= Weak-field photon-orbit validity
{title2=$u=r_S/r\ll1$}
The first-order null-orbit equation $(u')^2=r_S^2E^2/h^2-u^2(1-u)$ formally has a circular solution at $u=2/3$, with $|h/E|=3\sqrt3r_S/2$. Its second-order equation gives a perturbation $\delta u''=\delta u$, hence instability. However, $u=2/3$ is outside the weak-field expansion. The exact <Schwarzschild spacetime> has a corresponding unstable <photon sphere>, but that physical result needs the exact geometry; extrapolating the linearized metric is not a controlled proof.
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