Wormhole shadows in rotating dust

As an extension of our previous work, which investigated the shadows of the Ellis wormhole surrounded by nonrotating dust, in this paper we study wormhole shadows in a rotating dust flow. First, we derive steady-state solutions of slowly rotating dust surrounding the wormhole by solving relativistic Euler equations. Solving null geodesic equations and radiation transfer equations, we investigate the images of the wormhole surrounded by dust for the above steady-state solutions. Because the Ellis wormhole spacetime possesses unstable circular orbits of photons, a bright ring appears in the image, just as in Schwarzschild spacetime. The bright ring looks distorted due to rotation. Aside from the bright ring, there appear weakly luminous complex patterns by the emission from the other side of the throat. These structure could be detected by high-resolution very-long-baseline-interferometry observations in the near future.

Figure 1

Diagram of a two-dimensional Ellis wormhole curve surface embedded in three-dimensional Euclidean space: plot of $z$ vs $r^{*}$ in Eq. (2.5) with the $\phi$ direction. It shows that two various spaces are connected by the Ellis wormhole like a tunnel.

Figure 2

Effective potential of photons in the Ellis wormhole spacetime. The maximum point at $r=0$, which indicates the throat of the wormhole, corresponds to the unstable circular orbits. The region for by $r<0$ is the other side of the spacetime and we use a dashed line for the trajectory in this region.

Figure 3

Photon trajectories around the Ellis wormhole. The coordinates $(x,y)$ are defined as in (3.7). The dashed circle denotes the throat of the wormhole. All trajectories end up at the observer at $r=300a$, $\phi =0$. Three labels A, B, and C correspond to those in Fig. 2. The dashed line of the label A represents the trajectory on the other side region of the wormhole $(r<0)$.

Figure 4

Contour map of the dust density $n(r,\theta )/n_a$. The rectangle coordinates $(x,z)$ are defined as (4.30) and hence the wormhole throat corresponds to the circle $x^2+z^2=a^2$. No real spacetime exists in the black region of $x^2+z^2<a^2$. We set $u_0^r=-0.1$ and $C(\theta )=0$ and illustrate four cases $(f_0^{\theta },k)=(-0.02,2)$, (0.02, 2), $(-0.02,4)$ and (0.02, 4), in (a1), (a2), (b1), and (b2), respectively.

Figure 5

Positional relationship among the wormhole, the apparent position of a light source, and the observer. (a) Plane A is defined as the plane which is normal to ${\overrightarrow{r}}_{\mathrm{obs}}$ and contains the origin. We denote the intersection of plane A with the tangent to the ray at the observer by $\overrightarrow{\alpha }$, which corresponds the apparent position of a light source. (b) is the same diagram as (a) but the standpoint is on the $y$ axis. We accord the rotational axis of dust to the $z$ axis and denote the angle between it and ${\overrightarrow{r}}_{\mathrm{obs}}$ as $i$. We put the observer on the $x\text{−}z$ plane.

Figure 6

Optical images of the wormhole surrounded by rotating dust for several cases of $(f_0^{\theta },k)$: (a1), (a2), (b1), and (b2) show the cases of $(-0.02,2)$, (0.02, 2), $(-0.02,4)$ and (0.02,4), respectively. A bright ring looks distorted like the high-density region of dust. Additionally, there appears a weakly luminous pattern differently depending on the value of $(f_0^{\theta },k)$.

Figure 7

Optical images on the case of $i=\pi /4$: (a) is $(f_0^{\theta },k)=(-0.02,2)$, (b) is $(f_0^{\theta },k)=(-0.02,4)$. Other parameters are same values as the case of Fig. 6. The patterns are more complex than on the case of Fig. 6.