Relativistic invariance of dispersion-relations and their associated wave-operators and Green-functions

Identifying invariance properties helps in simplifying calculations and consolidating concepts. Presently the special relativistic invariance of dispersion relations and their associated scalar wave operators is investigated for general dispersive homogeneous linear media. Invariance properties of the four-dimensional Fourier transform integrals is demonstrated, from which the invariance of the scalar Green function is inferred. Dispersion relations and the associated group velocities feature in Hamiltonian ray tracing theory. The derivation of group velocities for moving media from the dispersion relation for these media at rest is discussed. It is verified that the group velocity concept is a proper velocity, satisfying the relativistic transformation formula for velocities. Conversely, if we assume the group velocity to be a true (e.g., mechanical) velocity, then it follows that the dispersion relation must be a relativistic invariant.