Pulsating circular string
= Pulsating circular string
{title2=$L(t)=2\pi R|\cos t|,\quad E=2\pi TR$}
A circular <closed string> contracts and re-expands with $X^0=Rt$ and $X^1+iX^2=R\cos t\,e^{i\sigma}$. Its <proper length> is $2\pi R|\cos t|$, while its <energy> remains $2\pi TR$. At collapse the <induced worldsheet metric> degenerates, but the <Nambu-Goto phase-space action> supplies a regular continuation.