Quasi-normal modes of naked singularities in presence of non-linear scalar fields

TL;DR

This paper investigates the stability and quasi-normal mode frequencies of naked singularities in General Relativity with non-linear scalar fields, revealing significant differences from Schwarzschild and linear scalar field cases.

Contribution

It provides the first analysis of axial perturbations and QNM frequencies for naked singularities with non-linear scalar fields, demonstrating linear stability and highlighting the impact of non-linearity.

Findings

Positive effective potential ensures linear stability.

Significant differences in QNM frequencies compared to Schwarzschild.

Distinct frequencies from linear scalar field backgrounds.

Abstract

We study linear perturbations against static spherically symmetric background configurations of General Relativity with a real scalar field (SF), which is minimally coupled with gravity; it is non-linear due to the presence of the self-action potential. The background solutions have a naked singularity at the center of the configuration. The focus is on the stability of the background and fundamental frequencies of the quasi-normal modes (QNM) of the axial perturbations in the Regge-Wheeler gauge. The problem is reduced to one hyperbolic master equation with an effective potential $W_{\rm eff}$, which turns out to be positive for a general non-negative SF potential; this ensures the linear stability with respect to this kind of perturbations. For numerical simulations, the SF potential was chosen in the power-law form $V(\phi)\sim\phi^{2n}$ with $2<n\le 40$. We extracted the fundamental frequencies of QNM for different $n$ and various sets of the background configuration parameters. The results show that even for a small background SF, there is a significant difference between the fundamental frequencies and ones in case of the Schwarzschild background. The results are also compared with the case of the Fisher-Janis-Newman-Winicour background dealing with a massless linear scalar field.