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

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SUMMARY

The discussion centers on the exploration of solutions to Einstein's Field Equations (EFEs) that resemble Helmholtz's vortex ring solutions in fluid dynamics. Notably, Emparan and Reall have developed a "black ring" solution in five dimensions characterized by horizon topology ##S^1 \times S^2##, which has been further elaborated to incorporate various angular momenta and charges. In contrast, stationary black holes in four dimensions are constrained to spherical topology, indicating that any toroidal black hole must be non-stationary and would ultimately collapse into a standard Kerr black hole.

PREREQUISITES
  • Understanding of Einstein's Field Equations (EFEs)
  • Familiarity with black hole topology, specifically Kerr black holes
  • Knowledge of fluid dynamics principles, particularly Helmholtz's vortex ring solutions
  • Basic grasp of higher-dimensional physics, especially in relation to black hole solutions
NEXT STEPS
  • Research the "black ring" solution by Emparan and Reall in five-dimensional space
  • Study the implications of horizon topology in black hole physics
  • Explore time-dependent solutions to Einstein's Field Equations
  • Investigate the relationship between fluid dynamics and gravitational theories
USEFUL FOR

Physicists, mathematicians, and researchers focused on general relativity, black hole physics, and the intersection of fluid dynamics with gravitational theories.

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

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