Energy and Angular Momentum of a Relativistic String

[dodelson]

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.

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

 

[dodelson]

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...