tensors: why is contraction on a pair of contravariant indices not possible in general?

Question

tensors: why is contraction on a pair of contravariant indices not possible in general?

Wikipedia states:

"contraction on a pair of indices that are either both contravariant or both covariant is not possible in general. However, in the presence of an inner product (also known as a metric) g, such contractions are possible. One uses the metric to raise or lower one of the indices, as needed, and then one uses the usual operation of contraction. The combined operation is known as metric contraction." Tensor Contraction on Wikipedia

Isn't the following called contraction?

Let's contract $V^{ii}_j$

$\large V^{ii}_j = V^{11}_j + V^{22}_j + V^{33}_j = W_j$

Therefore $W_j$ is the contraction of $V^{ii}_j$...

Now my question is this: what's wrong with this contraction in light of wikipedia quotation above that says you can't contract a pair of contravariant index in general? I was just curious if there was something i'm missing in my understanding of contraction.

Answer 1

When a covariant index is "contracted" against a contravariant index their transformation properties cancel each other out. The result is a tensor of lesser rank because of this. If you try to perform the same operation on two contravariant indices their transformation rules do not cancel, and the result is probably not still a tensor anymore. The purpose of the metric in contracting a pair of contravariant indices is to provide two covariant indices which "cancels out" the contravariant transformation.

In your example you are implicitly assuming that $\delta_{ij}$ is a tensor on your manifold. If it isn't then $W_j$ isn't a tensor.

Answer 2

The operation you defined depends on the cordinates the tensor is presented in. The idea is that contraction is, when done according to the rules of that article, independent of coordinates. You are effectively implicitly using the metric where the basis of presentation is orthonormal to do that contraction, which may be what you want, but is not necessarily what you want.

Answer 3

When you read "contraction" one model to have in mind is vector multiplication. In general, you can multiple a row $n$-vector with a column $n$-vector, but there is no such multiplication between row and row or between column and column. In the more general setting, this requires you to pick a "row index" and a "column index", i.e. a contravariant index and a covariant index.

Your operation is not a tensor contraction. In the above language, it is an inner product between column vectors. While one can define such an inner product, it does not correspond to the usual multiplication. As explained in the article, you can use a gadget to first convert a column to a row and then use the normal multiplication. Note that this conversion is capable of substantially altering the components of your tensor.

Answer 4

Consider the metric tensor $g_{ab}$ on the plane $\mathbb{R}^2$.

In Cartesian coordinates we have $g_{ab} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$ with $g_{11} + g_{22} = 1+1 = 2.$

In polar coordinates we instead have $g_{ab} = \begin{pmatrix} 1 & 0 \\ 0 & r^2 \\ \end{pmatrix}$ with $g_{11} + g_{22} = 1 + r^2.$

These only coincide when $r=1.$