Deflection angle in the strong deflection limit: a perspective from local geometrical invariants and matter distributions

In static, spherically symmetric spacetimes, the deflection angle of photons in the strong deflection limit exhibits a logarithmic divergence. We introduce an analytical framework that clarifies the physical origin of this divergence by employing local, coordinate-invariant geometric quantities alongside the properties of the matter distribution. In contrast to conventional formulations -- where the divergence rate ā is expressed via coordinate-dependent metric functions -- our approach relates ā to the components of the Einstein tensor in an orthonormal basis adapted to the spacetime symmetry. By applying the Einstein equations, we derive the expression {align*} ā={1}{√{1-8π Rₘ^2(ρₘ+Πₘ)}}, {align*} where ρₘ and Πₘ denote the local energy density and tangential pressure evaluated at the photon sphere of areal radius Rₘ. This result reveals that ā is intrinsically governed by the local matter distribution, with the universal value ā=1 emerging when ρₘ+Πₘ=0. Notably, this finding resolves the long-standing puzzle of obtaining ā=1 in a class of spacetimes supported by a massless scalar field. Furthermore, these local properties are reflected in the frequencies of quasinormal modes, suggesting a profound connection between strong gravitational lensing and the dynamical response of gravitational wave signals. Our framework, independent of any specific gravitational theory, offers a universal tool for testing gravitational theories and interpreting astrophysical observations.