# How special relativity leads to Lorentz Invariance of the Maxwell Equations. Differential form will suffice.

How special relativity leads to Lorentz Invariance of the Maxwell Equations. Differential form will suffice.
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Kenley Rasmussen
Due to the constraint $\mathrm{\nabla }\cdot B=0$, there exists a vector potential $A$ such that $B=\mathrm{\nabla }×A$ and ${E}_{j}={\mathrm{\partial }}_{0}{A}_{j}-{\mathrm{\partial }}_{j}{A}_{0}$. In other words, $E$ and $B$ assemble into a "field strength tensor" ${F}_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }=d{A}_{\mu \nu }$. This is the correct object to reason about when thinking relativistically. It's just a $2$-form, so its transformation rules are simple. We can write Maxwell's equations as
$d\ast F=0$
here $\ast$ is the Hodge star. This is clearly invariant under isometries (Lorentz transformations).
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Lance Liu
Another way to see it is when deriving the EM wave equation from Maxwell equations. The Lorentz Invariance means that the amplitude should be symmetric under translations of space and time and rotations. In the wave eq. you have only 2nd derivatives (time and space), hence symmetry under Lorentz is preserved.