- #1
BucketOfFish
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Homework Statement
Given that ##x_\mu x^\mu = y_\mu y^\mu## under a Lorentz transform (##x^\mu \rightarrow y^\mu##, ##x_\mu \rightarrow y_\mu##), and that ##x^\mu \rightarrow y^\mu = \Lambda^\mu{}_\nu x^\nu##, show that ##x_\mu \rightarrow y_\mu = \Lambda_\mu{}^\nu x_\nu##.
Homework Equations
$$g_{\rho\sigma} = g_{\mu\nu}\Lambda^\mu{}_\rho\Lambda^\nu{}_\sigma$$
The Attempt at a Solution
So this isn't actually a homework problem, it's just an exercise in Lahiri and Pal that I was looking at. Seems like this thing would be really simple, but I can't work it out for some reason.
I get, for example, to the point where ##x_\mu x^\mu = g_{\mu\nu}x^\mu x^\nu = g_{\rho\sigma}\Lambda^\rho{}_\mu\Lambda^\sigma{}_\nu x^\mu x^\nu = y_\mu y^\mu##. Then, switching labels and using the definition of ##y^\mu##, we get that ##y_\mu = g_{\rho\mu}\Lambda^\mu{}_\nu x^\nu##. But I go in circles from there.
Part of my confusion is that I don't really know what the relation is between ##\Lambda_\mu{}^\nu## and ##\Lambda^\mu{}_\nu##. Seems like I'm missing something really obvious. Can anyone help?