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dodelson
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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.
For X^0=B\tau, X^1=B\cos \tau\cos\sigma, X^2=B\sin\tau\cos\sigma, X^i=0 for i>2, compute the energy and angular momentum and show that E^2|J|^{-1}=2\pi T.
E=P^0
P^\mu=T\dot{X}^\mu
J^{\mu\nu}=T\int_0^\pi d\sigma\, {X}^\mu\dot{X}^\nu-{X}^\nu\dot{X}^\mu
E=T\dot{X}^0=BT,
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
\frac{E^2}{|J|}=\frac{2T}{\pi},
which is off by a factor of \pi^2. Any ideas?
Thanks,
Matthew
Homework Statement
For X^0=B\tau, X^1=B\cos \tau\cos\sigma, X^2=B\sin\tau\cos\sigma, X^i=0 for i>2, compute the energy and angular momentum and show that E^2|J|^{-1}=2\pi T.
Homework Equations
E=P^0
P^\mu=T\dot{X}^\mu
J^{\mu\nu}=T\int_0^\pi d\sigma\, {X}^\mu\dot{X}^\nu-{X}^\nu\dot{X}^\mu
The Attempt at a Solution
E=T\dot{X}^0=BT,
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
\frac{E^2}{|J|}=\frac{2T}{\pi},
which is off by a factor of \pi^2. Any ideas?
Thanks,
Matthew
