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I am curious to see a simple demonstration of how special relativity leads to Lorentz Invariance of the Maxwell Equations. Differential form will suffice.

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up vote 3 down vote accepted

Due to the constraint $\nabla \cdot B =0$, there exists a vector potential $A$ such that $B = \nabla \times A$ and $E_j = \partial_0 A_j - \partial_j A_0$ (up to a sign I forget). In other words, $E$ and $B$ assemble into a "field strength tensor" $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu = dA_{\mu \nu}$. 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

$d * F = 0$,

here $*$ is the Hodge star. This is clearly invariant under isometries (Lorentz transformations).

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I feel this answer, although accepted, doesn't relate to the role of the Lorentz transformations, does it? – NikolajK Aug 27 '12 at 21:30
Lorentz transformations are just isometries. What I meant above is that the equations can be written so that they only depend on the metric structure, so a diffeomorphism fixing the metric gives an automorphism of the space of solutions. These equations have this property for any metric, not just Minkowski's. Did I misunderstand your comment? – Ryan Thorngren Aug 27 '12 at 21:40
"a diffeomorphism fixing the metric"? Anyway, you rewrote it geometrically, and so it's coordinate intedepend, i.e. diffeomorphism invariant. But this doesn't in any way highlight the role of Lorentz transformations for the Maxwell equations in special relativity. (Btw. the isometries are actually the whole Poincare group.) – NikolajK Aug 27 '12 at 22:23
Arbitrary diffeomorphisms don't induce automorphisms of the space of solutions. Take for example a diffeomorphism reversing the signature. I think this does not even fix the dimension. This is just a notational difficulty in the physics literature. If you make a change of coordinates and say that the metric transforms in the way you'd expect, this is an isometry generated by a diffeomorphism. Lorentz invariant (or Poincare invariant, if it matters) should just mean isometries => automorphisms of the space of solutions. Sometimes the easiest way to show it is to just rewrite the equations. – Ryan Thorngren Aug 28 '12 at 9:44

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.

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