Re: Numerical General Relativity packages

*To*: mathgroup at smc.vnet.net*Subject*: [mg93938] Re: Numerical General Relativity packages*From*: magma <maderri2 at gmail.com>*Date*: Sat, 29 Nov 2008 04:30:04 -0500 (EST)*References*: <gglt37$8lh$1@smc.vnet.net> <ggofv9$s42$1@smc.vnet.net>

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