Re: Order of indices in tensors...
From: Igor Khavkine (igor.kh_at_gmail.com)
Date: 08/25/04
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Date: Wed, 25 Aug 2004 07:46:50 +0000 (UTC)
"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh$2rv$1@news.Stanford.EDU>...
> Hello--I'm doing some intro-GR (using Carroll's new book), and I am confused
> about a subtle point:
>
> What is the significance of the order of indices in a tensor? I understand
> that the convention is for upper indices to sum with lower indices and vice
> versa when the tensor acts on the appropriate object, however, what is the
> significance of having the upper index listed first or the lower index
> listed first (horizontally)?
The order of upper indices matters, the order of lower indices matters,
the order of the lower indices relative to the upper ones (and vice versa)
does not matter. However, when you raise and lower indices using the metric
tensor, you need to care where the new upper (lower) index will be placed
relative to the other upper (lower) indices. Sometimes people fix an
absolute order for all lower and upper indices to avoid ambiguity. this
convention can be denoted by leaving a blank right above or below each
index.
The reason the order of the indices matters at all is simple. A 2 index
tensor a_{ij} can always be written as a linear combination of terms
of the form b_i c_j, but clearly b_i c_j != c_i b_j.
Hope this helps.
Igor
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