The Clockwork Universe Page 15

  Or think about traveling by plane. Drop your phone in a jet and during the fraction of a second it takes to hit the floor, the plane will have traveled perhaps a hundred yards. How is it that it falls at your feet and not a football field behind you? For that matter, how can the flight attendants dare to pour coffee? While the coffee is in midair, on its way to the cup but not yet there, the cup itself will have moved hundreds of feet. How can the crew serve first-class without scalding everyone in economy?

  “A company of chessmen standing on the same squares of the chessboard where we left them, we say are all in the same place or unmoved: though perhaps the chessboard has been in the meantime carried out of one room into another.” So wrote the philosopher John Locke in 1690, in one of the earliest discussions of relativity. Whether the board sits on a table or is carried from here to there makes no difference to how the game is played. As for the chess pieces, so for us. Whether the Earth sits immobile at the center of the cosmos or speeds around the sun, all our activities go on in their customary ways.

  Chapter Twenty-Nine

  Sputnik in Orbit, 1687

  In a story called “The Red-Headed League,” Dr. Watson looks hard at Sherlock Holmes’s latest visitor, but nothing strikes him as noteworthy. He turns toward the great detective. Perhaps Holmes has seen more? “Beyond the obvious facts that he has at some time done manual labour, that he takes snuff, that he is a Freemason, that he has been in China, and that he has done a considerable amount of writing lately, I can deduce nothing else,” says Holmes.

  Galileo and his fellow scientists favored a similar technique. By paying close attention to what others had overlooked, they could find their way to utterly unexpected conclusions. Galileo’s analysis of life on shipboard showed, for instance, that a marble that rolled off a table would take precisely the same time to reach the floor whether the ship was moving at a steady speed or standing still. The ship’s horizontal motion has no effect on the rock’s vertical fall. In Galileo’s hands, that seemingly small observation had momentous consequences.

  Picture any projectile moving through the air—a baseball soaring toward the outfield, a penny flipped into the air, a dancer leaping across the stage. In all such cases, the moving object’s horizontal motion and its vertical motion take place independently and can be examined separately. The horizontal movement is steady and unchanging, in line with Galileo’s law of motion. Ball and coin and dancer travel a certain distance horizontally in the first second, the same distance in the next second, and so on, moving at a constant speed from liftoff until touchdown.38 At the same time, the projectile’s vertical progress—its height above the ground—changes according to a different rule. At the moment of launch, the projectile rises quickly but then it rises slower and slower, stops rising altogether, and sits poised for an instant neither rising nor falling, and then plummets earthward faster and faster. The change in speed follows a simple, precise rule, and the upward part of the flight and the downward part are exactly symmetrical.

  Any object launched into the air—arrow, bullet, cannonball—travels in a curved path like this one. The moving object covers the same horizontal distance during each second of its flight.

  Mathematically, it’s easy to show that the combination of steady horizontal motion and steadily changing vertical motion makes for a parabolic path. (A parabola is an arch-shaped curve, but it is not just a generic arch; it is one that satisfies specific technical conditions, just as an ellipse is not a generic oval but one of a specific sort.) Parabolas had been painted against the sky ever since the first caveman threw a rock, but no one before Galileo had ever recognized them, and he was immensely proud of his discovery. “It has been observed that missiles and projectiles describe a curved path of some sort,” he wrote. “However no one has pointed out the fact that this path is a parabola. But this and other facts, not few in number or less worth knowing, I have succeeded in proving.”

  God had once again shown his taste for geometry. The planets in the heavens traveled not in haphazard curves but in perfect ellipses, and objects here on Earth traced exact parabolas.

  Concealed within the same observation about the independence of horizontal motion and vertical motion was a further surprise. Galileo might have found it, but he didn’t. Isaac Newton did. Imagine someone firing a gun horizontally, and at the same instant someone standing next to the shooter and dropping a bullet from the same height as the gun. When the two bullets reach the ground, they will be far apart. The one shot from the gun will have traveled hundreds of yards; the other will rest in the grass directly below the spot where it was dropped. Which bullet will hit the ground first?

  Surprisingly, both reach the ground at exactly the same moment. That’s what it means for the bullet’s vertical motion—its fall—to be independent of its horizontal motion. For Newton, that was enough to draw a remarkable conclusion.

  Suppose it takes one second for a bullet dropped from a certain height to hit the ground. That means that a bullet shot horizontally from the same height would also hit the ground in one second. A more powerful gun would send the bullet faster and farther, but—if the ground was perfectly flat—that bullet, too, would fall to the ground in one second.

  Bullets shot horizontally with different force travel different distances before they come to rest, but they all fall at the same rate. Each second a bullet is in the air it falls 16 feet toward the ground.

  Newton preferred to imagine a cannon blasting away horizontally. He imagined faster and faster cannonballs, covering greater and greater distances in their one-second journey. But the Earth is round, not flat.

  That makes all the difference. Since the Earth isn’t flat, it curves away beneath the speeding cannonball. In the meantime, the cannonball is falling toward the ground. Suppose you fired a cannonball from high above the atmosphere, horizontally. With nothing to slow it down, it would continue at the same speed forever, falling all the while. If you launched it at just the right speed, then by the time the cannonball had fallen, say, four feet, the ground itself would have fallen four feet below horizontal.

  And then what? The cannonball would continue on its journey forever, always falling but never coming any closer to the ground. Why? Because the cannonball always falls at the same rate, and the ground always curves beneath it at the same rate, so the cannonball falls and falls, and the Earth curves and curves, and the picture never changes. We’ve launched a satellite.

  Newton pictured it all in 1687.

  Chapter Thirty

  Hidden in Plain Sight

  Kepler had taken the first giant steps toward showing that mathematics governed the heavens. Galileo showed that mathematics reigned here on Earth. Newton’s great achievement, to peek ahead for a moment, was to demonstrate that Kepler’s discoveries and Galileo’s fit seamlessly together, and to explain why.

  It was Kepler who spelled out explicitly the credo that all the great seventeenth-century scientists endorsed. When he began studying astronomy he had talked of planets as if they had souls. He soon recanted. The planets surely moved, but their motion had nothing in common with that of galloping horses or leaping porpoises. “My aim is to show that the machine of the universe is not similar to a divine animated being,” Kepler declared, “but similar to a clock.”

  Galileo was the first to grasp, in detail, the workings of the cogs and gears of that cosmic clock. He liked to tell a story, perhaps invented, about how he had made his first great discovery. He had been young and bored, in church, daydreaming. An attendant had lit the candles on a giant chandelier and inadvertently set it swinging. Rather than listen to the service, Galileo watched the chandelier. It swung widely at first and then gradually in smaller and smaller arcs. Using his pulse beat to measure the time (in his day no one had yet built a clock with a second hand), Galileo discovered what has ever since been known as the law of pendulums—a pendulum takes the same time to swing through a small arc as through a large one.

  Perhaps it was bec
ause Galileo had been raised in a musical household—his father was a renowned composer and musician—that counting time came naturally to him.39 Eventually his counting would lead to one of history’s profound discoveries. What Galileo did, and what no one before him had ever done, was find a new way to think about time. It was an accomplishment akin to a fish’s finding a new way to think about water. “Galileo spent twenty years wrestling with the problem before he got free of man’s natural biological instinct for time as that in which he lives and grows old,” wrote the historian Charles C. Gillispie. “Time eluded science until Galileo.”

  Galileo’s solution was so successful and so radical that everyone today—even those without the slightest knowledge of physics—takes his insight for granted. The breakthrough was to identify time—not distance or temperature or color or any of a thousand other possibilities—as the essential variable that governs the world. For years Galileo had tried to find a relationship between the speed of a falling object and the distance it had fallen. All his efforts failed. Finally he turned away from distance and focused on time. Suddenly everything fell into place. Galileo had found a way to pin numbers to the world.

  The crucial experiments might have occurred only to a musician. Once again they involved rolling a ball down a ramp. The setup was bare-bones: a wooden ramp with a thin groove down the middle, a bronze ball to roll down the groove, and a series of movable catgut strings. The strings lay on the surface of the ramp, at a right angle to the groove, like frets on the neck of a guitar. When the ball crossed a string, it made an audible click but its speed continued almost unchanged.

  Galileo may actually have dropped rocks from the Leaning Tower of Pisa, as legend has it, but if he did they fell too quickly to study. So he picked up a ball, released it at the top of the ramp, and cocked his ears.

  Now the strings came into play. Galileo could hear the ball cross each string in turn, and he painstakingly rolled the ball again and again, each time trying to position the strings so that the travel time between each pair of strings was the same. He needed to arrange the strings, in other words, so that the time it took the ball to move from the top of the ramp to string A was the same as the time it took to move from string A to string B, which was the same as the time from B to C, and C to D, and so on. (He measured time intervals by weighing the water that leaked through a hole in the bottom of a jug. Twice as much water meant twice as much time.) It was finicky, tedious work.

  Finally satisfied, Galileo measured the distance between strings. That yielded this little table.

  The pattern in the right-hand column was easy to spot, but Galileo looked at the numbers again and recast the same data into a new table. Instead of looking at the ball as it traveled from one string to the next, he focused on the total distance the ball had traveled from the starting line. (All he had to do was add up the distances in the right-hand column.) This time he saw something more tantalizing.

  Each number in the right column of this new table represented the distance the ball had traveled in a certain amount of time—in one second, in two seconds, in three seconds, and so on. That distance, Galileo saw, could be expressed as a function of time. In t seconds, a ball rolling down a ramp at gravity’s command traveled precisely t2 inches.40 In 1 second, a ball rolled 12 inches, in 2 seconds 22 inches, in 5 seconds 52 inches, and so on.

  What was just as surprising was what the law didn’t say—it didn’t say anything about how much the ball weighed. Roll a cannonball and a BB down a ramp in side-by-side grooves, and they would travel alongside one another all the way and reach the bottom at precisely the same moment. For a given ramp, the same tidy law always held—the distance the ball traveled was proportional to time squared. All that counted was the height above the ground of the point where the ball was released.

  Repeat the experiment on a steeper ramp, and the cannonball and the BB would both travel faster, but they would still travel side by side every inch of the way. That was enough. Galileo made a daring leap: what held for a steep ramp and for an even steeper ramp would also hold for the steepest “ramp” of all, a free fall through the air. All objects, regardless of their weight, fall at exactly the same rate.

  Chapter Thirty-One

  Two Rocks and a Rope

  Tradition has it that Galileo discovered how objects fall by dropping weights from the top of the Leaning Tower of Pisa. Unlike most legends—Archimedes and his bathtub, Columbus and the flat Earth, George Washington and the cherry tree—historians believe this one might possibly be true. A tower drop would have disproved Aristotle’s claim that heavy objects fall faster than light ones. But it took the ramp experiments, which Galileo indisputably carried out, to yield the quantitative law about distance and time.

  Whether he really climbed a tower or not, Galileo did propose a thought experiment to test Aristotle’s claim. Imagine for a moment, said Galileo, that it was true that the heavier the object, the faster its fall. What would happen, he asked, if you tied a small rock and a big rock together, with some slack in the rope that joined them? On the one hand, the tied-together rocks would fall slower than the big rock alone, because the small rock would lag behind the big one and bog it down, just as a toddler tied to a sprinter would slow him down. (That was where the slack in the rope came into play.) On the other hand, the tied-together rocks would fall faster than the big rock alone, because they constituted a new, heavier “object.”

  Which meant, Galileo concluded triumphantly, that Aristotle’s assumption led to an absurd conclusion and had to be abandoned. Regardless of what Aristotle had decreed, logic forced us to conclude that all objects fall at the same rate, regardless of their weight. This is a story with a curious twist. Galileo, the great pioneer of experimental science, may never have bothered to perform his most famous experiment. No one is sure. What we know with certainty is that, like the Aristotelians he scorned, Galileo sat in a chair and deduced the workings of the world with no tool but the power of logic.

  Since Galileo’s day, countless tests have confirmed his Leaning Tower principle (including some at the Leaning Tower itself ). In ordinary circumstances, air resistance complicates the picture—feathers flutter to the ground and arrive long after cannonballs. Not until the invention of the air pump, which came after Galileo’s death, could you drop objects in a vacuum. A century after Galileo, the demonstration retained its power to surprise. King George III demanded that his instrument makers arrange a test for him, featuring a feather and a one-guinea coin falling in a vacuum. “In performing the experiment,” one observer wrote, “the young optician provided the feather, the King supplied the guinea and at the conclusion the King complimented the young man on his skill as an experimenter but frugally returned the guinea to his waistcoat pocket.”

  Today we’ve all seen the experiment put to the test, at every Olympic games. When television shows a diver leaping from the ten-meter board, thirty feet above the pool, how does the camera stick with her as she plummets toward the water? Galileo could have solved the riddle—just as a small stone falls at exactly the same rate as a heavy one, a camera falls at exactly the same rate as a diver. The trick is to set up a camera near the diver, at exactly the same height above the water. Attach the camera to a vertical pole and release the camera at the instant the diver starts her fall poolward. Gravity will do the rest.

  Galileo exulted in his discovery that “distance is proportional to time squared.” The point was not merely that nature could be described in numbers but that a single, simple law—in operation since the dawn of time but unnoticed until this moment (just as the Pythagorean theorem had been true but unknown before its discovery)—applied to the infinite variety of falling objects in the world. A geranium knocked off a windowsill, a painter tumbling off his ladder, a bird shot by a hunter, all fell according to the same mathematical law.

  The difference between Galileo’s world and Aristotle’s leaps out, as we have seen. Galileo had stripped away the details that fascinated Aristotle—t
he color of the bird’s plumage, the motives behind the painter’s absentmindedness—and replaced the sensuous, everyday world with an abstract, geometric one in which both a bird and a painter were simply moving dots tracing a trajectory against the sky. Ever since, we have been torn between celebrating the bounty that science and technology provide and lamenting the cost of those innovations.

  Chapter Thirty-Two

  A Fly on the Wall

  The mathematical patterns that Kepler had found in the heavens looked different from those Galileo had found on Earth. Perhaps that was to be expected. What did falling rocks have to do with endlessly circling planets, which plainly were not falling at all?

  Isaac Newton’s answer to that question would make use of mathematical tools that Kepler and Galileo did not know. Both astronomers were geniuses, but everything they found might conceivably have been discovered in Greece two thousand years before. To go further would require a breakthrough the Greeks never made.

  The insight that eluded Euclid and Archimedes (and Kepler and Galileo as well) supposedly came to René Descartes when he was lying in bed one morning in 1636, idly watching a fly crawl along the wall. (“I sleep ten hours every night,” he once boasted, “and no care ever shortens my slumber.”) The story—so claimed one of Descartes’ early biographers—was that Descartes realized that the path the fly traced as it moved could be precisely described in numbers. When the fly first caught Descartes’ eye, for instance, it was 10 inches above the floor and 8 inches from the left-hand edge of the wall. A moment later it was 11 inches above the floor and 9 inches from the left edge. All you needed were two lines at right angles—the horizontal line where the wall met the floor, say, and the vertical line from floor to ceiling where two walls met. Then at any moment the fly’s position could be pinpointed—this many inches from the horizontal line, that many from the vertical.