Even perturbations of the self-similar Vaidya space-time

We study even parity metric and matter perturbations of all angular modes in self-similar Vaidya space-time. We focus on the case where the background contains a naked singularity. Initial conditions are imposed, describing a finite perturbation emerging from the portion of flat space-time preceding the matter-filled region of space-time. The most general perturbation satisfying the initial conditions is allowed to impinge upon the Cauchy horizon (CH), where the perturbation remains finite: There is no “blue-sheet” instability. However, when the perturbation evolves through the CH and onto the second future similarity horizon of the naked singularity, divergence necessarily occurs: This surface is found to be unstable. The analysis is based on the study of individual modes following a Mellin transform of the perturbation. We present an argument that the full perturbation remains finite after resummation of the (possibly infinite number of) modes.

Figure 1

Conformal diagram for Vaidya space-time admitting a globally naked singularity. The region preceding $\mathcal{N}$ is a portion of Minkowski space-time, and the region to the future of $\mathcal{N}$ is filled with null dust. There are three similarity horizons at which the similarity coordinate $x$ is null: $x=0$ denoted $\mathcal{N}$, $x=x_c$ shown as a dashed line, and $x=x_e$ shown as a double line. We identify $x=x_c$ as the Cauchy horizon and will call $x=x_e$ the SFSH. The apparent horizon given by $x=1/\lambda$ is shown as a bold curve.

Figure 2

Conformal diagram for Vaidya space-time with censored singularity, corresponding to the case $\lambda >1/8$. A spacelike singularity forms at $r=0$ for $v\ge 0$.