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- Having trouble deriving the electric field of a plane wave from the four-potential.

I'm trying to derive the electric and magnetic fields of a plane wave from the four-potential ##\mathbf{A} = (A^t , \mathbf{a}) ## in the Lorenz gauge. Given:

##\mathbf{A}(\mathbf{R}) = \Re \left( \mathbf{C} e^{i \mathbf{K} \cdot \mathbf{R}} \right)##

for constant future-pointing lightlike ##\mathbf{K} = (K^t, \mathbf{k})## and constant ##\mathbf{C} = (C^t , \mathbf{c})## orthogonal to ##\mathbf{K}##, I

##\nabla \times \mathbf{a} = \nabla \times \left( \mathbf{c} e^{i \mathbf{K} \cdot \mathbf{R}} \right) = \nabla e^{i \mathbf{K} \cdot \mathbf{R}} \times \mathbf{c}##,

which gives me ##\mathbf{B} = \sin (\mathbf{K} \cdot \mathbf{R}) (\mathbf{k} \times \mathbf{c}) ##. That seems reasonable, and if it's right then I expect ##\mathbf{E}## to have magnitude ##\sin (\mathbf{K} \cdot \mathbf{R}) \Vert \mathbf{k} \times \mathbf{c} \Vert## and be perpendicular to both ##\mathbf{B}## and ##\mathbf{k}## (and perhaps parallel to ##\mathbf{c}##?). But I'm having trouble:

##- \nabla A^t - \partial^t \mathbf{a} = - \nabla \left( C^t e^{i \mathbf{K} \cdot \mathbf{R}} \right) - \partial^t \left( \mathbf{c} e^{i \mathbf{K} \cdot \mathbf{R}} \right) = -C^t \nabla e^{i \mathbf{K} \cdot \mathbf{R}} - \mathbf{c} \partial^t e^{i \mathbf{K} \cdot \mathbf{R}}##.

If the first term vanishes, taking the real part of the second term I get ##\mathbf{E} = \sin (\mathbf{K} \cdot \mathbf{R}) k \mathbf{c}##, which works if it's true that ##\mathbf{c} \parallel \mathbf{E}## (correct magnitude, and correct right-handed set for ##\mathbf{E}##, ##\mathbf{B}##, and ##\mathbf{k}##). Otherwise, for the first term I get:

##-C^t \nabla e^{i \mathbf{K} \cdot \mathbf{R}} = -i e^{i \mathbf{K} \cdot \mathbf{R}} C^t \mathbf{k}##,

whose real part I think is ##\sin (\mathbf{K} \cdot \mathbf{R}) C^t \mathbf{k}##, giving:

##\mathbf{E} = \sin (\mathbf{K} \cdot \mathbf{R}) \left( C^t \mathbf{k} + k \mathbf{c} \right) = \sin (\mathbf{K} \cdot \mathbf{R}) \left( (\mathbf{c} \cdot {\hat{\mathbf{k}}}) \mathbf{k} + k \mathbf{c} \right)##,

which doesn't seem right at all, unless it's true that ##\mathbf{c} \perp \mathbf{k}## in all frames (the condition under which the first term drops out). But I haven't been able to prove to myself that ##\mathbf{c} \perp \mathbf{k}## is a Lorentz-invariant statement. In fact, the notion seems silly, because unless I'm missing something it's equivalent to saying that I can't boost to a frame in which ##C^t \neq 0##.

Given that ##\mathbf{K}## is lightlike and ##\mathbf{C}## is orthogonal to it (and obviously spacelike), does it follow that their spatial three-vector components ##\mathbf{k}## and ##\mathbf{c}## are perpendicular in all Lorentz frames? If so, is it easily demonstrated? And if not, can you spot where I've gone off the rails?

##\mathbf{A}(\mathbf{R}) = \Re \left( \mathbf{C} e^{i \mathbf{K} \cdot \mathbf{R}} \right)##

for constant future-pointing lightlike ##\mathbf{K} = (K^t, \mathbf{k})## and constant ##\mathbf{C} = (C^t , \mathbf{c})## orthogonal to ##\mathbf{K}##, I

*think*I correctly get the magnetic field by taking the real part of this:##\nabla \times \mathbf{a} = \nabla \times \left( \mathbf{c} e^{i \mathbf{K} \cdot \mathbf{R}} \right) = \nabla e^{i \mathbf{K} \cdot \mathbf{R}} \times \mathbf{c}##,

which gives me ##\mathbf{B} = \sin (\mathbf{K} \cdot \mathbf{R}) (\mathbf{k} \times \mathbf{c}) ##. That seems reasonable, and if it's right then I expect ##\mathbf{E}## to have magnitude ##\sin (\mathbf{K} \cdot \mathbf{R}) \Vert \mathbf{k} \times \mathbf{c} \Vert## and be perpendicular to both ##\mathbf{B}## and ##\mathbf{k}## (and perhaps parallel to ##\mathbf{c}##?). But I'm having trouble:

##- \nabla A^t - \partial^t \mathbf{a} = - \nabla \left( C^t e^{i \mathbf{K} \cdot \mathbf{R}} \right) - \partial^t \left( \mathbf{c} e^{i \mathbf{K} \cdot \mathbf{R}} \right) = -C^t \nabla e^{i \mathbf{K} \cdot \mathbf{R}} - \mathbf{c} \partial^t e^{i \mathbf{K} \cdot \mathbf{R}}##.

If the first term vanishes, taking the real part of the second term I get ##\mathbf{E} = \sin (\mathbf{K} \cdot \mathbf{R}) k \mathbf{c}##, which works if it's true that ##\mathbf{c} \parallel \mathbf{E}## (correct magnitude, and correct right-handed set for ##\mathbf{E}##, ##\mathbf{B}##, and ##\mathbf{k}##). Otherwise, for the first term I get:

##-C^t \nabla e^{i \mathbf{K} \cdot \mathbf{R}} = -i e^{i \mathbf{K} \cdot \mathbf{R}} C^t \mathbf{k}##,

whose real part I think is ##\sin (\mathbf{K} \cdot \mathbf{R}) C^t \mathbf{k}##, giving:

##\mathbf{E} = \sin (\mathbf{K} \cdot \mathbf{R}) \left( C^t \mathbf{k} + k \mathbf{c} \right) = \sin (\mathbf{K} \cdot \mathbf{R}) \left( (\mathbf{c} \cdot {\hat{\mathbf{k}}}) \mathbf{k} + k \mathbf{c} \right)##,

which doesn't seem right at all, unless it's true that ##\mathbf{c} \perp \mathbf{k}## in all frames (the condition under which the first term drops out). But I haven't been able to prove to myself that ##\mathbf{c} \perp \mathbf{k}## is a Lorentz-invariant statement. In fact, the notion seems silly, because unless I'm missing something it's equivalent to saying that I can't boost to a frame in which ##C^t \neq 0##.

Given that ##\mathbf{K}## is lightlike and ##\mathbf{C}## is orthogonal to it (and obviously spacelike), does it follow that their spatial three-vector components ##\mathbf{k}## and ##\mathbf{c}## are perpendicular in all Lorentz frames? If so, is it easily demonstrated? And if not, can you spot where I've gone off the rails?