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

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.