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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.

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This isn't really answerable in the form you have it. The question I would ask is: what boundary condition are you putting on at the surface of the shell? It's not enough to just assert that the exterior solution is the Vaidya solution, you do need to apply some sort of conditions from the (supposedly Minkowski) interior of the shell and the exterior of the shell, as well as some equation R(u) giving the location of the shell as a function of (presumably null) time.

If you're stuck, refer to Eric Poisson's excellent book, A relativists' cookbook, which has a chapter on this very type of problem.

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  • $\begingroup$ 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. $\endgroup$ – RMAndre Mar 23 '16 at 9:25

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