On Integrability of the Geodesic Deviation Equation

TL;DR

This paper presents a method to integrate the geodesic deviation (Jacobi) equation in spacetimes with integrable geodesics, using conserved charges and invariant Wronskians, demonstrated on Kerr black hole geometries.

Contribution

It introduces a novel approach to solve the Jacobi equation by linearizing geodesic equations and conserved charges, applicable to integrable spacetimes, with explicit examples in Kerr geometries.

Findings

Explicit integration method for Jacobi equation in integrable spacetimes

Application to Kerr black hole and higher-dimensional geometries

Discussion of phase space and covariant Hamiltonian formulations

Abstract

The Jacobi equation for geodesic deviation describes finite size effects due to the gravitational tidal forces. In this paper we show how one can integrate the Jacobi equation in any spacetime admitting completely integrable geodesics. Namely, by linearizing the geodesic equation and its conserved charges, we arrive at the invariant Wronskians for the Jacobi system that are linear in the `deviation momenta' and thus yield a system of first-order differential equations that can be integrated. The procedure is illustrated on an example of a rotating black hole spacetime described by the Kerr geometry and its higher-dimensional generalizations. A number of related topics, including the phase space formulation of the theory and the derivation of the covariant Hamiltonian for the Jacobi system are also discussed.