Considering this answer, it is mentioned that the range of indices $a, b, c,\dots$ are seen as abstract and coordinate-free and linear operations can be represented with them; and the range of indices $i, j, k,\dots$ have numerical values and are subject to Einstein summation. What would be the problem of using the same range of indices, say $i, j, k,\dots$ as abstract as in $T_{ijkl}$ and refering to components only when a coordinate frame is applied, like $T_{ijkl}{\bf E}_{ijkl}$ since I always need a coordinate frame to know the values of the scalar components of the tensor?
In other words, what would be the problem of considering $T_{ijkl}$ both as abstract tensor and multidimensional array of scalars depending on the context? Or why abstract index notation should not be confused with Ricci calculus as said in the first paragraph of the article?
1 Answers
I don't think there are any issues in doing this. The abstract index notation is more a shift of perspective than anything else. In effect, it allows one to do Ricci-style computations without picking a basis, by deciding that the indices are just labels for the different factors in the tensor product. A contraction in the abstract notation would be written the same way, structurally, as the same contraction with a coordinate system fixed and the indices concrete and no longer abstract.
In the Ricci calculus, a contraction indicates a literal summation. Since this requires numbers, it also requires a coordinate system to be chosen. In the abstract notation, a contraction indicates a basis-independent trace operation being applied, which reduces to the aforementioned summation whenever a specific basis is fixed.
Really, the abstract index notation is nothing more than the observation that almost all of the Ricci calculus remains intact if one does not choose a basis. There's a great deal of meaning in the structure of the index expressions which is not basis dependent.
- 5,043