On the electromagnetic constitutive laws that are equivalent to   spacetime metrics
  The raising of both indices in the components of the Minkowski electromagnetic field strength 2-form to give the components of the electromagnetic excitation bivector field can be regarded as being equivalent to an electromagnetic constitutive law, as well as being defined by the components of the spacetime metric. This notion is clarified, and the nature of the equivalent dielectric tensors and magnetic permeability tensors that are defined by some common spacetime metrics is discussed. The relationship of the basic construction to effective metrics is discussed, and, in particular, the fact that this effective metric is more general than the Gordon metric. 
D. H. Delphenich
gr-qc