I'm trying to figure out if there is an deep or simple relationship between the following two facts.(adsbygoogle = window.adsbygoogle || []).push({});

(1) A stick, in its own rest frame, makes an angle θ with the x axis. In a frame boosted in the x direction, it makes an angle with the x' axis that is given by [itex]\tan\theta'=\gamma\tan\theta[/itex].

(2) A charge, in its own rest frame, has an electric field line that makes an angle θ with the x axis. In a frame boosted in the x direction, the field line makes an angle with the x' axis that is given by [itex]\tan\theta'=\gamma\tan\theta[/itex]. (One way of unambiguously defining the notion that it's the "same field line" is that if the charge is suddenly accelerated from rest, the field lines inside and outside the radiation front have angles related in this way.)

Purcell remarks on the similarity at the end of section 5.7: See http://www.lightandmatter.com/purcell/ [Broken]

The simplest way I know in which to derive fact 1 is by taking the world-lines of the two ends of the stick, putting them through a Lorentz transformation, and then slicing through them with a plane of simultaneity in the new frame.

For fact 2, I can take the electromagnetic field tensor of an electric field, and transform into the new frame.

So although I know how to derive both facts, my derivations seem fairly unrelated. It seems spooky that the two results come out the same. The only relation I can see is that the timelike row of the EM field tensor is a vector whose timelike component is zero, and after we transform to the new frame, the antisymmetry of the tensor is preserved, so that the timelike component is still zero. This is sort of similar to the idea of describing the ends of the stick by a displacement vector, transforming, and then projecting out the spacelike part again. But the analogy doesn't seem especially close, since the transformation is different for a rank 1 tensor than for rank 2.

Is there any elementary derivation of this fact, other than the kind of long, tedious thing Purcell does? It would be cute to have something I could use with students who don't know anything about tensors. I've fiddled around for a long time trying to imagine something involving charged test particles in the shape of beads, sliding on sticks.

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# Direction of field line transforms like direction of a stick

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