Geodesic deviation at higher orders via covariant bitensors

TL;DR

This paper develops a covariant formalism using covariant bitensors to derive higher-order geodesic deviation equations, enabling precise analysis of neighboring worldlines in curved spacetime.

Contribution

It introduces a systematic method to compute higher-order corrections to geodesic deviation using covariant bitensor expansions, extending the classical deviation equation to all orders.

Findings

Derived geodesic deviation equation valid to all orders.

Explicitly expanded the deviation equation up to fourth order.

Presented formal solutions and action principles for higher-order deviations.

Abstract

We review a simple but instructive application of the formalism of covariant bitensors, to use a deviation vector field along a fiducial geodesic to describe a neighboring worldline, in an exact and manifestly covariant manner, via the exponential map. Requiring the neighboring worldline to be a geodesic leads to the usual linear geodesic deviation equation for the deviation vector, plus corrections at higher order in the deviation and relative velocity. We show how these corrections can be efficiently computed to arbitrary orders via covariant bitensor expansions, deriving a form of the geodesic deviation equation valid to all orders, and producing its explicit expanded form through fourth order. We also discuss the generalized Jacobi equation, action principles for the higher-order geodesic deviation equations, results useful for describing accelerated neighboring worldlines, and the formal general solution to the geodesic deviation equation through second order.