The Science of Interstellar

The second handle is the wormhole’s length, as measured equally well by Cooper or by an engineer in the bulk.

The third handle determines how strongly the wormhole lenses the light from objects behind it. The details of the lensing are fixed by the shape of space near the wormhole’s mouths. I chose that shape similar to the shape of space outside the horizon of a nonspinning black hole. My chosen shape has just one adjustable handle: the width of the region that produces strong lensing. I call this the lensing width[29] and depict it in Figure 15.1.

How the Handles Influence the Wormhole’s Appearance

Just as I had done for Gargantua (Chapter 8), I used Einstein’s relativistic laws to deduce equations for the paths of light rays around and through the wormhole, and I worked out a procedure for manipulating my equations to compute the wormhole’s gravitational lensing and thence what a camera sees when it orbits the wormhole or travels through it. After checking that my equations and procedure produced the kinds of images I expected, I sent them to Oliver and he wrote computer code capable of creating the quality IMAX images needed for the movie. Eugénie von Tunzelmann added background star fields and images of astronomical objects for the wormhole to lens, and then she, Oliver, and Paul began exploring the influence of my handles. Independently, I did my own explorations.

Eugénie kindly provided the pictures in Figures 15.2 and 15.4 for this book, in which we look at Saturn through the wormhole. (The resolutions of her pictures are far higher than my own crude computer code can produce.)

The Wormhole’s Length

We first explored the influence of the wormhole’s length, with modest lensing (small lensing width); see Figure 15.2.

When the wormhole is short (top picture), the camera sees one distorted image of Saturn through the wormhole, the primary image, filling the right half of the wormhole’s crystal-ball-like mouth. There is an extremely thin secondary, lenticular image on the left edge of the crystal ball. (The lenticular structure at the lower right is not Saturn; it is a distorted part of the external universe.)

^{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

When Chris saw the various possibilities, with varying wormhole length and lensing width, his choice was unequivocal. For medium and large length the multiple images seen through the wormhole would be confusing to a mass audience, so he made Interstellar’s wormhole very short: 1 percent of the wormhole radius. And he gave Interstellar’s wormhole a modest lensing width, about 5 percent of the wormhole radius, so the lensing of stars around it would be noticeable and intriguing, but much smaller than Gargantua’s lensing.

The resulting wormhole is the one at the top of Figure 15.2. And in Interstellar, after the Double Negative team had created for its far side a galaxy with beautiful nebulae, dust lanes, and star fields, it is marvelous to behold (Figure 15.5). To me it is one of the movie’s grandest sights.

^{Fig. 15.5. The wormhole as seen in a trailer for Interstellar. The Endurance is in front of the wormhole, near the center. Around the wormhole in pink I have drawn the Einstein ring, like that in Figure 8.4 for a nonspinning black hole. Primary and secondary images of gravitationally lensed stars move in the same way here as there. Looking at the trailer, can you identify some and trace their motion? [From Interstellar, used courtesy of Warner Bros. Entertainment Inc.]}

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Most of the lensing occurs in the region where the wormhole’s shape in the bulk is strongly curved. This is the region where its outward slope is steeper than 45 degrees, so I define the lensing width to be the radial distance, in the bulk, from the wormhole throat to the location with 45-degree outward slope (Figure 15.1).