by Brian Greene
To illustrate the contradiction, imagine you've had a really disappointing evening (hometown ball club lost, no one remembered your birthday, someone ate the last chunk of Velveeta) and need a little time alone, so you take the family skiff out for some relaxing midnight boating. With the moon overhead, the water is at high tide (it's the moon's gravity pulling up on bodies of water that creates the tides), and beautiful moonlight reflections dance on its waving surface. But then, as if your night hadn't already been irritating enough, hostile aliens zap the moon and beam it clear across to the other side of the galaxy. Now, certainly, the moon's sudden disappearance would be odd, but if Newton's law of gravity was right, the episode would demonstrate something odder still. Newton's law predicts that the water would start to recede from high tide, because of the loss of the moon's gravitational pull, about a second and a half before you saw the moon disappear from the sky. Like a sprinter jumping the gun, the water would seem to retreat a second and a half too soon.
The reason is that, according to Newton, at the very moment the moon disappears its gravitational pull would instantaneously disappear too, and without the moon's gravity, the tides would immediately start to diminish. Yet, since it takes light a second and a half to travel the quarter million miles between the moon and the earth, you wouldn't immediately see that the moon had disappeared; for a second and a half, it would seem that the tides were receding from a moon that was still shining high overhead as usual. Thus, according to Newton's approach, gravity can affect us before light—gravity can outrun light—and this, Einstein felt certain, was wrong. 12
And so, around 1907, Einstein became obsessed with the goal of formulating a new theory of gravity, one that would be at least as accurate as Newton's but would not conflict with the special theory of relativity. This turned out to be a challenge beyond all others. Einstein's formidable intellect had finally met its match. His notebook from this period is filled with half-formulated ideas, near misses in which small errors resulted in long wanderings down spurious paths, and exclamations that he had cracked the problem only to realize shortly afterward that he'd made another mistake. Finally, by 1915, Einstein emerged into the light. Although Einstein did have help at critical junctures, most notably from the mathematician Marcel Grossmann, the discovery of general relativity was the rare heroic struggle of a single mind to master the universe. The result is the crowning jewel of pre-quantum physics.
Einstein's journey toward general relativity began with a key question that Newton, rather sheepishly, had sidestepped two centuries earlier. How does gravity exert its influence over immense stretches of space? How does the vastly distant sun affect earth's motion? The sun doesn't touch the earth, so how does it do that? In short, how does gravity get the job done? Although Newton discovered an equation that described the effect of gravity with great accuracy, he fully recognized that he had left unanswered the important question of how gravity actually works. In his Principia, Newton wryly wrote, "I leave this problem to the consideration of the reader." 13 As you can see, there is a similarity between this problem and the one Faraday and Maxwell solved in the 1800s, using the idea of a magnetic field, regarding the way a magnet exerts influence on things that it doesn't literally touch. So you might suggest a similar answer: gravity exerts its influence by another field, the gravitational field. And, broadly speaking, this is the right suggestion. But realizing this answer in a manner that does not conflict with special relativity is easier said than done.
Much easier. It was this task to which Einstein boldly dedicated himself, and with the dazzling framework he developed after close to a decade of searching in the dark, Einstein overthrew Newton's revered theory of gravity. What is equally dazzling, the story comes full circle because Einstein's key breakthrough was tightly linked to the very issue Newton highlighted with the bucket: What is the true nature of accelerated motion?
The Equivalence of Gravity and Acceleration
In special relativity, Einstein's main focus was on observers who move with constant velocity—observers who feel no motion and hence are all justified in proclaiming that they are stationary and that the rest of the world moves by them. Itchy, Scratchy, and Apu on the train do not feel any motion. From their perspective, it's Martin and everyone else on the platform who are moving. Martin also feels no motion. To him, it's the train and its passengers that are in motion. Neither perspective is more correct than the other. But accelerated motion is different, because you can feel it. You feel squeezed back into a car seat as it accelerates forward, you feel pushed sideways as a train rounds a sharp bend, you feel pressed against the floor of an elevator that accelerates upward.
Nevertheless, the forces you'd feel struck Einstein as very familiar. As you approach a sharp bend, for example, your body tightens as you brace for the sideways push, because the impending force is inevitable. There is no way to shield yourself from its influence. The only way to avoid the force is to change your plans and not take the bend. This rang a loud bell for Einstein. He recognized that exactly the same features characterize the gravitational force. If you're standing on planet earth you are subject to planet earth's gravitational pull. It's inevitable. There is no way around it. While you can shield yourself from electromagnetic and nuclear forces, there is no way to shield yourself from gravity. And one day in 1907, Einstein realized that this was no mere analogy. In one of those flashes of insight that scientists spend a lifetime longing for, Einstein realized that gravity and accelerated motion are two sides of the same coin.
Just as by changing your planned motion (to avoid accelerating) you can avoid feeling squeezed back in your car seat or feeling pushed sideways on the train, Einstein understood that by suitably changing your motion you can also avoid feeling the usual sensations associated with gravity's pull. The idea is wonderfully simple. To understand it, imagine that Barney is desperately trying to win the Springfield Challenge, a monthlong competition among all belt-size-challenged males to see who can shed the greatest number of inches. But after two weeks on a liquid diet (Duff Beer), when he still has an obstructed view of the bathroom scale, he loses all hope. And so, in a fit of frustration, with the scale stuck to his feet, he leaps from the bathroom window. On his way down, just before plummeting into his neighbor's pool, Barney looks at the scale's reading and what does he see? Well, Einstein was the first person to realize, and realize fully, that Barney will see the scale's reading drop to zero. The scale falls at exactly the same rate as Barney does, so his feet don't press against it at all. In free fall, Barney experiences the same weightlessnessthat astronauts experience in outer space.
Indeed, if we imagine that Barney jumps out his window into a large shaft from which all air has been evacuated, then on his way down not only would air resistance be eliminated, but because every atom of his body would be falling at exactly the same rate, all the usual external bodily stresses and strains—his feet pushing up against his ankles, his legs pushing into his hips, his arms pulling down on his shoulders—would be eliminated as well. 14 By closing his eyes during the descent, Barney would feel exactly what he would if he were floating in the darkness of deep space. (And, again, in case you're happier with nonhuman examples: if you drop two rocks tied by a rope into the evacuated shaft, the rope will remain slack, just as it would if the rocks were floating in outer space.) Thus, by changing his state of motion—by fully "giving in to gravity"— Barney is able to simulate a gravity-free environment. (As a matter of fact, NASA trains astronauts for the gravity-free environment of outer space by having them ride in a modified 707 airplane, nicknamed the Vomit Comet, that periodically goes into a state of free fall toward earth.)
Similarly, by a suitable change in motion you can create a force that is essentially identical to gravity. For example, imagine that Barney joins astronauts floating weightless in their space capsule, with the bathroom scale still stuck to his feet and still reading zero. If the capsule should fire up its boosters and accelerate, things will change sig
nificantly. Barney will feel pressed to the capsule's floor, just as you feel pressed to the floor of an upward accelerating elevator. And since Barney's feet are now pressing against the scale, its reading is no longer zero. If the captain fires the boosters with just the right oomph, the reading on the scale will agree precisely with what Barney saw in the bathroom. Through appropriate acceleration, Barney is now experiencing a force that is indistinguishable from gravity.
The same is true of other kinds of accelerated motion. Should Barney join Homer in the outer space bucket, and, as the bucket spins, stand at a right angle to Homer—feet and scale against the inner bucket wall—the scale will register a nonzero reading since his feet will press against it. If the bucket spins at just the right rate, the scale will give the same reading Barney found earlier in the bathroom: the acceleration of the spinning bucket can also simulate earth's gravity.
All this led Einstein to conclude that the force one feels from gravity and the force one feels from acceleration are the same. They are equivalent. Einstein called this the principle of equivalence.
Take a look at what it means. Right now you feel gravity's influence. If you are standing, your feet feel the floor supporting your weight. If you are sitting, you feel the support somewhere else. And unless you are reading in a plane or a car, you probably also think that you are stationary—that you are not accelerating or even moving at all. But according to Einstein you actually are accelerating. Since you're sitting still this sounds a little silly, but don't forget to ask the usual question: Accelerating according to what benchmark? Accelerating from whose viewpoint?
With special relativity, Einstein proclaimed that absolute spacetime provides the benchmark, but special relativity does not take account of gravity. Then, through the equivalence principle, Einstein supplied a more robust benchmark that does include the effects of gravity. And this entailed a radical change in perspective. Since gravity and acceleration are equivalent, if you feel gravity's influence, you must be accelerating. Einstein argued that only those observers who feel no force at all—including the force of gravity—are justified in declaring that they are not accelerating. Such force-free observers provide the true reference points for discussing motion, and it's this recognition that requires a major turnabout in the way we usually think about such things. When Barney jumps from his window into the evacuated shaft, we would ordinarily describe him as accelerating down toward the earth's surface. But this is not a description Einstein would agree with. According to Einstein, Barney is not accelerating. He feels no force. He is weightless. He feels as he would floating in the deep darkness of empty space. He provides the standard against which all motion should be compared. And by this comparison, when you are calmly reading at home, you are accelerating. From Barney's perspective as he freely falls by your window—the perspective, according to Einstein, of a true benchmark for motion—you and the earth and all the other things we usually think of as stationary are accelerating upward. Einstein would argue that it was Newton's head that rushed up to meet the apple, not the other way around.
Clearly, this is a radically different way of thinking about motion. But it's anchored in the simple recognition that you feel gravity's influence only when you resist it. By contrast, when you fully give in to gravity you don't feel it. Assuming you are not subject to any other influences (such as air resistance), when you give in to gravity and allow yourself to fall freely, you feel as you would if you were freely floating in empty space—a perspective which, unhesitatingly, we consider to be unaccelerated.
In sum, only those individuals who are freely floating, regardless of whether they are in the depths of outer space or on a collision course with the earth's surface, are justified in claiming that they are experiencing no acceleration. If you pass by such an observer and there is relative acceleration between the two of you, then according to Einstein, you are accelerating.
As a matter of fact, notice that neither Itchy, nor Scratchy, nor Apu, nor Martin was truly justified in saying that he was stationary during the duel, since they all felt the downward pull of gravity. This has no bearing on our earlier discussion, because there, we were concerned only with horizontal motion, motion that was unaffected by the vertical gravity experienced by all participants. But as an important point of principle, the link Einstein found between gravity and acceleration means, once again, that we are justified only in considering stationary those observers who feel no forces whatsoever.
Having forged the link between gravity and acceleration, Einstein was now ready to take up Newton's challenge and seek an explanation of how gravity exerts its influence.
Warps, Curves, and Gravity
Through special relativity, Einstein showed that every observer cuts up spacetime into parallel slices that he or she considers to be all of space at successive instants of time, with the unexpected twist that observers moving relative to one another at constant velocity will cut through spacetime at different angles. If one such observer should start accelerating, you might guess that the moment-to-moment changes in his speed and/or direction of motion would result in moment-to-moment changes in the angle and orientation of his slices. Roughly speaking, this is what happens. Einstein (using geometrical insights articulated by Carl Friedrich Gauss, Georg Bernhard Riemann, and other mathematicians in the nineteenth century) developed this idea—by fits and starts—and showed that the differently angled cuts through the spacetime loaf smoothly merge into slices that are curved but fit together as perfectly as spoons in a silver-ware tray, as schematically illustrated in Figure 3.8. An accelerated observer carves spatial slices that are warped.
With this insight, Einstein was able to invoke the equivalence principle to profound effect. Since gravity and acceleration are equivalent, Einstein understood that gravity itself must be nothing but warps and curves in the fabric of spacetime. Let's see what this means.
If you roll a marble along a smooth wooden floor, it will travel in a straight line. But if you've recently had a terrible flood and the floor dried with all sorts of bumps and warps, a rolling marble will no longer travel along the same path. Instead, it will be guided this way and that by the warps and curves on the floor's surface. Einstein applied this simple idea to the fabric of the universe. He imagined that in the absence of matter or energy—no sun, no earth, no stars—spacetime, like the smooth wooden floor, has no warps or curves. It's flat. This is schematically illustrated in Figure 3.9a, in which we focus on one slice of space. Of course, space is really three dimensional, and so Figure 3.9b is a more accurate depiction, but drawings that illustrate two dimensions are easier to understand, so we'll continue to use them. Einstein then imagined that the presence of matter or energy has an effect on space much like the effect the flood had on the floor. Matter and energy, like the sun, cause space (and spacetime 5 ) to warp and curve as illustrated in Figures 3.10a and 3.10b. And just as a marble rolling on the warped floor travels along a curved path, Einstein showed that anything moving through warped space—such as the earth moving in the vicinity of the sun—will travel along a curved trajectory, as illustrated in Figure 3.11a and Figure 3.11b.
It's as if matter and energy imprint a network of chutes and valleys along which objects are guided by the invisible hand of the spacetime fabric. That, according to Einstein, is how gravity exerts its influence. The same idea also applies closer to home. Right now, your body would like to slide down an indentation in the spacetime fabric caused by the earth's presence. But your motion is being blocked by the surface on which you're sitting or standing. The upward push you feel almost every moment of your life—be it from the ground, the floor of your house, the corner easy chair, or your kingsize bed—is acting to stop you from sliding down a valley in spacetime. By contrast, should you throw yourself off the high diving board, you are giving in to gravity by allowing your body to move freely along one of its spacetime chutes.
Figure 3.8 According to general relativity, not only will the spacetime loaf be sliced in
to space at moments of time at different angles (by observers in relative motion), but the slices themselves will be warped or curved by the presence of matter or energy.
Figures 3.9, 3.10, and 3.11 schematically illustrate the triumph of Einstein's ten-year struggle. Much of his work during these years aimed at determining the precise shape and size of the warping that would be caused by a given amount of matter or energy. The mathematical result Einstein found underlies these figures and is embodied in what are called the Einstein field equations. As the name indicates, Einstein viewed the warping of spacetime as the manifestation—the geometrical embodiment—of a gravitational field. By framing the problem geometrically,
Figure 3.9 ( a ) Flat space (2-d version). ( b ) Flat space (3-d version).
Figure 3.10 ( a ) The sun warping space (2-d version). ( b ) The sun warping space (3-d version).
Einstein was able to find equations that do for gravity what Maxwell's equations did for electromagnetism. 16 And by using these equations, Einstein and many others made predictions for the path that would be followed by this or that planet, or even by light emitted by a distant star, as it moves through curved spacetime. Not only have these predictions been confirmed to a high level of accuracy, but in head-to-head competition with the predictions of Newton's theory, Einstein's theory consistently matches reality with finer fidelity.
Of equal importance, since general relativity specifies the detailed mechanism by which gravity works, it provides a mathematical framework for determining how fast it transmits its influence. The speed of transmission comes down to the question of how fast the shape of space can change in time. That is, how quickly can warps and ripples—ripples like those on the surface of a pond caused by a plunging pebble—race from place to place through space? Einstein was able to work this out, and the answer he came to was enormously gratifying. He found that warps and ripples—gravity, that is—do not travel from place to place instantaneously, as they do in Newtonian calculations of gravity. Instead, they travel at exactly the speed of light. Not a bit faster or slower, fully in keeping with the speed limit set by special relativity. If aliens plucked the moon from its orbit, the tides would recede a second and a half later, at the exact same moment we'd see that the moon had vanished. Where Newton's theory failed, Einstein's general relativity prevailed.
