Is There a Difference Between Covariant and Contravariant Tensor Notations?

[Markus Hanke]

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

$$A_{\mu }^{\nu }$$

and being "spaced" as in

$$A{^{\nu }}_{\mu }$$

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]

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]

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.

 

[guest1234]

Non-spaced indices represent symmetric tensor (in respective components).