Gravitational birefringence of light in Schwarzschild spacetime

We compute the gravitational birefringence of light as it undergoes gravitational lensing. To this end we re-derive the Souriau-Saturnini equations in the Schwarzschild metric and solve them numerically and perturbatively. Our main result is an offset between the trajectories of the photons of opposite polarisations, which grows with time. We also find an intriguing instability of the spin component transverse to the momentum.