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

In summary, there are currently no known solutions to EFEs that are analogous to Helmholtz's vortex ring solutions in fluid dynamics. However, in 5 dimensions, a "black ring" solution with horizon topology ##S^1 \times S^2## has been proposed by Emparan and Reall, and expanded upon by others. In 4 dimensions, it is believed that any toroidal black hole would have to be non-stationary and quickly collapse into a standard Kerr black hole due to its required spherical topology.
  • #1
cuallito
95
1
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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  • #2
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
Thanks Ben, just generic, smooth solutions with some matter distribution.
 

1. What are Einstein's Field Equations?

Einstein's Field Equations are a set of ten equations that form the core of Einstein's theory of General Relativity. They describe the relationship between the curvature of spacetime and the distribution of matter and energy within it.

2. What is the "Smoke Ring" Solution for Einstein's Field Equations?

The "Smoke Ring" Solution is a hypothetical solution to Einstein's Field Equations that describes a rotating, cylindrical object with a circular cross-section. This solution is often used as a simplified model for understanding the properties of black holes.

3. How does the "Smoke Ring" Solution relate to black holes?

The "Smoke Ring" Solution is a simplified model for understanding the properties of black holes. It helps to illustrate how a rotating black hole can create a spinning vortex of spacetime, much like a smoke ring. However, this solution is not an exact representation of a black hole and should not be taken as a literal description.

4. Are there other solutions to Einstein's Field Equations?

Yes, there are many other solutions to Einstein's Field Equations, including the Schwarzschild Solution, the Reissner-Nordström Solution, and the Kerr Solution. Each of these solutions describes a different type of spacetime curvature and can be used to model different physical phenomena.

5. How do Einstein's Field Equations impact our understanding of the universe?

Einstein's Field Equations are a cornerstone of modern physics and have greatly influenced our understanding of the universe. They have been used to successfully predict the behavior of objects in extreme environments, such as near black holes, and have helped to shape our current understanding of gravity and the structure of spacetime.

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