Accurate closed-form trajectories of light around a Kerr black hole using asymptotic approximants

TL;DR

This paper develops highly accurate closed-form formulas for photon trajectories around Kerr black holes using asymptotic approximants, improving computational efficiency and accuracy over numerical methods.

Contribution

It introduces novel asymptotic approximants for photon trajectories in Kerr spacetime, extending previous work on bending angles and providing comprehensive expressions for arbitrary spin and impact parameters.

Findings

Achieved accurate closed-form trajectory expressions

Provided new coefficients for weak-field bending angles

Demonstrated significant time savings over numerical methods

Abstract

Highly accurate closed-form expressions that describe the full trajectory of photons propagating in the equatorial plane of a Kerr black hole are obtained using asymptotic approximants. This work extends a prior study of the overall bending angle for photons (Barlow, et al. 2017, Class. Quantum Grav., 34, 135017). The expressions obtained provide accurate trajectory predictions for arbitrary spin and impact parameters, and provide significant time advantages compared with numerical evaluation of the elliptic integrals that describe photon trajectories. To construct approximants, asymptotic expansions for photon deflection are required in various limits. To this end, complete expansions are derived for the azimuthal angle as a function of radial distance from the black hole in the far-distance and closest-approach (pericenter) limits, and new coefficients are reported for the bending angle in the weak-field limit (large impact parameter).