Slepian’s Faster-Than-Light Wave

where ω = vk and k± = k x̂ ± kΓ ŷ. Slepian also noted that if v < c, then cos(kΓy) = cosh(kΓ′y) where Γ′ = iΓ = √ 1− v2/c2, and the waveform (1) again propagates with speed v. The vector potential (1), together with scalar potential V = 0, satisfy the conditions for the Coulomb, Gibbs, Lorenz and velocity gauges. See, for example, [2]. For completeness, the electric field is E = iωA ≡ E+ + E− where E± = iωA±, and the magnetic field is, using eq. (6),