Note that each of these tensors is orthogonal, or transverse, to u , meaning that u ⋅ ⋅ α α = α α ⋅ ⋅ u ♭ ♭ = 0 for each α α ∈ ∈ { ε ε c , ξ ξ , γ γ b c , γ γ e c } , which can be seen from the antisymmetry of χ χ on each pair of indices. Since each of the 1-form fields defined above is also transverse to u , we may conclude that each α α is an automorphism of a subspace of the cotangent space defined by orthogonality with respect to the observer. In other words, everything operates in the observer's purely spatial 3-dimensional space. In terms of these parameters, χ χ is found to be