Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 1 / e / i

When $r_{-}=0$, $\Omega =0$ and $f=(r^2-r_{+}^2)/L^2$. A circle $S$ at fixed $v,r$ has circumference $2\pi r$. Choose its future null normals asFor a one-dimensional transverse surface, each null expansion is the logarithmic derivative of its length:Inside the horizon, $0<r<r_{+}$ implies $f<0$, so both expansions are negative. Every such circle is therefore a trapped surface, specifically a Trapped circle inside a nonrotating BTZ black hole.