Einstein Field Equation Solution - How?

  1. Hi,
    I just wonder is there good reference to show how to solve Einstein's Field Equation? It seems that his equation can generate many possibilities but what techniques that require us to study and solve it???

    Thanks
    Alex
     
  2. jcsd
  3. I'm not sure if this is what you are looking for, but there is the well-known book; "Exact Solutions to Einstein's Field Equations".

    http://www.amazon.com/Exact-Solutio...1?ie=UTF8&s=books&qid=1268855356&sr=8-1-spell
     
  4. Thanks! and I read this before, it is too far beyond my understandings. Is there any elementary source that we can grab the idea before I go any further??

    Thanks
    Alex
     
  5. Stingray

    Stingray 674
    Science Advisor

    What are you looking for? Einstein's equation can be solved analytically only in very special cases. You usually start by assuming some symmetry or special form for the metric.

    You can also look at perturbation theory in various approximations or at numerical simulations. Both of these methods can get very complicated.
     
  6. Thanks!

    I am looking for what mathematical techniques that require to solve the equation. Any paper/reference that shows how to solve it??

    thanks
    Alex
     
  7. Stingray

    Stingray 674
    Science Advisor

    The main mathematical technique for finding exact solutions is essentially "guess and check." You can also look up things like the Kerr-Schild ansatz as well as transformations that take you from one axisymmetric solution to another. See standard textbooks like Wald. As with almost any nonlinear equation in physics, these methods are very limited.
     
  8. oic...that's why not many books mention that the meaningful solution are not many too.
     
  9. Matterwave

    Matterwave 3,816
    Science Advisor
    Gold Member

    In order to understand how to solve the EFE's even in very special cases, one needs to understand Tensor analysis. This is a main reason why General Relativity is not generally taught at the undergraduate level.
     
  10. Hi Matterwave,
    I even read a book mentioned that in order to solve E's equation, you should have much more imagination rather than mathematics skills.
    Since I feel that in order to have some breakthrough, solving E's equation is a key point. It may have other strange but meaningful results that may change our view to the world.

    Alex
     
  11. Matterwave

    Matterwave 3,816
    Science Advisor
    Gold Member

    Yes, imagination is important, and Einstein always said Imagination is more important than Knowledge; however, one has to at least have the basic mathematical tools ready to deal with these equations in order to do real science.

    I wouldn't try to redo Newton's laws without at least understanding arithmetic, and algebra (and probably not until I understood differential equations)! In this case, tensor analysis is required in 2 ways.

    1) Almost all formulations of the EFE's you're going to come across are in tensor form. And almost every step in obtaining the EFE's are in tensor form. It'd be very hard to do anything without at least understanding that the equation means and where it comes from.

    2) Tensors allow you to write one equation instead of 10 (or 6, if you're real clever) for just the basic EFE's. And, the number of terms you need to solve grow exponentially as the number of dimensions increases. One would not want to solve 256 different equations just to get anywhere. With tensors, you can compact those 256 equations into 1. Calculating Christoffel symbols and the like would be very very annoying without tensor analysis (even with tensors, it's annoying!).
     
  12. Absolutely agree with your points.
    I am a self-learner and at the beginning, I found headaches with those tensor symbols. After a few months effort (almost stay in library everyday), now feel better and the concept gradually becomes clear. However, there is still far away to understand deeply the EFE and hope that one day I can hear a new solution is found by physicists that allows us to view our world in different perspective.

    Alex
     
  13. hey Matterwave, why are the EFE in tensor form. :)
     
  14. Matterwave

    Matterwave 3,816
    Science Advisor
    Gold Member

    That's due to point 2. Tensor analysis is a very powerful tool that can shorten notation from several hundred equations into 1. If we have access to this tool, why not use it?
     
  15. ... For the same reason that a penguin wears a tuxedo. :rofl: ... just kidding!
     
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