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.
Highlights
The geodesic deviation has been used to give a geometrical and physical interpretation of spacetimes in ordinary four dimensions and higher [10,11,12], while using first and higher order it has been used to construct approximations to generic geodesics starting from simple ones [13,14], that can be used to model extreme mass-ratio systems [15]
This is the focus of the present work, where we show that integrals of geodesic motion give rise, through linearization, to integrals of the Jacobi equation that are expressed in the form of invariant Wronskians
We show that any symmetry of the geodesic motion yields two solutions of the Jacobi system – twice as many as envisioned in references [26,27,28]
Summary
The geodesic deviation has been used to give a geometrical and physical interpretation of spacetimes in ordinary four dimensions and higher [10,11,12], while using first and higher order it has been used to construct approximations to generic geodesics starting from simple ones [13,14], that can be used to model extreme mass-ratio systems [15]. A Lagrangian formulation of the geodesic deviation equations, including an electromagnetic field and spin, and a treatment of higher order deviation equations can be found in [7,13,30], and is obtained by an expansion of that of geodesic motion Given this natural connection, it is reasonable to expect that symmetries of dynamics of the geodesic motion, when present, descend to symmetries of the Jacobi equation. Given a solution of the Jacobi equation (in particular generated by a Killing vector or a Killing tensor) we obtain a linearized conserved quantity, given by the Wronskian. Appendix 1 presents the Jacobi equation, the linearization procedure and the Wronskians from the point of view of a general phase space and a general set of equations of motion, before the introduction of a cotangent bundle or a specific Hamiltonian, and Appendix 1 details the construction of a covariant Lagrangian and Hamiltonian
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
