Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 37D / b / iii / Solution

Let $\phi =\varphi -F(R)$, soThe new coefficient of $d{R}^2$ isIt vanishes ifOne choice that tends to zero as $R\to \infty$ and also has a regular $a\to 0$ limit isSubstitution gives the outgoing null oblate spheroidal coordinates formThere is no $d{R}^2$ term, so $g_{RR}=0$ and ${\partial }_R$ is null for every $a\ge 0$.