Light rays in gravitating and refractive media: A comparison of the field to particle and Hamiltonian approaches

The field-to-particle method of H. P. Robertson as applied by Noonan, in order to obtain the general relativistic equations describing the trajectory of a photon in a refractive medium, is compared with Synge’s general relativistic Hamiltonian theory of waves and rays. For a photon in vacuum it is known that both approaches yield the same equation for the trajectory, i.e., a null geodesic. However for a photon in a medium, in contradistinction to the Hamiltonian theory, the field-to-particle method (a) yields equations of the photon trajectory valid only in a nondispersive medium, (b) the time component u0of the tangent to the ray remains an undetermined quantity, (c) agreement with the Hamiltonian theory is achieved by substituting into Noonan’s equations the Hamiltonian expression for u0.