Getting the Ricci and metric tensor from T ?

[cuallito]

Okay, we have Einstein's field equation:

R_ab + 1/2 R g_ab = 8pi T_ab

Let's say we have T_ab defined for some region of space, and we want to calculate the spacetime from that. How would you calculate R_ab, R and g_ab? Supposedly you can write it as a system of PDEs but I cannot find them anywhere!

I'm just starting to learn GR, if I could see how you'd put it in a computer to solve them it would help me "see" it better. Thanks.

 

[malawi_glenn]

But it all depends on who T_ab looks, there is no "general" solution to ALL.

The easiest solution is obtained by T_ab as static (no timedep.) and spherical symmetric, that solution for the metric is called the Schwarzschild metric.

http://en.wikipedia.org/wiki/Schwarzschild_metric
here is an outline of it's derivation:
http://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

I mean HOW to solve it depends from case to case, you are just after the form of the PDE's? But that is trivial to find, just plug in whatever you have for the Ricci tensor and Ricci scalar in terms of the metric (your multidimensional function which depends on several variables)

 

[cuallito]

Okay, how do you calculate the Ricci scaler and tensor in the first place then?

 

[malawi_glenn]

Are you doing a class in GR?

The Ricci tensor are in terms of the cristoffel symbols:
http://upload.wikimedia.org/math/8/8/0/88070e13792efa80fbf539132420d523.png
and the cristoffel symbols in terms of metric:
http://upload.wikimedia.org/math/0/f/e/0feb8ec87e7564490a3716aa8743483e.png

the Ricci scalar is just the trace of the Ricci tensor:
$$R = g^{\mu \nu}R_{\mu \nu}$$

Are you learning GR from a book or? Your question is quite.. hmm.. rare and strange :P

 

[cuallito]

Just learning it on my own. I have Wald's GR right now, I'm getting Einstein's populist book and GR A to B soon thru ILL hopefully.

 

[malawi_glenn]

Just learning it on my own. I have Wald's GR right now, I'm getting Einstein's populist book and GR A to B soon thru ILL hopefully.

ok cool!

But how come you asked about solutions to Einsteins equation if you don't know how how ricci tensor and ricci scalar is related to the metric?... I strongly recommend to study in logical order :-)