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I think I can help with some explanation here, as metamaterials are not fringe or pseudo science but hard science on the cutting edge of our understanding of chemistry, quantum physics, and engineering. The doppler effect is difficult to understand in terms of metamaterials as it requires one to think in ways which we are not used to. The viewer is stationary, yes, but the photons of light or phonons of sound are moving, and moving though a medium with unique refractive properties.  This means that the wave itself will be curved in bizare ways. One of the effects of this is that the apparent wavelength of the photon or phonon will appear to lower as it approaches, and rise as it recedes from, a viewer.  The infinite inertia refers to the fact that a metamaterial can, in precise circumstances, temporarily halt light by trapping it in the lattice of electromagentic fields within the material or within the particles themselves as a quantum spin state charge. Another way to phrase this would be infintite internal inertia. Such effects have been thoroughly demonstrated, and you can see the article slow light for more information. You are correct in saying that very little in science is infinite, but the catch in a metamaterial is that it is not 'infinite' infinite inertia. The infinite inertia is an induced effect that lasts only as long as the proper conditions remain, whether that be an external current applied to the metamaterial, a temperature to which the material is raised or lowered, or internal quantum state which can be thrown off balance. At the point that these conditions end, the light trapped within the metamaterial is freed from its electromagnetic 'cage' and re-emmited. I hope that I have cleared up some of the misunderstandings about the properties of these materials. rant "