic analysis of the solution near the developing singularity.
• Employs energy estimates and decay estimates for the scalar field and its derivatives to assess stability.

Experimental results

Research questions

• RQ1 Can naked singularities form in the gravitational collapse of a scalar field under generic initial conditions?
• RQ2 Are naked singularities stable under small perturbations of the initial data?
• RQ3 Does the formation of a naked singularity require fine-tuning of initial data, implying non-genericity?
• RQ4 What is the role of the Einstein-scalar field system in testing the cosmic censorship hypothesis?
• RQ5 How do the dynamics of the scalar field and curvature evolve near the singularity in the presence of a naked singularity?

Key findings

• Naked singularities that form in the spherically symmetric collapse of a scalar field are unstable under generic perturbations of the initial data.
• The set of initial data leading to naked singularities has measure zero in the space of all regular initial data, implying non-generic formation.
• Small perturbations of initial data that produce a naked singularity instead lead to a black hole, indicating instability.
• The solution exhibits strong curvature blow-up at the singularity, but this singularity is not robust under perturbations.
• Energy and decay estimates confirm that the scalar field and its derivatives remain bounded in regions away from the singularity, supporting the stability of the outer spacetime.
• The results support the weak cosmic censorship conjecture in the context of spherically symmetric scalar field collapse.