A "Smoke Ring" solution to EFEs?

Tags:
1. Dec 27, 2016

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?

2. Dec 28, 2016

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.

3. Dec 28, 2016

cuallito

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