1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: 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


    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


    User Avatar
    Homework Helper

    Are you so sure it's zero? Let's look at the first factor:
    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
    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


    User Avatar
    Homework Helper

    I don't think that you're making a mistake at all, I think the answer really is zero.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook