Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 1 / d / i / Solution

Choose a coordinate sphere at late conformal time ${\eta }_0$ and comoving radius $r_0$. Its areal radius is $R=a(\eta )r$, so its area is $A=4\pi a^2{r}^2$. Future-directed radial null generators have $dr/d\eta =\pm 1$. Up to a common positive normalization of their affine tangents, their expansions therefore have the signs ofThe outgoing sign is positive because $a^{\prime }>0$. The ingoing sign is also positive wheneverAn arbitrarily large sphere exists because the spatial topology is ${\mathbb{R}}^3$. Both future null expansions are positive on such a sphere, so it is an anti-trapped surface.

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