Analytical solutions of bound timelike geodesic orbits in Kerr spacetime

TL;DR

This paper derives the first analytical solutions for bound timelike geodesics in Kerr spacetime using elliptic integrals and Mino time, enabling precise calculations of fundamental frequencies and coordinates relevant for gravitational wave research.

Contribution

It provides the first analytical expressions for all coordinates and fundamental frequencies of bound timelike geodesics in Kerr spacetime using Mino time.

Findings

Derived analytical solutions in terms of elliptic integrals

First expressions for fundamental frequencies of Kerr geodesics

Explicit coordinate formulas for bound timelike orbits

Abstract

We derive the analytical solutions of the bound timelike geodesic orbits in Kerr spacetime. The analytical solutions are expressed in terms of the elliptic integrals using Mino time $\lambda$ as the independent variable. Mino time decouples the radial and polar motion of a particle and hence leads to forms more useful to estimate three fundamental frequencies, radial, polar and azimuthal motion, for the bound timelike geodesics in Kerr spacetime. This paper gives the first derivation of the analytical expressions of the fundamental frequencies. This paper also gives the first derivation of the analytical expressions of all coordinates for the bound timelike geodesics using Mino time. These analytical expressions should be useful not only to investigate physical properties of Kerr geodesics but more importantly to applications related to the estimation of gravitational waves from the extreme mass ratio inspirals.