Optical-area minimum method for static spherical black hole shadows

We formulate a global optical-area method for shadows of static, spherically symmetric black holes. For the metric $d{s}^2=-A(r)d{t}^2+B(r)d{r}^2+C(r)d{\mathrm{\Omega }}^2$, spherical sections of the optical geometry have area ${\mathcal{A}}_{\mathrm{o}\mathrm{p}\mathrm{t}}=4\pi C\mathrm{/}A$. A null ray with impact parameter $b$ can cross a spherical section only if $b^2\le C\mathrm{/}A$. The capture threshold is fixed by the infimum of $C\mathrm{/}A$ on the connected interval between the observer and the black hole horizon. When attained at an interior point, the infimum gives $b_{\mathrm{s}\mathrm{h}}^2=\min (C\mathrm{/}A)$, while a static observer outside the controlling minimum, on the inward-sky branch, measures ${\sin }^2{\alpha }_{\mathrm{s}\mathrm{h}}={\mathcal{A}}_{\ast }\mathrm{/}{\mathcal{A}}_{\mathrm{o}\mathrm{p}\mathrm{t}}(r_{\mathrm{o}})$. The usual photon-sphere equation follows when the minimum occurs at a smooth interior point. Exponential instability additionally requires the minimum to be nondegenerate. The local optical-radius and photon-sphere formulas are established results. Our contribution is to organize them into an observer-to-horizon global selection rule that compares all stationary candidates and relevant endpoint limits. The radial function $B(r)$ does not affect the shadow angle, although it enters the coordinate-time instability rate, whose numerical value also depends on the normalization of the static time coordinate. We derive compact first- and second-order formulas for deformed metrics, demonstrate candidate comparison with a synthetic two-minimum profile, and apply the construction to Reissner–Nordström, Bardeen, charged dilaton, and Kottler black holes. An explicit transformation of the charged-dilaton example from a nonareal to an areal radial coordinate verifies radial-coordinate invariance, while the Kottler example probes a nonasymptotically flat static region.