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Tensor indices (proving Lorentz covariance)

  1. May 6, 2015 #1
    1. The problem statement, all variables and given/known data

    So, I need to show Lorentz covariance of a Proca field E-L equation, conceptually I have no problems with this, I just have to make one final step that I cannot really justify.

    2. Relevant equations

    "Proca" (quotation marks because of the minus next to the mass part, I saw on the internet there is also the plus convention) field is defined as:
    [tex]{\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}m^2V_{\mu}V^{\mu}[/tex]
    where [itex]V_{\mu}[/itex] is the massive field, and [itex]F_{\mu\nu}[/itex] the appropriate analogy to the EM field tensor. This leads to E-L:
    [tex]\partial^{\mu}F_{\mu\nu}-m^2V_{\nu}=0[/tex]

    3. The attempt at a solution

    So when I transform the equation according to: [itex] V^{\mu}(x) \rightarrow V'^{\mu}(x')=\Lambda^{\mu}_{\,\, \nu}V^{\nu}(x) [/itex], everything turns out okay but this one part that looks like: [itex]-\partial^{\mu}\Lambda_{\nu}^{\,\, \alpha}\partial_{\alpha}V_{\mu}(x)[/itex], and fr the proof to be over I need it to look like:

    [tex]-\partial^{\mu}\Lambda_{\nu}^{\,\, \alpha}\partial_{\alpha}V_{\mu}(x)=-\partial^{\mu}\partial_{\nu}V'_{\mu}(x')[/tex]

    and I can't seem to wrap my head around it, there must me something I'm not seeing...

    EDIT: initialy I transformed the derivatives as well, these are derivatives of the field over the "old" coordinates (x not x')
     
  2. jcsd
  3. May 6, 2015 #2

    Orodruin

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    You should keep doing that. Otherwise your equations are expressed in some weird combination of frames.
     
  4. May 6, 2015 #3
    I just figured it out, for some reason I was approaching the equation as though it was the Lagrangian density... Thanks, solved.
     
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