Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 52 / 2 / b / i

For the round sphere, every normal has only $v,r$ components because the Reissner-Nordstrom metric has no mixed angular terms. The vector $X=-{\partial }_r$ is normal and null: $g_{rr}=0$. Write the second null vector normal as $Y=A{\partial }_v+B{\partial }_r$. The normalization gives $X\cdot Y=-A=-1$, so $A=1$. Its null condition then reads $-f+2B=0$. ThusThe specified future orientation of $X$, together with $X\cdot Y<0$, makes $Y$ future-directed as well. This choice fixes the reciprocal scaling freedom of the two null vectors on the sphere.