= Solution
An <event horizon> is a <null hypersurface>. For its spherical defining function $H(v,r)=r-r_H(v)$, the inverse radial <metric tensor> has $g^{vv}=0$, $g^{vr}=1$ and $g^{rr}=f=1-2m(v)/r$. The normal's <null condition> is
$$
0=g^{ab}\partial_aH\partial_bH=f(v,r_H)-2\dot r_H.
$$
It follows that the <event horizon> follows the outgoing <radial null geodesics of the Vaidya metric>:
$$
\boxed{2r_H\dot r_H=r_H-2m(v).}
$$
This <ordinary differential equation> also describes other outgoing <null hypersurfaces>; the global <event horizon> is selected by its future boundary condition, not by this local equation alone.
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