In Peskin and Schroeder page 37, it is written that
- Using vector and tensor fields, we can write a variety of Lorentz-invariant equations.
- Criteria for Lorentz invariance: In general, any equation in which each term has the same set of uncontracted Lorentz indices will naturally be invariant under Lorentz transformations.
I would like to explicitly show that the above criteria is valid for Maxwell's equations $\partial^{\mu} F_{\mu \nu} = 0$ or $\partial^{2}A_{\nu}-\partial_{\nu}\partial^{\mu}A_{\mu}=0$.
Solution 1: Maxwell's equations follow from the Lagrangian $$\mathcal{L}_{MAXWELL}=-\frac{1}{4}(F_{\mu \nu})^{2} = -\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})^{2}$$ which is a Lorentz scalar, so this means that the equation of motion is Lorentz-invariant as well. That's one way to convince yourself that the above Maxwell's equations are, in fact, Lorentz invariant. Is this correct?
Solution 2: I would like to actively transform the electromagnetic field strength tensor $F_{\mu \nu}$ and show that the Maxwell's equations $\partial^{\mu} F_{\mu \nu} = 0$ or $\partial^{2}A_{\nu}-\partial_{\nu}\partial^{\mu}A_{\mu}=0$ remain Lorentz invariant.
I can see that $\partial^{2}$ and $\partial^{\mu}A_{\mu}$ will not Lorentz transform as they are Lorentz scalars.
Under an active Lorentz transformation, $V^{\mu}(x) \rightarrow \Lambda^{\mu}_{\nu}V^{\nu}(\Lambda^{-1}x)$. So, will $A_{\nu}$ and $\partial_{\nu}$ Lorentz transform in the same way?