Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 59 / 2 / f / Solution

A real spacetime point $p$ is represented by its twistor line $L_p\cong {\mathbb{CP}}^1$, lying entirely in projective null twistor space. Each point of $L_p$ is a null twistor specifying one real null geodesic through $p$. Their union is the null cone with vertex $p$. Thus the twistor description of the null cone is the family of null geodesics parametrized by the single line $L_p\subset PN$.

Equivalently, the spacetime points on that cone are precisely those whose twistor lines meet $L_p$:For distinct finite points this follows immediately from the incidence equations: a shared nonzero $\pi$ obeys $(q-p{)}^{A{A}^{\prime }}{\pi }_{A^{\prime }}=0$, which is equivalent to $\det (q-p)=0$, the null-separation condition. The flag description extends it to the compact spacetime. Projective null twistors encode whole unoriented null geodesics, so this includes both future and past generators; a time orientation and an affine patch distinguish the two halves. The line itself is the sphere of null directions at the vertex, not a three-dimensional image of the cone embedded pointwise into twistor space.