Einstein's Field Equations: "Smoke Ring" Solution?

[cuallito]

Hello, does anyone know if a solution to EFEs (Einstein's Field Equations) has been found that's analogous to Helmholtz's vortex ring solutions in fluid dynamics?

 

[Ben Niehoff]

Are you specifically interested in black hole solutions? Or just generic, smooth solutions with some matter distribution (i.e., a fluid with a vortex ring, for example)?

As far as black holes are concerned, in 5 dimensions, Emparan and Reall have written down a "black ring" solution with horizon topology $S^1 \times S^2$. It has been expanded upon by others to include all sorts of angular momenta and charges.

However, in 4 dimensions, stationary black holes must have spherical topology. Therefore if any sort of toroidal black hole exists, it must be non-stationary; i.e. time-dependent. It would quickly collapse into a standard Kerr black hole.

 

[cuallito]

Thanks Ben, just generic, smooth solutions with some matter distribution.