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

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.

The Penrose singularity theorem concludes future null-geodesic incompleteness; it does not require a divergent curvature invariant. This spacetime supplies exactly that distinction. For a null geodesic in the static coordinates, the Killing constantsand the null condition giveAs $r\to 0$, $f\to -r_{+}^2/L^2$. A radial geodesic with $J=0$ reaches $r=0$ linearly in finite affine parameter. If $J\ne 0$, then $|\dot{r}|\sim r_{+}|J|/(Lr)$, so it again reaches zero in finite affine parameter. The anti-de Sitter quotient ends these geodesics at $r=0$ even though local curvature remains finite. Thus the trapped circles lead to the BTZ causal singularity, satisfying the theorem through geodesic incompleteness rather than curvature blowup.