Tensor Index Notation Question

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45
I am just wondering, is there a difference in meaning/definition between the indices of a tensor being right on top of each other

[tex]A_{\mu }^{\nu }[/tex]

and being "spaced" as in

[tex]A{^{\nu }}_{\mu }[/tex]

I seem to remember that I once read that there is indeed a difference, but I can't remember what it was.

Thanks in advance.
 

WannabeNewton

Science Advisor
5,774
529
The spacing is very important if you are contracting with the metric tensor using the abstract index notation. For example if ##A^{a}{}{}_{b}## is a tensor then ##g^{bc}A^{a}{}{}_{c} = A^{ab}## but if we consider ##A_{c}{}{}^{a}## then ##g^{bc}A_{c}{}{}^{a} = A^{ba}## which will not equal ##A^{ab}## unless the tensor is symmetric. The notation ##A^{a}_{b}## makes the contraction with the metric tensor ill-defined in abstract index notation (index free notation is a different story of course).

The action of the tensor on a covector and a vector will subsequently be ambiguous if all you write down is ##A^{a}_{b}## because ##A_{b}{}{}^{a}v^{b}\omega_{a} = g^{ac}A_{bc}v^{b}\omega_{a} \neq g^{ac}A_{cb}v^{b}\omega_{a} = A^{a}{}{}_{b}v^{b}\omega_{a}## in general, so the spacing is important.
 

lurflurf

Homework Helper
2,417
122
You just need a convention for the order.
$$A{_\mu }^{\nu } \\
A{^{\nu }}_{\mu } \\
A_{\mu }^{\nu }$$

So the third one can be substituted for one of the others as long as you always know which one, or that it does not matter which.
 
41
1
Non-spaced indices represent symmetric tensor (in respective components).
 

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