How Can You Simplify General Relativity Equations Without Using Tensors?

[Vardonir]

Homework Statement

Essentially, my adviser just told me to get the equation, then expand it so that it doesn't have any tensors. That's it. Just get rid of the tensors.

He said that this would be normally found in some textbook...? I've searched a lot of books, but I never saw anything that didn't have the equations in tensor form.

Homework Equations

I got G_μv = 8πT_μv.

Reading further, I also got the following:
G_μv = R_μv - (1/2)g_μvR
R = g^αβR_αβ

And I also found the Ricci tensor in terms of the Christoffel symbols but that's too long to type.

The Attempt at a Solution

What I tried doing is just plug in values of the greek indices and treat each one as a separate equation. However, when I use g_αβ = diag(-1, 1, 1, 1), the left hand side always becomes zero.

Is there any other value of the metric tensor that I can use?

 

[HallsofIvy]

The whole point of the Einstein tensor is that it is trivial in flat space, where there is no gravity. What you get for the Einstein tensor depends upon the metric tensor which, in turn, depends upon the distribution of mass in space. What situation are you assuming? You are trying to find the Einstein tensor for what space?

 

[Vardonir]

The whole point of the Einstein tensor is that it is trivial in flat space, where there is no gravity. What you get for the Einstein tensor depends upon the metric tensor which, in turn, depends upon the distribution of mass in space. What situation are you assuming? You are trying to find the Einstein tensor for what space?

I wasn't really given that much information, aside from "find the equations in terms of x, y, z, and t." I'm guessing Euclidean space, then.

 

[Muphrid]

I wouldn't plug in anything for the metric. Just leave the formulas in terms of $g_{tt}, g_{tx}$ and so on.