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whatisreality
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Homework Statement
We've been told there's this operation called 'contraction' where if you have a superscript and a subscript that are the same they cancel. I don't understand how that works, partly in the sense that we haven't got round to what the superscripts and subscripts actually mean, and partly because at first, having a superscript and subscript that were repeated meant summing over that index. So I'm not sure now when something has to be summed over and when you can just drop them... it came up in the context of this question:
Given that ##S_{\mu\nu}^{\mu} T^{\nu\rho}_{\rho\sigma} = U_{\sigma}##
How many terms are in each equation on spacetime (i.e. 4 for dimensions)?
Homework Equations
The Attempt at a Solution
If all those indices that are the same can be contracted then I'm left with just ##T_{\sigma} = U_{\sigma}## which is a very reasonable looking equation but there'd only be one term in each equation (and four equations). So which indices are the ones I can't actually contract? Is it the ##\nu## that doesn't go away because it's attached to different tensors? Because in our lecture notes there is an example where the ##\mu## in ##S^{\mu}T_{\mu}## gets contracted.
Thank you for any help, I really appreciate it. :)
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