1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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:

    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


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted