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Energy and Angular Momentum of a Relativistic String

  1. May 26, 2010 #1
    The problem is 2.1.b in Becker, Becker, and Schwarz. I can't figure out what I'm doing wrong... any help would be appreciated, I'm probably missing something dumb.

    1. The problem statement, all variables and given/known data

    For [tex]X^0=B\tau[/tex], [tex]X^1=B\cos \tau\cos\sigma[/tex], [tex]X^2=B\sin\tau\cos\sigma[/tex], [tex]X^i=0[/tex] for i>2, compute the energy and angular momentum and show that [tex]E^2|J|^{-1}=2\pi T[/tex].

    2. Relevant equations

    [tex]E=P^0[/tex]
    [tex]P^\mu=T\dot{X}^\mu[/tex]
    [tex]J^{\mu\nu}=T\int_0^\pi d\sigma\, {X}^\mu\dot{X}^\nu-{X}^\nu\dot{X}^\mu[/tex]

    3. The attempt at a solution

    [tex]E=T\dot{X}^0=BT[/tex],
    [tex]J^{12}=B^2 T\int_0^\pi d\sigma\,\cos^2\tau\cos^2\sigma+\sin^2\tau\cos^2\sigma=\frac{\pi}{2}B^2T[/tex]
    [tex]\frac{E^2}{|J|}=\frac{2T}{\pi}[/tex],

    which is off by a factor of [tex]\pi^2[/tex]. Any ideas?

    Thanks,
    Matthew
     
  2. jcsd
  3. May 26, 2010 #2
    I'm actually fairly convinced that this is a typo in the book, so don't spend as much time on it as I did...
     
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