Wormhole shadows

We propose a new method of detecting Ellis wormholes by use of the images of wormholes surrounded by optically thin dust. First, we derive steady solutions of dust and a more general medium surrounding the wormhole by solving relativistic Euler equations. We find two types of dust solutions: one is a static solution with arbitrary density profile, and the other is a solution of dust which passes into the wormhole and escapes into the other side with constant velocity. Next, solving null geodesic equations and radiation transfer equations, we investigate the images of the wormhole surrounded by dust for the above steady solutions. Because the wormhole spacetime possesses unstable circular orbits of photons, a bright ring appears in the image, just as in Schwarzschild spacetime. This indicates that the appearance of a bright ring solely confirms neither a black hole nor a wormhole. However, we find that the intensity contrast between the inside and the outside of the ring are quite different. Therefore, we could tell the difference between an Ellis wormhole and a black hole with high-resolution very-long-baseline-interferometry observations in the near future.

Figure 1

Embedding diagram of the Ellis wormhole: plot of $z$ vs $r^{*}$ in (2.4) with the $\phi$ direction.

Figure 2

Density distribution of dust for $u\ne 0$, which is in accordance with (3.17). ${\rho }_{\mathrm{a}}$ is dust density at the throat, $r=0$. The region $r<0$ corresponds to the opposite side of the wormhole.

Figure 3

Effective potential of the Ellis wormhole. The maximum point at $r=0$ corresponds to the unstable circular orbits of photons.

Figure 4

Photon trajectories around (a) the Ellis wormhole and (b) the Schwarzschild black hole. The coordinates $(x,y)$ are defined by (4.9) on the $\theta =\pi /2$ plane. These trajectories end up at the observer at $r=300a$, $\phi =0$. In (a) labels A, B, and C correspond to those in the effective potential in Fig. 3; the dashed line of A represents the trajectory on the other side of the wormhole ($r<0$). In (b) $r_g$ denotes the Schwarzschild radius; the trajectory ${\mathrm{B}}^{\prime }$ represents the photon which approaches and winds many times in the vicinity of the unstable circular orbit.

Figure 5

Light orbit in the $\theta =\pi /2$ plane. We adopt the rectangle coordinates $(x,y)$ defined by (4.9). The center of the wormhole is located at the origin and the observer at $(x_o,0)(\phi =0)$. We denote the intersection of the $y$ axis with the tangent to the ray at the observer by $y=\alpha$. Therefore, we can regard $\alpha$ as the apparent length from the center.

Figure 6

Radiation intensity for case 1: static top-hat distribution. (a) is the graph of $I_{\nu }(\alpha )/I_{\nu }(0)$, which does not depend on $\nu$ for the flat spectrum, $H(\nu )=\text{const}$. ${\alpha }_{\max }$ denotes the value of $\alpha$ where $I_{\nu }(\alpha )$ is the maximum; it corresponds to the photon which winds around the throat infinite times. (b) is the optical image, which is depicted based on (a). The color shows intensity and is irrelevant to the frequency.

Figure 7

Radiation intensity for case 2: nonstatic dust that flows into the wormhole and escapes into the other side. As in Fig. 6, (a) shows intensity as a function of $\alpha$, and (b) displays the optical image.

Figure 8

Radiation intensity for dust flowing into the Schwarzschild black hole. In (a), for comparison, we plot $I_{\nu }$ for the wormhole (case 2) as well as that for the black hole. They are rescaled so that the difference in shape between them is clear. (b) displays the optical image for the black hole.