# An Easy Metric for Einstein Field Equations

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?

bcrowell
Staff Emeritus
Gold Member
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

Thank you for the posts.

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.

Those equations look right for an orthogonal metric.

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

$$ds^2=-dt^2+a(t)(dx^2+dy^2+dz^2)$$

$$0<t<\infty,\ \ \ -\infty<x,y,z<\infty$$

which is a variant of the Robertson-Walker metric.

The Christoffel symbols are

$${\Gamma^t}_{xx}={\Gamma^t}_{yy}={\Gamma^t}_{zz}= \frac{\dot{a}}{2},\ \ \ {\Gamma^t}_{tx}={\Gamma^t}_{ty}={\Gamma^t}_{tz}= \frac{\dot{a}}{2a}$$

The Einstein tensor components are

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

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

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