 Solution

The absorption question concerns the geometric-optics capture cross-section, assuming rays reaching the central singular boundary are absorbed. Null geodesic motion alone does not specify a wave boundary condition at a singularity.

By spherical symmetry choose an equatorial ray. Conservation of energy and angular momentum gives $E=f\dot{t}$, $L=C\dot{\varphi }$, and null normalization givesHere $B$ is the impact parameter measured at infinity. A returning ray has an exterior turning point; capture occurs when none exists. The critical impact parameter obeys $B_c^2=1/\sup H$, and the cross-section is the area $\pi B_c^2$ in the incident impact-parameter plane.

For $Q=0$, $H=(r-2M)/r^3$ andThe maximum is at the unstable photon sphere $r=3M$, with $H=1/(27M^2)$. Therefore

For $Q^2=2M^2$, cancellation for $r>2M$ gives $H=1/r^2$. There is no regular exterior circular photon orbit: the supremum occurs at the singular limiting surface, with $\sup H=1/(4M^2)$. Rays with $B>2M$ turn at $r=B$; those with $B<2M$ reach the central boundary. Thus the extremal dilaton metric photon capture result isThe critical ray is a measure-zero boundary of the capture set. The finite capture cross-section is not the horizon area, which vanishes in this limit. Nor is it a claim about a low-frequency quantum absorption cross-section or an unspecified reflection law at the singularity.