by Thorne, Kip
In the relativistic description of the Moon’s tides (Figure 4.9), the tidal forces are produced by blue tendex lines that squeeze the Earth’s lateral sides and red tendex lines that stretch toward and away from the Moon. This is just like a black hole’s tendex lines (Figure 4.7). The Moon’s tendex lines are visual embodiments of the Moon’s warping of space and time. It is remarkable that a warping so tiny can produce forces big enough to cause the ocean tides!
Fig. 4.9. Relativistic viewpoint on tides: they are produced by the Moon’s tendex lines.
On Miller’s planet (Chapter 17) the tidal forces are enormously larger and are key to the huge waves that Cooper and his crew encounter.
We now have three points of view on tidal forces:
• Newton’s viewpoint (Figure 4.8): The Earth does not feel the Moon’s full gravitational pull, but rather the full pull (which varies over the Earth) minus the average pull.
• The tendex viewpoint (Figure 4.9): The Moon’s tendex lines stretch and squeeze the Earth’s oceans; also (Figure 4.7) a black hole’s tendex lines stretch and squeeze the paths of planets and stars around the black hole.
• The straightest-route viewpoint (Figure 4.6): The paths of stars and planets around a black hole are the straightest routes possible in the hole’s warped space and time.
Having three different viewpoints on the same phenomenon can be extremely valuable. Scientists and engineers spend most of their lives trying to solve puzzles. The puzzle may be how to design a spacecraft. Or it may be figuring out how black holes behave. Whatever the puzzle may be, if one viewpoint doesn’t yield progress, another viewpoint may. Peering at the puzzle first from one viewpoint and then from another can often trigger new ideas. This is what Professor Brand does, in Interstellar, when trying to understand and harness gravitational anomalies (Chapters 24 and 25). This is what I’ve spent most of my adult life doing.
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8 See Some Technical Notes at the end of this book.
9 But see the first section of Chapter 24.
10 Unchanging after well-understood corrections for a bit of slowdown due to interaction with electrons in interplanetary space—so-called “plasma corrections.”
11 In 1907, Einstein realized that if he were to fall, off the roof of his house for example, then as he fell he would feel no gravity. He called this the “happiest thought of my life,” because it got him started on his quest to understand gravity, the quest that led to his concepts of warped time and space and the laws that govern the warping.
5
Black Holes
The black hole Gargantua plays a major role in Interstellar. Let’s look at the basic facts about black holes in this chapter and then focus on Gargantua in the next.
First, a weird claim: Black holes are made from warped space and warped time. Nothing else—no matter whatsoever.
Now some explanation.
Ant on a Trampoline: A Black Hole’s Warped Space
Imagine you’re an ant and you live on a child’s trampoline—a sheet of rubber stretched between tall poles. A heavy rock bends the rubber downward, as shown in Figure 5.1. You’re a blind ant, so you can’t see the poles or the rock or the bent rubber sheet. But you’re a smart ant. The rubber sheet is your entire universe, and you suspect it’s warped. To determine its shape, you walk around a circle in the upper region measuring its circumference, and then walk through the center from one side of the circle to the other, measuring its diameter. If your universe were flat, then the circumference would be π = 3.14159... times the diameter. But the circumference, you discover, is far smaller than the diameter. Your universe, you conclude, is highly warped!
Fig. 5.1. An ant on a warped trampoline. [My own hand sketch.]
Space around a nonspinning black hole has the same warping as the trampoline: Take an equatorial slice through the black hole. This is a two-dimensional surface. As seen from the bulk, this surface is warped in the same manner as the trampoline. Figure 5.2 is the same as Figure 5.1, with the ant and poles removed and the rock replaced by a singularity at the black hole’s center.
Fig. 5.2. The warped space inside and around a black hole, as seen from the bulk. [My own hand sketch.]
The singularity is a tiny region where the surface forms a point and thus is “infinitely warped,” and where, it turns out, tidal gravitational forces are infinitely strong, so matter as we know it gets stretched and squeezed out of existence. In chapters 26, 28, and 29, we see that Gargantua’s singularity is somewhat different from this one, and why.
For the trampoline, the warping of space is produced by the rock’s weight. Similarly, one might suspect, the black hole’s space warp is produced by the singularity at its center. Not so. In fact, the hole’s space is warped by the enormous energy of its warping. Yes, that’s what I meant to say. If this seems a bit circular to you, well, it is, but it has deep meaning.
Just as it requires a lot of energy to bend a stiff bow in preparation for shooting an arrow, so it requires a lot of energy to bend space; to warp it. And just as the bending energy is stored in the bent bow (until the string is released and feeds the bow’s energy into the arrow), so the warping energy is stored in the black hole’s warped space. And for a black hole, that energy of warping is so great that it generates the warping.
Warping begets warping in a nonlinear, self-bootstrapping manner. This is a fundamental feature of Einstein’s relativistic laws, and so different from everyday experience. It’s somewhat like a hypothetical science-fiction character who goes backward in time and gives birth to herself.
This warping-begets-warping scenario does not happen in our solar system hardly at all. Throughout our solar system the space warps are so weak that their energy is minuscule, far too small to produce much bootstrapped warping. Almost all the space warping in our solar system is produced directly by matter—the Sun’s matter, the Earth’s matter, the matter of the other planets—by contrast with a black hole where the warping is fully responsible for the warping.
Event Horizon and Warped Time
When you first hear mention of a black hole, you probably think of its trapping power as depicted in Figure 5.3, not its warped space.
Fig. 5.3. Signals I send after crossing the event horizon can’t get out. Note: Because one space dimension is removed from this diagram, I am a two-dimensional Kip, sliding down the warped two-dimensional surface, part of our brane. [My own hand sketch.]
If I fall into a black hole carrying a microwave transmitter, then once I pass through the hole’s event horizon, I’m pulled inexorably on downward, into the hole’s singularity. And any signals I try to transmit in any manner whatsoever get pulled down with me. Nobody above the horizon can ever see the signals I send after I cross the horizon. My signals and I are trapped inside the black hole. (See Chapter 28 for how this plays out in Interstellar.)
This trapping is actually caused by the hole’s time warp. If I hover above the black hole, supporting myself by the blast of a rocket engine, then the closer I am to the horizon, the more slowly my time flows. At the horizon itself, time slows to a halt and, therefore, according to Einstein’s law of time warps, I must experience an infinitely strong gravitational pull.
What happens inside the event horizon? Time is so extremely warped there that it flows in a direction you would have thought was spatial: it flows downward toward the singularity. That downward flow, in fact, is why nothing can escape from a black hole. Everything is drawn inexorably toward the future,12 and since the future inside the hole is downward, away from the horizon, nothing can escape back upward, through the horizon.
Space Whirl
Black holes can spin, just as the Earth spins. A spinning hole drags space around it into a vortex-type, whirling motion (Figure 5.4). Like the air in a tornado, space whirls fastest near the hole’s cent
er, and the whirl slows as one moves outward, away from the hole. Anything that falls toward the hole’s horizon gets dragged, by the whirl of space, into a whirling motion around and around the hole, like a straw caught and dragged by a tornado’s wind. Near the horizon there is no way whatsoever to protect oneself against this whirling drag.
Fig. 5.4. Space around a spinnning black hole is dragged into whirling motion. [My own hand sketch.]
Precise Depiction of the Warped Space and Time Around a Black Hole
These three aspects of spacetime warping—the warp of space, the slowing and distortion of time, and the whirl of space—are all described by mathematical formulas. These formulas have been deduced from Einstein’s relativistic laws, and their precise predictions are depicted quantitatively in Figure 5.5 (by contrast with Figures 5.1–5.4, which were only qualitative).
The warped shape of the surface in Figure 5.5 is precisely what we would see from the bulk, when looking at the hole’s equatorial plane. The colors depict the slowing of time as measured by someone who hovers at a fixed height above the horizon. At the transition from blue to green, time flows 20 percent as fast as it flows far from the hole. At the transition from yellow to red, time is slowed to 10 percent of its normal rate far away. And at the black circle, the bottom of the surface, time slows to a halt. This is the event horizon. It is a circle, not a sphere, because we are looking only at the equatorial plane, only at two dimensions of our universe (of our brane). If we were to restore the third space dimension, the horizon would become a flattened sphere: a spheroid. The white arrows depict the rate at which space whirls around the black hole. The whirl is fast at the horizon, and decreases as we climb upward in a spacecraft.
Fig. 5.5. Precise depiction of the warped space and time around a rapidly spinning black hole: one that spins at 99.8 percent of the maximum possible rate. [Drawing by Don Davis based on a sketch by me.]
In the fully accurate Figure 5.5, I don’t depict the hole’s interior. We’ll get to that later, in Chapters 26 and 28.
The warping in Figure 5.5 is the essence of a black hole. From its details, expressed mathematically, physicists can deduce everything about the black hole, except the nature of the singularity at its center. For the singularity, they need the ill-understood laws of quantum gravity (Chapters 26).
A Black Hole’s Appearance from Inside Our Universe
We humans are confined to our brane. We can’t escape from it, into the bulk (unless an ultra-advanced civilization gives us a ride in a tesseract or some such vehicle, as they do for Cooper in Interstellar; see Chapter 29). Therefore, we can’t see a black hole’s warped space, as depicted in Figure 5.5. The black-hole funnels and whirlpools so often shown in movies, for example, Disney Studios’ 1979 movie The Black Hole, would never be seen by any creature that lives in our universe.
Fig. 5.6. A fast-spinning black hole (left) moving in front of the star field shown on the right. [From a simulation for this book by the Double Negative visual-effects team.]
Interstellar is the first Hollywood movie to depict a black hole correctly, in the manner that humans would actually see and experience it. Figure 5.6 is a example, not taken from the movie. The black hole casts a black shadow on the field of stars behind it. Light rays from the stars are bent by the hole’s warped space; they are gravitationally lensed, producing a concentric pattern of distortion. Light rays coming to us from the shadow’s left edge move in the same direction as the hole’s whirling space. The space whirl gives them a boost, letting them escape from closer to the horizon than light rays on the shadow’s right edge, which struggle against the whirl of space. That’s why the shadow is flattened on the left and bulges out on the right. In Chapter 8 I talk more about this and other aspects of what a black hole really looks like, as seen up close in our universe, in our brane.
How Do We Know This Is True?
Einstein’s relativistic laws have been tested to high precision. I’m convinced they are right, except when they confront quantum physics. For a big black hole like Interstellar’s Gargantua, quantum physics is relevant only near its center, in its singularity. So if black holes exist at all in our universe, they must have the properties that Einstein’s relativistic laws dictate, the properties I described above.
These properties and others have been deduced from Einstein’s equations by a large number of physicists standing intellectually on each others’ shoulders (Figure 5.7); most importantly, Karl Schwarzschild, Roy Kerr, and Stephen Hawking. In 1915, shortly before his tragic death on World War I’s German/Russian front, Schwarzschild deduced the details of the warped spacetime around a nonspinning black hole. In physicists’ jargon, those details are called the “Schwarzschild metric.” In 1963, Kerr (a New Zealand mathematician) did the same for a spinning black hole: he deduced the spinning hole’s “Kerr metric.” And in the early 1970s Stephen Hawking and others deduced a set of laws that black holes must obey when they swallow stars, collide and merge, and feel the tidal forces of other objects.
Black holes surely do exist. Einstein’s relativistic laws insist that, when a massive star exhausts the nuclear fuel that keeps it hot, then the star must implode. In 1939, J. Robert Oppenheimer and his student Hartland Snyder used Einstein’s laws to discover that, if the implosion is precisely spherical, the imploding star must create a black hole around itself, and then create a singularity at the hole’s center, and then get swallowed into the singularity. No matter is left behind. None whatsoever. The resulting black hole is made entirely from warped space and time. Over the decades since 1939, physicists using Einstein’s laws have shown that if the imploding star is deformed and spinning, it will also produce a black hole. Computer simulations reveal the full details.
Astronomers have seen compelling evidence for many black holes in our universe. The most beautiful example is a massive black hole at the center of our Milky Way galaxy. Andrea Ghez of UCLA, with a small group of astronomers that she leads, has monitored the motions of stars around that black hole (Figure 5.8). Along each orbit, the dots are the star’s position at times separated by one year. I marked the black hole’s location by a white, five-pointed symbol. From the stars’ observed motions, Ghez has deduced the strength of the hole’s gravity. Its gravitational pull, at a fixed distance, is 4.1 million times greater than the Sun’s pull at that distance. This means the black hole’s mass is 4.1 million times greater than the Sun’s!
Fig 5.7. Black-hole scientists. Left to right: Karl Schwarzschild (1873–1916), Roy Kerr (1934– ), Stephen W. Hawking (1942– ), J. Robert Oppenheimer (1904–1967), and Andrea Ghez (1965– ).
Figure 5.9 shows where this black hole is on the night sky in summer. It is to the lower right of the constellation Sagittarius, the teapot, at the × labeled “Galactic Center.”
A massive black hole inhabits the core of nearly every big galaxy in our universe. Many of these are as heavy as Gargantua (100 million Suns), or even heavier. The heaviest yet measured is 17 billion times more massive than the Sun; it resides at the center of a galaxy whose name is NGC1277, 250 million light-years from Earth—roughly a tenth of the way to the edge of the visible universe.
Fig. 5.8. Observed orbits of stars around the massive black hole at the center of our Milky Way galaxy, as measured by Andrea Ghez and colleagues.
Fig. 5.9. The location of our galaxy’s center on the sky. A giant black hole resides there.
Inside our own galaxy, there are roughly 100 million smaller black holes: holes that typically are between about three and thirty times as heavy as the Sun. We know this not because we’ve seen evidence for all these, but because astronomers have made a census of heavy stars that will become black holes when they exhaust their nuclear fuel. From that census, astronomers have inferred how many such stars have already exhausted their fuel and become black holes.
So black holes are ubiquitous in our
universe. Fortunately, there are none in our solar system. If there were, the hole’s gravity would wreak havoc with the Earth’s orbit. The Earth would be thrown close to the Sun where it boils, or far from the Sun where it freezes, or even out of the solar system or into the black hole. We humans would survive for no more than a year or so!
Astronomers estimate that the nearest black hole to Earth is roughly 300 light-years away: a hundred times farther than the nearest star (other than the Sun), Proxima Centauri.
Now armed with a basic understanding of the universe, fields, warped time and space and especially black holes, we are ready, at last, to explore Interstellar’s Gargantua.
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12 If it is possible to go backward in time, you can only do so by traveling outward in space and then returning to your starting point before you left. You cannot go backward in time at some fixed location, while watching others go forward in time there. More on this in Chapter 30.
II
GARGANTUA
6
Gargantua’s Anatomy
If we know the mass of a black hole and how fast it spins, then from Einstein’s relativistic laws we can deduce all the hole’s other properties: its size, the strength of its gravitational pull, how much its event horizon is stretched outward near the equator by centrifugal forces, the details of the gravitational lensing of objects behind it. Everything.