> 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
I am not sure I can agree with you on this point. The Riemann tensor has
three covariant indices in the usual definition, but different authors put
these indices in different order, so "second covariant index" is not a very
meaningful concept without knowing a specific author's conventions. It is
much more important to know the sign of the Ricci tensor or Ricci scalar.
What I mean is that, regardless of index positions, the Ricci tensor is
defined as the contraction of the contravariant index with one of the
antisymmetric covariant indices, but it is not prescribed which one.
Assuming only that the Christoffel symbols' definition is unambiguous, the
Riemann tensor is defined as
\Gamma^m_{nb,a}-\Gamma^m_{na,b}+\Gamma^m_{sa}\Gamma^s_{nb}-\Gamma^m_{sb}\Gam
ma^s{na}.
It is easy to see that this expression is antisymmetric in the a,b indices.
The Ricci tensor is the contraction of m with one of either a or b. Which
one? MTW say it's a, and put this index in the second (middle) covariant
position. Weinberg says it's b, and puts that index in the second covariant
position, so you are right in this sense: both authors put the contraction
index in the second covariant position. As does Wald, Hawking & Ellis, and
Landau & Lifschitz, so you're certainly in good company there.
But, for instance, Ohanian and Ruffini do not; Narlikar does not; nor does
Peacock. So it's by no means universal.
> So ctensor is not "correct" at this point
I don't think we can speak of "correctness" when the question is a matter of
convention, and not one that is universally agreed upon.
> Sorry that I don't know whom you are referring to by MTW, has he used
> that different contracting convention?
MTW is the acronym for the book Gravitation by Misner, Thorne, and Wheeler,
aka the "Big Black Book". It is a commonly used, dare I say conventional
acronym, but obviously, the convention is far from universal :-)
Viktor