= Winding-supported circular string
{title2=$X^0=2R\tau,\quad X^1+iX^2=Re^{i(\sigma+\tau)},\quad X^3=R(\sigma-\tau)$}
In flat noncompact spacetime a fixed circular loop's cylindrical worldsheet is not a stationary point of the <Nambu–Goto action>. A compact coordinate changes the possibilities. The displayed embedding in $\mathbb R^{2,1}\times S^1_R$ has unit winding, satisfies the <wave equation>, and obeys $\dot X\cdot X'=0$ and $\dot X^2+X'^2=0$. Its projection is a circle with static shape, while its motion in the compact direction supplies the second chiral null tangent. The <induced worldsheet metric> is $2R^2\operatorname{diag}(-1,1)$, so the full worldsheet is nondegenerate.
Back to article page
