# Einstein Field Equation Solution - How?

## Main Question or Discussion Point

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

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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
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".

https://www.amazon.com/dp/0521461367/?tag=pfamazon01-20

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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".

https://www.amazon.com/dp/0521461367/?tag=pfamazon01-20
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

Last edited by a moderator:
Stingray
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.

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

Stingray
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.

oic...that's why not many books mention that the meaningful solution are not many too.

Matterwave
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.

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

Matterwave
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!).

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

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!).
hey Matterwave, why are the EFE in tensor form. :)

Matterwave