New non-linear modified massless Klein–Gordon equation

The massless Klein–Gordon equation on arbitrary curved backgrounds allows for solutions which develop “tails” inside the light cone and, therefore, do not strictly follow null geodesics as discovered by DeWitt and Brehme almost 60 years ago. A modification of the massless Klein–Gordon equation is presented, which always exhibits null geodesic propagation of waves on arbitrary curved spacetimes. This new equation is derived from a Lagrangian which exhibits current–current interaction. Its non-linearity is due to a self-coupling term which is related to the quantum mechanical Bohm potential.