Effects of rotating gravitational waves

We study gravitational waves to first and second order in amplitude in vacuum asymptotically flat spacetimes. The Einstein equations are solved to first order and these solutions are superposed to form a time-symmetric ingoing and then outgoing pulse regular everywhere. The waves are assumed to have odd parity and a nonvanishing angular momentum which keeps them away from the axis at all times. The averaged energy of the waves is evaluated. The relevant Einstein equation is then solved to second order in the amplitude. The influence of the angular momentum of the waves on the rotation of local inertial frames with respect to the frames at great distances is analyzed. The rotation of the frames occurs even in the region around the origin where spacetime is almost flat. The rotation is without time delay as it follows from the constraint equation. The results are illustrated graphically for various values of the “harmonic index $m$” corresponding to azimuthal rotation and the “harmonic index $l$” describing the latitudinal rotation of the waves. The apparent motions of the fixed stars on the celestial sphere as seen through rotating waves from the local inertial frame are calculated and displayed.

Figure 1

Snapshots of function ${\psi }_{lm}$ for $l=27$, $m=5$, and times $\tilde{t}=-6$, $-4$, $-2$, 0 (upper image) or $\tilde{t}=0$, 2, 4, 6 (lower image). For high values of $l$ the wave is concentrated near a shell with radius $r^2=a^2+t^2$, so we can combine waves at several moments into one figure. The wave rotates anticlockwise around the $z$ axis. A careful observation reveals that the wave comes inwards in the form of a leading spiral, i.e., with the outside ahead of the inside (see “white arms” at $\tilde{t}=-2$ and $-4$); at $\tilde{t}=0$ the spiral structure has changed to a cartwheel but rotation keeps the wave away from the origin; at $\tilde{t}>0$ the spiral becomes trailing (see “white arms” at $\tilde{t}=2$ and 4). At large $\tilde{t}$ the spiral becomes tighter (cf. snapshots at $\tilde{t}=6$).

Figure 2

Angular velocity of the central inertial frame ${\omega }_0(l,m;t)$ for $l=10$ and $m=1,2,\dots 10$ (from bottom to top). The vertical axis is scaled in units of ${\omega }_0(10,1;0)$.

Figure 3

The dependence of normalized angular velocity of the central inertial frame ${\omega }_0(l,1;t)/{\omega }_0(l,1;t=0)$ on the parameter $l=2$, 3, 10, 20, 30 (from inside to out). Also the function $(1+{\tilde{t}}^2{)}^{-3/2}$ is shown as a dashed line to indicate the limit for large $l$ [cf. Eq. (7.14)].

Figure 4

Since light from distant stars is influenced by the gravitational waves the observed positions of the stars change. An observer at the origin can record the apparent position of the stars on the celestial sphere on a photographic plate. When appropriately scaled and rotated, the trajectories of all stars are the same. A star starts at the origin of the plate ($x=y=0$ in the planes above) and moves along closed trajectories the structure of which becomes more complicated with increasing $l$. At the moment of time symmetry, $\tilde{t}=0$, the image is located at maximal value of $x$ (with $y=0$). This curve is an image of a straight line in the complex plane $z=1+is$, $s\in (-\infty ,\infty )$ mapped by the function $f(z)=z^{-l-2}$ [cf. Eq. (8.15)]. Together with the trajectory, positions at time $\tilde{t}=0,\pm 0.05,\pm 0.1,\dots$ are shown as circles.