# [Maxima] ctensor riemann puzzlement

Viktor T. Toth vttoth at vttoth.com
Fri Aug 22 18:54:06 CDT 2008

```I am looking at the book you mentioned on Google Books; page 103 is not
available in preview, but page 104 is, and it seems that this is where the
results are I believe.

Given the metric you describe, it seems that Kay may have omitted a few
terms from the Riemann tensor, presumably as they are implied by symmetry.
The full set of nonzero terms in R^i_{jkl}, which you can work out by hand
(it's tedious but not hard, you don't need Maxima) is

R^1_{212} = -R^1_{221} = 1/2x^1,
R^1_{323} = -R^1_{332} = 1/2x^1,
R^2_{112} = -R^2_{121} = -1/4(x^1)^2,
R^2_{313} = -R^2_{331} = 1/4(x^1)^2,
R^2_{323} = -R^2_{332} = 1/4x^1x^2,
R^3_{123} = -R^3_{132} = -1/4x^1x^2,
R^3_{213} = -R^3_{231} = -1/4x^1x^2,
R^3_{223} = -R^3_{232} = -1/4(x^2)^2.

In covariant form, this gives

R_{1212} = -R_{1221} = g_{11}R^1_{212} = 1/2x^1,
R_{1323} = -R_{1332} = g_{11}R^1_{323} = 1/2x^1,
R_{2112} = -R_{2121} = g_{22}R^2_{112} = -1/2x^1,
R_{2313} = -R_{2331} = g_{22}R^2_{313} = 1/2x^1,
R_{2323} = -R_{2332} = g_{22}R^2_{323} = 1/2x^2,
R_{3123} = -R_{3132} = g_{33}R^3_{123} = -1/2x^1,
R_{3213} = -R_{3231} = g_{33}R^3_{213} = -1/2x^1,
R_{3223} = -R_{3232} = g_{33}R^3_{223} = -1/2x^2.

This is exactly what Maxima produces (the ordering of the indices differs,
notably in the case of Maxima, the Riemann tensor is set up so that it's
antisymmetric in its middle two indices, but that's just a matter of
convention.) It is also what one obtains using Maple's tensor package.

Note that one should not confuse the number of nonzero terms with the number
of independent terms. When you take the trivial symmetries of the covariant
Riemann tensor into account: antisymmetry in the first two indices,
antisymmetry in the second two indices, and symmetry between the first and
second pair of indices, you get three independent components:

R_{1212} = -R_{1221} = -R_{2112} = R_{2121} = 1/2x^1,
R_{1323} = -R_{1332} = R_{2313} = -R_{2331} = -R_{3123} = R_{3132} =
-R_{3213} = R_{3231} = 1/2x^1,
R_{2323} = -R_{2332} = -R_{3223} = R_{3232} = 1/2x^2.

Viktor

-----Original Message-----
From: maxima-bounces at math.utexas.edu [mailto:maxima-bounces at math.utexas.edu]
On Behalf Of Rob Burns
Sent: Friday, August 22, 2008 3:33 PM
To: maxima at math.utexas.edu
Subject: [Maxima] ctensor riemann puzzlement

Could someone help me understand the following please?
Like some previous mailing list posters I am working through Schaum's
'Tensor Calculus' (TC) by David Kay.
Without Maxima to remove some of the algebraic grind I would not have got as
far as I have.
When I got to Example 8.4 on page 103 I typed in the following to check the
example:-
lg:matrix(
[1,0,0],
[0,2*x1,0],
[0,0,2*x2]
);
ct_coords:[x1,x2,x3];
dim: 3;
cmetric();
christof(all);
/* The Christoff symbols matched those in TC */
riemann(true);
This is where things seem to go a bit awry...
The riemann tensor components returned by Maxima do not match those in TC
and also TC claims that for 3 dimensions there are only 6 independant
components whereas Maxima returns 8!
Any clarification would be gratefully received.
Regards, Rob.

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