Gravitational lensing by using the 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere

TL;DR

This paper introduces a more accurate method for calculating gravitational lensing deflection angles near photon spheres using the 0th order affine perturbation series, improving precision especially in retro lensing scenarios.

Contribution

It develops and applies a 0th order affine perturbation series approach for deflection angles, reducing hidden errors present in traditional strong deflection limit methods.

Findings

The 0th order affine perturbation series provides more accurate deflection angles.

In usual lensing, hidden errors are negligible, validating existing methods.

In retro lensing, the new method significantly improves accuracy and brightness predictions.

Abstract

The 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere is more accurate than a deflection angle in a strong deflection limit, which is used often, because the later has hidden error terms. We investigate gravitational lensing by using 0th order affine perturbation series of the deflection angle in a general asymptotically-flat, static, and spherical symmetric spacetime with the photon sphere. We apply our formula to Schwarzschild black hole, Reissner-Nordstr\"om black hole, and Ellis-Bronnikov wormhole spacetimes as examples. By comparing observables by using the deflection angles, we show that we can ignore the effect of the hidden error terms in the deflection angle in the strong deflection limit on the observables in a usual lens configuration with the photon sphere since the hidden error terms are tiny. On the other hand, in a retro lensing configuration, the deflection angle in the strong-deflection-limit analysis have error of several percent and the 0th order of affine perturbation series of the deflection angle has almost half of the error. Thus, in the retro lensing configuration, we should use the 0th order of affine perturbation series of the deflection angle rather than the deflection angle in the strong-deflection-limit analysis. The 0th order of affine perturbation series of the deflection angle can give a brighter magnification by a dozen percent than the one by using the deflection angle in the strong-deflection-limit analysis.