[Maxima] question about ctensor
evanxxx at gmail.com
Mon Dec 3 20:42:58 CST 2007
On Dec 3, 2007 10:29 PM, Viktor T. Toth <vttoth at vttoth.com> wrote:
> It seems to me that Weinberg (Eq. 6.1.5 vs. 6.2.4) and MTW (Eq. 8.47) use
> opposite sign conventions for the Ricci tensor. Whereas MTW (and we) form
> the Ricci tensor by summing over the contravariant index and that covariant
> index which appears as a derivative index with a positive sign, Weinberg
> sums over the contravariant index and the covariant derivative index with a
> negative sign, which produces a Ricci tensor with a sign opposite to MTW's.
> Which means that using Weinberg's conventions, the scalar curvature of the
> sphere would be negative. In that sense, then, perhaps the convention of MTW
> is preferable. But that is purely a matter of convention of course.
> Please correct me if I am wrong, but changing the conventions of the Riemann
> tensor without an accompanying change in the Ricci tensor would change the
> sign of the latter, yielding negative scalar curvature for the sphere.
> Changing both would take us back to square one, since after all indices are
> swapped, our definition for the Ricci tensor would still agree with that of
> MTW, and disagree with that of Weinberg.
You are right. What I meant is that no matter what sign conventions
are used for the Riemann tensor, the Ricci tensor is always (as far as
I know) defined by contracting the Riemann tensor's contravariant
index and its second covariant index. There is no other convention
here. So ctensor is not "correct" at this point. After "correcting"
this, if you insist a positive scalar curvature for a sphere, the sign
convention of the Riemann tensor used in ctensor need to be changed.
What made me confused is the contracting convention used in ctensor
when forming the Ricci tensor. Contracting the contravariant index
with the last covariant index is different with what is usually used.
Sorry that I don't know whom you are referring to by MTW, has he used
that different contracting convention?
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