A covariant theory of negative phase velocity propagation

We develop a covariant criterion for negative phase velocity propagation, one that is robust to coordinate transformations in both space and spacetime, and can be applied to moving media. Conversely, the criterion used widely for static media, namely (1/2)Re(E×H∗)⋅k<0, is not covariant and can be satisfied by viewing a simple medium (characterized by rest-frame constitutive parameters ϵ,μ>0ϵ,μ>0) in a frame that moves faster than (ϵμ)−1/2(ϵμ)−1/2, the speed of light in the medium’s rest frame. Our covariant criterion also confirms that in the geometric optics limit negative refraction cannot occur in curved vacuum.