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Lorentz Transform on Covariant Vector (Lahiri QFT 1.5)

  1. Dec 5, 2014 #1
    1. The problem statement, all variables and given/known data

    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##.

    2. Relevant equations

    $$g_{\rho\sigma} = g_{\mu\nu}\Lambda^\mu{}_\rho\Lambda^\nu{}_\sigma$$

    3. 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?
     
  2. jcsd
  3. Dec 8, 2014 #2

    stevendaryl

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    [edit]
    I see that you were actually sort of close, except that you made a slight mistake.

    [itex]y_\mu = g_{\rho\mu}\Lambda^\mu{}_\nu x^\nu[/itex]

    should be
    [itex]y_\rho = g_{\rho\mu}\Lambda^\mu{}_\nu x^\nu[/itex]

    At this point, you can operate on both sides by [itex]\Lambda^\rho{}_\lambda[/itex].
     
    Last edited: Dec 8, 2014
  4. Dec 9, 2014 #3

    Fredrik

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    $$\Lambda_{\mu}{}^\nu =g_{\mu\rho}\Lambda^\rho{}_\sigma g^{\sigma\nu}.$$ What does this tell you about a combo like ##\Lambda_\rho{}^\mu \Lambda^\rho{}_\nu##?

    You may also find my post in this thread useful: https://www.physicsforums.com/threads/einstein-notation-notes.770129/#post-4847943
     
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