Manipulation of tensor indices

[Andy_X]

I would welcome advice on the practical issues of manipulating tensor indices, in particular how to determine the order of indices during calculations. Some of the following questions are probably simplistic but I have been unable so far to find a consistent answer.
1) Some texts use a unique order for the complete set of indices on a tensor, leaving spaces in the subscript list where superscripts appear and vice versa. Others have superscripts and subscripts above one another. What is the difference and when is it important to have an ordering for the combined set?
2) How is the order of indices determined for a tensor product? E.g. if W=ST are the indices of W formed from the indices of S followed by the indices of T and is this also the case for mixed tensors?
3) How is the answer to 2) affected if the product of two tensors also involves a dummy index? i.e. does the dummy index just disappear from the combined list of subscripts and superscripts?
4) in raising (or lowering) indices using a metric tensor (or the inverse metric) some texts always put the metric to the left of the tensor being changed (and the inverse to the right) while other texts don’t make this distinction. Which is correct? Also where does the raised (lowered) index fit into the pre-existing list of superscripts (subscripts)?
Any help greatly appreciated

 

[Ben Niehoff]

1) Some texts use a unique order for the complete set of indices on a tensor, leaving spaces in the subscript list where superscripts appear and vice versa. Others have superscripts and subscripts above one another. What is the difference and when is it important to have an ordering for the combined set?

It is best to stick to the first convention, where ALL indices (up or down) are put in a definite order. The reason is because you might want to raise or lower indices. There are a few occasions where it's ok to forget about ordering, such as $\delta^a_b$.

2) How is the order of indices determined for a tensor product? E.g. if W=ST are the indices of W formed from the indices of S followed by the indices of T and is this also the case for mixed tensors?

Really, index notation allows you to mix up the order a bit. Strictly speaking, W should have the S indices first, and the T indices second, but nothing prevents you from writing

$$W_{abcd} = S_{ac}T_{db}$$.

The notation explicitly tells you what the new ordering of indices is. Strictly speaking, this is not merely $W = S \otimes T$, but it is $W = \mathcal{P} (S \otimes T)$, where $\mathcal{P}$ is a map that permutes the indices in a specific way.

3) How is the answer to 2) affected if the product of two tensors also involves a dummy index? i.e. does the dummy index just disappear from the combined list of subscripts and superscripts?

Yes. For example, one can write

$$W_{abcd} = S_a{}^e{}_c T_{dbe}$$.

4) in raising (or lowering) indices using a metric tensor (or the inverse metric) some texts always put the metric to the left of the tensor being changed (and the inverse to the right) while other texts don’t make this distinction. Which is correct? Also where does the raised (lowered) index fit into the pre-existing list of superscripts (subscripts)?

It doesn't matter where you put the metric when you write out the indices explicitly. The order is simply whatever you write down. For example, you might write

$$W_{abcd} = g_{ce} S_a{}^e{} T_{db}$$.

It is important to give a definition of the symbol $W_{abcd}$ and remain consistent, but it doesn't matter exactly what order you write all the factors, because the index notation tells you exactly which parts slots are contracted, and which slots get mapped to where.