On the tensorial properties of the generalized Jacobi equation

TL;DR

This paper investigates the tensorial transformation properties of the generalized Jacobi equation, demonstrating its tensor nature in Fermi coordinates and highlighting limitations in general coordinate systems.

Contribution

It proves that solutions to the generalized Jacobi equation transform tensorially in Fermi coordinates and shows that this property does not hold in arbitrary coordinate systems for dimensions n≥3.

Findings

Solutions form tensors under affine coordinate changes in Fermi coordinates.

The tensor transformation rule preserves solutions in Fermi coordinates.

In dimensions n≥3, the transformation rule generally does not preserve solutions.

Abstract

The generalized Jacobi equation is a differential equation in local coordinates that describes the behavior of infinitesimally close geodesics with an arbitrary relative velocity. In this note we study some transformation properties for solutions to this equation. We prove two results. First, under any affine coordinate changes we show that the tensor transformation rule maps solutions to solutions. As a consequence, the generalized Jacobi equation is a tensor equation when restricted to suitable Fermi coordinate systems along a geodesic. Second, in dimensions n\ge 3, we explicitly show that the transformation rule does not in general preserve solutions to the generalized Jacobi equation.