- #1
MattRob
- 211
- 29
Homework Statement
Take the Covariant Derivative
[itex]∇_{c} ({∂}_b X^a)[/itex]
Homework Equations
[itex]∇_{c} (X^a) = ∂_c X^a + Γ_{bc}^a X^b[/itex]
[itex]∇_{c} (X^a_b) = ∂_c X^a_b + Γ_{dc}^a X^d_b - Γ^d_{bc} X^a_d[/itex]
The Attempt at a Solution
Looking straight at
[itex]∇_{c} ({∂}_b X^a)[/itex]
I'm seeing two indices. However, the b is nothing more than a reference to the derivative with respect to what, so I'm not sure whether this counts as a one or two-index object. Namely, do I solve it as
[itex]∇_{c} ({∂}_b X^a) = ∂_c {∂}_b X^a + Γ_{dc}^a {∂}_b X^d[/itex]
since the index in the partial differential operator isn't the same as an index on a tensor, or is it?
So I'm thinking since the indices merely indicate you're working with a variety of objects ([itex]x^a = k => x^1 = k, x^2 = k, x^3 = k, ...[/itex]), then the index on the differential operator should be treated the same, so I can treat [itex]{∂}_b X^a[/itex] as a two-index tensor [itex]T^a_b = {∂}_b X^a[/itex]
[itex]∇_{c} ({∂}_b X^a) = ∂_c {∂}_b X^a + Γ_{dc}^a {∂}_b X^d - Γ^d_{bc} {∂}_d X^a[/itex]
So is it the latter case, former, or something else entirely?
I suppose I have every reason to think it should be the latter case, but I guess I'm just uncertain and want some clarification.
Thanks!