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 $\bar{a}$ is expressed via coordinate-dependent metric functions – our approach relates $\bar{a}$ 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 \begin{align*} \bar{a}=\frac{1}{\sqrt{1-8πR_{$\mathrm{m}$}^2\left(ρ_{$\mathrm{m}$}+Π_{$\mathrm{m}$}\right)}}, \end{align*} where ${\rho }_{\mathrm{m}}$ and ${\mathrm{\Pi }}_{\mathrm{m}}$ denote the local energy density and tangential pressure evaluated at the photon sphere of areal radius $R_{\mathrm{m}}$. This result reveals that $\bar{a}$ is intrinsically governed by the local matter distribution, with the universal value $\bar{a}=1$ emerging when ${\rho }_{\mathrm{m}}+{\mathrm{\Pi }}_{\mathrm{m}}=0$. Notably, this finding resolves the long-standing puzzle of obtaining $\bar{a}=1$ 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.