The Transformation Properties of the Energy?Momentum Tensor for Dispersive Waves

Abstract

A covariant formulation of wave dispersion is extended to include dissipation and hence to identify the energy-momentum tensor for dispersive waves. The resulting form is equivalent to that obtained from an averaged-Lagrangian approach. The transformation properties of the wave energy-momentum tensor are discussed. The appearance of unacceptable negative wave energies in frames moving at greater than the phase speed of the waves may be avoided by choosing a physically equivalent second branch of the solution of the dispersion equation in such frames. A quantum mechanical description of dispersive waves as a collection of wave quanta with energy liw and momentum lik is compatible with the form of the energy-momentum tensor discussed here. The quantum description also involves the occupation number which plays the role of the distribution function of the wave quanta. The classical counterpart of the occupation number is defined, shown to be an invariant and shown to obey an equation which may be interpreted directly as a transfer equation for a distribution of wave quanta.