Talk:Raising and lowering indices

$$g_{ij}g^{ij}$$
I have corrected the article which was stating that the product of $$g_{ij}g^{ij}$$ was $$\operatorname{Tr}(g)=N$$. Since $$g^{ij}$$ is the inverse of $$g_{ij}$$ their product is the identity matrix and thus one obtains the trace of the identity which is, correctly, N in a N-dimensional manifold. -- CristianCantoro (talk) 16:01, 30 July 2010 (UTC)

My own reversion
Quondum: just in case you're wondering why I reverted myself, its because I copied + pasted + edited the markup separately, then pasted back into the edit panel + saved then realized you had edited. Your tweaks were better than mine so I undid my paste. Apologies for that. F = q(E+v×B) ⇄ ∑ici 13:18, 31 March 2012 (UTC)


 * Considering that recently I made made only a very minor change to this article (substituting "tensor field" for "tensor"), I guess you must be refering to slightly earlier edits by someone else. Keep up the good work.  — Quondum☏✎ 13:57, 31 March 2012 (UTC)


 * I thought you did more at a first glance (don't ask...). I'll revert my own change back. Thanks for your encouragement. =) F = q(E+v×B) ⇄ ∑ici 14:52, 31 March 2012 (UTC)

Dual tensor
The new section (written entirely by me) will be split off to a new article: Dual tensor with redirects Dual of a tensor and Tensor dual. It doesn't really fit too well here (at first I thought it would); just because it involves raising indices, so is this done everywhere else, rather it's another operation worthy of it's own article.

It's astonishing that there is no article on this, for instance in classical electromagnetism and special relativity, where to link the sentence


 * "There is another way of merging the electric and magnetic fields into an antisymmetric tensor, by replacing E/c → B and B → − E/c, to get the dual tensor Gμν."

?? And no - this is not the same thing as a dual space in the context of vector spaces of covectors. F = q(E+v×B) ⇄ ∑ici 15:12, 10 June 2012 (UTC)


 * But it is almost certainly the same thing as the Hodge dual, and should be called that. Though I prefer the (very) closely related Clifford dual, which is defined by Lounesto and is better-behaved.  — Quondum☏ 15:37, 10 June 2012 (UTC)
 * (ec)PS: Beware, both these duals are (almost certainly) only defined on the same subspace of the tensor algebra as the wedge product is defined; i.e. not on general tensors. So to refer to the Hodge dual of a tensor may be misleading.  — Quondum☏ 15:42, 10 June 2012 (UTC)


 * That article is so abstract, and less followable... Its on anyone who would like to merge. F = q(E+v×B) ⇄ ∑ici 15:40, 10 June 2012 (UTC)


 * I think that Ricci calculus should deal with it in a section on Levi-Civita symbol and Hodge dual. These are significant aspects of the Ricci calculus not yet covered.  Put it in there and we can streamline it later.  — Quondum☏ 15:50, 10 June 2012 (UTC)
 * When Ricci calculus was first started, I did want to write about it there, but thought it was off-topic so left it. Lets go with your plan though, its better than nothing. =) F = q(E+v×B) ⇄ ∑ici 15:54, 10 June 2012 (UTC)


 * It has been transplanted, and all redirects and double redirects eliminated. F = q(E+v×B) ⇄ ∑ici 16:00, 10 June 2012 (UTC)

Example in Minkowski spacetime
The example is correct, but unnecessarily confusing, because after contraction of $$\mu$$, $$\nu$$ is silently renamed to $$\mu$$.

It would be easier to understand if it would say $$\eta_{\mu\nu} \, X^\mu \, Y^\nu = X_\nu \, Y^\nu$$ instead, as then one would clearly see that the index of $$X$$ was lowered.

Alternatively, lower the index of $$Y$$ instead, but **keep the order** in the rest of the example (as commutativity is not obvious to a beginner):

$$\eta_{\mu\nu} \, X^\mu \, Y^\nu = X^\mu \, Y_\mu$$

PointedEars (talk) 07:51, 6 June 2023 (UTC)