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Four vectors and Lorentz invariance

  1. May 9, 2004 #1
    Does anyone know where I can find a mathematical proof that the norm of any four-vector is Lorentz invaraint?
  2. jcsd
  3. May 9, 2004 #2


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    This sounds like a homework problem, but I am feeling generous. First verify by direct Lorentz transfomation of the special relativistic covariant metric tensor that it is unchanged in a Lorentz transformation.
    Then consider the quantity [tex]\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu}[/tex].
    By the transformation property definition of a four vector:
    [tex]\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu} = \eta' _{\mu}_{ \nu}\Lambda ^{\mu}_{ \kappa}T^{\kappa}\Lambda ^{\nu}_{ \lambda} T^{\lambda}[/tex]
    Regroup so as to work the transformation on the metric tensor first in the summations.
    [tex]\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu} = (\Lambda ^{\mu}_{ \kappa}\Lambda ^{\nu}_{ \lambda}\eta' _{\mu}_{ \nu})T^{\kappa} T^{\lambda}[/tex]
    At this point you should have already verified the following step as I mentioned:
    [tex]\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu} = \eta _{\kappa}_{ \lambda}T^{\kappa} T^{\lambda}[/tex]
    Last edited: May 9, 2004
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