Einstein Field Equation Solution - How?

In summary: The EFE are written in tensor form because they are geometric relationships that describe the curvature of spacetime. Tensors are the mathematical objects used to describe the geometry of spacetime. So, it's a natural fit to write them in tensor form. Additionally, as mentioned before, using tensors allows us to write those relationships in a more compact and elegant form.
  • #1
physics.alex
28
0
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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  • #2
physics.alex said:
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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  • #3
elect_eng said:
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:
  • #4
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.
 
  • #5
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
 
  • #6
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.
 
  • #7
oic...that's why not many books mention that the meaningful solution are not many too.
 
  • #8
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.
 
  • #9
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
 
  • #10
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!).
 
  • #11
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
 
  • #12
Matterwave said:
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. :)
 
  • #13
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?
 
  • #14
Gatchaman said:
hey Matterwave, why are the EFE in tensor form. :)

... For the same reason that a penguin wears a tuxedo. :rofl: ... just kidding!
 

1. How did Einstein come up with the field equations?

Einstein developed the field equations as part of his theory of general relativity, which he published in 1915. He was inspired by the work of mathematician David Hilbert and physicist Hermann Minkowski, and also drew on his own previous work on special relativity.

2. What do the Einstein field equations represent?

The Einstein field equations are a set of ten nonlinear partial differential equations that describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. In simpler terms, they explain how gravity works on a large scale.

3. How are the Einstein field equations solved?

The Einstein field equations can be solved using a variety of mathematical techniques, including numerical simulations and perturbative methods. However, exact solutions to the equations are rare and often only apply to specific conditions or idealized scenarios.

4. What is the significance of finding a solution to the Einstein field equations?

Finding a solution to the Einstein field equations can provide valuable insights into the behavior of gravity and the structure of spacetime. It can also help us understand the behavior of massive objects, such as black holes, and make predictions about the evolution of the universe.

5. Are there any unsolved problems or limitations of the Einstein field equations?

While the Einstein field equations have been extremely successful in explaining many phenomena, there are still some unsolved problems and limitations. For instance, the equations do not fully incorporate the principles of quantum mechanics and cannot fully explain the behavior of black holes. Scientists continue to work on improving our understanding of gravity and the universe through further research and experimentation.

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