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 a ¯ is expressed via coordinate-dependent metric functions—our approach relates a ¯ 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 a ¯ = 1 1 − 8 π R m 2 ( ρ m + Π m ) , where ρ m and Π m denote the local energy density and tangential pressure evaluated at the photon sphere of areal radius R m . This result reveals that a ¯ is intrinsically governed by the local matter distribution, with the universal value a ¯ = 1 emerging when ρ m + Π m = 0 . Notably, this finding resolves the long-standing puzzle of obtaining a ¯ = 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.