Thread: Learning GR
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Aug7-04, 11:30 AM
P: 78
Quote Quote by pmb_phy
Quote Quote by homology
Be aware that some books will treat covariant and contravariant tensors the same, which they are in an orthonormal basis, ...
I think I disagree with that statement big time. But first, can you please clarify what you mean by this statement? Thanks.
Since I'm not a mind reader, I don't know for sure what "homology" meant by that but I think I understand and I think I agree.

First, a qualifier: No general relativity text, nor any tensor calculus text, ever treats them as "the same". But that leaves a lot of other textbooks!

In any basis, the dual vectors (covariant rank 1 tensors) are isomorphic to the vectors (contravariant rank 1 tensors), via the fact that both are vector spaces of the same dimension. The isomorphism is through the correspondence between the basis dual vectors and the basis vectors.

However, there is in general nothing unique or "natural" about this isomorphism, because it depends entirely on the basis chosen.

But -- and here's where the weasel gets in -- if you restrict yourself to orthogonal changes of basis, then ordinary dot product is preserved, and the isomorphism between dual vectors and vectors is fixed and "natural". And in this limited case, the distinction between covariant and contravariant tensors largely melts away.

Note, however, that this means you must use a basis at every point which is orthonormal under ordinary dot-product. But that means that you will often find yourself using a non-coordinate basis! The integrability conditions may not generally be met, in which case you can't extend the basis to a full coordinate system on the manifold.

Now, go pick up just about any intermediate mechanics text, and look at the way they write basis vectors. You'll most likely find they're wearing hats (^). They're using unit basis vectors, even though this means they must give up the convenience of having a coordinate system to back them with. One very large reason for doing this is that it allows them to retain the natural correspondence between vectors and dual vectors, and in consequence, they can totally ignore the distinction between covariant and contravariant tensors.

Symon's "Mechanics" text, for instance, is something of a standard among intermediate-level texts and it does indeed make no distinction between contravariant and covariant tensors.