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Is there any solution to Einstein's field equation left to be discovered?

Is it still possible to get a solution of Einstein's field equation?

And is every mathematical solution of Einstein's field equation physically valid?

hft
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Keshav Shrestha
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  • Welcome to Physics SE. We recommend that you search first and tell us what you have found so far (you can edit the question anytime). As for the third point, the answer is no, because even if the solutions were physically valid, we do not know how to extend Einstein's equation to quantum theory so it cannot be a complete theory. – Mauricio Sep 01 '22 at 12:28
  • Thank you, and apology for this question but I searched these question a lot on Google mainly on Quora but those answers for this question are not free and ask to pay the money to access those answers given by professional so i asked these questions here i know Physics SE is not a site for discussion but couldn't wait myself to ask such questions. One more question by me is that given a specific energy momentum tensor on R. H. S of Einstein's field equation can't we solve for Einstein tensor which is on L.H. S of Einstein's field equation and get a required solution or required metric? – Keshav Shrestha Sep 01 '22 at 12:36
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    In principle yes. In practice it may be very hard depending of the form of the tensor. There are not many analytical solutions. – Mauricio Sep 01 '22 at 12:49
  • So is Schwarzschild metric, Vaidya metric, kerr metric, Vaidya-kerr metric an analytical solution of Einstein's field equation? – Keshav Shrestha Sep 01 '22 at 12:57
  • Here is a reference: http://www.scholarpedia.org/article/Exact_solutions_of_Einstein%27s_equations – Mauricio Sep 01 '22 at 12:57
  • Thank you for your generous help, one more question for you: Is there any important research and discovery on black hole after Hawking? If so would you please provide me a link to those research paper? – Keshav Shrestha Sep 01 '22 at 13:05
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    Sabina you are asking too many questions. Please post a formal question for each of your doubts. – Mauricio Sep 01 '22 at 13:08
  • Ok i will post formal questions. – Keshav Shrestha Sep 01 '22 at 13:14

3 Answers3

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Quick answer is no.

There is no general analytical technique to take a given stress-energy tensor and solve for the metric.

Treated as a numerical task (solving by computer numerical integration methods) the task is also extremely challenging. For many cases of interest the numerical methods on large modern computer networks can only yield rough answers.

A further difficulty is that one might not even know how to get the stress-energy tensor until the metric is known. This happens for example when the gravitational effects are very large (typically neutron stars and black holes).

Of course even in Newtonian physics the three-body problem is known to be also very challenging.

Andrew Steane
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  • Thank you for your answer, will you please tell me how much effective is perturbation method to get solution to Einstein's field equation? – Keshav Shrestha Sep 01 '22 at 13:00
  • @SabinaShrestha perturbation methods are effective and widely used. They still require a lot of care to include everything correctly, because the field equation is both non-linear and also deals with a lot of variables. – Andrew Steane Sep 02 '22 at 09:23
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Is there any solution to Einstein's field equation left to be discovered?

Probably most of the interesting solutions have been found yet. Anyway, all the time someone is publishing new solutions, for example two months ago this new exact solution was found.

Is it still possible to get solution of Einstein's field equation?

The above recent solution corresponds to an anisotropic matter distribution. If you try to find solutions for exotic or uncommon matter distribution, possibly you can find solutions previously unnoticed. However, the general procedures that have been used have already generated hundreds of qualitatively different solutions and it will be increasingly difficult to find new solutions, especially if they correspond to particularly simple matter distributions.

And do every mathematical solution of Einstein's field equation can be physically valid?

Almost certainly not, the white holes are an exact solution of Einstein's equations, but almost no one believes they are physically possible solutions for a host of reasons. Likewise, some of the best-known solutions lead to mathematical singularities, which ultimately physically could be something else (a future theory of quantum gravity should make it clear to us what those surely unphysical singularities we see in GR correspond to).

Urb
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Davius
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Is there any solution to Einstein's field equation left to be discovered?

Yes, infinitely many.

Is it still possible to get a solution of Einstein's field equation?

Yes, for example, if you restrict yourself to static spherically symmetric perfect fluid solutions you can find a new one by solving that second order linear differential equation https://physics.stackexchange.com/a/679431. Personally, I have found some new solutions that have been not yet published.

And is every mathematical solution of Einstein's field equation physically valid?

The consensus among relativists is no. I think otherwise.

JanG
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