## 2 Results
### Transverse Hidden Momentum and Ray-Wave Tilt as Differential Aberration of Ray and Wave Sources
Transforming back from \(S^{\prime\prime}\) into \(S_{mat}\) leads to an aberration of both \(g_{Min}\) and \(g_{Abr}\); however, this aberration is differential and results in different directions of \(g_{Min}\) and \(g_{Abr}\) in \(S_{mat}\). This can be understood in terms of sources of waves and rays, which appear to coincide in \(S^{\prime\prime}\), but are differentially aberrated and appear from different directions in \(S_{mat}\). It should be  
Figure 5: **(a)** The fields and apparent sources of rays and waves in frames \(S=S_{mat},S^{\prime}=S_{n}\), and \(S^{\prime\prime}\) for a material with \(\xi=diag\{2,2-1\}\). In S frame, the Abraham and Minkowski momenta have different longitudinal components and are at angle \(\theta\) to each other, which corresponds to refraction of the wave because of Fresnel–Fizeau drag with respect to frame \(S^{\prime}\) and the divergence of apparent sources for rays and waves as the result of aberration from frame \(S^{\prime\prime}\), where the ray and wave sources converge. **(b)** The hidden momentum breakdown in \(S\) frame, the ray and wave sources shown in reference to the Fresnel wave surface in k-space. noted that if the fields \(\mathbf{E}\) and \(\mathbf{H}\) in \(S_{mat}\) are not linearly polarized, the \(S^{\prime\prime}\) frame is different for different times within the period of wave oscillations and at different locations. In a general case of arbitrarily polarized waves, we turned to the \"light\" frames \(S_{\text{N}}\) as was described above, in \(S_{\text{N}}\)\(\mathbf{n}_{\mathbf{\beta}}\to-\mathbf{\beta}\) and \(\mathbf{n}_{\mathbf{\beta}}\to 1\), so that the waves appear as if they were under optical neutrality conditions [58]. Since \(\gamma\gg 1\), the field components parallel \(\mathbf{\beta}\) are the same as in \(S_{mat}\), while the field components transverse to \(\mathbf{\beta}\) are proportional to \(\gamma\), \(\mathbf{E}_{t},\mathbf{H}_{t},\mathbf{D}_{t},\mathbf{B}_{t}\propto\gamma\to\infty\), which means that the ray-wave tilt disappears at high \(\gamma\). Note that even though \(n\to 1\), the transverse field polarization is different from what it would be if the wave propagated in a vacuum, meaning that in the \(S_{\text{N}}\) frame, the wave appears in an optical neutrality state and not as a pure vacuum field configuration. The optical neutrality here is in the sense that while the material parameters matrix \(\hat{M}\) is not an identity matrix, the eigenvalues of the index of refraction operator are \(n\to\pm 1\). In Figure 6, we show the fields, the Roentgen and Shockley hidden momentum breakdown and the apparent ray and wave sources in frames \(S_{mat},S^{\prime}=S_{\text{N}},S^{\prime\prime}=S_{\text{N}}\) for the material described in Figure 3. We considered a wave with \(n=1.915\) propagating along the \(z\)-axis, i.e., \(\mathbf{g}_{Min}\propto\hat{z}\), as shown in Figure 6b in reference to the Fresnel wave surface. In Figure 6b, we see that the fields \(\mathbf{E},\mathbf{H}\) in \(S_{mat}\) are elliptically polarized and the instantaneous \(\mathbf{g}_{Abr}\) follows the green ellipse. The time-average \(\mathbf{g}_{Abr}\) points into the center of the green ellipse and is normal to the Fresnel wave surface. Transforming this wave from \(S_{mat}\) to \(S_{\text{N}}\) with \(\beta=\frac{1}{n}=0.522\) in the direction of the wave vector, we find that the wave becomes a quasistatic high-k wave with \(\mathbf{g}_{Abr}=0\) with longitudinally polarized fields \(\mathbf{E},\mathbf{H}\), while \(\mathbf{D},\mathbf{B}\) are still elliptically polarized. Reverse transformation into \(S_{mat}\) leads to non-zero hidden momentum and refraction of the wave as described above. Transformation from \(S_{mat}\) into \(S^{\prime\prime}\) results in \(\mathbf{g}_{Min}\) and \(\mathbf{g}_{Abr}\) tending to the same direction due to \(\mathbf{E}_{t},\mathbf{H}_{t},\mathbf{D}_{t},\mathbf{B}_{t}\propto\gamma\to\infty\).