## 2 Results
### Index of Refraction in Moving Frames and Fresnel-Fizeau Drag  
To bring further insights into this, we considered the following: The index of refraction of water is 1.33. For most glasses, it is around 1.5, etc. We assigned indices of refraction to materials, but this is only valid for isotropic media, where all waves have the same index. This is no longer true in isotropy-broken media, where different waves have different indices of refraction and the index of refraction is no longer a material parameter, but a property of the wave. The Lorentz transformations for the k-vector \(\mathbf{k}\) and frequency  
Figure 3: Topological transformation of a tetra-hyperbolic material; (**a**) Fresnel waves surface \(\mathcal{H}=0\) in the material rest frame \(S_{mat}\) (tetra-hyperbolic class); (**b**) material parameters matrix \(\hat{M}\) in \(S_{mat}\); (**c**) Fresnel waves surface in the moving frame \(S_{\mathbf{\beta}=0.992}\) (bi-hyperbolic material); (**d-i**) Topological phase transformations of Fresnel waves surfaces \(\mathcal{H}=0\) at high-k limit in moving frames from tetra-hyperbolic in \(S_{mat}\) in panel (**d**), to tri-hyperbolic in panel (**e**), and bi-hyperbolic in frames (**f-i**). \(\omega=k_{0}c\) from the material rest frame \(S_{mat}\) into a frame \(S_{\mathbf{\beta}}\) moving with velocity \(\mathbf{V}=c\mathbf{\beta}\) are [57]. \[\mathbf{k}_{\mathbf{\beta}}=\mathbf{k}+\left\{(\gamma-1)\frac{1}{\beta^{2}}(\mathbf{\beta}\cdot \mathbf{k})-k_{0}\gamma\right\}\mathbf{\beta}\]  
\[k_{0\mathbf{\beta}}=\gamma(k_{0}-\mathbf{\beta}\cdot\mathbf{k})\]  
Correspondingly, the Fresnel wave surface \(\mathcal{H}(\mathbf{k},k_{0})=0\) is modified to \(\mathcal{H}_{\mathbf{\beta}}\big{(}\mathbf{k}_{\mathbf{\beta}},k_{0\mathbf{\beta}}\big{)}=0\) as shown in Figures 2 and 3. More fundamentally, any electromagnetic wave with index of refraction \(\mathbf{n}=\mathbf{k}/k_{0}\) acquires a new index of refraction \(\mathbf{n}_{\mathbf{\beta}}\) in moving frames according to  
\[\mathbf{n}_{\mathbf{\beta}}=\frac{\mathbf{k}_{\mathbf{\beta}}}{k_{0\mathbf{\beta}}}=\frac{\mathbf{n}+ \left\{(\gamma-1)\frac{1}{\beta^{2}}(\mathbf{\beta}\cdot\mathbf{n})-\gamma\right\} \mathbf{\beta}}{\gamma(1-\mathbf{\beta}\cdot\mathbf{n})} \tag{15}\]  
Note that no material parameters enter Equation (15). This means that any wave with a certain index of refraction will appear to have its index of refraction changed in moving frames according to Equation (15), independently of the material medium that hosts this. For example, in Figure 4, we plot indices of refraction \(n_{\mathbf{\beta}}\) as seen in all possible moving frames as a function of \(\mathbf{\beta}\) for two waves with \(n_{\mathbf{\beta}=0}=1.33\) (Figure 4a) and \(n_{\mathbf{\beta}=0}=3.45\) (Figure 4b) propagating the \(z\)-direction in the material rest frame \(S_{mat}\). As can be seen from Figure 4, there are two universal features in the function given by Equation (15). Firstly, for the frames moving with \(\beta\to 1,\gamma\to\infty\) indicated by the red circles in Figure 4, \(\mathbf{n}_{\mathbf{\beta}}\to-\hat{\mathbf{\beta}}\). Note that in this case, \(k_{\mathbf{\beta}}\to\infty\), but this does not result in a high-k waves, since \(k_{0\mathbf{\beta}}\to\infty\), meaning that in these frames, \(n_{\mathbf{\beta}}\to 1\) and the waves appear vacuum-like as if they were under optical neutrality conditions [58]. We call these frames N-frames for \(\mathfrak{Na}-\)light in Hebrew or \(S_{\mathfrak{R}}\)-frames [1, 2]. Secondly, in frames moving with \(\beta_{x}\to 1/n_{\beta=0}\), indicated by the green lines in Figure 4, the index of refraction tends to infinity, since \(k_{0\mathbf{\beta}}\to 0\) and the waves appear as a quasistatic high-k dark field. We call these frames -nframes for \(\mathfrak{Na}-\)darkness in Hebrew or \(S_{\mathfrak{R}}\)-frames [1, 2]. As we mentioned above, all waves in water or silicon have the same index of refraction 1.33 or 3.45 in the material rest frame \(S_{mat}\). Their indices transform according to Equation (15) into moving frames.