Finite-azimuth Schwarzschild photon plunge ($r\sim \frac{\mu }{2}({\varphi }_{\ast }-\varphi {)}^2$)

For a captured nonradial null geodesic with $u=1/r$, the large-$u$ orbit equation is $(du/d\varphi {)}^2\sim 2\mu u^3$. Integration on the inward branch givesThe curvature singularity is reached at finite azimuth. This does not prevent a near-critical ray from winding repeatedly near the photon sphere before its final plunge. A radial ray has constant azimuth, so this asymptotic coordinate parametrization is inapplicable to it.

For nonzero angular constant $h$, the interior null radial equation gives ${\dot{r}}^2\sim 2\mu h^2/r^3$. Its inward branch integrates to $r^{5/2}\sim \frac{5}{2}\sqrt{2\mu }|h|({\lambda }_{\ast }-\lambda )$. Thus the singularity is at finite affine parameter and $r\propto ({\lambda }_{\ast }-\lambda {)}^{2/5}$. The zero-angular-momentum case has a different affine scaling.