Eur. Phys. J. C (2018) 78: 661
https://doi.org/10.1140/epjc/s10052-018-6133-1

Regular Article - Theoretical Physics

On integrability of the geodesic deviation equation

¹ Departamento de Física, Universidade Federal de Ouro Preto, Ouro Preto, MG, 35400-000, Brazil
² National Institute of Technology, Maizuru College, 234 Shiraya, Maizuru, Kyoto, 625-8511, Japan
³ Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, Prague, Czech Republic
⁴ Perimeter Institute, 31 Caroline St. N., Waterloo, N2L 2Y5, ON, Canada

^* e-mail: marco.cariglia@ufop.edu.br

Received: 23 May 2018
Accepted: 2 August 2018
Published online: 18 August 2018

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.