General Relativity: Curvature and Stress Energy Tensor

[StephenD420]

Hello all,

I have a quick question regarding the relation of the space-time metric and the curvature. I have determined the space-time metric, g_(alpha beta), but I am unsure as how to go from the line element ds^2 = [ 1 + (dz/dr)^2] dr^2 + r^2 dtheta^2
and the space-time metric g to the curvature R_(alpha beta)
which I can then use R_(alpha beta) = (G/c^4) T_(alpha beta) to find the stress energy tensor. So, stated in another way, my question is how do I go from line element and space-time metric to the curvature? Do I have to go through all of the Christoffel symbols and is there a formula to help with this?

Thanks for any help you guys can give.
Stephen

 

[stevendaryl]

Hello all,

I have a quick question regarding the relation of the space-time metric and the curvature. I have determined the space-time metric, g_(alpha beta), but I am unsure as how to go from the line element ds^2 = [ 1 + (dz/dr)^2] dr^2 + r^2 dtheta^2
and the space-time metric g to the curvature R_(alpha beta)
which I can then use R_(alpha beta) = (G/c^4) T_(alpha beta) to find the stress energy tensor. So, stated in another way, my question is how do I go from line element and space-time metric to the curvature? Do I have to go through all of the Christoffel symbols and is there a formula to help with this?

Thanks for any help you guys can give.
Stephen

There are a bunch of steps, and I don't know of any shortcuts.

First, from the line element, you can read off the metric components: the quantity multiplying $dr^2$ is $g_{rr}$, and the quantity multiplying $d\theta^2$ is $g_{\theta \theta}$.

Next, from $g_{\alpha \beta}$ and its inverse $g^{\alpha \beta}$ and its derivatives, you can compute the Christoffel symbols $\Gamma^\mu_{\alpha \beta}$.

Finally, from $\Gamma^\mu_{\alpha \beta}$, its derivatives, and $g$ and its inverse, you can compute $R^\mu_{\alpha \beta \gamma}$. As far as I know, there is no shorter way.