The 4-dimensional inverse of these tensors does not exist, but the bar notation ξ ξ ¯ ¯ denotes an inverse defined with respect to the subspace orthogonal to u , which exists and is a valid operation since it was noted above that ξ ξ is an automorphism of this subspace. In Minkowski space-time, the space-space part (relative to observer u ) of each of these tensors is equivalent to the traditional 3 × × 3 constitutive matrices of 3-vector electrodynamics. In terms of this alternative set of constitutive tensors, χ χ is found to be