DOI: https://doi.org/10.36347/sjpms.2026.v13i03.004

Pages: 118-132

We present a variational formulation for photon spheres in static, spherically symmetric spacetimes that unifies existence, localization, stability, and observables under minimal metric assumptions. By identifying the photon sphere as the stationary point of b(r)=r/√(A(r)), we derive an elementary condition rA^' (r)=2A(r)that depends only on the temporal potential A(r)(independent of B(r)). From this, we obtain closed-form, two-sided bounds for the photon-sphere radius r_phand the critical impact parameter b_phusing only interval information about Aand A^'; a curvature-informed refinement incorporating ∣A^''∣further tightens these bounds. A simple stability test via the second variation of F(r)=A(r)/r^2(sign of F^'' (r_ph)) matches the geodesic-instability/Lyapunov picture and connects directly to the eikonal quasinormal-mode regime. On the observational side, we link the bounds to the black-hole shadow seen by a finite-distance observer through sin⁡α=b_ph √(A(r_0))/r_0, yielding rigorous, frequency-agnostic constraints on the shadow angle that reduce to α≃b_ph/r_0in the far field. In canonical tests, the bounds are exact for Schwarzschild and near-tight for Reissner–Nordström at moderate charge, and they propagate seamlessly to simple beyond-GR parameterizations of A(r). The framework is algebraic, transparent, and data-ready: it requires no field equations, complements ray-tracing, and provides priors for inference pipelines that combine shadow size, photon-ring structure, and ringdown information. We outline extensions to slow rotation, finite-distance systematics (including plasma), and model-agnostic metrics for strong-field gravity tests.