Freely falling bodies in a standing-wave spacetime

We study the motion of free masses subject to the influence of standing gravitational waves in the polarized Gowdy cosmology with a three-torus topology. We show that antinodes attract freely falling particles and we trace the velocity memory effect.

Figure 1

The components of the Riemann tensor: ${R^{\widehat{1}}}_{\widehat{0}\widehat{1}\widehat{0}}$ (solid line), ${R^{\widehat{2}}}_{\widehat{0}\widehat{2}\widehat{0}}$ (dashed line), ${R^{\widehat{3}}}_{\widehat{0}\widehat{3}\widehat{0}}$ (dashed gray line) at the antinode which determine [via Eq. (7)] the force acting on test particles along directions ${\partial }_z$, ${\partial }_x$, ${\partial }_y$, respectively. The positive values imply attraction, negative repulsion. (The parameters: $\lambda =1/10$, $\beta =1/5$.)

Figure 2

The components of the Riemann tensor. A different range with all remaining details as in Fig. 1. After initial phase, the transverse components of the force oscillate about zero with no clear preference for a sign (Fig. 1). In contrast to that, the attraction toward the antinodes dominates in the longitudinal direction as the spacetime expands.

Figure 3

The Tissot diagram. Two rings (solid and dashed) of test particles were initially at rest relative to the central observer in his freely falling frame. Since the solid ring corresponds to a different phase of the standing gravitational wave than of the dashed ring, then their evolution looks differently. The interaction with the first wave is decisive. The test particles will move along presented profiles through a caustic point (line) only slightly perturbed by subsequent waves. The long-range effect of the first wave on the bundle of test particles trajectories resembles the velocity memory effect. The parameters: $\lambda =1/10$, $\beta =1/5$, the radius at $t_0=1.173$ of the solid ring is $r=0.1$. The dashed ring of test particles is released at $t_0^{\prime }=t_0+0.31$. The rings are evolved for $\mathrm{\Delta }\tau =5$.

Figure 4

The Tissot diagram for a contracting model ($t$ is a decreasing function of the proper time $\tau$). The parameters: $t_0=6.51$ (the solid ring), $t_0^{\prime }=t_0-0.31$ (the dashed ring). Remaining details as in Fig. 3.

Figure 5

Trajectories of initially stationary test particles near antinode (the expanding model). Parameters: $\lambda =1/10$, $\beta =1/5$, $t_0$ varies uniformly from 0.38 to 1.02, ${\xi }^{\widehat{1}}(t_0)=0.1$, $d{\xi }^{\widehat{1}}/d\tau (t_0)=0$.

Figure 6

Trajectories of a typical initially stationary test particle near antinode in the contracting model. The initial time $t_0=5$. All remaining details as in Fig. 5.

Figure 7

Geodesics. Parameters: $\lambda =1/10$, $\beta =1/5$. The initial time $t(\tau =0)=0.1$. The final time $t(\tau =40)\simeq 32.4$.

Figure 8

Geodesics in contracting model. Parameters: $\lambda =1/10$, $\beta =1/5$. The initial time $t(\tau =0)=10$.