Tensor indices (proving Lorentz covariance)

[VintageGuy]

Homework Statement

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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.

Homework 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:
$${\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}m^2V_{\mu}V^{\mu}$$
where $V_{\mu}$ is the massive field, and $F_{\mu\nu}$ the appropriate analogy to the EM field tensor. This leads to E-L:
$$\partial^{\mu}F_{\mu\nu}-m^2V_{\nu}=0$$

The Attempt at a Solution

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

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

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')

 

[Orodruin]

EDIT: initialy I transformed the derivatives as well, these are derivatives of the field over the "old" coordinates (x not x')

You should keep doing that. Otherwise your equations are expressed in some weird combination of frames.

 

[VintageGuy]

You should keep doing that. Otherwise your equations are expressed in some weird combination of frames.

I just figured it out, for some reason I was approaching the equation as though it was the Lagrangian density... Thanks, solved.