Perturbations of spacetime around a stationary rotating cosmic string

We consider the metric perturbations around a stationary rotating Nambu-Goto string in Minkowski spacetime. By solving the linearized Einstein equations, we study the effects of azimuthal frame-dragging around the rotation axis and linear frame-dragging along the rotation axis, the Newtonian logarithmic potential, and the angular deficit around the string as the potential mode. We also investigate gravitational waves propagating off the string and propagating along the string, and show that the stationary rotating string emits gravitational waves toward the directions specified by discrete angles from the rotation axis. Waveforms, polarizations, and amplitudes which depend on the direction are shown explicitly.

Figure 1

Helical strings: $l+q=1$ ($q\ne 0$). The three-dimensional snapshots are given in the left panel, and the projection of strings to the $x\mathrm{\text{−}}y$ plane are given in the middle. The dashed circle in the middle figure represents the light cylinder $\rho =1/\Omega$. The parameters on the $l\mathrm{\text{−}}q$ plane are plotted in the right panel.

Figure 2

Three-dimensional snapshots and projections of string are shown in the case $l=0$, $1/5$, $1/3$, and $1/2$ as the same as Fig. 1.

Figure 3

Three cases of ${\kappa }_{ns}^2$ in the $n\mathrm{\text{−}}s$ plane.

Figure 4

The pairs $(n,s)$ of nonvanishing modes specified by (3.32) are shown by dots for $l=0$ and $l=1/3$ cases, as examples. The dots in the shadowed regions are short-range modes, while the dots in the unshaded region represent the gravitational wave modes. The dots on the border, thick broken lines, are traveling wave modes, and $(n,s)=(0,0)$ is the gravitational potential mode.

Figure 5

Waveforms of gravitational waves emitted to $\cos {\theta }_{s/n}=0$, $2/3$, $2/5$, $2/7$, and $6/7$ from a planar string $(l,q)=(0,1/2)$. In each row, the left panel shows the $(n,s)$ for nonvanishing modes. The dots on the solid lines correspond to the modes which propagate in the direction of ${\theta }_{s/n}$, i.e., fundamental mode and overtone modes for ${\theta }_{s/n}$. The amplitudes $|{\widehat{\tilde{h}}}_{+,\times }^{(n,s)}|$ of the fundamental mode and the overtone modes are shown in the middle two panels. The right panel shows the waveforms of the plus-mode (solid line) and cross modes (broken line).

Figure 6

Waveforms of gravitational waves emitted to the directions $\cos {\theta }_{s/n}=2/3$, 0, $-1/3$, $4/15$, $-8/15$, $-2/3$, and $-4/21$ from the string with $(l,q)=(1/3,1/2)$.