# Negative mass thin shell collapse

Suppose we have a collapsing light-like (ingoing) shell with negative mass and decreasing further. The shell is radiating and so the exterior region is that of the outgoing Vaidya solution.

$$ds^2 = -(1-\frac{2m(u)}{r})du^2-2 du dr +r^2d\Omega^2$$

By the time the shell reaches the origin, the mass diverges to negative infinity. But the Vaidya mass function is given by $m=m(u)$, where $u$ is a null coordinate and it is constant on an outgoing light ray.

I am having trouble finding out what results from this collapse configuration:

1. Would a timelike singularity with infinite negative mass form from this sort of collapse? If so, what kind of exterior spacetime would it have?

2. Once the shell collapses to $r=0$ (or, say, at some finite $u=u_1$), the mass function as seen from an outgoing lightray at $u_1$ is infinite (negative), so the Kretschmann scalar $\left(K=\frac{48m^2(u)}{r^6}\right)$ would diverge and yield a light-like curvature singularity, correct?

I thought light-like singularities in this spacetime would have no mass, since the event horizon would be at the singularity itself and so $m=r/2$ at $r=0$.

A hand would be appreciated. Open for brainstorming.

• Thank you for your reply. I am actually following work from (arxiv.org/pdf/gr-qc/9608063.pdf). Here, the authors match an interior spacetime which is that of Roberts spacetime of a self-similar massless scalar field, to an exterior outgoing Vaidya solution. The matching can be done smoothly (no thin shell), but the authors also consider the case of a nonvanishing shell, so the mass function as seen from the outside is given by $(M_{\phi}(u)+M_{shell}(u))$. Their approach on the negative mass shell is rather questionable (Fig.5), so I was checking a different approach. – RMAndre Mar 23 '16 at 9:25