Certain relativistic phenomena in crystal optics
Authors: Chee-Seng Lim 
Published: March 2003

Relativistic unsteady phenomena are established for a crystalline medium with unaligned sets of permittivity and permeability principal axes, but incorporating a compounded uniaxiality about some nonprincipal direction. All effects originate from a suddenly activated, arbitrarily oriented, maintained line current conducted with a finite velocity v. Integral representations studied in another paper (Chee-Seng) are applied. The original coordinate system is subjected to a series of rotational and translational, scaled and unscaled transformations. No specific coordinate frame is strictly adhered to. Instead, it is often expedient and advantageous to exploit several reference frames simultaneously in the course of the analysis and interpretations. The electric field is directly related to a net scalar field Δ involving another scalar Ψ and its complement Ψ̄ which can be deduced from Ψ; Ψ and Ψ̄ are associated with two expanding, inclined ellipsoidal wavefronts ξ and ξ̄; these are cocentered at the current origin and touch each other twice along the uniaxis. Elsewhere, ξ leads ξ̄. For a source current faster than ξ:vt ∈ extξ, Ψ≢0 within a finite but growing ’’ice-cream cone’’ domain, its nontrivial composition being χ−1/2 inside ξ and 2χ−1/2 inside part of a tangent cone from the advancing current edge vt to, and terminating at, ξ; the function χ vanishes along such a tangent cone. Alternatively, for a source current slower than ξ:vt∈ intξ, if vt is avoided, χ≳0 everywhere, while Ψ=χ−1/2 inside ξ but vanishes identically outside ξ. However, the crucial scalar field Δ depends on three separate current–velocity regimes. Over a slow regime: vt∈ intξ̄, Δ is nontrivial inside ξ wherein it is discontinuous across ξ̄. Over an intermediate regime: vt ∈ intξ  extξ̄, Δ takes four distinct forms on 12 adjacent domains bounded by ξ, ξ̄ and a double-conical tangent surface linking vt to ξ̄. But for a fast regime: vt∈ extξ, Δ assumes six distinct forms on 18 adjacent domains bounded by ξ, ξ̄ plus two double-conical tangent surfaces, convertexed at vt, to both ξ and ξ̄. Singularities are normally confined to these boundaries. Relative to a moving frame, χ is time-independent. Nevertheless, Ψ and, consequently, Δ evolve unsteadily, principally because of transitions across the expanding ellipsoids ξ and ξ̄ which also acquire a relative retreat from the current edge vt. An evolution scheme is discussed in detail. This produces, among other things, a steady state corollary which, in turn, covers Čerenkov radiation. A quadrical symmetry exists with respect to a family {Qν} of constant χ-surfaces. These are quadric surfaces cocentered at vt and having principal axes inclined to those of ξ (and ξ̄). Their interactions with ξ are closely examined. If vt ∈ extξ, each Qν is a hyperboloid of two sheets which are asymptotic to the double-conical tangent surface connecting ξ to vt; Ψ can become nontrivial on only one sheet, viz., that which is approached by ξ as the latter retreats from vt; eventually, two permanent intersections, one following the other, occur along two expanding and travelling parallel plane circuits. But if vt ∈ intξ, each Qν is an ellipsoid inside which ξ initially evolves until an encounter occurs, first as a point contact which immediately grows into a plane circuit; as this traverses Qν, it expands and then contracts to a diametrically opposite point where contact breaks off. Finally, an elliptical axisymmetry about a principal direction of {Qν} is demonstrated. Corresponding behaviors hold in relation to ξ̄.