On Dec 4, 2007 11:31 AM, Viktor T. Toth <vttoth at vttoth.com> wrote:
> > 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}.
>
I think this is the convention for the Riemann tensor. This formula
can be defined as $R^m_{nba}$ or $R^m_{nab}$
> 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.
>
Yes, once the Riemann tensor is defined, the contravariant index is
contracted with the middle covariant index to get the Ricci tensor.
> But, for instance, Ohanian and Ruffini do not; Narlikar does not; nor does
> Peacock. So it's by no means universal.
>
Ok, I didn't read so many books and I wonder if I can find all these
books in our library:-)
Since there is freedom to choose which covariant index to contract
with, I think it need to be point out explicitly which one ctensor has
chosen in the document.
> > 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.
>
I totally agree, that's why I use a quotation mark and I use the word
"correct" following you :p
> > 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 :-)
>
I learn general relativity mainly by reading the books written by
Schutz and Weinberg.
Thank you.
>
> Viktor
>
>
Thank you for your patience.