Epicycles and Poincar\'{e} Resonances in General Relativity

TL;DR

This paper discusses the method of geodesic deviations in general relativity, highlighting subtleties in its application to bound orbits, and demonstrates its effectiveness in accurately approximating orbits in Schwarzschild space-time.

Contribution

It provides a detailed analysis of second-order geodesic deviations in Schwarzschild space-time, improving analytical approximations of bound orbits.

Findings

Second-order deviations yield accurate orbit approximations.

Method remains effective up to the innermost stable circular orbit.

Highlights subtleties in secular motion analysis.

Abstract

The method of geodesic deviations provides analytic approximations to geodesics in arbitrary background space-times. As such the method is a useful tool in many practical situations. In this note we point out some subtleties in the application of the method related to secular motions, in first as well as in higher order. In particular we work out the general second-order contribution to bound orbits in Schwarzschild space-time and show that it provides very good analytical results all the way up to the innermost stable circular orbit.