Example of cross-polarized standing gravitational waves

We use a cosmological counterpart of the cylindrical Halilsoy solution to illustrate properties of cross-polarized standing gravitational waves.

Figure 1

The rotation of shear axes of geodesic null congruences for several values of the $\alpha$ parameter. The antinodes and nodes correspond to solid and dashed vertical lines, respectively. The initial conditions are set to have at the first antinode $z=\pi \lambda /2$: $\arg (\sigma )=\pi$. The remaining parameters are $\lambda =1/10$, $\beta =3$. We set $t_0=0.383171$, which gives $z_0=0.226091$.

Figure 2

The rotation of shear axes of geodesic null congruences for several values of the $\alpha$ parameter. The antinodes and nodes correspond to solid and dashed vertical lines, respectively. The initial conditions are set to have at the second antinode $z=3\pi \lambda /2$: $\arg (\sigma )=\pi$. The remaining parameters are $\lambda =1/10$, $\beta =3$. We set $t_0=0.383171$, which gives $z_0=-0.0880683$.

Figure 3

The cumulative effect of rotation of shear axes as measured at subsequent antinodes. The points above and at $\pi$ correspond to $k=0$. The points below and at $\pi$ correspond to $k=1$. The parameters are $\lambda =1/10$, $\beta =3$.

Figure 4

The photon’s flight time between subsequent antinodes (numbered by $N$) measured in the proper time of stationary observers at antinodes. The photon flies along the ${\partial }_z$ direction. The remaining parameters are $t_0=1$, $\lambda =1/10$.

Figure 5

The superenergy density for $\lambda =1/10$ and several sets of remaining parameters. The amplitude of the solid black function was multiplied by a factor of 300. The antinodes and nodes are indicated with solid and dashed vertical lines, respectively.

Figure 6

The $S^{\widehat{1}}$ component of the super-Poynting vector for $\lambda =1/10$ and several sets of remaining parameters. It shows the superenergy transfer in the ${\partial }_z$ direction. The amplitude of the solid black function was multiplied by a factor of 300. The antinodes and nodes are indicated with solid and dashed vertical lines, respectively.