What is the practical difference between abstract index notation and “ordinary” index notation

I understand that in “normal” index notation the indexes can be thought of as coordinates of scalar values inside a tabular data structure, while in the abstract index notation they can not. However, I am not clear on what practical difference this makes when actually doing math. If your are doing numerical calculations then you need to plug actual components into your tensors, so that is not abstract, but is there any difference if you are doing symbolic/algebraic computations? The notations look identical, and even though the interpretation is different, expressions in both cases ultimately denote tensors. As far as I know the algebraic laws are the same. Are there manipulations that are valid in one but not in the other? If you see some tensor calculations, how can you tell if abstract index notation is being used? If you are doing differential geometry with indexes do you need to decide if your indexes are abstract or not? Or am I just missing something?

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The expressions in the abstract index notation and the normal index notation look identical on purpose. This is done in order to retain the calculational flexibility of indices but have a coordinate-free treatment of the subject. The Wikipedia’s article is well enough written, but it would also make sense to read the original Penrose’s book where one finds most of the details. Examples are given throughout that book, by the way. Be sure, that you have read the digest of the first volume’s chapter in the beginning of Volume 2 (I find that brief summary very illuminating). To be honest, I have to admit that Penrose’s monograph is rather hard for the first reading, and a beginner should better consult with the first three chapters of the textbook of R.Wald “General relativity” where I personally found the cure against the fear of the abstract indices.

The conventions for abstract indexes are made so that the calculations’ appearance is indistinguishable from the same calculations in concrete indices. Indeed, everything must be preserved if one introduces a frame to convert abstract indices into “normal” tensor indices.

So what would be the advantages of the abstract index notation? The main thing is that it is coordinate free. For instance, the Riemann curvature operator in abstract indices can be denoted by $R_{a b}{}^{c}{}_{d}$ at any point and the manifold, but if a coordinate chart or just a frame $\{E_{i}|i=1,\dots,n\}$ has been chosen (which is usually can be done only locally), one can pass to the components of the tensor $R_{a b}{}^{c}{}_{d}$ in this chart as follows
$$
R_{i j}{}^{k}{}_{l} = R_{a b}{}^{c}{}_{d} E^{a}{}_{i} E^{b}{}_{j} E_{c}{}^{k} E^{d}{}_{l}
$$
where the RHS is understood as
$$
R(E_{i},E_{j},E_{l})E_{k},
$$
Notice that in the above equations the indices from the range $a,b,c,\dots$ are seen as abstract, so whenever the same such index appears twice an action of a linear operator on an element of a vector space is assumed. In contrast with that, the indices form the range $i,j,k,\dots$ have numerical values ($1,2,\dots,n$), so when they pair up, the Einstein summation convention takes place.

Another advantage is that the abstract index is quite economical, and the expressions often look much shorter that in the usual coordinate-free notation, especially when one deals with tensor symmetries (compare, for instance the Bianchi identity in both notations).

Caveat. One needs to be told that the abstract indices are used, and also the index ranges must be specified, for instance, $a,b,c,\dots$ would be tensor indices, whereas $A,B,C,\dots$ can represent spinor or tractor indices, and so on.

A simple example of a calculation with abstract indices one can find, for instance, in this answer

Slightly more advanced examples are given here and also here.