The Science of Interstellar
Page 4
Fig. 3.5. Our universe, in the vicinity of the Sun, depicted as a two-dimensional surface or brane, residing in a three-dimensional bulk. In reality, our brane has three space dimensions and the bulk has four. This figure is explained further in Chapter 4; see especially Figure 4.4.
Fig. 3.6. Relativity equations on Professor Brand’s blackboard, describing possible foundations for gravitational anomalies. For details see Chapter 25.
Throughout this book, when discussing the science of Interstellar, I explain the status of that science—truth, educated guess, or speculation—and I label it so at the beginning of a chapter or section with a symbol:
for truth
for educated guess
for speculation
Of course, the status of an idea—truth, educated guess, or speculation—can change; and you’ll meet such changes occasionally in the movie and in this book. For Cooper, the bulk is an educated guess that becomes a truth when he goes there in the tesseract (Chapter 29); and the laws of quantum gravity are a speculation until TARS extracts them from inside a black hole so for Cooper and Murph they become truth (Chapters 28 and 30).
For nineteenth-century physicists, Newton’s inverse square law for gravity was an absolute truth. But around 1890 it was revolutionarily upended by a tiny observed anomaly in the orbit of Mercury around the Sun (Chapter 24). Newton’s law is very nearly correct in our solar system, but not quite. This anomaly helped pave the way for Einstein’s twentieth-century relativistic laws, which—in the realm of strong gravity—began as speculation, became an educated guess when observational data started rolling in, and by 1980, with ever-improving observations, evolved into truth (Chapter 4).
Revolutions that upend established scientific truth are exceedingly rare. But when they happen, they can have profound effects on science and technology.
Can you identify in your own life speculations that became educated guesses and then truth? Have you ever seen your established truths upended, with a resulting revolution in your life?
* * *
5 In these realms, for example, the energy of light has huge quantum fluctuations. They are so huge that they warp space and time enormously and randomly. The fluctuating warpage is beyond the scope of Einstein’s relativistic laws, and the warpage’s influence on the light is beyond the scope of the light’s quantum laws.
6 The phrase “fiery marriage” was coined by my mentor John Wheeler, who was superb at naming things. John also coined the words “black hole” and “wormhole” and the phrase “a black hole has no hair”; Chapters 14 and 5. He once described to me lying in a warm bath for hours on end, letting his mind soar in a search for just the right word or phrase.
7 Chapters 5, 6, and 8.
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Warped Time and Space, and Tidal Gravity
Einstein’s Law of Time Warps
Einstein struggled to understand gravity on and off from 1907 onward. Finally in 1912 he had a brilliant inspiration. Time, he realized, must be warped by the masses of heavy bodies such as the Earth or a black hole, and that warping is responsible for gravity. He embodied this insight in what I like to call “Einstein’s law of time warps,” a precise mathematical formula8 that I describe qualitatively this way: Everything likes to live where it will age the most slowly, and gravity pulls it there.
The greater the slowing of time, the stronger gravity’s pull. On Earth, where time is slowed by only a few microseconds per day, gravity’s pull is modest. On the surface of a neutron star, where time is slowed by a few hours per day, gravity’s pull is enormous. At the surface of a black hole, time is slowed to a halt, whence gravity’s pull is so humungous that nothing can escape, not even light.
This slowing of time near a black hole plays a major role in Interstellar. Cooper despairs of ever seeing his daughter Murph again, when his travel near Gargantua causes him to age only a few hours while Murph, on Earth, is aging eight decades.
Human technology was too puny to test Einstein’s law until nearly half a century after he formulated it. The first good test came in 1959 when Bob Pound and Glen Rebca used a new technique called the Mössbauer effect to compare the rate of flow of time in the basement of a 73-foot tower at Harvard University with time in the tower’s penthouse. Their experiment was exquisitely accurate: good enough to detect differences of 0.0000000000016 seconds (1.6 trillionths of a second) in one day. Remarkably, they found a difference 130 times larger than this accuracy and in excellent agreement with Einstein’s law: Time flows more slowly in the basement than in the penthouse by 210-trillionths of a second each day.
The accuracy improved in 1976, when Robert Vessot of Harvard flew an atomic clock on a NASA rocket to a 10,000-kilometer height, and used radio signals to compare its ticking rate with clocks on the ground (Figure 4.1). Vessot found that time on the ground flows more slowly than at a height of 10,000 kilometers by about 30 microseconds (0.00003 seconds) in one day, and his measurement agreed with Einstein’s law of time warps to within his experimental accuracy. That accuracy (the uncertainty in Vessot’s measurement) was seven parts in a hundred thousand: 0.00007 of 30 microseconds in a day.
Fig. 4.1. Atomic clocks measure slowing of time on Earth. [Reproduced from Was Einstein Right? Putting General Relativity to the Test, by Clifford M. Will (Basic Books, 1993).]
The global positioning system (GPS), by which our smart phones can tell us where we are to 10 meters’ accuracy, relies on radio signals from a set of 27 satellites at a height of 20,000 kilometers (Figure 4.2). Typically only four to twelve satellites can be seen at once from any location on Earth. Each radio signal from a viewable satellite tells the smart phone where the satellite is located and the time the signal was transmitted. The smart phone measures the signal’s arrival time and compares it with its transmission time to learn how far the signal traveled—the distance between satellite and phone. Knowing the locations and distances to several satellites, the smart phone can triangulate to learn its own location.
This scheme would fail if the signal transmission times were the true times measured on the satellite. Time at a 20,000-kilometer height flows more rapidly than on Earth by forty microseconds each day, and the satellites must correct for this. They measure time with their own clocks, then slow that time down to the rate of time flow on Earth before transmitting it to our phones.
Fig. 4.2. The global positioning system.
Einstein was a genius. Perhaps the greatest scientist ever. This is one of many examples where his insights about the laws of physics could not be tested in his own day. It required a half century for technology to improve enough for a test with high precision, and another half century until the phenomena he described became part of everyday life. Among other examples are the laser, nuclear energy, and quantum cryptography.
The Warping of Space: The Bulk and Our Brane
In 1912 Einstein realized that if time can be warped by massive bodies, then space must also be warped. But despite the most intense mental struggle of his life, the full details of space warps long eluded him. From 1912 to late 1915 he struggled. Finally in November 1915, in a great Eureka moment, he formulated his “field equation of general relativity,” which encapsulated all his relativistic laws including space warps.
Again, human technology was too puny for high-precision tests.9 This time the needed improvements took sixty years, culminating in several key experiments. The one I liked best was led by Robert Reasenberg and Irwin Shapiro of Harvard. In 1976–77 they transmitted radio signals to two spacecraft in orbit around Mars. The spacecraft, called Viking 1 and Viking 2, amplified the signals and sent them back to Earth, where their round-trip travel time was measured. As the Earth and Mars moved around the Sun in their orbits, the radio signals traversed paths that were changing. At first, the paths were far from the Sun, then they passed near the Sun, and then far again,
as shown in the bottom half of Figure 4.3.
Fig. 4.3. Travel time for radio signals from Earth to Viking to Earth.
If space were flat, the round-trip travel time would have changed gradually and steadily. It did not. When the radio waves passed near the Sun, their travel time was longer than expected, longer by hundreds of microseconds. The extra travel time is shown, as a function of the spacecraft’s location at the top of Figure 4.3; it went up and then back down. Now, one of Einstein’s relativistic laws says that radio waves and light travel at an absolutely constant, unchanging speed.10 Therefore, the distance from Earth to the spacecraft had to be longer than expected when passing near the Sun, longer by hundreds of microseconds times the speed of light: about 50 kilometers.
This greater length would be impossible if space were flat, like a sheet of paper. It is produced by the Sun’s space warp. From the extra time delay and how it changed as the spacecraft moved relative to Earth, Reasenberg and Shapiro inferred the shape of the space warp. More precisely, they inferred the shape of the two-dimensional surface formed by the paths of the Viking radio signals. That surface was very nearly the Sun’s equatorial plane, so I describe it that way here.
The shape that the team measured, for the Sun’s equatorial plane, is shown in Figure 4.4 with the magnitude of the warping exaggerated. The measured shape was precisely what Einstein’s relativistic laws predict—precise to within the experimental error, which was 0.001 of the actual warping, that is, a part in a thousand. Around a neutron star, the space warp is far greater. Around a black hole, it is enormously greater.
Now, the Sun’s equatorial plane divides space into two identical halves, that above the plane and that below. Nonetheless, Figure 4.4 shows the equatorial plane as warped like the surface of a bowl. It bends downward inside and near the Sun, so that diameters of circles around the Sun, when multiplied by π (3.14159 . . . ), are larger than circumferences—larger, in the case of the Sun, by roughly 100 kilometers. That’s not much, but it was easily measured by the spacecraft, with a precision of a part in a thousand.
How can space “bend down”? Inside what does it bend? It bends inside a higher-dimensional hyperspace, called “the bulk,” that is not part of our universe!
Let's make that more precise. In Figure 4.4 the Sun’s equatorial plane is a two-dimensional surface that bends downward in a three-dimensional bulk. This motivates the way we physicists think about our full universe. Our universe has three space dimensions (east-west, north-south, up-down), and we think of it as a three-dimensional membrane or brane for short that is warped in a higher-dimensional bulk. How many dimensions does the bulk have? I discuss this carefully in Chapter 21, but for the purposes of Interstellar, the bulk has just one extra space dimension: four space dimensions in all.
Now, it’s very hard for humans to visualize our three-dimensional universe, our full brane, living and bending in a four-dimensional bulk. So throughout this book I draw pictures of our brane and bulk with one dimension removed, as I did in Figure 4.4.
Fig. 4.4. Paths of Viking radio signals through the Sun’s warped equatorial plane.
In Interstellar, the characters often refer to five dimensions. Three are the space dimensions of our own universe or brane (east-west, north-south, up-down). The fourth is time, and the fifth is the bulk’s extra space dimension.
Does the bulk really exist? Is there truly a fifth dimension, and maybe even more, that humans have never experienced? Very likely yes. We’ll explore this in Chapter 21.
The warping of space (warping of our brane) plays a huge role in Interstellar. For example, it is crucial to the very existence of the wormhole connecting our solar system to the far reaches of the universe, where Gargantua lives. And it distorts the sky around the wormhole and around the black hole Gargantua; this is the gravitational lensing we met in Figure 3.3.
Figure 4.5 is an extreme example of space warps. It is a fanciful drawing by my artist friend Lia Halloran, depicting a hypothetical region of our universe that contains large numbers of wormholes (Chapter 14) and black holes (Chapter 5) extending outward from our brane into and through the bulk. The black holes terminate in sharp points called “singularities.” The wormholes connect one region of our brane to another. As usual, I suppress one of our brane’s three dimensions, so the brane looks like a two-dimensional surface.
Fig. 4.5. Black holes and wormholes extending out of our brane into and through the bulk. One space dimension is removed from both our brane and the bulk. [Drawing by the artist Lia Halloran.]
Tidal Gravity
Einstein’s relativistic laws dictate that planets, stars, and unpowered spacecraft near a black hole move along the straightest paths permitted by the hole’s warped space and time. Figure 4.6 shows examples of four such paths. The two purple paths headed into the black hole begin parallel to each other. As each path tries to remain straight, the two paths get driven toward each other. The warping of space and time drives them together. The green paths, traveling circumferentially around the hole, also begin parallel. But in this case, the warping drives them apart.
Fig. 4.6. Four paths for planetary motion in the vicinity of a black hole. The picture of the hole is extracted from Lia Halloran’s drawing, Figure 4.5.
Several years ago, my students and I discovered a new point of view about these planetary paths. In Einstein’s relativity theory there is a mathematical quantity called the Riemann tensor. It describes the details of the warping of space and time. We found, hidden in the mathematics of this Riemann tensor, lines of force that squeeze some planetary paths together and stretch others apart. “Tendex lines,” my student David Nichols dubbed them, from the Latin word tendere meaning “to stretch.”
Figure 4.7 shows several of these tendex lines around the black hole of Figure 4.6. The green paths begin, on their right ends, parallel to each other, and then the red tendex lines stretch them apart. I draw a woman lying on a red tendex line. It stretches her, too; she feels a stretching force between her head and her feet, exerted by the red tendex line.
Fig. 4.7. Tendex lines around a black hole. The picture of the hole is extracted from Lia Halloran’s drawing, Figure 4.5.
The purple paths begin, at their top ends, running parallel to each other. They are then squeezed together by the blue tendex lines, and the woman whose body lies along a blue tendex line is also squeezed.
This stretching and squeezing is just a different way of thinking about the influence of the warping of space and time. From one viewpoint, the paths are stretched apart or squeezed together due to the planetary paths moving along the straightest routes possible in the warped space and time. From another viewpoint it is the tendex lines that do the stretching and squeezing. Therefore, the tendex lines must, in some very deep way, represent the warping of space and time. And indeed they do, as the mathematics of the Riemann tensor taught us.
Black holes are not the only objects that produce stretching and squeezing forces. Stars and planets and moons also produce them. In 1687 Isaac Newton discovered them in his own theory of gravity and used them to explain ocean tides.
The Moon’s gravity pulls more strongly on the near face of the Earth than on the far face, Newton reasoned. And the direction of pull on the Earth’s sides is slightly inward, because it is toward the Moon’s center, a slightly different direction on the Earth’s two sides. This is the usual viewpoint about the Moon’s gravity depicted in Figure 4.8.
Fig. 4.8. Newton’s explanation for the tides on the Earth’s oceans.
Now, the Earth does not feel the average of these gravitational pulls, because it is falling freely along its orbit.11 (This is like the Endurance’s crew not feeling Gargantua’s gravitational pull when they are in the Endurance, in its parking orbit above the black hole. They only feel centrifugal forces due to the Endurance’s rotation.) What the
Earth does feel is the red-arrowed lunar pulls in the left half of Figure 4.8, with their average subtracted away; that is, it feels a stretch toward and away from the Moon, and a squeeze on its lateral sides (right half of Figure 4.8). This is qualitatively the same as around a black hole (Figure 4.7).
These felt forces stretch the ocean away from the Earth’s surface on the faces toward and away from the Moon, producing high tides there. And the felt forces squeeze the oceans toward the Earth’s surface on the Earth’s lateral sides, producing low tides there. As the Earth turns on its axis, one full turn each twenty-four hours, we see two high tides and two low tides. This was Newton’s explanation of ocean tides, aside from a slight complication: The Sun’s tidal gravity also contributes to the tides. Its stretch and squeeze get added to the Moon’s stretch and squeeze.
Because of their role in ocean tides, these gravitational squeezing and stretching forces—the forces the Earth feels—are called tidal forces. To extremely high accuracy, these tidal forces, computed using Newton’s laws of gravity, are the same as we compute using Einstein’s relativistic laws. They must be the same, since the relativistic laws and the Newtonian laws always make the same predictions when gravity is weak and objects move at speeds much slower than light.