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

TL;DR

This work recasts the strong deflection limit of photon trajectories near photon spheres in static, spherically symmetric spacetimes into a coordinate-invariant framework. By expressing the divergence rate $\bar{a}$ and the constant offset $\bar{b}$ in terms of local, tetrad components of the Einstein tensor at the photon sphere, the authors connect strong lensing observables directly to local curvature and matter fields, yielding $\bar{a} = 1/\sqrt{1 - 8\pi R_m^2(\rho_m + \Pi_m)}$ (GR) and its invariant form $\bar{a} = 1/\sqrt{1 - R_m^2(G_{(0)(0)}^m + G_{(2)(2)}^m)}$. The framework shows universal behavior $\bar{a}=1$ when $\rho_m + \Pi_m = 0$ and links the strong deflection data to quasi-local mass and matter content, with implications for quasinormal modes via $\omega_{QNM} = \Omega_c l - i(n + 1/2)|\lambda_L|$, $\lambda_L = 1/(b_c \bar{a})$. Through explicit examples (vacuum, electrovacuum, scalar fields, and wormholes), the results illuminate how local physics at the photon sphere shapes strong lensing and dynamical responses, offering a robust path to testing gravity with lensing and gravitational-wave observations.

Abstract

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 $ρ_{\mathrm{m}}$ and $Π_{\mathrm{m}}$ 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 $ρ_{\mathrm{m}}+Π_{\mathrm{m}}=0$. 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.