Scalar perturbations to naked singularities of perfect fluid

TL;DR

This work addresses the stability of naked singularities in the Einstein--Euler system with an isothermal fluid by analyzing a scalar-field perturbed, spherically symmetric, self-similar model. It employs double-null coordinates, null structure equations, and robust a priori estimates to prove that such naked singularities are unstable to trapped surface formation under $C^{1,\alpha}$ perturbations of an external massless scalar field, with the instability mechanism driven by the scalar field's blueshift along the past null cone. The results extend prior instability findings beyond exact self-similarity to a broader class of solutions and establish a rigorous framework for gravitational perturbations in the Einstein--Euler context, supporting a form of weak cosmic censorship in this toy model. The analysis highlights the critical role of the lapse function $\Omega$ and energy flux $|rL\phi|^2$ on the past null cone in triggering horizon formation, and suggests a broader applicability to matter fields with subluminal sound speeds.

Abstract

In this paper, we study the instability of naked singularities arising in the Einstein equations coupled with isothermal perfect fluid. We show that the spherically symmetric self-similar naked singularities of this system, are unstable to trapped surface formation, under $C^{1,α}$ perturbations of an external massless scalar field. We viewed this as a toy model in studying the instability of these naked singularities under gravitational perturbations in the original Einstein--Euler system which is non-spherically symmetric.

Figures (2)

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Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Lemma 2.1
• Remark 2.1
• proof
• Theorem 3.1