Energy-Momentum Tensor of Electromagnetic Field in Moving Dispersive Media and Instability: Relativistic Formulation

Abstract

Based on the Lagrangian principle, the energy-momentum tensor of the electromagnetic field is obtained for general linear anisotropic dispersive media whose components are moving with different uniform velocities. The field is assumed to change in space and time with a center wave-number 4-vector ${\ensuremath{\omega}}_{\ensuremath{\nu}}$. The energy-momentum tensor obtained can be classified into three parts of the pure electromagnetic field, the medium and the interaction with the external system, and the explicit expression for each part is given in terms of the two 4-forces acting on unit electric and magnetic charges. When $\ensuremath{\omega}_{\ensuremath{\nu}}^{}{}_{}{}^{2}>0$, the energy density is not positive definite, depending upon the frame of reference, while the "rest" energy of the field is nonvanishing (in contrast with the case of nondispersive media) and has a positive sign at least in isotropic media with no external source; thus the energy density is positive definite in the rest frame of the wave itself. The direction of the power flow 4-vector agrees with that of the wave packet. The instability of the wave is discussed from a general point of view based on the energy-momentum tensor and the covariance of equations, and the wave is found to become unstable when the group velocity reaches the velocity of light in vacuum, over a finite range of the wave number; the two beam instability in a plasma belongs to this type of instability. The wave becomes unstable also when, in any frame of reference, the conductivities of one or more of the medium components become negative and the field energy is positive, so that the total energy increases in timelike direction.