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

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}}+Î _{\mathrm{m}}\right)}}, \end{align*} where $\ddot{I}{}_{\mathrm{m}}$ and ${\hat{I}}_{\mathrm{m}}$ 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 $\ddot{I}{}_{\mathrm{m}}+{\hat{I}}_{\mathrm{m}}=0$. 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.

21 pages, no figures. v2: references added, minor revisions. v3: accepted version. v4: matches the published version in Physical Review D

Cited by in corpus (4)

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• Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits
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• Strong deflection of massive particles via the geodesic deviation equation