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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.

The area of a string worldsheet is computed from its induced worldsheet metric. The Nambu–Goto action is minus string tension times Lorentzian worldsheet area.

The intrinsic Lorentzian metric in the Polyakov action is independent of the string embedding map. Its equation relates it conformally to the induced worldsheet metric. A Weyl transformation changes the intrinsic metric without changing the classical action.