1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Dual vector is the covariant derivative of a scalar?

  1. Apr 21, 2012 #1

    NSH

    User Avatar

    1. The problem statement, all variables and given/known data

    In Wald's text on General Relativity he makes an assertion that I'm not sure why it is allowed mathematically. Here's the basic setup:

    Let [itex]\omega_{b}[/itex] be a dual vector, [itex]\nabla_{b}[/itex] and [itex]\tilde{\nabla}_{b}[/itex] be two covariant derivatives and [itex]f\in\mathscr{F}[/itex]. Then we may let [itex]\omega_{b}=\nabla_{b}f=\tilde{\nabla}_{b}f[/itex]

    This is in chapter 3 on curvature between equations 3.1.7 and 3.1.8...

    2. Relevant equations

    If it is relevant he is using this assertion to show:

    [itex]\nabla_{a}\omega_{b}=\tilde{\nabla}_{a}\omega_{b}-C^{c}_{ab}\nabla_{c}f[/itex]

    implies
    [itex]\nabla_{a}\nabla_{b}f=\tilde{\nabla}_{a}\tilde{ \nabla}_{b}f-C^{c}_{ab}\nabla_{c}f[/itex]

    3. The attempt at a solution

    I know how to plug in his assertion I just don't get why the heck it is allowed? I've tried reading up on dual vector spaces but I haven't found what I'm looking for... any help would be appreciated.
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Dual vector is the covariant derivative of a scalar?
  1. Calculating covariance (Replies: 0)

Loading...