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Simple problem - contracting tensors

  1. Aug 16, 2011 #1
    1. The problem statement, all variables and given/known data
    Hello guys, hope you'll help me out with this!!
    I'm asked to calculate [tex]g^{\alpha\beta}g^{\sigma\rho}(g_{\alpha\sigma}g_{\beta\rho}-g_{\alpha\rho}g_{\beta\sigma})[/tex]
    where g is the metric tensor on a n-dimensional manifold but I can't get to the right result, i keep on getting zero!! (i know that's wrong although i dont know the exact solution -sorry- but it should depend on the dimension of the manifold)


    2. Relevant equations

    none

    3. The attempt at a solution

    Well, i contracted firs with respect to [itex]g^{\sigma\rho}[/itex] and i end up with [tex]g^{\alpha\beta}(g_{alpha\beta}+g_{alpha\beta}-g_{alpha\beta}-g_{alpha\beta})[/tex]which obviously vanish.
    I must be missing something!!!
     
  2. jcsd
  3. Aug 16, 2011 #2

    hunt_mat

    User Avatar
    Homework Helper

    Are you so sure it's zero? Let's look at the first factor:
    [tex]
    g^{\alpha\beta}g^{\sigma\rho}g_{\alpha\sigma}g_{ \beta\rho}=g^{\alpha\beta}(g^{\sigma\rho}g_{\alpha\sigma})g_{\beta\rho}=g^{\alpha\beta}\delta^{\rho}_{\alpha}g_{\beta\rho}=g^{\rho\beta}g_{\beta\rho}=\delta^{\rho}_{\rho}=n
    [/tex]
    What is the other factor?
     
  4. Aug 17, 2011 #3
    Thanks for the answer hunt_mat but... I keep on getting zero!!

    That's becaouse contracting the second factor like you did for the first i get
    [tex]-g^{\alpha\beta} g^{\sigma\rho}g_{\alpha\rho}g_{\beta\sigma}=-g^{\alpha\beta} (g^{\sigma\rho}g_{\alpha\rho})g_{\beta\sigma}=-g^{\alpha\beta} \delta^{\sigma}_{\alpha}g_{\beta\sigma}=-g^{\sigma\beta}g_{\beta\sigma}=-n[/tex]

    which added to the first factor gives zero!
    They seems equal to me, becaouse when i get the kronecker delta it doesent matter which index it has, it will be a mute index anyhow! (maybe the error is here?)


    where am i mistaking?
     
  5. Aug 17, 2011 #4

    hunt_mat

    User Avatar
    Homework Helper

    I don't think that you're making a mistake at all, I think the answer really is zero.
     
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