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- 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##.

I can see that ##\partial^{2}## and ##\partial^{\mu}A_{\mu}## will not Lorentz transform as they are Lorentz scalars.

- 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.

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?

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# Proof than an equation is Lorentz invariant

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