Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 1

Write $f(r)=1-r_{+}^2/r^2$ and let a dot denote differentiation with respect to an affine parameter $\tau$. The time-translation and $z$-translation Killing vector fields, together with rotational symmetry of the unit $S^3$, give the conserved quantitiesDefine $\epsilon =-g_{ab}{\dot{x}}^a{\dot{x}}^b$, so $\epsilon =1,0,-1$ for timelike, null and spacelike geodesics respectively. The normalization equation becomesIt therefore has the effective potential form

For radial null geodesics, $d{\Omega }_3=dz=0$ and $d{s}^2=0$, soIntroduce the tortoise coordinate byup to an additive constant. The plus sign gives $d(t-r_{\star })=0$ on outgoing rays, while the minus sign gives $d(t+r_{\star })=0$ on ingoing rays. Henceare respectively constant on the stated radial null families.

The normal to a surface of constant $r$ has squared normwhich vanishes at $r=r_{+}$. Thus this surface is a null hypersurface. The stationary Killing vector fieldhas $K^2=g_{tt}=-f$, so it becomes null there and generates a Killing horizon. Since $f$ has a simple zero, its surface gravity isA horizon cross-section has topology $S^3\times S^1$, where the circle is the periodic $z$ direction. Including a complete generator, the null hypersurface has topology

Using $v=t+r_{\star }$ puts the black string metric into regular ingoing form,Its inverse metric givesFor $r<r_{+}$, $f<0$, so $\nabla r$ is timelike. It is future-directed by continuity from the future horizon, where it equals the future generator ${\partial }_v$. If $X$ is any future-directed causal tangent, thenThus $r$ decreases strictly along every future-directed causal curve in the interior. No such curve can cross back through $r=r_{+}$ or reach future null infinity. The region $r<r_{+}$ is consequently outside the causal past of future null infinity and is part of the black hole region.