Hard General Relativity question, help ?

[hamjam9]

Hard General Relativity question, help :(!?

I cannot come up with a solution to the following, can some one please help me? I would greatly apprecieate it! Thank you!

Question 1

The Kretschmann curvature scalar is defined to be K = R_abcdR^abcd where Rabcd is the Riemann tensor. Compute the Kretschmann scalar for each of the following solutions of Einstein’s equtions:
(a) Schwarzschild exterior (vacuum) solution
(b) Schwarzschild interior (constant density) solution
(c) Friedman-LeMaitre-Robertson-Walker solution

In each case, express the Kretschmann scalar both in terms of the quantities occurring in the metric tensor, and in terms of the physical quantities that occur in the energy-momentum tensor, where possible.

Question 2

Describe the conditions under which a singularity exists, and the nature of the singularity for each of the solutions that you considered in problem 1 for the Kretschmann scalar. Ensure that you describe the appropriate physics in each case. Where possible, relate your conclusions to conclusions reached by other lines of argument for these metrics.

 

[gabbagabbahey]

What have you tried? You must show some attempt n order to get assistance here; it is forum policy.

A good place to start would be to use the definition of 'Kretschmann scalar'.

 

[Pacopag]

Gabba is right. It seems like it should be a straight forward matter of defining the K-scalar, and plugging everything in. I'm not saying that it won't take many hours to do this, and you'll probably make a few index errors along the way, but it is straight-forward. If you have Maple, you can download and install the GRTensorII package for free. Take a couple hours to learn to use GRTensorII. Then you'll be able to do this question in about five minutes. Also, the rest of your course will be much easier once you know GRTensorII because these sort of "turn the crank" questions are handled in milliseconds by GRTensorII. And IT doesn't make mistakes (I love computers and hate 'em at the same time).
Of course, it is best that you convince yourself that you COULD do the computation by hand if you had to before you put your faith in a computer program.

 

[cristo]

Of course, it is best that you convince yourself that you COULD do the computation by hand if you had to before you put your faith in a computer program.

Exactly: what's he going to do in an exam when Maple isn't there?

 

[Pacopag]

Ya. That's what I meant. I never use maple for anything that I'm not sure that I COULD do by hand if I had to. Or better yet, DO it by hand, then use maple to check your answer.

 

[quasar_4]

PS: There will be a new general relativity package in upcoming versions of Maple. It will be located in the Differential Geometry package. Keep your eyes open for it...

and incidentally, you don't have to download anything new, just go to the DG package in Maple. You can compute curvature tensors, Einstein tensors, etc. right there. Just my suggestion.