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I'm studying the component representation of tensor algebra alone. There is a exercise question but I cannot solve it and cannot deduce answer from the text. The text is concise, I think it assumes a bit of familiarity with the knowledge.

(a) Convert the following expressions and equations into geometric, index-free notation: $A^αB_{γβ} ;\; A_αB_γ^{\;β} ;\; S_α^{\;βγ}=S^{γβ}_{\quadα} ;\; A^αB_\alpha=A_αB^βg^α_{\;β}$

In this problem, I can't see any difference between first two expressions except for the index position, and my only solution for the expression of index position is using metric tensor $g$, which I think is included in slot-naming notation. Is "index-free" notation able to express the difference? Other expressions are similarly confusing for me.

(b) Convert $\mathbf T(\_,\mathbf S(\mathbf R(\vec C,\_),\_),\_)$ into slot-naming index notation.

I think this notation would be not so universal notation. These problem are from http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1202.1.K.pdf (Ex 2.7, pg 19) and the help of anyone who is familiar with the notation would be appreciated.

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  • $\begingroup$ Equation 2.23f seems to suggest that the slot notation cannot distinguish upper and lower indices, so it seems to me that some of these might be the same in (a). $\endgroup$
    – Kyle Kanos
    Commented May 21, 2017 at 11:44
  • $\begingroup$ Upward indices are contravariant, downward indices covariant. It is possible to convert between the two by replacing the term with it tensor multiplied by the metric and then contracted, I think it is called musical isomorphism. I am self learned as well though so might be wrong. $\endgroup$
    – Emil
    Commented May 21, 2017 at 14:25
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    $\begingroup$ I don't know what they want for (a). To me these seem like exactly the sort of expressions that justify the index notation because they can't be expressed without it. The answer to (b) is $T_{abc}S^b_{\;de}R^d_{\;fg}C^f$. $\endgroup$
    – Oscar Cunningham
    Commented May 21, 2017 at 23:00
  • $\begingroup$ @OscarCunningham: Shouldn't $S$ and $R$ be rank-2 tensors in that expression, not rank-3? $\endgroup$
    – Michael Seifert
    Commented May 22, 2017 at 17:08
  • $\begingroup$ @MichaelSeifert Good question! In the notes provided it did seem like a three index tensor (for example) was always treated as a function that took three arguments and gave a scalar. But in the expression (b) we see that $T$ takes $S$ as an argument, and $S$ takes $R$ as an argument. So I was assuming that they were vector-valued functions. I would say that the notes aren't exactly clear in this section and that Heptacle has a right to be puzzled. $\endgroup$
    – Oscar Cunningham
    Commented May 22, 2017 at 18:00

3 Answers 3

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There are several reasons why we have indices on tensors:

  • to indicate their rank
  • to indicate the variance of a component, either covariant (lower index), or contravariant (upper index).
  • to indicate contractions

Now, so long as a tensor $T^{ij}_{klm}$ is understood as being of a particular rank and variance you can simply write $T$.

However, there is no simple notation that covers all the kinds of contractions that are possible on general tensors. For particular classes of tensors there are index free notations. One that you should be familiar with is matrices, vectors and covectors. These are tensors of type (1,1), (1,0) & (0,1). And contraction here is just multiplication of matrices, or matrices acting on vectors, or matrices being acted on by covectors and we also have convectors acting on vectors. Quite a list, and here we have only three types of tensors!

Hence, the modern way of viewing tensors with their indices is to view them as geometric objects with the indices indicating possible contractions. This appears to be what they are calling ‘slot-naming index notation’ in your question, it’s also been called ‘abstract index notation’.

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a) I think that the writers want us to use notation without indices to express the tensor or the equation, like the first one

$A^{\alpha}B_{\gamma\beta}=A^{\alpha}B(\vec e_{\gamma},\vec e_{\beta})=A(\underline{})\cdot \vec e_{\alpha} B(\vec e_{\gamma},\vec e_{\beta})$

I guess

b) I think that R should be a two-rank tensor but as for S, whether it is a 2-rank tensor or a 3-rank tensor, I'm not sure.

I guess.

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For b) I agree with Michael Siefert that R and S should be rank 2. If I ask for an expression with indices for 4 slots, the only way I've been able to see to do that is $$T_{\alpha\zeta\beta}S^\zeta_{\,\gamma} R^\epsilon_{\,\delta} C_\epsilon$$

The examples they give in the text are irritatingly overly simplified and so ambiguous and because it isn't a widely used notation it's hard to find guidance.

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