# Is there an exact solution to the Vacuum Einstein Equations for a universe with a single massless particle?

One can think of the Kerr metric as a solution to the EE's with an empty universe and a single non-charged, rotating, massive particle. The stress energy tensor is 0 because we remove the world line of the single particle, so it's really a vacuum solution on $$\mathbb{R}\times\mathbb{R_+}\times S^2$$. In the same way can we try to solve the vacuum EE's on $$3+1$$ dimensional space where we remove a null-line? Perhaps the path of a photon?

I'm not sure if photons have a stress energy tenser, but I'm in mathematical GR so I don't really care if they do. I'm looking for Vacuum Solutions, not Electro-Vacuum solutions. The purely mathematical question is as follows:

Does there exist a (non-trivial) Ricci flat, Lorentzian metric on $$(\mathbb{M_+}\sqcup \mathbb{M_-})\times S^2$$ with $$\mathbb{R}\times SO(2)$$ symmetry? Here $$\mathbb{M}$$ is $$1+1$$ dimensional Minkowski space, and $$M_\pm$$ is the region above/below a fixed null-geodesic. The $$SO(2)$$ symmetry is spacial rotation about the trajectory of the null-line. The $$\mathbb{R}$$ symmetry is Lorentz boosts' parallel to the removed null-line.

Possible methods of answering the question:

1) Use symmetries to solve the system. This appears to have the same number of symmetries as Kerr. But Kerr was only solvable because there was an extra hidden symmetry' coming from Noether's theorem. This one may or may not have that extra symmetry.

2) Take a limit of Lorentz boosts (an ultra boost?) of Kerr or even of Schwarschild metrics.

So if anyone has heard of such a metric, can do one of these methods, or can offer another method, please let me know.