Instability of naked singularities of perfect fluid under $C^{1,\alpha }$ gravitational perturbations

We study the instability of the naked singularities arising in the spherically symmetric self-similar collapsing of the Einstein–Euler system under gravitational perturbations. We show that small $C^{1,\alpha }$ perturbations (without any symmetries) of the initial conformal metric lead to trapped surface formation. One key point is that the speed of sound is strictly slower the the speed of light, so the gravitational radiation can become sufficiently concentrated to form a trapped surface before the fluid develops any singularity.