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 [itex]dr^2[/itex] is [itex]g_{rr}[/itex], and the quantity multiplying [itex]d\theta^2[/itex] is [itex]g_{\theta \theta}[/itex].

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

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