The response of a medium to an electromagnetic perturbation can be dispersive in both time and space: the response at time t and position x depends on the disturbance at earlier times, t′ < t, and other positions, x′ ≠ x. In describing such a response it is appropriate to Fourier transform so that both the disturbance and the response are regarded as functions of frequency, ω, and wavevector, k. A variety of quantities may be chosen to describe the response and the disturbance. In the covariant description developed here, the response is described by the induced 4-current, J μ (k), and the disturbance is described by the 4-potential, A μ (k), where the argument k denotes the 4-vector k μ = [γ,k]. Provided that any nonlinearity is weak, one may expand the response in powers of the disturbance. The linear term defines the linear response 4-tensor, π μν (k), and the nonlinear terms define a hierarchy of nonlinear response 4-tensors. The response tensor completely characterizes the electromagnetic properties of the medium, and various physical requirements are reflected in mathematical constraints on π μν (k).
KeywordsRest Frame Lorentz Transformation Covariant Form Transverse Part Tensor Equation
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