= Solution
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 $-f+2B=0$. Thus
$$
\boxed{X=-\partial_r,\qquad Y=\partial_v+\frac{f(r_0)}2\partial_r\quad\text{on }S.}
$$
The 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.
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