Why Does Tensor Contraction Yield Zero in This Calculation?

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Homework Statement


Hello guys, hope you'll help me out with this!
I'm asked to calculate g^{\alpha\beta}g^{\sigma\rho}(g_{\alpha\sigma}g_{\beta\rho}-g_{\alpha\rho}g_{\beta\sigma})
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 don't know the exact solution -sorry- but it should depend on the dimension of the manifold)


Homework Equations



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The Attempt at a Solution



Well, i contracted firs with respect to g^{\sigma\rho} and i end up with g^{\alpha\beta}(g_{alpha\beta}+g_{alpha\beta}-g_{alpha\beta}-g_{alpha\beta})which obviously vanish.
I must be missing something!
 
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Are you so sure it's zero? Let's look at the first factor:
<br /> 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<br />
What is the other factor?
 
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
-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

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?
 
I don't think that you're making a mistake at all, I think the answer really is zero.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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