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From: psycho_dad on 28 Nov 2008 05:08
Numerically solve the field equations? The field equations give
different differential equations for different metrics. So, I guess
you need a program that given a metric will give you the ODEs that
govern the evolution of the metric components etc. Then you can use
Mathematica's DSolve or NDsolve to solve these equations analytically
(if you are lucky) or numerically respectively.

One such package is RGTC. To download it and see a complete list of
its features, have a look at:

http://www.inp.demokritos.gr/~sbonano/RGTC/

Cheers,
psycho_dad

From: magma on 29 Nov 2008 04:29
Thank you for pointing at RGTC, but that is not what I needed.
Packages like RGTC, MathTensor or David Park's Tensorial normally
tackle the opposite problem:
given a metric tensor, calculate the curvature tensors.
This is conceptually easy: just take the metric and calculate al lot
of derivatives.

What I need is a package which solves the opposite problem: given the
Ricci tensor (or the Einstein tensor, or the stress-energy tensor),
calculate the underlining metric.
More exactly: given the initial conditions (metric and fields and
derivatives) on an initial spacelike surface (the cosmos NOW),
calculate the evolution of the cosmos in the future (or in the good
old past if you prefer) using the field eqs as your time machine.
There are 10 field eqs and 10 metric components, but only 6 eqs are
functionally independent, so indeed the system is under determined.
This is because for any metric solution that you find, you may find
another metric solution related to the first via an arbitrary
coordinate transformation. So the 4 degrees of freedom left by the eqs
reflect the 4 degrees left in choosing the coordinates system.
You fix that by giving suitable initial conditions.
The Einstein's eqs can be solved symbolically for idealized highly
symmetric spacetimes, but not for example, for a black hole swalloing
a star and consequently emitting gravitational waves. These
calculations are done numerically.
Some software exists to perform these calculations, but AFAIK not for
Mathematica.
I have recently looked at the NSolve capabilities in 6.0. 3 and they
are very impressive and well documented.
So perhaps it won't be too difficult to write a solver package from
scratch. We'll see.

Assuming one knows general relativity, a good introductory text is:
Elements of numerical relativity
by Bona and Palenzuela-Luque,
Springer Verlag



> Numerically solve the field equations? The field equations give
> different differential equations for different metrics. So, I guess
> you need a program that given a metric will give you the ODEs that
> govern the evolution of the metric components etc. Then you can use
> Mathematica's DSolve or NDsolve to solve these equations analytically
> (if you are lucky) or numerically respectively.
>
> One such package is RGTC. To download it and see a complete list of
> its features, have a look at:
>
> http://www.inp.demokritos.gr/~sbonano/RGTC/
>
> Cheers,
> psycho_dad


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