= Free-ended straight-string Regge relation
{title2=$J=\alpha'M^2$}
With <string tension> $T=1/(2\pi\alpha')$, the free-ended <open string> $X^0=a\tau$, $X^1+iX^2=a\cos\sigma\,e^{i\tau}$, $0\leq\sigma\leq\pi$, is straight at every target time. Its <Nambu–Goto equations of motion> and <Virasoro constraints> hold, and its ends move at light speed. Integrating the <target-space Noether charges of a string> gives $M=\pi Ta$ and $J=\pi Ta^2/2$. Thus $J=\alpha'M^2$. Extending the same map to $0\leq\sigma\leq2\pi$ describes a double-covered folded <closed string>, with twice the mass and twice the <angular momentum>, hence $J=\alpha'M^2/2$. The different slopes record the different boundary topology.
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