# Tensor indices (proving Lorentz covariance)

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1. May 6, 2015

### VintageGuy

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

3. 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')

2. May 6, 2015

### Orodruin

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

3. May 6, 2015

### VintageGuy

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