For the circular embedding,
$$
\gamma_{\tau\tau}=-1+\dot R^2,
\qquad
\gamma_{\tau\sigma}=0,
\qquad
\gamma_{\sigma\sigma}=R^2.
$$
On $R=R_0\cos(\tau/R_0)$ this becomes
$$
(\gamma_{\alpha\beta})
=\cos^2\left(\frac{\tau}{R_0}\right)
\begin{pmatrix}
-1&0\\
0&R_0^2
\end{pmatrix}.
$$
Thus the worldsheet metric is conformally flat between collapse events and becomes degenerate at the instant when the loop shrinks to zero size.

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 1 / c / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-306/1/c/ii/solution