A note on the geodesic deviation equation for null geodesics in the Schwarzschild black-hole

TL;DR

This paper explores the integrability of null geodesics in Schwarzschild spacetime using Hamiltonian formulation and differential Galois theory, providing new insights into geodesic deviation and its mathematical properties.

Contribution

It introduces a Hamiltonian approach to the geodesic deviation equation and applies Morales-Ramis theorem to analyze integrability of null geodesics in Schwarzschild spacetime.

Findings

Link between geodesic integrability and variational equation integrability

Explicit analysis of null geodesics and their variational equations

Application of differential Galois theory to geodesic equations

Abstract

We use the Hamiltonian formulation of the geodesic equation in the Schwarzschild space-time so as to get the variational equation as the counterpart of the Jacobi equation in this approach. In this context we are able to apply the Morales-Ramis theorem to link the integrability of the geodesic equation to the integrability, in the sense of differential Galois theory, of the variational equation. This link is strong enough to hold even on geodesics for which the usual conserved quantities fail to be independent, as is the case of circular geodesics. We show explicitly the particular cases of some null geodesics and their variational equations.