Winding-supported circular string ($X^0=2R\tau ,X^1+i{X}^2=R{e}^{i(\sigma +\tau )},X^3=R(\sigma -\tau )$)

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 ${\mathbb{R}}^{2,1}\times S_R^1$ has unit winding, satisfies the wave equation, and obeys $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$. 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 $2R^2\mathrm{diag}(-1,1)$, so the full worldsheet is nondegenerate.