Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 66 / 2 / a / Solution

Let $x^a(\lambda ,s)$ be a smooth family of affinely parametrized null geodesics. The tangent is $U=\partial x/\partial \lambda$, and the deviation vector is $Z=\partial x/\partial s$, comparing nearby rays at the same affine parameter. The two parameter derivatives commute, soThus the separation vector is Lie-transported along the null geodesic congruence; it is not required to be parallel-transported. Differentiating once more gives the geodesic deviation equation. With curvature convention $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$,For optical measurements one uses transverse deviations with $U\cdot Z=0$ and identifies longitudinal additions proportional to $U$. The scalar $U\cdot Z$ is constant along affine null rays by the displayed transport relation, so an initially transverse deviation remains transverse.