Strong-deflection expansion of the deflection angle near a degenerate photon sphere

TL;DR

This paper develops a precise strong-deflection expansion for light rays near a degenerate photon sphere in certain spacetimes, revealing universal and local factors that influence the deflection angle.

Contribution

It introduces a novel prescription that isolates the divergent part of the deflection angle, providing a universal leading term and invariant local factors in the expansion.

Findings

The leading coefficient factorizes into a universal branch constant and a local factor.

The local factor admits an invariant representation via the Weyl tensor's electric part.

Analytic examples confirm the factorization and provide explicit coefficients.

Abstract

We present a strong-deflection expansion for the deflection angle of light rays scattered near a degenerate photon sphere in asymptotically flat, static, and spherically symmetric spacetimes. Our prescription isolates the divergent contribution to the deflection-angle integral arising from the ray's passage near the marginal orbit in a way that remains well defined at marginality, thereby yielding a unique leading power-law term. When expressed in terms of the radius of closest approach, the leading coefficient in the strong deflection limit factorizes into a universal branch constant and a local factor determined by the third derivative of the effective potential at the degenerate photon sphere. Passing to the expansion in terms of the impact parameter then only multiplies the coefficient by an additional local conversion factor. We show that the local factor in the closest-approach expansion admits an invariant representation through the areal-radius derivative of a dimensionless tidal measure constructed from the electric part of the Weyl tensor. In general relativity, we further relate this quantity to the areal-radius derivative of a weighted null-energy density profile. Analytic examples validate this factorization and yield closed-form expressions for the leading divergent coefficients in representative marginal configurations.