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Feb23-12, 04:49 PM
Sci Advisor
Matterwave's Avatar
P: 2,951
Ok I think I figured out the difficulty here. Basically it all has to deal with the fact that the basis vectors "dual" to the basis one forms by the usual requirement that they give the delta function when one acts on the other are not the vectors which are "dual" to the basis one forms as picked out by the metric.

Basically, even though the vector gradient is normal to the level sets, the basis vector is not. If the basis vector t were normal to the level sets then my metric would have a form of the first column and row being (1,0,0,0) which it doesn't need to have depending on my choice of coordinates x y and z.

EDIT: eendavid, it seems like this is what you were talking about in the first part of your post?

Addendum: If my analysis is correct, then it seems that there MUST be a way to choose coordinates such that N=1 and N_a=0 right? We just choose not to do it this way because we don't want to introduce those constraints at this level correct?