The Einstein equation

The well-known solutions ( well known to me, that is ) were not derived by starting with a EMT and solving the Einstein field equations for the metric. The Schwarzschild metric has zero Ricci tensor and so represents a vacuum near a spherically symmetric, non-rotating source. The spacetime derives its properties from spherical symmetry alone. I highly recommend the English translation of Scharzschilds paper.

The basic cosmological solution is derived from the principles of homogeneity ( same to all observers) and isotropy ( no preferred direction). It is a happy accident that the Einstein tensor of this line element ( Robertson-Walker) turns out to describe a space filled with non-interacting, pressureless ( no collisions) dust. This leads to the Friedmann equations which link the expansion rate to the energy density, ( and look where that got us - Dark Matter ).

There are probably numerical methods available to solve the EFE, but it's doubtful whether closed algebraic solutions even exist for most general EMTs.

 

The Riemann tensor is only related to the metric and its derivatives. Isn't this in your book ?
[tex]R^{r}_{mqs} = \Gamma ^{r}_{mq,s}-\Gamma ^{r}_{ms,q}+\Gamma ^{r}_{ns}\Gamma ^{n}_{mq}-\Gamma ^{r}_{nq}\Gamma ^{n}_{ms}[/tex]
where
[tex]\Gamma^m_{ab}=\frac{1}{2}g^{mk}(g_{ak,b}+g_{bk,a}-g_{ab,k})[/tex]
The comma before an index indicates partial differentiation. The Ricci tensor is the contraction of the Riemann tensor on the first and third indices, and appears in the EFE.

 

I can't see how the expression could be simplified.

I don't know enough about numerical relativity to answer your question. There ought to be information available on the internet, so a search would probably get something useful.

 

I'm afraid the equation doesn't become any simpler than that; The Einstein's equations are quite horrible to write down explicitly in terms of the metric.

 

ghc,

I forgot to mention that there's a very good free math program called Maxima, which runs on most OSs. It has a package called ctensor that can calculate most of the GR tensors, when given a metric. It also does frame calculations ( something you should investigate if you're not familiar with the concept ).

You can get Maxima and it's GUI interfaces xMaxima and wxMaxima here.
http://maxima.sourceforge.net/

Coming back to solving the EFE. I'm told by a correspondent who doesn't post here that the solution of the EFE is greatly simplified if you use the method of frame fields, aka "local Lorentz frames", aka "using Cartan's method of moving frames") . He gives a 'recipe' for this -

I haven't studied this myself yet, and I may have made mistakes in transcription.

M

 

I've always wondered what the EFE looks like in terms of just the metric and the stress-energy tensor. Figured I might try it out when I have some free time :) Anybody want to give it a go and TeX it for our amusem--erm, edification?