Penrose hypothesis and instability of naked singularities in static spherically symmetric systems with scalar fields

TL;DR

This paper investigates the stability of static spherically symmetric scalar field configurations with naked singularities, confirming Penrose's hypothesis that such singularities are generally unstable, especially for small scalar charge Q.

Contribution

It provides a numerical analysis of linear radial stability of scalar field configurations, demonstrating instability for small Q and stability for large Q, supporting Penrose's conjecture.

Findings

Configurations are unstable for small Q, with divergent perturbation modes.

Configurations are stable for large Q, with no divergent modes found.

Supports Penrose hypothesis regarding naked singularity instability.

Abstract

General relativistic static spherically symmetric (SSS) asymptotically flat configurations with scalar fields typically contain naked singularities at the center. We consider minimally coupled scalar fields with power-law potentials leading to the Coulomb asymptotic of the field $\phi(r)\approx Q/r$ for large values of the radial variable r. The configurations are uniquely defined by total mass and a Q-parameter characterizing the strength of the scalar field at spatial infinity. The focus is on the linear stability against radial (monopole) perturbations of the SSS configurations satisfying conditions of asymptotic flatness. Our numerical investigations show the existence of divergent modes of small perturbations against the static background, at least for sufficiently small values of Q. This means instability of the configurations, confirming the well-known Penrose conjecture about the nonexistence of naked singularities - in this particular case. On the other hand, we have not found divergent modes of linear radial perturbations for sufficiently large Q.