At the end you say:<p>>Let me remind you that these are not the real equations of motion for the photon. The real equations of motion would have been achieved by calculating all of the geodesic equations from the Lagrangian. There are four of them instead of the three equations (in vector form) above. However, the three spatial equations will generate the exact same spatial curve as the real geodesic equations would, and they were relatively easily achieved.<p>What do you mean that these are not the real equations of motion for the photon, but that they give the exact same spatial curve as the real geodesic equations? Are there cases where these equations would not give the exact same curve as the real geodesic equations?