Higher-order geodesic deviations applied to the Kerr metric

TL;DR

This paper introduces a method to generate approximate geodesics in Kerr spacetime by summing higher-order deviations from a simple geodesic, avoiding Newtonian approximations and providing explicit solutions for low-eccentricity orbits.

Contribution

It presents a novel approach to approximate geodesics in Kerr spacetime using higher-order deviations, with explicit solutions for low-eccentricity orbits without relying on Newtonian methods.

Findings

Effective for low-eccentricity orbits with arbitrary (GM/Rc^2) < 1/6

Explicit harmonic oscillator solutions for geodesic deviations

Applicable to closed orbital motion in Kerr spacetime

Abstract

Starting with an exact and simple geodesic, we generate approximate geodesics by summing up higher-order geodesic deviations within a General Relativistic setting, without using Newtonian and post-Newtonian approximations. We apply this method to the problem of closed orbital motion of test particles in the Kerr metric space-time. With a simple circular orbit in the equatorial plane taken as the initial geodesic we obtain finite eccentricity orbits in the form of Taylor series with the eccentricity playing the role of small parameter. The explicit expressions of these higher-order geodesic deviations are derived using successive systems of linear equations with constant coefficients, whose solutions are of harmonic oscillator type. This scheme gives best results when applied to the orbits with low eccentricities, but with arbitrary values of $(GM/Rc^2)$, smaller than 1/6 in the Schwarzschild limit.