Re: Visualizing a curved space
- From: tessel@xxxxxx
- Date: Wed, 19 Jul 2006 06:35:03 +0000 (UTC)
On Sat, 15 Jul 2006, David Park wrote:
If you are willing to invest a little money, $150 or $180 to buy the required Mathematica packages then you could easily investigate and visualize cases such as you describe. And if you wanted to you could join a small group (3 persons at present) working on learning these subjects.
They are the Tensorial 4.0 package for tensor calculus and the DrawGraphics package for making diagrams and animations. The ExtendUnits package might also be useful for implementing geometric units when calculating practical cases and for working with units in general.
To the OP: I am not familiar with Tensorial but I think I can suggest an alternative approach which will probably be more convenient and will certainly be significantly cheaper, at least for readers who are registered university students or who already own Mathematica or Maple.
For investigating specific (semi)-Riemannian manifolds, you will almost certainly get more bang for your time/money with a mature, powerful and FREE Maple package (compatible with Maple 9.5-10.0)
http://grtensor.org/
I am not affiliated in any way with the GRTensorII team, BTW, but I have used this software extensively and know that it is convenient and very easy to learn and use. In his recent book, Eric Poisson also sings the praises of GRTensorII.
Some specific comments: GRTensorII fully supports the convenient use of NP formalism and frame fields (orthonormal bases, rather than coordinate bases) for tensor computations. Even for working with coordinate bases, the built in Maple tensor package is incredibly clunky by comparision. Datasharing with other Maple packages can sometimes be a bit problematic with GRTensorII, but I might be able to help with specific problems of this nature. For example, computing Killing vectors is a breeze with GRTensorII and the casesplit command from Maple's built-in PDETools package.
In fact, as I see it, there is only one possible caveat: Maple itself is not cheap. But registered students can get it (legally) for a hundred dollars or so in the US, so for students Maple plus GRTensorII is definitely the way to go. Similarly, Mathematica is not cheap---unless you are a registered university student. (Last time I checked, at least.)
For working with completely general Lorentzian manifolds, e.g. symbolic index gymnastics, other packages may offer significant advantages, but these tend to be more specialized.
For Mathematica users unwilling to try Maple (not that there is much new syntax or many new commands to learn if you just want to start using GRTensorII), there is GRTensorM, an older and somewhat less capable counterpart of GRTensorII developed by the same team (come to think of it, GRTensorM is so old that it might not work with the most recent versions of Mathematica). Another good choice for Mathematica users might be the package
http://www.math.washington.edu/~lee/Ricci/
which is mature, reliable, well documented, and free. I am slightly acquainted with Jack Lee but have no affiliation with his software.
HTH!
"T. Essel"
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