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.
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.)
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.
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?