The Science of Interstellar

^{Fig. 15.2. Left: The wormhole, with small lensing width (just 5 percent of the wormhole’s radius), viewed from the bulk. Right: What the camera sees. Top to bottom: Increasing wormhole length: 0.01, 1, and 10 times the wormhole’s radius. [From simulations by Eugénie von Tunzelmann’s team using Oliver James’ code based on my equations.]}

As the wormhole is lengthened (middle picture), the primary image shrinks and moves inward, the secondary image also moves inward, and a very thin lenticular tertiary image emerges from the right edge of the crystal ball.

With further lengthening (bottom picture), the primary image shrinks further, all the images move inward, a fourth image emerges from the left edge of the crystal ball, a fifth from the right, and so forth.

These behaviors can be understood by drawing light rays on the wormhole as seen from the bulk (Figure 15.3). The primary image is carried by the black light ray (1), which travels on the shortest possible path from Saturn to the camera, and by a bundle of rays tightly surrounding it. The secondary image is carried by a bundle surrounding the red ray (2), which travels around the wormhole’s wall in the opposite direction to the black ray: counterclockwise. This red ray is the shortest possible counterclockwise ray from Saturn to the camera. The tertiary image is carried by a bundle surrounding the green ray (3), which is the shortest possible clockwise ray that makes more than one full trip around the wormhole. And the fourth image is carried by a bundle surrounding the brown ray (4): the shortest possible counterclockwise ray that makes more than one full trip around the wormhole.

^{Fig. 15.3. Light rays from Saturn, through the wormhole, to the camera.}

Can you explain the fifth and sixth images? and explain why the images shrink when the wormhole is lengthened? and explain why the images appear to emerge from the edge of the wormhole’s crystal-ball mouth and move inward?

The Wormhole’s Lensing Width

Having understood how the wormhole’s length affects what the camera sees, we then fixed the length to be fairly short, the same as the wormhole’s radius, and varied the gravitational lensing. We increased the wormhole’s lensing width from near zero to about half the wormhole’s diameter and computed what that did to the images the camera sees. Figure 15.4 shows the two extremes.

With very small lensing width, the wormhole shape (upper left) has a sharp transition from the external universe (horizontal sheets) to the wormhole throat (vertical cylinder). As seen by the camera (upper right), the wormhole distorts the star field and a dark cloud in the upper left only slightly, near the wormhole’s edge. Otherwise it simply masks the star field out, as would any opaque body with weak gravity, for example a planet or a spacecraft.

In the lower half of Figure 15.4, the lensing width is about half the wormhole’s radius, so there is a slow transition from the throat (vertical cylinder) to the external universe (asymptotically horizontal sheet).

With this large lensing width, the wormhole strongly distorts the star field and dark cloud (lower right in Figure 15.3) in nearly the same way as does a nonspinning black hole (Figures 8.3 and 8.4), producing multiple images. And the lensing also enlarges the secondary and tertiary images of Saturn. The wormhole looks bigger in the lower half of Figure 15.3 than in the upper half. It subtends a larger angle as seen by the camera. This is not because the camera is closer to the mouth; it is not closer. The camera is the same distance in both pictures. The enlargement is entirely due to the gravitational lensing.

^{Fig. 15.4. Wormhole’s gravitational lensing of a star field and Saturn: influence of the lensing width which is 0.014 and 0.43 times the wormhole radius in the top and bottom images. [From simulations by Eugénie von Tunzelmann’s team using Oliver James’ code based on my equations.]}

Interstellar’s Wormhole