Deflection angle in the strong deflection limit for static and axisymmetric spacetimes: Local curvature, matter fields, and quasinormal modes

We investigate the deflection of photons in the strong deflection limit within static and axisymmetric spacetimes possessing reflection symmetry. As the impact parameter approaches its critical value, the deflection angle exhibits a logarithmic divergence. This divergence is characterized by a logarithmic coefficient and a constant offset, which we express in terms of the coordinate-invariant curvature quantities evaluated at the unstable circular photon orbit. The curvature contribution is encoded in the electric part of the Weyl tensor, reflecting tidal effects, and the matter contribution is encoded in the Einstein tensor, capturing the influence of local energy and pressure. We also express these coefficients using the Newman-Penrose scalars. By exploiting the relationship between the strong deflection limit and the quasinormal modes, we derive a new expression for the quasinormal mode frequency in the eikonal limit in terms of the curvature scalars. Our results provide a unified and coordinate-invariant framework that connects observable lensing features and quasinormal modes to the local geometry and matter distribution near compact objects.