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Following Schwartz's argument re: measuring (g-2)

  1. Dec 13, 2007 #1
    I've been reading Melvin Schwartz's mostly excellent book, Principles of Electrodynamics. He has a section on measuring (g-2) for the muon, and I'm trying to follow the argument. I've attached the figure for reference.

    He takes the approach of considering an instantaneously comoving frame of reference (this is all within the context of special relativity) and measuring the various effects. Basically, for illustration sake, the problem we're considering is that of a particle in a uniform magnetic field [itex] B [/itex], moving at right angles to the magnetic field, so in the lab frame it should be in uniform circular motion. In the boosted frame, there is an electric field [itex] \bar{E} = \gamma \beta B [/itex], and the magnetic field there is [itex] \bar{B} = \gamma B [/itex] , and if we stayed in this frame (instead of moving to the next instantaneously comoving frame) the motion would be a cycloid.

    Schwartz analyzes an infinitesimal portion of the orbit in each case, as follows. He says that in each frame, the path would appear to be a segment of arc, but in the comoving frame that segment is 'foreshortened'. In the attached figure, he's drawn both arc segments on top of each other.

    Here's my problem: it seems to me that in the instantaneously comoving frame, at the instant when t=0, the particle has zero velocity. (We're measuring the velocity in a frame which has the same velocity as the particle, ergo it starts from rest in the primed frame.) Now the electric field is at right angles to the boost direction, so the initial motion should all be perpendicular to the boost. But in the lab frame, that infinitesimal portion of the arc should be along the boost direction, right? So it seems like the arc in one frame is perpendicular to the arc segment in the other frame, and so I don't understand why he draws them on top of each other.

    It's not just a small point, either. Based on this diagram, he equates [itex] s =s^\prime[/itex], since dimensions perpendicular to the boost don't contract. Since [itex] d = \gamma d^\prime[/itex], he can relate [itex] d\theta [/itex] to [itex] d\theta^\prime [/itex], and thereby relate the (instantaneous) angular velocity in the primed frame, to the cyclotron frequency in the lab frame. It seems pretty central to his argument.

    Can anyone help me out here? i.e. is my basic analysis -- that the arc segments are perpendicular, and that this invalidates his argument about [itex] s=s^\prime [/itex] -- essentially correct?

    Best of all would be if someone who has access to the book (it's Dover and highly rated, so probably several people here have it) can explain to me what the heck he's talking about in section 4-11. It seems like really interesting physics and I'd love to learn it. :)

    Thanks!
    Chip
     

    Attached Files:

  2. jcsd
  3. Dec 17, 2007 #2
    No replies? In retrospect I can understand that -- I made it hard for people without the book to follow the argument. Let me now reproduce it here; hopefully, along with the attached figure, that should grant everything you need to help me out.

    Now that it's placed in a bit of context, does that make it any easier for people to help me evaluate the argument? Again: I don't get how he can draw the arc segments to be parallel at their centres, when in an instantaneously comoving frame, the force at that instant should be perpendicular to the boost direction. (And obviously, the boost direction is the direction of motion in the lab frame.)

    His argument for the length of the arc being contracted seems to be based on length contraction along the direction of the boost, where the perpendicular dimensions (here, [itex] s = s ^\prime [/itex] ) don't contract. Can anyone help me understand what he's trying to say?

    Thanks in advance!
     
  4. Dec 17, 2007 #3

    pervect

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    It sounds like Schwarz is trying to explain Thomas precession.

    I'm not sure I particularly care for or follow his explanation. Online, you might try http://bohr.physics.berkeley.edu/classes/221/0708/notes/thomprec.pdf for an alternate explanation. Jackson also has a section about Thomas Precession in "Classical Electrodynamics", and many GR books will disuss it (for example, MTW's "Gravitation").

    A very simple overview would be this. A particle held in a circular orbit can be modelled as being subjected to a large number of Lorentz boosts. The effect of multiple Loretnz boosts can be described as taking a matrix product of the Lorentz transformation matrices (or the equivalent in tensor notation). The matrix product of the infinite number of Loretnz boosts of an object accelerated around in a circle is, however, not the identity matrix, but a rotation. This is easy to say, but it gets a bit more involved to prove.
     
  5. Dec 19, 2007 #4
    I think I've come to understand this problem to my satisfaction now.

    If we stayed in one instantaneously comoving frame (instead of always jumping to the next frame, etc.) we would see the path of the particle as following an ellipse, which itself is moving at a constant velocity. If we 'correct' for that motion (in the Galilean sense), we do indeed see that the length of the arc, at the point where this frame is the comoving one, is 'foreshortened' by the Lorentz contraction, just as Schwartz says. Then we can follow his whole argument.

    Basically, the problem is that he doesn't mention that the contracted path which the particle follows is itself moving, and this led to all my difficulties in understanding. Without taking that into account, the particle indeed stays still to first order in time, but that's not what's relevant.

    Thanks for the great reference. I skimmed it, and I'll probably check it out in greater detail when I have some time.
     
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