Basis vectors and abstract index notation

[branislav]

First of all, I'd like to say hi to all the peole here on the forum!

Now to my question:

When reading some general relativity articles, I came upon this strange notation:

T$^{a}$$_{b}$ = C(dt)$^{a}$(∂$_{t}$)$_{b}$ + D(∂$_{t}$)$^{a}$(dt)$_{b}$. Can someone please explain to me what this means? Clearly the author is trying to use the abstract index notation but I'm used to think of dx$^{i}$ as the covector basis and ∂$_{i}$ as the vector basis thus you're not allowed to change the co- or contravariance of these in an expression.

Thank you,
Branislav

 

[DrGreg]

Follow the usual rules for raising and lowering indices, e.g.$$(dt)_a = g_{ab}\,(dt)^b$$