I have been self-studying the concept of index of refraction in relativity and encountered the Abraham-Minkowski Dilemma. In particular I found this article that appears to approach my ideas on this topic, with the Abraham relation being the kinetic momentum and the Minkowski representation being the canonical momentum.

http://www.ncbi.nlm.nih.gov/pubmed/20366861

Would it be incorrect to alternately express this idea as a difference in the covariant and contravariant representation of the momentum tensor using an invariant scalar index of refraction n? (Using the rest frame of a refractive medium as a preferred frame would even allow us to introduce a frequency dependence to n without risking Lorentz invariance.)

Minkowski:
Abraham:

Physics rests on many relations that can be expressed as products of tensors. Let A be a standard tensor, and A* be that tensor with the components scaled by n to the power of the time-like covariant indices minus time-like contravariant indices. It is straightforward to show that AB=C implies A*B*=C* for any tensors, including contractions on indices.

Derivatives become complicated because this representation implies that the metric depends on n.
and
It should be noted that photons bending as a result of gradients in index of refraction is a well known phenomenon.

My question is, does this relativistic representation of an index of refraction fit within the mainstream?