The Science of Interstellar

by Thorne, Kip

  Within minutes they are at the critical orbit and all hell breaks loose.

  In this chapter, I describe my scientist’s interpretation of this.

  Tidal Gravity: Breaking the Endurance Away from Mann’s Planet

  In my interpretation Mann’s planet is on a highly elongated orbit (Chapter 19). When the Endurance arrived at the planet, it was rather far from Gargantua but zooming inward. The Endurance’s explosion (Chapter 20) occurred when the planet was nearing the black hole (Figure 27.1).

  Cooper rescues the Endurance after the explosion and lifts it upward, away from the planet. In my interpretation, he lifts the Endurance high enough for Gargantua’s huge tidal forces to pry it away from the planet, sending it on a separate trajectory (Figure 27.2).

  Centrifugal forces fling Mann’s planet outward on its next distant excursion, while the Endurance heads onto the critical orbit.49

  Fig. 27.1. The orbit of Mann’s planet and its location at the moment of the Endurance’s explosion.

  Fig. 27.2. The Endurance is pried away from Mann’s planet by Gargantua’s tidal forces. [Image of the Endurance is from Interstellar.]

  The Critical Orbit and the Volcano Analogy

  I discuss the critical orbit using a different type of picture than I’ve used before: Figure 27.3. I first describe this picture heuristically, and then I explain it in physicists’ language.

  Fig. 27.3. The Endurance’s trajectory on a volcano-like surface that represents its gravitational and centrifugal energies.

  Think of the surface in Figure 27.3 as that of a smooth, granite sculpture sitting on the floor in your home. It sinks down to a deep moat that surrounds a sculpted volcano.

  The Endurance, after being pried away from Mann’s planet, is like a tiny marble that rolls freely on this granite surface. As it rolls inward toward the moat, the marble picks up speed, because of the surface’s downward slope. It then rolls up the volcano’s side, slowing as it goes, and arrives on the volcano’s rim with some residual circumferential motion. And it then rolls around and around on the rim, delicately and unstably balanced between falling inward, into the volcano, and falling back outward and down to the moat.

  The volcano’s interior is Gargantua, and the volcano’s rim is the critical orbit, from which the Endurance launches toward Edmunds’ planet.

  The Meaning of the Volcano: Gravitational and Circumferential Energy

  To explain the volcano’s meaning—how it relates to the laws of physics—I have to get a bit technical.

  For the sake of simplicity, let’s pretend the Endurance is moving in Gargantua’s equatorial plane. (For the Endurance’s nonequatorial trajectory the ideas are the same but because the black hole is not spherical, the details are more complicated.) The volcano analogy neatly encapsulates the true physics of the critical orbit and Gargantua’s trajectory. To explain how, I need two physics concepts: the Endurance’s angular momentum, and its energy.

  After tidal forces pry it apart from Mann’s planet, the Endurance has a certain amount of angular momentum (its circumferential speed around Gargantua times its distance from Gargantua). The relativistic laws tell us that this angular momentum remains fixed (conserved) along the Endurance’s trajectory; see Chapter 10. This means that, as the Endurance plunges toward Gargantua, with its distance from Gargantua decreasing, its circumferential speed increases. This is similar to an ice-skater, whose whirling speed increases when she moves her arms in (Figure 27.4).

  Fig. 27.4. Ice skater.

  The Endurance heads toward Gargantua with a certain amount of energy, which like its angular momentum remains constant along its trajectory. This energy consists of three parts: the Endurance’s gravitational energy, which gets more and more negative as the Endurance plunges toward Gargantua; its centrifugal energy (its energy of circumferential motion around Gargantua), which increases as the Endurance plunges because the circumferential motion is speeding up; and its radial kinetic energy (its energy of motion toward Gargantua).

  The surface in Figure 27.3 is the Endurance’s gravitational energy plus its centrifugal energy plotted vertically, and location in Gargantua’s equatorial plane plotted horizontally. Wherever the surface dips downward, the Endurance’s gravitational plus centrifugal energy decreases, so its radial kinetic energy must increase (since the total energy is unchanged); its radial motion must speed up. This is precisely what happens in our intuitive, volcano analogy.

  Outside the moat of Figure 27.3, the surface’s height is controlled by the Endurance’s negative gravitational energy (see the “gravitational energy” label on the figure). By comparison, there the positive centrifugal energy is unimportant. On the outer edge of the volcano, by contrast, the height is controlled by the rising centrifugal energy, which has come to dominate over the gravitational energy. On the inside of the volcano, near Gargantua’s horizon, the gravitational energy has grown hugely negative and overwhelms the centrifugal energy, so the surface plunges downward (Figure 27.5). The critical orbit is on the volcano’s rim.

  Fig. 27.5. The Endurance’s critical orbit on the rim of the volcano, with centrifugal energy and force dominating outside the rim and gravitational energy and force dominating inside. [Image of the Endurance is from Interstellar.]

  The Critical Orbit: Balance of Centrifugal and Gravitational Forces

  Upon reaching the volcano’s rim, the Endurance, ideally, would travel around and around it, at constant speed. Because it moves neither inward nor outward, the inward pull of gravity on the rim must precisely be counterbalanced by the outward centrifugal force that arises from the ship’s fast circumferential motion.

  This indeed is the case, as shown in Figure 27.6—an analog of the force balance plot for Miller’s planet (Figure 17.2). At the Endurance’s critical orbit, the red curve (the inward gravitational pull on the Endurance) and the blue curve (the outward centrifugal force) cross, so the two forces are in balance.

  Fig. 27.6. The gravitational and centrifugal forces acting on the Endurance, and how they change with changing distance from Gargantua.

  However, the balance is unstable, as our volcano-rim analogy suggests.50 If the Endurance is randomly pushed inward just a bit, then gravity overwhelms the centrifugal force (the red curve rises above the blue curve), so the Endurance is pulled on inward toward Gargantua’s horizon. If the Endurance is pushed outward just a bit, then the centrifugal force wins the battle with gravity (the blue curve is above the red curve), so the Endurance is pushed on outward, escaping Gargantua’s tight grip.

  By contrast (as we saw in Chapter 17), on the orbit of Miller’s planet, the balance between the gravitational and centrifugal forces is stable.

  Disaster on the Rim: Ejection of TARS and Cooper

  In my science interpretation of the movie, the volcano’s rim is very narrow, so the critical orbit on the rim is exceedingly unstable. Tiny errors in navigation will send the Endurance careening down toward Gargantua (down into the volcano) or away from Gargantua (down toward the moat).

  Errors are inevitable, so the Endurance’s course must be corrected, continually, by a well-designed feedback system, like an automobile’s cruise control but much better.

  In my interpretation, the feedback system is not quite good enough and the Endurance winds up dangerously far down the inside lip of the volcano. The Endurance must use all the thrust at its disposal to climb back up to the critical orbit.

  But this is too subtle and technical for action-packed scenes and a hugely diverse audience, so Christopher Nolan chose a simpler, more in-your-face approach. No mention of instability. No mention of feedback. The Endurance simply plunges too close to Gargantua, and Cooper responds with all the thrust he can muster to climb back out and escape Gargantua’s grip.

  The result is the same: lander 1, piloted by TARS, and Ranger 2, piloted b
y Cooper, fire their rockets while attached to the Endurance, pushing the Endurance back out of Gargantua’s gravitational grip. Then, to get the last possible kick, explosive bolts blow the Endurance apart from lander 1 and Ranger 2. The lander and Ranger go plunging downward toward Gargantua, carrying TARS and Cooper with them, and the Endurance is saved (Figures 27.7 and 27.8).

  In the movie, there is a tragic, parting conversation between Brand and Cooper. Brand doesn’t understand why Cooper and TARS must accompany the lander and Ranger into the black hole. Cooper gives her a rather lame though poetic excuse: “Newton’s third law. The only way humans have ever figured out for getting somewhere is to leave something behind.”

  Fig. 27.7. The Endurance is thrown back up to the critical orbit by firing of rockets, followed by ejection of lander 1 and Ranger 2. [Image of the Endurance is from Interstellar.]

  Fig. 27.8. Ranger 2 descending toward Gargantua, as seen by Brand in the Endurance, with portions of two Endurance modules in the foreground. The Ranger is the faintly seen object in the picture’s lower center, surrounded by Gargantua’s accretion disk. [From Interstellar, used courtesy of Warner Bros. Entertainment Inc.]

  This surely is true. But the additional thrust on the Endurance, from Cooper and TARS accompanying the lander and Ranger into the hole, is awfully small. The greater truth, of course, is that Cooper wants to go into Gargantua. He hopes that he and TARS can learn the quantum gravity laws from a singularity inside Gargantua, and somehow transmit them back to Earth. It is his last, desperate hope for saving all of humanity.

  The Endurance’s Launch Toward Edmunds’ Planet

  The critical orbit is an ideal spot for Brand and the robot Case to launch the Endurance in any desired direction, in particular, toward Edmunds’ planet.

  How do they control their launch direction? Because the critical orbit is so unstable, a small rocket blast is sufficient to send the Endurance off it. And if the blast is ignited at precisely the right location along the critical orbit and has precisely the right strength, it will send the Endurance in precisely the desired direction (Figure 27.9).

  Fig. 27.9. The Endurance’s trajectory off the critical orbit, toward Edmunds’ planet. [Image of the Endurance is from Interstellar.]

  Actually, Figure 27.9 may leave you unconvinced that Brand and Case can launch in any direction they wish. That’s because it doesn’t capture the critical orbit’s three-dimensional structure. For that, see Figure 27.10.

  Fig. 27.10. A three-dimensional picture of the Endurance’s critical orbit and its launch toward Edmunds’ planet. The critical orbit wraps around a sphere that surrounds Gargantua.

  This convoluted critical orbit is a close analog of the trajectories of temporarily trapped light rays inside Gargantua’s shell of fire (Figures 6.5 and 8.2). Like those light rays, the Endurance is temporarily trapped when on its critical orbit. Unlike the light rays, the Endurance has a control system and rockets, so its launch off the critical orbit is in Brand’s and Case’s hands. And because of the orbit’s convoluted three-dimensional structure, the launch can be in any direction they wish.

  But their launch leaves behind Cooper and TARS, plunging through Gargantua’s horizon. Plunging toward Gargantua’s singularities.

  * * *

  49 This big difference is due to the Endurance’s having slightly less angular momentum than Mann’s planet, after tidal forces have done their thing. In Figure 27.3 the Endurance climbs up onto the volcano’s rim, but Mann’s planet does not quite make it up to the rim; it spirals back down the volcano’s side (centrifugal forces push it outward) and then up the gravitational energy surface, away from Gargantua.

  50 The agreement between our volcano-rim analogy and these force arguments is due to a key fact: The net force (gravitational plus centrifugal) on the Endurance is proportional to the slope of the energy surface (Figures 27.3 and 27.5). Can you figure out why?

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  Into Gargantua

  Some Personal History

  In 1985, when Carl Sagan wanted to send his heroine, Eleanore Arroway (Jodie Foster), through a black hole to the star Vega, I told him NO! Inside a black hole she will die. The singularity in the hole’s core will tear her apart, chaotically and painfully. I suggested he send Dr. Arroway through a wormhole instead (Chapter 14).

  In 2013, I encouraged Christopher Nolan to send Cooper into the black hole Gargantua.

  So what happened in the quarter century between 1985 and 2013? Why did my attitude toward falling into a black hole change so dramatically?

  In 1985, we physicists thought the cores of all black holes were inhabited by chaotic, destructive BKL singularities, and everything that entered a black hole would be destroyed by the singularity’s stretch and squeeze (Chapter 26). That was our highly educated guess. We were wrong.

  In the intervening quarter century, two additional singularities were discovered, mathematically, inside black holes: gentle singularities, to the extent that any singularity can be gentle (Chapter 26). Gentle enough that Cooper, falling into one, might possibly survive. I’m dubious of survival, but we can’t be sure. So I now think it respectable, in science fiction, to posit survival.

  Also in the intervening quarter century, we have learned that our universe is probably a brane in a higher-dimensional bulk (Chapter 21). So it’s respectable, I think, to posit living beings that inhabit the bulk—a very advanced civilization of bulk beings—who might save Cooper from the singularity at the last moment. That’s what Christopher Nolan chose.

  Through the Event Horizon

  In Interstellar, when Ranger 2 piloted by Cooper (and lander 1, piloted by TARS) eject from the Endurance, they spiral down toward Gargantua’s event horizon and then through it. What do Einstein’s relativistic laws say about this downward spiral?

  According to those laws, and hence my interpretation of the movie, Brand, watching from the Endurance, can never see the Ranger penetrate the horizon. No signal Cooper tries to send her from inside the horizon can ever get out. The flow of time inside the horizon is downward, and that downward time flow drags Cooper and all signals he sends downward with itself, away from the horizon. See Chapter 5.

  So what does Brand see (if she and Case can stabilize the Endurance long enough for her to watch)? Because the Endurance and the Ranger are both deep in the cylindrical part of Gargantua’s warped space (Figure 28.1), they are both dragged circumferentially by Gargantua’s whirling space with almost the same angular velocity (the same orbital period). So as seen by Brand, in her orbiting reference frame, the Ranger drops away from the Endurance almost straight downward toward the horizon (Figure 28.1). That’s what the movie depicts.

  Fig. 28.1. The Ranger’s trajectory through Gargantua’s warped space, as seen in the Endurance’s orbiting reference frame. The Endurance is drawn far larger than it should be, so you can see it. Inset: A larger portion of Gargantua’s warped space. [Image of the Endurance is from Interstellar.]

  As Brand watches the Ranger approach the horizon, she must see time on the Ranger slow and then freeze relative to her time, Einstein’s laws say. This has several consequences: She sees the Ranger slow its downward motion and then freeze just above the horizon. She sees light from the Ranger shift to longer and longer wavelengths (lower and lower frequencies, becoming redder and redder), until the Ranger turns completely black and unobservable. And bits of information that Cooper transmits to Brand one second apart as measured by his time on the Ranger arrive with larger and larger time separations as measured by Brand. After a few hours Brand receives the last bit that she will ever receive from Cooper, the last bit that Cooper emitted before piercing the horizon.

  Cooper, by contrast, continues receiving signals from Brand even after he crosses the horizon. Brand’s signals have no trouble entering Gargantua and reaching Cooper. Cooper’s signals can’t get
out to Brand. Einstein’s laws are unequivocal. This is how it must be.

  Moreover, those laws tell us that Cooper sees nothing special as he crosses the horizon. He can’t know, at least not with any ease, which bit that he transmits is the last one Brand will receive. He can’t tell, by looking around himself, precisely where the horizon is. The horizon is no more distinguishable to him than the Earth’s equator is to you as you cross it in a ship.

  These seemingly contradictory observations by Brand and Cooper are a result of two things: The warping of time, and the finite travel time for the light and information that they send to each other. When I think carefully about both of these things, I don’t see any contradiction at all.

  Sandwiched Between Singularities

  As the Ranger carries Cooper deeper and deeper into the bowels of Gargantua, he continues to see the universe above himself. Chasing the light that brings him that image is an infalling singularity. The singularity is weak at first, but it grows stronger rapidly, as more and more stuff falls into Gargantua and piles up in a thin sheet (Chapter 27). Einstein’s laws dictate this.

  Below the Ranger is an outflying singularity, created by stuff that fell into the black hole long ago and was backscattered upward toward the Ranger (Chapter 27).

  The Ranger is sandwiched between the two singularities (Figure 28.2). Inevitably, it will be hit by one or the other.