Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 37E / d / Solution

For the outgoing family,When $r>2M$, this derivative is positive, while ingoing rays move to smaller $r$. The future light cone therefore has one outward and one inward radial edge, and a future-directed massive particle may move either inward or outward, provided its worldline remains inside that cone.

At $r=2M$, the outgoing edge has $dr/dv=0$ and lies along the horizon. When $0<r<2M$, even this nominally outgoing edge has $dr/dv<0$. Both future null directions point toward decreasing $r$, so every future-directed timelike direction between them does too. A massive particle inside the horizon cannot remain at fixed $r$ or return to $r>2M$; it must continue toward the curvature singularity at $r=0$.