Cosmic string lensing and closed timelike curves

In an analysis of the gravitational lensing by two relativistic cosmic strings, we argue that the formation of closed timelike curves proposed by Gott is unstable in the presence of particles (e.g. the cosmic microwave background radiation). Because of the attractorlike behavior of the closed timelike curve, we argue that this instability is very generic. A single graviton or photon in the vicinity, no matter how soft, is sufficient to bend the strings and prevent the formation of closed timelike curves. We also show that the gravitational lensing due to a moving cosmic string is enhanced by its motion, not suppressed.

Figure 1

Gravitational lensing by a straight cosmic string. The two images are identical. Note that they are not mirror reflections of each other, but one is a translational displacement of the other. As the source moves to the left (or equivalently, the cosmic string moves to the right), part of the right image is cut off, eventually leaving only a single image.

Figure 2

A straight cosmic string introduces a deficit angle $\delta$. Here, the sources $s$ and $s^{\prime }$ are to be identified.

Figure 3

The path of a photon coming from the source at the lower left and reaching the observer at the right. The straight cosmic string is moving with speed $v_{cs}$ to the left. The photon’s initial velocity ${\widehat{\mathbf{n}}}_1$ makes an angle $\alpha$ with respect to $-{\mathbf{v}}_{cs}$. At A the photon strikes the leading edge of the deficit wedge (a distance $r$ from the position of the cosmic string).

Figure 4

The Gott spacetime, before the strings are moving. This spacetime will be severed at the $y=0$ hyperplane (line AB), boosted, and then smoothly glued together again. A and B are separated by a distance of $2x$, while the cosmic strings are separated by a distance of $2d$.

Figure 5

This is the critical case where $\gamma {\delta }_0=2$. The deflection angle is calculated using Eq. (6) and the discontinuity in the world line using Eq. (3). We have drawn $d>0$ for clarity. The particle travels along the path labeled 1 to 7 and back to 1, arriving at the same point in space and time, i.e., a closed timelike curve. In this case, the particle is neither blueshifted nor redshifted.

Figure 6

This plot derives entirely from Eq. (6). Here, $\delta (\alpha )$ is the deflection angle of the photon as a function of the angle $\alpha$ between the photon direction and (minus) the cosmic string velocity (see Fig. 3). $\gamma {\delta }_0=2$ is the critical case, when the peak of $\delta (\alpha )$ touches the value $\pi$ (the solid horizontal line). A $\gamma {\delta }_0>2$ curve crosses $\delta (\alpha )=\pi$ at 2 points. Notable is where $\delta (\alpha )$ crosses $\pi$ (see e.g. the $\gamma {\delta }_0=3$ curve) with positive slope at ${\alpha }_s$. A positive slope crossing implies a stable fixed point. Photons with different initial $\alpha$ not far from ${\alpha }_s$ will follow closed timelike paths and approach $\alpha ={\alpha }_s$. Photons are blueshifted at all positive slope crossings. The amount of blueshift each time is given by Eq. (7).

Figure 7

The supercritical case. Cosmic string #1 is moving to the left and cosmic string #2 is moving to the right. The particle enters from the bottom right and reaches cs1’s upper deficit angle at 1. Its world line appears to jump to the right to 2, now directed down and slightly to the right. This process continues indefinitely, 3, 4, $\dots \infty$, the world line spiraling clockwise outward as it falls into a stable orbit (the two long, outermost parallel segments). The coordinate discontinuity seen here does not reflect any actual discontinuity. The numbers on the world line satisfy $2n+1\equiv 2n+2$ where the equivalence is identification via Eq. (2). The coordinate discontinuities plotted here are calculated with Eq. (3), and the trajectory angle is determined by Eq. (6). The initial conditions of the above trajectory are not tuned, but rather generic. Approximately half of all initial particle trajectories will end up in a CTC.