Gravitational radiation from a cylindrical naked singularity

We construct an approximate solution which describes the gravitational emission from a naked singularity formed by the gravitational collapse of a cylindrical thick shell composed of dust. The assumed situation is that the collapsing speed of the dust is very large. In this situation, the metric variables are obtained approximately by a kind of linear perturbation analysis in the background Morgan solution which describes the motion of cylindrical null dust. The most important problem in this study is what boundary conditions for metric and matter variables should be imposed at the naked singularity. We find a boundary condition that all the metric and matter variables are everywhere finite at least up to the first order approximation. This implies that the spacetime singularity formed by this high-speed dust collapse is very similar to that formed by the null dust and the final singularity will be a conical one. Weyl curvature is completely released from the collapsed dust.

Figure 1

Morgan’s cylindrical null dust solution. There is the null dust in the shaded region; $D(v)>0$ for $0<v<v_{\mathrm{w}}$. The dashed line on $t$-axis corresponds to the intermediate p.p. singularity at which an observer suffers infinite tidal force although any scalar polynomials of the Riemann tensor do not vanish. On the other hand, the dotted line on $t$-axis is the conical singularity. The Cauchy horizon is short dashed line of $t=r$. We estimate $C$-energy transfered through the null hypersurface $v=v_{\mathrm{w}}$ from $u=0$ to $u=v_{\mathrm{w}}$, which is depicted by dot-dashed line.