Covariant theory of wave dispersion

The wave equation follows from the Fourier transform of Maxwell’s equations, with the current separated into an induced part, that describes the response of the medium, and an extraneous part, that acts as a source term. General solutions of the inhomogeneous wave equation may be written down in terms of the Green’s function, sometimes also called the photon propagator. The natural wave modes of the medium correspond to poles in the photon propagator. In the absence of a medium, the only waves are transverse waves, with dispersion relation k = 0. In the presence of a medium, there can be a variety of different wave modes. The properties of a natural wave mode include its dispersion relation, its polarization vector and the ratio of electric to total energy in the waves. The energetics of waves in a specific wave mode includes the form of the energy-momentum tensor, and the separation of the energy density and energy flux in the waves into electric, magnetic and nonelectromagnetic components. The energetics also includes the damping of the waves.