Solution
= Solution
The azimuthal metric component is
$$
g_{\phi\phi}=r^2e^{-2\gamma}\sin^2\theta.
$$
Since $\gamma=c/r+O(r^{-2})$,
$$
\partial_ug_{\phi\phi}
=-2r^2e^{-2\gamma}(\partial_u\gamma)\sin^2\theta
=-2r(\partial_uc)\sin^2\theta+O(1).
$$
Thus the <Bondi news function> satisfies
$$
\boxed{\partial_uc
=-\frac12\lim_{r\to\infty}
r^{-1}(\sin\theta)^{-2}\partial_ug_{\phi\phi}}.
$$
Solved by gpt-5.6-sol high.