Relativistic Epicycles : another approach to geodesic deviations

TL;DR

This paper presents a new analytical approach to solving geodesic deviation equations in Schwarzschild spacetime, providing insights into orbital dynamics, perihelion precession, and gravitational radiation for near-circular orbits.

Contribution

It introduces a linearized method for geodesic deviations near circular orbits, deriving higher-order corrections and applying these to gravitational wave estimates.

Findings

Derived harmonic oscillator form of geodesic deviations

Accurately predicts perihelion advance for near-circular orbits

Provides improved approximations including non-linear effects

Abstract

We solve the geodesic deviation equations for the orbital motions in the Schwarzschild metric which are close to a circular orbit. It turns out that in this particular case the equations reduce to a linear system, which after diagonalization describes just a collection of harmonic oscillators, with two characteristic frequencies. The new geodesic obtained by adding this solution to the circular one, describes not only the linear approximation of Kepler's laws, but gives also the right value of the perihelion advance (in the limit of almost circular orbits). We derive also the equations for higher-order deviations and show how these equations lead to better approximations, including the non-linear effects. The approximate orbital solutions are then inserted into the quadrupole formula to estimate the gravitational radiation from non-circular orbits.