In a recent work PRD 106 L021501 (2022), a new geometric approach is proposed
to obtain the photon sphere (circular photon orbit) and black hole shadow
radius. In this approach, photon spheres and black hole shadow radius are
determined using geodesic curvature and Gaussian curvature in the optical
geometry of black hole spacetimes. However, the calculations in PRD 106 L021501
(2022) only restricted to a subclass of static and spherically symmetric black
holes with spacetime metric $g_{tt} \cdot g_{rr}=-1$, $g_{\theta\theta}=r^{2}$
and $g_{\phi\phi}=r^{2}\sin^{2}\theta$. In this work, we extend this approach
to more general spherically symmetric black holes (with spacetime metric
$ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$).
Furthermore, it can be proved that our results from the geometric approach are
completely equivalent to those from conventional approach based on effective
potentials of test particles.
Key Words: Photon Sphere, Black Hole Shadow, Optical Geometry, Gaussian
Curvature, Geodesic Curvature, Spherically Symmetric Black Hole
