What is the difference between abstract index notation and Ricci index notation?

Question

I'm reading Straumann's GR text and he talks about the difference between abstract index notation and Ricci index notation very briefly. So I read the wiki article, but that did not help much. Say we have the Ricci tensor and two vectors. What does the expression $R_{\mu\nu}u^\mu u^\nu$ mean in the two index notations? Is contraction not the same thing as summation over an index pair (this is what I was led to believe in undergrad and not-so-mathematically-rigorous grad texts).

Answer 1

Ricci calculus explicitly works with components with respect to a basis. Abstract index notation has the same look but means something very different: the indices are merely placeholders, signifying whether the arguments of tensors are vectors or covectors.

In Ricci calculus, for instance, the expression $K_{ab}$ would mean $K(e_a, e_b)$, where $e_a, e_b$ are basis vectors. For any particular $a, b$ this is a number. For the full range of indices, this is an indexed collection of values.

In abstract index notation, $K_{ab}$ means a tensor $K$ that is a function of two vectors: $K(\text{vector}, \text{vector})$.

One hangup with this idea is that it's less clear what's going on when you do a contraction: in abstract index notation, you're no longer summing over bases, so it takes a little more thought to realize what's going on. In that sense, the abstract notation for a contraction should be understood as corresponding to a more complicated mathematical operation that merely agrees with what you would get if you were to break into components and compute a trace. In particular: one way to compute a contraction is to differentiate with respect to an argument.

Answer 2

You can see this as a quadratic form where you are using the components of $u$, which are $u^{\mu}$, and the components of a matrix $R_{\mu\nu}$.