Tensor Index Notation Question

[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).

 

[blue_raver22]

Yes, there is a difference in meaning between the two notations. The first notation, A_{\mu }^{\nu }, is called the "covariant" notation, while the second notation, A{^{\nu }}_{\mu }, is called the "contravariant" notation.

In the covariant notation, the upper index (superscript) represents the "contravariant" transformation of the tensor, meaning how its components change when the coordinate system is transformed. The lower index (subscript) represents the "covariant" transformation, which describes how the basis vectors of the coordinate system change.

In the contravariant notation, the upper index represents the "covariant" transformation, while the lower index represents the "contravariant" transformation.

In general, the two notations are related by the metric tensor, which is used to raise and lower indices. This metric tensor describes the relationship between the basis vectors of the coordinate system and the basis vectors of the dual coordinate system.

It is important to note that the choice of notation does not change the physical meaning of the tensor itself, but it is a matter of convenience and convention in mathematical notation. Both notations are commonly used in tensor analysis and have their own advantages in different situations.