Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Covariant derivative of metric tensor

  1. Mar 2, 2008 #1
    Hi, I'm trying to verify that the covariant derivative of the metric tensor is D(g) = 0.
    But I have a few questions:
    1) This is a scalar 0 or a tensorial 0? Because it is suposed that the covariant derivative of a (m,n) tensor is a (m,n+1) tensor, and g is a (0,2) tensor so I think this 0 should be a (0,3) tensor. Am I right?

    2) Following the MTW book in the example of page 341 we have:
    [tex]\Gamma^{\theta}_{\phi\phi} = -sin(\theta)cos(\theta)[/tex]
    [tex]\Gamma^{\phi}_{\phi\theta} = cos(\theta)/sin(\theta)[/tex]
    and all the other [tex]\Gamma = 0[/tex]

    But when I try to verify the covariant derivative of the metric tensor for the component [tex]g_{\phi\phi;\theta}[/tex] it doesn't give me 0, but instead:

    [tex]g_{\phi\phi;\theta} = g_{\phi\phi,\theta} - \Gamma^{k}_{\theta\phi} g_{k\phi} - \Gamma^{k}_{\theta\phi} g_{\phi k} = 2a^2sin(\theta)cos(\theta) - (0+0) - (0+0) = 2a^2sin(\theta)cos(\theta) \neq 0 [/tex]

    I checked it a lot of times and am not sure if this is a conceptual error or a procedure error.
    Can someone clarify this to me?
  2. jcsd
  3. Mar 2, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Yup, your reasoning is correct: it would be the type (0,3) tensor with every component zero.

    I don't know what the actual question is (since I don't have the book to hand) but I can't quite follow your last line. I get [tex]g_{\phi\phi;\theta}=g_{\phi\phi,\theta}-g_{k\phi}\Gamma^{k}_{\phi\theta} - g_{\phi k}\Gamma^{k}_{\phi\theta}= g_{\phi\phi,\theta}-2g_{\phi\phi}\Gamma^\phi_{\phi\theta}[/tex]

    Is this what you get before plugging in the values?
  4. Mar 2, 2008 #3


    User Avatar

    I do get zero. Note that [tex] \Gamma^\phi_{\theta \phi} [/tex] is not zero, it is equal to
    [tex] \Gamma^\phi_{\phi \theta} [/tex] !
  5. Mar 2, 2008 #4
    Thanks a lot for your answers!!
    Indeed that was the mistake, I forgot the symetry of the chistoffel symbols, it gives 0 now!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook