# RE: [Axiom-mail] how to calculate Ricci tensor from a metric tensor?

On March 25, 2007 9:55 AM Ondrej Certik wrote:
>
> I am new to Axiom, I am studying theoretical physics (4th grade)
> in Prague and I want to use the computer algebra system as a
> physicist and I understand that mathematicians are looking at
> the mathematics from a completely different angle than physicists.

I have some doubts about that claim. But I guess this is not really
the issue you want to discuss. :-)

> But anyway, I want this:
>
> I have a diagonal matrix:
>
> gdd=Matrix((
>     (-exp(nu(r)),0,0,0),
>     (0, exp(lam(r)), 0, 0),
>     (0, 0, r**2, 0),
>     (0, 0, 0, r**2*sin(theta)**2)
>     ))
>
> that is a metric tensor on a 4 dimensional manifold with signature
> (-,+,+,+), this corresponds to g_\mu_\nu (i.e. the indices are
> lowered). No I want to calculate the Christoffel symbols -> riemann
> tensor -> ricci tensor.
>
> the x^\mu vector are variables (t, r, theta, phi). The nu(r) and
> lam(r) are unknown functions of "r".  I am interested in the
> (differential) equations for the unknown functions nu(r) and lam(r)
> that I get by setting:
>
> R_\mu_\nu = 0
>
> If you need some more explanation, I'll be glad to explain
> the details.
>
> In maple I can use the grtensor
>
> http://grtensor.org/
>
> package, but I find the maple not suitable for me, as I want to
> use the symbolic manipulation in my programs and I don't want to
> use the ugly maple language.
>

If you could suppress your revulsion of the Maple language, then I
could send you information about a package I wrote a few years ago
called TensorLite. I used it to do symbolic manipulation of tensors
in Maple V release 5 for calculations similar to that which you
describe. If you have Maple, you should be able to read this:

http://synthesis.anikast.ca/maple/mathematics/tensor5c.mws

If you have any interest, just let me know. With a little effort
I could probably remember what I was doing at that time...

> I found all the other symbolic packages unsuitable for me, so
> I wrote my own:
>
>

Yes, isn't that the way things go in computer algebra - everyone
seems to write their own? :-( That's a pity. If what you really
want to do is physics, then it's a trap that many people have
fallen into. It took me a long time to get over that stage.

> And in SymPy I can now do it quite easily:
>
>
> SymPy is just a general package and all I am using from it
> are just symbolic matrices. (I am lowering and raising indices
> by myself in the relativity.py example).

Actually I looked at SumPy a while back on Google Code. I think
its pretty cool. Do you know anything about Sage? Sage is another
computer algebra system that is written in Python, but they
incorporate a lot of other open source math software packages
such as Maxima, Gap, gmp, mpfr, etc. to do things not implemented
in native Python. Perhaps they would be interested in SymPy for
it's symbolic capabilities.

>
> I was curious - how could I do the same in Axiom?
>

I would recommend that you consult the Axiom book:

http://wiki.axiom-developer.org/Mirrors?go=/public/book2.pdf&it=Axiom+Book

section 9.7 CartesianTensor

for some examples of how tensors are currently implemented in
Axiom. In some respects the support is rather primitive but on
the other hand one can take advantage of the symbolic capabilities
of the rest of the Axiom system.

You will also need:

section 1.13 Differential Equations  (introduction)

and

section 8.10 Solution of Differential Equations

Using these two parts of Axiom in principle it should not be
too difficult to do the calculations you want to do.

On the other hand if I ever somehow find some more time, I would
really like to implement much better support for tensors,
differential forms and abstract multilinear operators on vector
spaces in general (see for example related sections 9.14
DeRhamComplex, and 9.10 CliffordAlgebra). But that might be me
just falling into the same old trap again. Maybe it would be
better to wait for someone else to come along and do it for
me. ;)

> ...

Kind Regards,
Bill Page.

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