An Easy Metric for Einstein Field Equations

In summary, the conversation discusses the difficulties of solving the Einstein Field equations for a wormhole metric, and the suggestion is made to use simpler metrics such as the spherically symmetric and axi-symmetric metrics. The benefits of using computer algebra programs are also mentioned. The conversation then shifts to a discussion of the basics of the Ricci Scalar, Einstein Tensor, and Field equations, with an example provided for a variant of the Robertson-Walker metric. It is suggested to use a symbolic math program such as Maxima to simplify the calculations.
  • #1
edgepflow
688
1
So I am an engineering graduate trying to teach myself some general relativity.

I have tried to solve the Einstein Field equations for a wormhole metric and some others.

After pages and pages of calculating Christoffel Symbols, Riemann Tensors, Ricci Tensors and Scalars, and so on, I end up with a mess that is not correct.

Is there a simple metric(s) I could start with that is less messy?
 
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  • #3
edgepflow said:
So I am an engineering graduate trying to teach myself some general relativity.

I have tried to solve the Einstein Field equations for a wormhole metric and some others.

After pages and pages of calculating Christoffel Symbols, Riemann Tensors, Ricci Tensors and Scalars, and so on, I end up with a mess that is not correct.

Is there a simple metric(s) I could start with that is less messy?

One of the easiest metrics to begin with is the spherically symmetric metric that leads to the Schwarzschild solution. Another relatively easy metric, but not quite as easy algebraically is the axi-symmetric metric that leads to the Weyl solution. As the previous post indicates, using computer algebra programs is a nice way to go. However, my experience is that there is some benefit to doing a few derivations out the long hard way. What you can learn is to see patterns in the symbols that allow cancellations and subsequently neat, pretty equations. I don't know if computer algebra programs are as adept at this as the human mind. Of course, I may be a little old fashioned that way... :).

Allan
 
  • #4
Thank you for the posts.

I downloaded the links. I have heard of GRTensor for mathmaticia. Is there anything like this available for MathCAD?

Please help me check that I have some basics straight.

For a coordinate basis (t,u,v,w), the Ricci Scalar is:

R = g^tt Rtt + g^uu Ruu + g^vv Rvv + g^ww R ww

The Einstein Tensor is:

Gtt = Rtt - (1/2) R gtt
Guu = Ruu - (1/2) R guu
Gvv = Rvv - (1/2) R gvv
Gww = Rww - (1/2) R gww

With no other terms included in each term.

The Field equations are now:

Gtt = 8 Pi G Ttt
Guu = 8 Pi G Tuu
Gvv = 8 Pi G Tvv
Gww = 8 Pi G Tww

With no other terms for each coordinate.

Let me know if I have these basics straight.
 
  • #5
Those equations look right for an orthogonal metric.

If you want to work out an earsy example by hand, try this

[tex]
ds^2=-dt^2+a(t)(dx^2+dy^2+dz^2)
[/tex]

[tex] 0<t<\infty,\ \ \ -\infty<x,y,z<\infty
[/tex]

which is a variant of the Robertson-Walker metric.

The Christoffel symbols are

[tex]
{\Gamma^t}_{xx}={\Gamma^t}_{yy}={\Gamma^t}_{zz}= \frac{\dot{a}}{2},\ \ \ {\Gamma^t}_{tx}={\Gamma^t}_{ty}={\Gamma^t}_{tz}= \frac{\dot{a}}{2a}
[/tex]

The Einstein tensor components are

[tex]
G_{tt}=\frac{3\,{\left( \dot{a}\right) }^{2}}{4\,{a}^{2}},\ \ \ G_{xx}=G_{yy}=G_{zz}= \frac{\left( \dot{a}\right) ^{2}-4a\ddot{a}}{4a}
[/tex]

You should get some symbolic math program to do this for you - I can recommend Maxima if you don't have Maple or Mathematica.
 
Last edited:

1. What is the "An Easy Metric for Einstein Field Equations"?

The "An Easy Metric for Einstein Field Equations" is a mathematical equation that describes the relationship between the curvature of space-time and the distribution of matter and energy in the universe. It was developed by physicist Albert Einstein in his theory of general relativity.

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Yes, the "An Easy Metric for Einstein Field Equations" has many current applications and implications. It has been used to develop technologies such as GPS and to understand the behavior of black holes. It also plays a crucial role in modern cosmology and the study of the origins and evolution of the universe.

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