# Direction of field line transforms like direction of a stick

1. Dec 28, 2013

### bcrowell

Staff Emeritus
I'm trying to figure out if there is an deep or simple relationship between the following two facts.

(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 $\tan\theta'=\gamma\tan\theta$.

(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 $\tan\theta'=\gamma\tan\theta$. (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.

Last edited by a moderator: May 6, 2017
2. Dec 28, 2013

### A.T.

Are you looking for an intuitive argument that makes it obvious? Maybe a stick that can rotate and has charged ends, so it aligns with the E-field lines. Obviously it must be the aligned with the E-field in every frame.

Last edited: Dec 28, 2013
3. Dec 28, 2013

### bcrowell

Staff Emeritus
D'oh -- is it really that simple!? I'm trying to see a flaw in this argument and not finding it.

4. Dec 29, 2013

### pervect

Staff Emeritus
An electric field can be represented by a 2-form, and a 2-form can be represented by a geometric object, a tubular like structure.

See MTW, or http://125.71.228.222/wlxt/ncourse/dccydcb/web/condition/9.pdf.

So we expect the two-form to transform, the electric field, and the representation of the two-form as "world tubes" to all transform in the same manner.

So we expect the field lines (which are just the center of the tube structure of the two-forms) to transform like sticks.

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5. Dec 29, 2013

### pervect

Staff Emeritus
Add importnat note: You need the B-field to be zoero for this arument to work.

6. Dec 29, 2013

### bcrowell

Staff Emeritus
The Warnick paper is nice -- thanks!

Well, what it requires is that there exist some frame in which B=0 (which I guess we can probably tell is true if, in some other frame, the invariant E.B is zero and the invariant B^2-E^2 is negative). In the old-fashioned tensor language, I find it easy to see why the transformation property we're talking about only works with respect to a preferred frame, and requires such a frame to exist. The transformation property treats the two frames asymmetrically, and this is also true in the case of the stick, which has a preferred rest frame.

Warnick's pictures of flux tubes, etc., are nice, but since the treatment isn't explicitly relativistic it becomes harder for me to see how you look at the pictures and see how they're affected by magnetic fields.

A.T.'s simple argument about a test dipole has some features that I'm now seeing are a little more complex than they appeared at first sight. In general, it is not true that an electric dipole p *in motion* points in the direction of the electric field. It experiences a torque $p\times(v\times B)$. So let's say that there's a preferred frame K in which B=0. In this frame, clearly an electric dipole in equilibrium points in the direction of E. If we now transform into some other frame K', then in this frame we have both an E and a B, and both of them make a torque on the moving dipole. So in the new frame, the dipole's equilibrium is due to a combination of electric and magnetic torques...and this makes it seem as if the whole property shouldn't even be true...??

7. Jan 15, 2014

### bcrowell

Staff Emeritus