Using spacetime algebra, the Doppler shift, aberration and the refractive index covariance, relating the co-moving and laboratory frames, are readily obtained. Lorentz transformations and the cumbersome constitutive relations in the laboratory frame are circumvented altogether. The theory of a moving double-negative medium (DNG) is developed by using alossless model to account for material dispersion. The normal surfaces and the Doppler effect are analyzed. We show that a complex index of refraction may arise in the laboratory frame as a result of a complex aberration effect. A physical interpretation for this new type of aberration is also presented.