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[Axiom-mail] how to calculate Ricci tensor from a metric tensor?
From: |
Ondrej Certik |
Subject: |
[Axiom-mail] how to calculate Ricci tensor from a metric tensor? |
Date: |
Sun, 25 Mar 2007 15:55:05 +0200 |
Hi,
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. 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.
I found all the other symbolic packages unsuitable for me, so I wrote my own:
http://code.google.com/p/sympy/
And in SymPy I can now do it quite easily:
http://sympy.googlecode.com/svn/trunk/examples/relativity.py
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).
I was curious - how could I do the same in Axiom?
Thanks,
Ondrej
BTW, the resulting equations are:
-1/4*exp(\nu(r))*exp(\lambda(r))**(-1)*\lambda'(r)*\nu'(r)+1/4*exp(\nu(r))*exp(\lambda(r))**(-1)*\nu'(r)**2+1/2*exp(\nu(r))*exp(\lambda(r))**(-1)*(\nu'(r))'+exp(\nu(r))*r**(-1)*exp(\lambda(r))**(-1)*\nu'(r)
= 0
1/4*\nu'(r)*\lambda'(r)-1/2*(\nu'(r))'+r**(-1)*\lambda'(r)-1/4*\nu'(r)**2 = 0
-1/2*r*exp(\lambda(r))**(-1)*\nu'(r)+1/2*r*exp(\lambda(r))**(-1)*\lambda'(r)-sin(\theta)**(-2)*cos(\theta)**2-(-1-sin(\theta)**(-2)*cos(\theta)**2)-exp(\lambda(r))**(-1)
= 0
-sin(\theta)**2*exp(\lambda(r))**(-1)+sin(\theta)**2-1/2*sin(\theta)**2*r*exp(\lambda(r))**(-1)*\nu'(r)+1/2*sin(\theta)**2*r*exp(\lambda(r))**(-1)*\lambda'(r)
= 0
- [Axiom-mail] how to calculate Ricci tensor from a metric tensor?,
Ondrej Certik <=
- RE: [Axiom-mail] how to calculate Ricci tensor from a metric tensor?, Bill Page, 2007/03/26
- Re: [Axiom-mail] how to calculate Ricci tensor from a metric tensor?, Ondrej Certik, 2007/03/26
- CAS for the "masses" (was RE: [Axiom-mail] how to calculate Ricci...), C Y, 2007/03/27
- Re: CAS for the "masses" (was RE: [Axiom-mail] how to calculate Ricci...), Ondrej Certik, 2007/03/27
- Re: CAS for the "masses" (was RE: [Axiom-mail] how to calculate Ricci...), Ondrej Certik, 2007/03/27
- Re: CAS for the "masses" (was RE: [Axiom-mail] how to calculate Ricci...), C Y, 2007/03/27
- Re: CAS for the "masses" (was RE: [Axiom-mail] how to calculate Ricci...), Gabriel Dos Reis, 2007/03/27