A new geometric shortcut to calculating black hole shadow sizes

A geometric framework built on geodesic and Gaussian curvature offers a fresh way to compute the radii of photon spheres, massive-particle orbits, and shadow sizes around black holes. Unlike prior perturbative approaches, it works for massive particles at leading order — a gap previously unaddressed. The mass-variation results could help interpret future observations if a black hole's mass changes detectably over time.