The ( 2 2 ) tensor χ χ = χ χ α α β β μ μ ν ν is antisymmetric in each pair of indices, and the vacuum is seen to be a trivial dielectric such that χ χ v a c F = F . This means that the distribution of dielectric material within the curved background space-time can be completely described functionally by giving χ χ and smooth transitions from vacuum into media can be described. The electric and magnetic fields E → → , B → → , D → → , and H → → , as they are commonly understood in the 3-vector representation, have no independent existence. They are merely different parts of the 2-forms F and G , as measured relative to a chosen observer. Let u be the contravariant velocity 4-vector of the observer. Then one may define the covariant 1-forms