The Science of Interstellar

by Thorne, Kip

  This is amazing. So different from everyday experience. It is as though knowing my weight and how fast I can run, you could deduce everything about me: the color of my eyes, the length of my nose, my IQ, . . .

  John Wheeler (my mentor, who gave “black holes” their name) has described this by the phrase “A black hole has no hair”—no extra, independent properties beyond its mass and its spin. Actually, he should have said, “A black hole has only two hairs, from which you can deduce everything else about it,” but that’s not as catchy as “no hair,” which quickly became embedded in black-hole lore and scientists’ lexicon.13

  From the properties of Miller’s planet, as depicted in Interstellar, a physicist who knows Einstein’s relativistic laws can deduce Gargantua’s mass and spin, and thence all else about it. Let's see how this works.14

  Gargantua’s Mass

  Miller’s planet (which I talk about at length in Chapter 17) is about as close to Gargantua as it can possibly be and still survive. We know this because the crew’s extreme loss of time can only occur very near Gargantua.

  At so close a distance, Gargantua’s tidal gravitational forces (Chapter 4) are especially strong. They stretch Miller’s planet toward and away from Gargantua and squeeze the planet’s sides (Figure 6.1).

  Fig. 6.1. Gargantua’s tidal gravitational forces stretch and squeeze Miller’s planet.

  The strength of this stretch and squeeze is inversely proportional to the square of Gargantua’s mass. Why? The greater Gargantua’s mass, the greater its circumference, and therefore the more similar Gargantua’s gravitational forces are on the various parts of the planet, which results in weaker tidal forces. (See Newton’s viewpoint on tidal forces; Figure 4.8.) Working through the details, I conclude that Gargantua’s mass must be at least 100 million times bigger than the Sun’s mass. If Gargantua were less massive than that, it would tear Miller’s planet apart!

  In all my science interpretations of what happens in Interstellar, I assume that this actually is Gargantua’s mass: 100 million Suns.15 For example, I assume this mass in Chapter 17, when explaining how Gargantua’s tidal forces could produce the giant water waves that inundate the Ranger on Miller’s planet.

  The circumference of a black hole’s event horizon is proportional to the hole’s mass. For Gargantua’s 100 million solar masses, the horizon circumference works out to be approximately the same as the Earth’s orbit around the Sun: about 1 billion kilometers. That’s big! After consulting with me, that’s the circumference assumed by Paul Franklin’s visual-effects team, when producing the images in Interstellar.

  Physicists attribute to a black hole a radius equal to its horizon’s circumference divided by 2π (about 6.28). Because of the extreme warping of space inside the black hole, this is not the hole’s true radius. Not the true distance from the horizon to the hole’s center, as measured in our universe. But it is the event horizon’s radius (half its diameter) as measured in the bulk; see Figure 6.3 below. Gargantua’s radius, in this sense, is about 150 million kilometers, the same as the radius of the Earth’s orbit around the Sun.

  Gargantua’s Spin

  When Christopher Nolan told me how much slowing of time he wanted on Miller’s planet, one hour there is seven years back on Earth, I was shocked. I didn’t think that possible and I told Chris so. “It’s non-negotiable,” Chris insisted. So, not for the first time and also not the last, I went home, thought about it, did some calculations with Einstein’s relativistic equations, and found a way.

  I discovered that, if Miller’s planet is about as near Gargantua as it can get without falling in,16 and if Gargantua is spinning fast enough, then Chris’s one-hour-in-seven-years time slowing is possible. But Gargantua has to spin awfully fast.

  There is a maximum spin rate that any black hole can have. If it spins faster than that maximum, its horizon disappears, leaving the singularity inside it wide open for all the universe to see; that is, making it naked—which is probably forbidden by the laws of physics (Chapter 26).

  I found that Chris’s huge slowing of time requires Gargantua to spin almost as fast as the maximum: less than the maximum by about one part in 100 trillion.17 In most of my science interpretations of Interstellar, I assume this spin.

  The crew of the Endurance could measure the spin rate directly by watching from far, far away as the robot TARS falls into Gargantua (Figure 6.2).18 As seen from afar, TARS never crosses the horizon (because signals he sends after crossing can’t get out of the black hole). Instead, TARS’ infall appears to slow down, and he appears to hover just above the horizon. And as he hovers, Gargantua’s whirling space sweeps him around and around Gargantua, as seen from afar. With Garantua’s spin very near the maximum possible, TARS’ orbital period is about one hour, as seen from afar.

  You can do the math yourself: the orbital distance around Gargantua is a billion kilometers and TARS covers that distance in one hour, so his speed as measured from afar is about a billion kilometers per hour, which is approximately the speed of light! If Gargantua were spinning faster than the maximum, TARS would whip around faster than the speed of light, which violates Einsteinʼs speed limit. This is a heuristic way to understand why there is a maximum possible spin for any black hole.

  Fig. 6.2. TARS, falling into Gargantua, is dragged around the hole’s billion-kilometer circumference once each hour, as seen from afar.

  In 1975, I discovered a mechanism by which Nature protects black holes from spinning faster than the maximum: When it gets close to the maximum spin, a black hole has difficulty capturing objects that orbit in the same direction as the hole rotates and that therefore, when captured, increase the hole’s spin. But the hole easily captures things that orbit opposite to its spin and that, when captured, slow the hole’s spin. Therefore, the spin is easily slowed, when it gets close to the maximum.

  In my discovery, I focused on a disk of gas, somewhat like Saturn’s rings, that orbits in the same direction as the hole’s spin: an accretion disk (Chapter 9). Friction in the disk makes the gas gradually spiral into the black hole, spinning it up. Friction also heats the gas, making it radiate photons. The whirl of space around the hole grabs those photons that travel in the same direction as the hole spins and flings them away, so they can’t get into the hole. By contrast, the whirl grabs photons that are trying to travel opposite to the spin and sucks them into the hole, where they slow the spin. Ultimately, when the hole’s spin reaches 0.998 of the maximum, an equilibrium is reached, with spin-down by the captured photons precisely counteracting spin-up by the accreting gas. This equilibrium appears to be somewhat robust. In most astrophysical environments I expect black holes to spin no faster than about 0.998 of the maximum.

  However, I can imagine situations—very rare or never in the real universe, but possible nevertheless—where the spin gets much closer to the maximum, even as close as Chris requires to produce the slowing of time on Miller’s planet, a spin one part in 100 trillion less than the maximum spin. Unlikely, but possible.

  This is common in movies. To make a great film, a superb filmmaker often pushes things to the extreme. In science fantasy films such as Harry Potter, that extreme is far beyond the bounds of the scientifically possible. In science fiction, it’s generally kept in the realm of the possible. That’s the main distinction between science fantasy and science fiction. Interstellar is science fiction, not fantasy. Gargantua’s ultrafast spin is scientifically possible.

  Gargantua’s Anatomy

  Having determined Gargantua’s mass and spin, I used Einstein’s equations to compute its anatomy. As in the previous chapter, here we focus solely on the external anatomy, leaving the interior (especially Gargantua’s singularities) for Chapters 26 and 28.

  In the top picture in Figure 6.3, you see the shape of Gargantua’s equatorial plane as viewed from the bulk. This is like Figure 5.5, but because Gargantua’s spin is much
closer to the maximum possible (one part in 100 trillion contrasted with two parts in a thousand in Figure 5.5), Gargantua’s throat is far longer. It extends much farther downward before reaching the horizon. The region near the horizon, as seen from the bulk, looks like a long cylinder. The length of the cylindrical region is about two horizon circumferences, that is, 2 billion kilometers.

  Fig. 6.3. Gargantua’s anatomy, when its spin is only one part in 100 trillion smaller than the maximum possible, as is required to get the extreme slowing of time on Miller’s planet.

  The cylinder’s cross sections are circles in the diagram, but if we were to restore the third dimension of our brane by moving out of Gargantua’s equatorial plane, the cross sections would become flattened spheres (spheroids).

  On Gargantua’s equatorial plane I marked several special locations that occur in my science interpretations of Interstellar: Gargantua’s event horizon (black circle), the critical orbit from which Cooper and TARS fall into Gargantua near the end of the movie (green circle; Chapter 27), the orbit of Miller’s planet (blue circle; Chapter 17), the orbit in which the Endurance is parked while the crew visit Miller’s planet (yellow circle), and a segment of the nonequatorial orbit of Mann’s planet, projected into the equatorial plane (purple circle). The outer part of Mann’s orbit is so far away from Gargantua (600 Gargantua radii or more; Chapter 19) that I had to redraw the picture on a much larger scale to fit it in (bottom picture), and, even so, I didn’t do it honestly: I only put the outer part at 100 Gargantua radii instead of 600 as I should. The red circles are labeled “SOF” for “shell of fire”; see below.

  How did I come up with these locations? I use the parking orbit as an illustration here and discuss the others later. In the movie, Cooper describes the parking orbit this way: “So we track a wider orbit of Gargantua, parallel with Miller’s planet but a little further out.” And he wants it to be far enough from Gargantua to be “out of the time shift,” that is, far enough from Gargantua that the slowing of time compared to Earth is very modest. This motivated my choice of five Gargantua radii (yellow circle in Figure 6.3). The time for the Ranger to travel from this parking orbit to Miller’s planet, two and a half hours, reinforced my choice.

  But there was a problem with this choice. At this distance, Gargantua would look huge; it would subtend about 50 degrees on the Endurance’s sky. Truly awe inspiring, but undesirable for so early in the movie! So Chris and Paul chose to make Gargantua look much smaller at the parking orbit: about two and a half degrees, which is five times the size of the Moon as seen from Earth—still impressive but not overwhelmingly so.

  The Shell of Fire

  Gravity is so strong near Gargantua, and space and time are so warped, that light (photons) can be trapped in orbits outside the horizon, traveling around and around the hole many times before escaping. These trapped orbits are unstable in the sense that the photons always escape from them, eventually. (By contrast, photons caught inside the horizon can never get out.)

  I like to call this temporarily trapped light the “shell of fire.” This fire shell plays an important role in the computer simulations (Chapter 8) that underlie Gargantua’s visual appearance in Interstellar.

  For a nonspinning black hole, the shell of fire is a sphere, one with circumference 1.5 times larger than the horizon’s circumference. The trapped light travels around and around this sphere on great circles (like the lines of constant longitude on the Earth); and some of it leaks into the black hole, while the rest leaks outward, away from the hole.

  When a black hole is spun up, its shell of fire expands outward and inward, so it occupies a finite volume rather than just the surface of a sphere. For Gargantua, with its huge spin, the shell of fire in the equatorial plane extends from the bottom red circle of Figure 6.3 to the upper red circle. The shell of fire has expanded to encompass Miller’s planet and the critical orbit, and much, much more! The bottom red circle is a light ray (a photon orbit) that moves around and around Gargantua in the same direction as Gargantua spins (the forward direction). The upper red circle is a photon orbit that moves in the opposite direction to Gargantua’s spin (the backward direction). Evidently, the whirl of space enables the forward light to be much closer to the horizon without falling in than the backward light. What a huge effect the space whirl has!

  The region of space occupied by the shell of fire above and below the equatorial plane is depicted in Figure 6.4. It is a large, annular region. I omit the warping of space from this picture; it would get in the way of showing the shell of fire’s full three dimensions.

  Fig. 6.4. The annular region around Gargantua, occupied by the shell of fire.

  Figure 6.5 shows some examples of photon orbits (light rays) trapped, temporarily, in the shell of fire.

  The black hole is at the center of each of these orbits. The leftmost orbit winds around and around the equatorial region of a small sphere, traveling always forward, in the same direction as Gargantua’s spin. It is nearly the same as the bottom (inner) red orbit in Figures 6.3 and 6.4. The next orbit in Figure 6.5 winds around a slightly larger sphere, in a nearly polar and slightly forward direction. The third orbit is larger still, but backward and nearly polar. The fourth is very nearly equatorial and backward, that is, nearly the upper (outer) red equatorial orbit of Figures 6.3 and 6.4. These orbits are actually inside each other; I pulled them apart so they are easier to see.

  Some photons that are temporarily trapped in the shell of fire escape outward; they spiral away from Gargantua. The rest escape spiraling inward; they spiral toward Gargantua and plunge through its horizon. The nearly trapped but escaping photons have a big impact on Gargantua’s visual appearance in Interstellar. They mark the edge of Gargantua’s shadow as seen by the Endurance’s crew, and they produce a thin bright line along the shadow’s edge: a “ring of fire” (Chapter 8).

  Fig. 6.5. Examples of light rays (photon orbits) temporarily trapped in the shell of fire, as computed using Einstein’s relativistic equations.

  * * *

  13 The literal French translation of “a black hole has no hair” is so obscene that French publishers resisted it vigorously, to no avail.

  14 For some quantitative details, see Some Technical Notes, at the end of this book.

  15 A more reasonable value might be 200 million times the Sun’s mass, but I want to keep the numbers simple and there’s a lot of slop in this one, so I chose 100 million.

  16 See Figure 17.2 and the discussion of it in Chapter 17.

  17 In other words, its spin is the maximum minus 0.00000000000001 of the maximum.

  18 When TARS falls in, the Endurance is not far, far away, but rather is on the critical orbit, quite near the horizon, whirling around the hole nearly as fast as TARS; so Amelia Brand, in the Endurance, does not see TARS swept around at high speed. For more on this, see Chapter 27.

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  Gravitational Slingshots

  Navigating a spacecraft near Gargantua is hard because the speeds are very high. To survive, a planet or star or spacecraft must counteract Gargantua’s huge gravity with a comparably huge centrifugal force. This means it must move at very high speed. Near the speed of light, it turns out. In my science interpretation of Interstellar, the Endurance, parked at ten Gargantua radii while the crew visit Miller’s planet, moves at one-third the speed of light: c/3, where c represents the speed of light. Miller’s planet moves at 55 percent the speed of light, 0.55c.

  To reach Miller’s planet from the parking orbit in my interpretation (Figure 7.1), the Ranger must slow its forward motion from c/3 to far less than that, so Gargantua’s gravity can pull it downward. And when it reaches the vicinity of the planet, the Ranger must turn from downward to forward. And, having picked up far too much speed while falling, it must slow by about c/4 to reach the planet’s 0.55c speed and rendezvous with it.
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  Fig. 7.1. The Ranger’s trip to Miller’s planet, in my interpretation of Interstellar.

  What mechanism can Cooper, the Ranger’s pilot, possibly use to produce these huge velocity changes?

  Twenty-First-Century Technology

  The required changes of velocity, roughly c/3, are 100,000 kilometers per second (per second, not per hour!).

  By contrast, the most powerful rockets we humans have today can reach 15 kilometers per second: seven thousand times too slow. In Interstellar, the Endurance travels from Earth to Saturn in two years at an average speed of 20 kilometers per second, five thousand times too slow. The fastest that human spacecraft are likely to achieve in the twenty-first century, I think, is 300 kilometers per second. That would require a major R&D effort on nuclear rockets, but it is still three hundred times too slow for Interstellar’s needs.

  Fortunately, Nature provides a way to achieve the huge speed changes, c/3, required in Interstellar: gravitational slingshots around black holes far smaller than Gargantua.

  Slingshot Navigation to Miller’s Planet

  Stars and small black holes congregate around gigantic black holes like Gargantua (more on this in the next section). In my science interpretation of the movie, I imagine that Cooper and his team make a survey of all the small black holes orbiting Gargantua. They identify one that is well positioned to gravitationally deflect the Ranger from its near circular orbit and send it plunging downward toward Miller’s planet (Figure 7.2). This gravity-assisted maneuver is called a “gravitational slingshot,” and has often been used by NASA in the solar system—though with the gravity coming from planets rather than a black hole (see the end of the chapter).