The Science of Interstellar

by Thorne, Kip

  Fig. 17.2. The gravitational and centrifugal forces on Miller’s planet.

  At the inner balance point, the planet’s orbit is unstable: If the planet gets pushed outward a tiny bit (for example, by the gravity of some passing comet), the centrifugal force wins the competition and pushes the planet further outward. If the planet is pushed inward, the gravitational force wins and the planet is pulled into Gargantua. This means Miller’s planet can’t live for long at the inner balance point.

  The outer balance point, by contrast, is stable: If Miller’s planet is there and gets pushed outward, gravity wins the competition and pulls the planet back in. If the planet gets pushed inward, centrifugal forces win and push it back out. So this is where Miller’s planet lives, in my interpretation of Interstellar.31

  The Slowing of Time, and Tidal Gravity

  Among all stable, circular orbits around Gargantua, the orbit of Miller’s planet is the closest to the black hole. This means it’s the orbit with the maximum slowing of time. Seven years on Earth is one hour on Miller’s planet. Time flows sixty thousand times more slowly there than on Earth! This is what Christopher Nolan wanted for his movie.

  But being so close to Gargantua, in my interpretation of the movie, Miller’s planet is subjected to enormous tidal gravity, so enormous that Gargantua’s tidal forces almost tear the planet apart (Chapter 6). Almost, but not quite. Instead, they simply deform the planet. Deform it greatly (Figure 17.3). It bulges strongly toward and away from Gargantua.

  Fig. 17.3. Tidal deformation of Miller’s planet.

  If Miller’s planet were to rotate relative to Gargantua (if it didn’t keep the same face toward Gargantua at all times), then as seen by the planet, the tidal forces would rotate. First the planet would be crushed east-west and stretched north-south. Then, after a quarter rotation, the crush would be north-south and the stretch east-west. These crushes and stretches would be enormous compared to the strength of the planet’s mantle (its solid outer layers). The mantle would be pulverized, and then friction would heat it and melt it, making the whole planet red hot.

  That’s not at all what Miller’s planet looks like! So the conclusion is clear: In my science interpretation, the planet must always keep the same face pointing toward Gargantua (Figure 17.4), or nearly so (as I discuss later).

  Fig. 17.4. The orbital motion and spin of Miller’s planet relative to distant stars. The red spot on the planet’s surface and the tidal bulge always face Gargantua.

  The Whirl of Space

  Einstein’s laws dictate that, as seen from afar, for example, from Mann’s planet, Miller’s planet travels around Gargantua’s billion-kilometer-circumference orbit once each 1.7 hours. This is roughly half the speed of light! Because of time’s slowing, the Ranger’s crew measure an orbital period sixty thousand times smaller than this: a tenth of a second. Ten trips around Gargantua per second. That’s really fast! Isn’t it far faster than light? No, because of the space whirl induced by Gargantua’s fast spin. Relative to the whirling space at the planet’s location, and using time as measured there, the planet is moving slower than light, and that’s what counts. That’s the sense in which the speed limit is enforced.

  In my science interpretation of the movie, since the planet always keeps the same face pointed toward Gargantua (Figure 17.4), it must spin at the same rate as it orbits, ten revolutions per second. How can it possibly spin so fast? Won’t centrifugal forces tear it apart? No; and again the savior is the whirl of space. The planet would feel no disruptive centrifugal forces if it were spinning at precisely the same rate as space near it whirls, which it is almost doing! So centrifugal forces due to its rotation, in fact, are weak. If, instead, it were nonspinning relative to the distant stars, it would turn at ten revolutions per second relative to whirling space and so would be torn apart by centrifugal forces. It’s weird what relativity can do.

  Giant Waves on Miller’s Planet

  What could possibly produce the two gigantic water waves, 1.2 kilometers high, that bear down on the Ranger as it rests on Miller’s planet (Figure 17.5)?

  I searched for a while, did various calculations with the laws of physics, and found two possible answers for my science interpretation of the movie. Both answers require that the planet be not quite locked to Gargantua. Instead it must rock back and forth relative to Gargantua by a small amount, from the orientation on the left of Figure 17.6, to that on the right, then back to the left, and so on.

  This rocking is a natural thing, as you can see by looking at Gargantua’s tidal gravity.

  In Figure 17.6, I describe the tidal gravity by tendex lines (Chapter 4). No matter which way the planet is tilted (left or right in Figure 17.6), Gargantua’s blue squeezing tendex lines push its sides in, which drives the planet back toward its preferred orientation, the one with its bulges nearest Gargantua and farthest away (Figure 17.3). Similarly, Gargantua’s red stretching tendex lines pull its bottom bulge toward Gargantua and push its top bulge away from Gargantua. This also drives the planet back toward its preferred orientation.

  Fig. 17.5. A giant water wave bearing down on the Ranger. [From Interstellar, used courtesy of Warner Bros. Entertainment Inc.]

  Fig. 17.6. The rocking of Miller’s planet in response to Gargantua’s tidal gravity: its stretching tendex lines (red) and squeezing tendex lines (blue).

  The result is a simple rocking of the planet, back and forth, if the tilts are small enough that the planet’s mantle isn’t pulverized. When I computed the period of this rocking, how long it takes to swing from left to right and back again, I got a joyous answer. About an hour. The same as the observed time between giant waves, a time chosen by Chris without knowing about my science interpretation.

  The first explanation for the giant waves, in my science interpretation, is a sloshing of the planet’s oceans as the planet rocks under the influence of Gargantua’s tidal gravity.

  A similar sloshing, called “tidal bores,” happens on Earth, on nearly flat rivers that empty into the sea. When the ocean tide rises, a wall of water can go rushing up the river; usually a tiny wall, but very occasionally respectably big. You can see an example in the top half of Figure 17.7: a tidal bore on the Qiantang River in Hangzhou, China, in August 2010. Though impressive, this tidal bore is very small compared to the 1.2-kilometer-high waves on Miller’s planet. But the Moon’s tidal gravity that drives this tidal bore is tiny—really tiny—compared to Gargantua’s huge tidal gravity!

  Fig. 17.7. Top: A tidal bore on the Qiantang River. Bottom: A tsunami in Miyako City.

  My second explanation is tsunamis. As Miller’s planet rocks, Gargantua’s tidal forces may not pulverize its crust, but they do deform the crust first this way and then that, once an hour, and those deformations could easily produce gigantic earthquakes (or “millerquakes,” I suppose we should call them). And those millerquakes could generate tsunamis on the planet’s oceans, far larger than any tsunami ever seen on Earth, such as the one that hit Miyako City, Japan, on March 11, 2011 (bottom half of Figure 17.7).

  Past History of Miller’s Planet

  It is interesting to speculate about the past and future history of Miller’s planet. Try it using as much physics as you know or can scrounge up from the web or elsewhere. (This is not easy!) Here are some things you might think about.

  How old is Miller’s planet? If, as an extreme hypothesis, it was born in its present orbit when its galaxy was very young (about 12 billion years ago), and Gargantua has had its same ultrafast spin ever since, then the planet’s age is about 12 billion years divided by 60,000 (the slowing of time on the planet): 200,000 years. This is awfully young compared to most geological processes on Earth. Could Miller’s planet be that young and look like it looks? Could the planet develop its oceans and oxygen-rich atmosphere that quickly? If not, how could the planet have formed el
sewhere and gotten moved to this orbit, so close to Gargantua?

  How long can the planet’s rocking continue until friction inside the planet converts all the rocking energy to heat? And how long could it have rocked in the past? If a lot shorter than 200,000 years, then perhaps something set it rocking. What could have done so?

  When friction converts rocking energy to heat, how hot does the planet’s interior get? Hot enough to trigger volcanos and lava flows?

  Jupiter’s moon Io is a remarkable example of this. Io, the large moon that orbits closest to Jupiter’s surface, doesn’t rock. But it does move closer and farther from Jupiter along an elliptical orbit, so it feels Jupiter’s tidal gravity strengthen then weaken then strengthen, much like Miller’s planet feels Gargantua’s tidal gravity oscillate. This heats Io enough to produce huge volcanos and lava flows (Figure 17.8).

  Fig. 17.8. Io as photographed by the Galileo spacecraft shows many volcanos and lava flows. Inset: A 50-kilometer-high volcanic plume.

  The Appearance of Gargantua from Miller’s Planet

  In Interstellar, as the Ranger approaches Miller’s planet carrying Cooper and his crew, we see Gargantua in the sky above, 10 degrees across (twenty times larger than the Moon as seen from Earth!) and surrounded by its bright accretion disk. See Figure 17.9. As startlingly impressive as this may be, Gargantua’s angular size has actually been reduced greatly from what it would really be at the location of Miller’s planet.

  If Miller’s planet is, indeed, close enough to Gargantua to experience extreme time slowing—as I chose for my interpretation of the movie—then it must be deep into the cylindrical region of Gargantua’s warped space, as depicted in Figure 17.1. It seems likely, then, that if you look down the cylinder from Miller’s planet you will see Gargantua, and if you look up the cylinder you will see the external universe; so Gargantua should encompass roughly half of the sky (180 degrees) around the planet and the universe the other half. Indeed, that is what Einstein’s relativistic laws predict.

  It also seems clear that, since Miller’s planet is the closest anything can live stably, without falling into Gargantua, the entire accretion disk should be outside the orbit of Miller’s planet. Therefore, as the crew approach the planet, they should see a giant disk above them and a giant black-hole shadow below. Again, that is what Einstein’s laws predict.

  If Chris had followed these dictates of Einstein’s laws, it would have spoiled his movie. To see such fantastic sights so early in the movie would make the movie’s climax, when Cooper falls into Gargantua, visually anticlimactic. So Chris consciously saved such sights for the end of the movie; and invoking artistic license, near Miller’s planet he depicted Gargantua and its disk together, “just” twenty times bigger than the Moon looks from Earth.

  Fig. 17.9. Gargantua and its disk, partially eclipsed by Miller’s planet, as the Ranger, in the foreground, descends toward landing. [From Interstellar, used courtesy of Warner Bros. Entertainment Inc.]

  Although I’m a scientist and aspire to science accuracy in science fiction, I can’t blame Chris at all. I would have done the same, had I been making the decision. And you’d have thanked me for it.

  * * *

  31 The centrifugal force depends on the planet’s orbital angular momentum, a measure of its orbital speed that is constant along its orbit (Chapter 10). In plotting how the force changes with distance from Gargantua in Figure 17.2, I hold that angular momentum constant. If the angular momentum were a bit smaller than the amount Miller’s planet actually has, then the centrifugal force would everywhere be smaller, and the two curves in Figure 17.2 would not cross. There would be no balance point, and the planet would fall into Gargantua. That’s why the location of Miller’s planet in Figures 17.1 and 17.2 is the closest to Gargantua that the planet can stably live—the location I want, in order to get maximum slowing of time. For more details see Some Technical Notes at the end of this book.

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  Gargantua’s Vibrations

  While Cooper and Amelia Brand are on Miller’s planet, Romilly stays behind in the Endurance, observing Gargantua. From exquisitely accurate observations, he hopes to learn more about gravitational anomalies. Above all (I presume), he hopes that quantum data from Gargantua’s singularity (Chapter 26) will leak out through the event horizon, bringing information about how to control the gravitational anomalies (Chapter 24). Or, in Romilly’s pithy language, bringing information for “solving gravity.”

  When Amelia Brand returns from Miller’s planet, Romilly tells her, “I learned what I could from studying the black hole, but I couldn’t send anything to your father. We’ve been receiving but nothing gets out.”

  What did Romilly observe? He’s not specific, but I presume he would focus on Gargantua’s vibrations, and I offer this chapter’s extrapolation of the movie for that.

  Vibrations of Black Holes

  In 1971 Bill Press, a student of mine at Caltech, discovered that black holes can vibrate at special, resonant frequencies, in much the same way as a violin string vibrates.

  When a violin string is plucked just right, it emits a very pure tone: sound waves with a single frequency. When plucked a little differently, it emits that pure tone and also higher harmonics of the pure tone. In other words (if the string is firmly clamped, with the clamping finger not moving around) its vibrations produce sound at only a discrete set of frequencies, the string’s resonant frequencies.

  The same is true of a wine glass whose rim you rub with your finger, and a bell struck by a hammer. And also a black hole disturbed by something falling into it, Press discovered.

  A year later Saul Teukolsky, another of my students, used Einstein’s relativistic laws to work out a mathematical description of these resonant vibrations for a spinning black hole. (That’s the best thing about teaching at Caltech; we get fabulous students!) By solving Teukolsky’s equations, we physicists can compute a black hole’s resonant frequencies. But solving them for an extremely fast spinning hole (like Gargantua) is very difficult. So difficult that it was not done successfully until forty years later—by a collaboration in which the lead players again were two Caltech students: Huan Yang and Aaron Zimmerman.

  In September 2013, Ritchie Kremer, the property master for Interstellar (the person in charge of props) asked me for observational data that Romilly could show to Brand. Of course, I turned to the world’s best experts for help: Yang and Zimmerman. They quickly produced tables of Gargantua’s resonant vibration frequencies and also of the rates that the vibrations die out by feeding energy into gravitational waves—tables based on their own calculations using Teukolsky’s equations. Then they added fake observational numbers to go along with the theoretical predictions and I added pictures of Gargantua’s event horizon (or rather, the edge of its shadow), pictures from simulations by the Interstellar visual-effects team at Double Negative. The result was Romilly’s observational data set.

  When Christopher Nolan filmed the scene where Romilly discusses his observations with Amelia Brand, Romilly wound up not actually showing her his data set. It was there on a table, but he didn’t pick it up. However, the data set is central to my science extrapolation of Interstellar.

  Gargantua’s Resonant Vibrations

  Figure 18.1 is the data set’s first page. Each line of data on that page refers to a single resonant frequency at which Gargantua vibrates.

  Fig. 18.1. The first page of the data that Yang and Zimmermann prepared for Romilly to show to Amelia Brand. [Prop from Interstellar, used courtesy of Warner Bros. Entertainment Inc.]

  The first column is a three-number code for the shape of Gargantua’s vibrations and the picture is a still from a movie Romilly took, in my extrapolation of Interstellar, which verified that the vibrations had the predicted shape. The second column of data is the vibration frequency and the third is the rate at which this vibration dies out, as p
redicted by Teukolsky’s equations.32 The fourth and fifth columns show the difference between Romilly’s observations and the theoretical predictions.

  In my extrapolation Romilly finds a few anomalies, severe disagreements between his observations and the theory. He prints the disagreements in red. On page one of the data set (Figure 18.1), there is just one anomaly, but the disagreement is severe: thirty-nine times larger than the uncertainty in his measurements!

  These anomalies might be helpful in “solving gravity” (learning how to harness the anomalies), Romilly thinks, in my extrapolation. He wishes he could transmit what he has learned to Professor Brand back on Earth, but the outbound communication link has been severed, so he’s frustrated.

  Even more, he wishes he could see inside Gargantua, to extract the crucial quantum data embedded in its singularity (Chapter 26). But he can’t.

  And he doesn’t know whether the anomalies he observed are encoding some of the quantum data or not. Perhaps, with the hole spinning so rapidly, some of the quantum data leaked out through the horizon and produced the anomalies. Maybe Professor Brand could figure that out, if only Romilly could transmit the data to him.

  I say a lot more later (Chapters 24–26) about gravitational anomalies, and quantum data from inside Gargantua as the key to harnessing the anomalies. But that’s later. For now, let’s continue our exploration of Gargantua’s environs, turning next to Mann’s planet.