Optical-area minimum method for static spherical black hole shadows

Abstract

We formulate a global optical-area method for shadows of static, spherically symmetric black holes. For the metric \(ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)d\Omega^{2}\), spherical sections of the optical geometry have area \(\mathcal{A}_{\rm opt}=4\pi C/A\). A null ray with impact parameter \(b\) can cross a spherical section only if \(b^{2}\leq C/A\). The capture threshold is fixed by the infimum of \(C/A\) on the connected interval between the observer and the black hole horizon. When attained at an interior point, the infimum gives \(b_{\rm sh}^{2}=\min(C/A)\), while a static observer outside the controlling minimum, on the inward-sky branch, measures \(\sin^{2}\alpha_{\rm sh}=\mathcal{A}_{*}/ \mathcal{A}_{\rm opt}(r_{\rm o})\). 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--Nordstr\"om, 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.