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Covariant Derivative

  1. Mar 8, 2017 #1
    1. The problem statement, all variables and given/known data
    Take the Covariant Derivative

    [itex]∇_{c} ({∂}_b X^a)[/itex]

    2. Relevant 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]

    3. 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!
     
  2. jcsd
  3. Mar 8, 2017 #2

    strangerep

    User Avatar
    Science Advisor

    Treat it as a 2-index object.
     
  4. Mar 9, 2017 #3
    Second formula, then?

    Thanks!
     
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