Then the condition P = 0 is satisfied by either of 1 2 g ± ± − − 1 ( k ⊗ ⊗ k ) = 0 (written with indices, 1 2 g ± ± μ μ ν ν k μ μ k ν ν = 0 ). What has been shown so far is that wave solutions of Maxwell's equations, in the ray limit, must satisfy one of these two polynomial conditions. The tensors g ± ± − − 1 therefore determine the lightcone structures. The fact that there are two of them implies a double light cone structure - one for each of the two polarization states, i.e. birefringence. In vacuum, it is readily found that g + − − 1 = g − − − − 1 = g − − 1 degenerates to the space-time metric. Since the g ± ± − − 1 determine the lightcones in media in the way that g − − 1 does for the vacuum, they are referred to as optical metrics. However, it is perhaps more appropriate to take the point of view that the space-time metric happens to also serve as the optical metric in vacuum, which is not so surprising considering that the space-time metric is the only available structure in vacuum. So far, no assumptions have been imposed on the form of ε ε , μ μ , γ γ e , or γ γ h , so there are currently 36 freely specifiable parameters. To determine the optical metrics, Thompson imposes the conditions that γ γ e and γ γ h are antisymmetric with respect to g (i.e. antisymmetric when the indices on γ γ e and γ γ h are either both up or both down). The antisymmetry condition allows them to be written in the forms