The Fabric of the Cosmos: Space, Time, and the Texture of Reality - Brian Greene (2004)

Part I. REALITY’S ARENA

Chapter 2. The Universe and the Bucket

IS SPACE A HUMAN ABSTRACTION OR A PHYSICAL ENTITY?

It’s not often that a bucket of water is the central character in a three-hundred-year-long debate. But a bucket that belonged to Sir Isaac Newton is no ordinary bucket, and a little experiment he described in 1689 has deeply influenced some of the world’s greatest physicists ever since. The experiment is this: Take a bucket filled with water, hang it by a rope, twist the rope tightly so that it’s ready to unwind, and let it go. At first, the bucket starts to spin but the water inside remains fairly stationary; the surface of the stationary water stays nice and flat. As the bucket picks up speed, little by little its motion is communicated to the water by friction, and the water starts to spin too. As it does, the water’s surface takes on a concave shape, higher at the rim and lower in the center, as in Figure 2.1.

That’s the experiment—not quite something that gets the heart racing. But a little thought will show that this bucket of spinning water is extremely puzzling. And coming to grips with it, as we have not yet done in over three centuries, ranks among the most important steps toward grasping the structure of the universe. Understanding why will take some background, but it is well worth the effort.

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Figure 2.1 The surface of the water starts out flat and remains so as the bucket starts to spin. Subsequently, as the water also starts to spin, its surface becomes concave, and it remains concave while the water spins, even as the bucket slows and stops.

Relativity Before Einstein

“Relativity” is a word we associate with Einstein, but the concept goes much further back. Galileo, Newton, and many others were well aware that velocity—the speed and direction of an object’s motion—is relative. In modern terms, from the batter’s point of view, a well-pitched fastball might be approaching at 100 miles per hour. From the baseball’s point of view, it’s the batter who is approaching at 100 miles per hour. Both descriptions are accurate; it’s just a matter of perspective. Motion has meaning only in a relational sense: An object’s velocity can be specified only in relation to that of another object. You’ve probably experienced this. When the train you are on is next to another and you see relative motion, you can’t immediately tell which train is actually moving on the tracks. Galileo described this effect using the transport of his day, boats. Drop a coin on a smoothly sailing ship, Galileo said, and it will hit your foot just as it would on dry land. From your perspective, you are justified in declaring that you are stationary and it’s the water that is rushing by the ship’s hull. And since from this point of view you are not moving, the coin’s motion relative to your foot will be exactly what it would have been before you embarked.

Of course, there are circumstances under which your motion seems intrinsic, when you can feel it and you seem able to declare, without recourse to external comparisons, that you are definitely moving. This is the case with accelerated motion, motion in which your speed and/or your direction changes. If the boat you are on suddenly lurches one way or another, or slows down or speeds up, or changes direction by rounding a bend, or gets caught in a whirlpool and spins around and around, you know that you are moving. And you realize this without looking out and comparing your motion with some chosen point of reference. Even if your eyes are closed, you know you’re moving, because you feel it. Thus, while you can’t feel motion with constant speed that heads in an unchanging straight-line trajectory —constant velocity motion, it’s called—you can feel changes to your velocity.

But if you think about it for a moment, there is something odd about this. What is it about changes in velocity that allows them to stand alone, to have intrinsic meaning? If velocity is something that makes sense only by comparisons—by saying that this is moving with respect to that—how is it that changes in velocity are somehow different, and don’t also require comparisons to give them meaning? In fact, could it be that they actually do require a comparison to be made? Could it be that there is some implicit or hidden comparison that is actually at work every time we refer to or experience accelerated motion? This is a central question we’re heading toward because, perhaps surprisingly, it touches on the deepest issues surrounding the meaning of space and time.

Galileo’s insights about motion, most notably his assertion that the earth itself moves, brought upon him the wrath of the Inquisition. A more cautious Descartes, in his Principia Philosophiae, sought to avoid a similar fate and couched his understanding of motion in an equivocating framework that could not stand up to the close scrutiny Newton gave it some thirty years later. Descartes spoke about objects’ having a resistance to changes to their state of motion: something that is motionless will stay motionless unless someone or something forces it to move; something that is moving in a straight line at constant speed will maintain that motion until someone or something forces it to change. But what, Newton asked, do these notions of “motionless” or “straight line at constant speed” really mean? Motionless or constant speed with respect to what? Motionless or constant speed from whose viewpoint? If velocity is not constant, with respect to what or from whose viewpoint is it not constant? Descartes correctly teased out aspects of motion’s meaning, but Newton realized that he left key questions unanswered.

Newton—a man so driven by the pursuit of truth that he once shoved a blunt needle between his eye and the socket bone to study ocular anatomy and, later in life as Master of the Mint, meted out the harshest of punishments to counterfeiters, sending more than a hundred to the gallows—had no tolerance for false or incomplete reasoning. So he decided to set the record straight. This led him to introduce the bucket.1

The Bucket

When we left the bucket, both it and the water within were spinning, with the water’s surface forming a concave shape. The issue Newton raised is, Why does the water’s surface take this shape? Well, because it’s spinning, you say, and just as we feel pressed against the side of a car when it takes a sharp turn, the water gets pressed against the side of the bucket as it spins. And the only place for the pressed water to go is upward. This reasoning is sound, as far as it goes, but it misses the real intent of Newton’s question. He wanted to know what it means to say that the water is spinning: spinning with respect to what? Newton was grappling with the very foundation of motion and was far from ready to accept that accelerated motion such as spinning—is somehow beyond the need for external comparisons.1

A natural suggestion is to use the bucket itself as the object of reference. As Newton argued, however, this fails. You see, at first when we let the bucket start to spin, there is definitely relative motion between the bucket and the water, because the water does not immediately move. Even so, the surface of the water stays flat. Then, a little later, when the water is spinning and there isn’t relative motion between the bucket and the water, the surface of the water is concave. So, with the bucket as our object of reference, we get exactly the opposite of what we expect: when there is relative motion, the water’s surface is flat; and when there is no relative motion, the surface is concave.

In fact, we can take Newton’s bucket experiment one small step further. As the bucket continues to spin, the rope will twist again (in the other direction), causing the bucket to slow down and momentarily come to rest, while the water inside continues to spin. At this point, the relative motion between the water and the bucket is the same as it was near the very beginning of the experiment (except for the inconsequential difference of clockwise vs. counterclockwise motion), but the shape of the water’s surface is different (previously being flat, now being concave); this shows conclusively that the relative motion cannot explain the surface’s shape.

Having ruled out the bucket as a relevant reference for the motion of the water, Newton boldly took the next step. Imagine, he suggested, another version of the spinning bucket experiment carried out in deep, cold, completely empty space. We can’t run exactly the same experiment, since the shape of the water’s surface depended in part on the pull of earth’s gravity, and in this version the earth is absent. So, to create a more workable example, let’s imagine we have a huge bucket—one as large as any amusement park ride—that is floating in the darkness of empty space, and imagine that a fearless astronaut, Homer, is strapped to the bucket’s interior wall. (Newton didn’t actually use this example; he suggested using two rocks tied together by a rope, but the point is the same.) The telltale sign that the bucket is spinning, the analog of the water being pushed outward yielding a concave surface, is that Homer will feel pressed against the inside of the bucket, his facial skin pulling taut, his stomach slightly compressing, and his hair (both strands) straining back toward the bucket wall. Here is the question: in totally empty space—no sun, no earth, no air, no doughnuts, no anything—what could possibly serve as the “something” with respect to which the bucket is spinning? At first, since we are imagining space is completely empty except for the bucket and its contents, it looks as if there simply isn’t anything else to serve as the something. Newton disagreed.

He answered by fixing on the ultimate container as the relevant frame of reference: space itself. He proposed that the transparent, empty arena in which we are all immersed and within which all motion takes place exists as a real, physical entity, which he called absolute space.2 We can’t grab or clutch absolute space, we can’t taste or smell or hear absolute space, but nevertheless Newton declared that absolute space is a something. It’s the something, he proposed, that provides the truest reference for describing motion. An object is truly at rest when it is at rest with respect to absolute space. An object is truly moving when it is moving with respect to absolute space. And, most important, Newton concluded, an object is truly accelerating when it is accelerating with respect to absolute space.

Newton used this proposal to explain the terrestrial bucket experiment in the following way. At the beginning of the experiment, the bucket is spinning with respect to absolute space, but the water is stationary with respect to absolute space. That’s why the water’s surface is flat. As the water catches up with the bucket, it is now spinning with respect to absolute space, and that’s why its surface becomes concave. As the bucket slows because of the tightening rope, the water continues to spin—spinning with respect to absolute space—and that’s why its surface continues to be concave. And so, whereas relative motion between the water and the bucket cannot account for the observations, relative motion between the water and absolute space can. Space itself provides the true frame of reference for defining motion.

The bucket is but an example; the reasoning is of course far more general. According to Newton’s perspective, when you round the bend in a car, you feel the change in your velocity because you are accelerating with respect to absolute space. When the plane you are on is gearing up for takeoff, you feel pressed back in your seat because you are accelerating with respect to absolute space. When you spin around on ice skates, you feel your arms being flung outward because you are accelerating with respect to absolute space. By contrast, if someone were able to spin the entire ice arena while you stood still (assuming the idealized situation of frictionless skates)—giving rise to the same relative motion between you and the ice—you would not feel your arms flung outward, because you would not be accelerating with respect to absolute space. And, just to make sure you don’t get sidetracked by the irrelevant details of examples that use the human body, when Newton’s two rocks tied together by a rope twirl around in empty space, the rope pulls taut because the rocks are accelerating with respect to absolute space. Absolute space has the final word on what it means to move.

But what is absolute space, really? In dealing with this question, Newton responded with a bit of fancy footwork and the force of fiat. He first wrote in the Principia “I do not define time, space, place, and motion, as [they] are well known to all,”3 sidestepping any attempt to describe these concepts with rigor or precision. His next words have become famous: “Absolute space, in its own nature, without reference to anything external, remains always similar and unmovable.” That is, absolute space just is, and is forever. Period. But there are glimmers that Newton was not completely comfortable with simply declaring the existence and importance of something that you can’t directly see, measure, or affect. He wrote,

It is indeed a matter of great difficulty to discover and effectually to distinguish the true motions of particular bodies from the apparent, because the parts of that immovable space in which those motions are performed do by no means come under the observations of our senses.4

So Newton leaves us in a somewhat awkward position. He puts absolute space front and center in the description of the most basic and essential element of physics—motion—but he leaves its definition vague and acknowledges his own discomfort about placing such an important egg in such an elusive basket. Many others have shared this discomfort.

Space Jam

Einstein once said that if someone uses words like “red,” “hard,” or “disappointed,” we all basically know what is meant. But as for the word “space,” “whose relation with psychological experience is less direct, there exists a far-reaching uncertainty of interpretation.”5 This uncertainty reaches far back: the struggle to come to grips with the meaning of space is an ancient one. Democritus, Epicurus, Lucretius, Pythagoras, Plato, Aristotle, and many of their followers through the ages wrestled in one way or another with the meaning of “space.” Is there a difference between space and matter? Does space have an existence independent of the presence of material objects? Is there such a thing as empty space? Are space and matter mutually exclusive? Is space finite or infinite?

For millennia, the philosophical parsings of space often arose in tandem with theological inquiries. God, according to some, is omnipresent, an idea that gives space a divine character. This line of reasoning was advanced by Henry More, a seventeenth-century theologian/philosopher who, some think, may have been one of Newton’s mentors.6 He believed that if space were empty it would not exist, but he also argued that this is an irrelevant observation because, even when devoid of material objects, space is filled with spirit, so it is never truly empty. Newton himself took on a version of this idea, allowing space to be filled by “spiritual substance” as well as material substance, but he was careful to add that such spiritual stuff “can be no obstacle to the motion of matter; no more than if nothing were in its way.”7 Absolute space, Newton declared, is the sensorium of God.

Such philosophical and religious musings on space can be compelling and provocative, yet, as in Einstein’s cautionary remark above, they lack a critical sharpness of description. But there is a fundamental and precisely framed question that emerges from such discourse: should we ascribe an independent reality to space, as we do for other, more ordinary material objects like the book you are now holding, or should we think of space as merely a language for describing relationships between ordinary material objects?

The great German philosopher Gottfried Wilhelm von Leibniz, who was Newton’s contemporary, firmly believed that space does not exist in any conventional sense. Talk of space, he claimed, is nothing more than an easy and convenient way of encoding where things are relative to one another. Without the objects in space, Leibniz declared, space itself has no independent meaning or existence. Think of the English alphabet. It provides an order for twenty-six letters—it provides relations such as is next to b, d is six letters before j, x is three letters after u, and so on. But without the letters, the alphabet has no meaning—it has no “supra-letter,” independent existence. Instead, the alphabet comes into being with the letters whose lexicographic relations it supplies. Leibniz claimed that the same is true for space: Space has no meaning beyond providing the natural language for discussing the relationship between one object’s location and another. According to Leibniz, if all objects were removed from space—if space were completely empty—it would be as meaningless as an alphabet that’s missing its letters.

Leibniz put forward a number of arguments in support of this so-called relationist position. For example, he argued that if space really exists as an entity, as a background substance, God would have had to choose where in this substance to place the universe. But how could God, whose decisions all have sound justification and are never random or haphazard, have possibly distinguished one location in the uniform void of empty space from another, as they are all alike? To the scientifically receptive ear, this argument sounds tinny. However, if we remove the theological element, as Leibniz himself did in other arguments he put forward, we are left with thorny issues: What is the location of the universe within space? If the universe were to move as a whole—leaving all relative positions of material objects intact—ten feet to the left or right, how would we know? What is the speed of the entire universe through the substance of space? If we are fundamentally unable to detect space, or changes within space, how can we claim it actually exists?

It is here that Newton stepped in with his bucket and dramatically changed the character of the debate. While Newton agreed that certain features of absolute space seem difficult or perhaps impossible to detect directly, he argued that the existence of absolute space does have consequences that are observable: accelerations, such as those at play in the rotating bucket, are accelerations with respect to absolute space. Thus, the concave shape of the water, according to Newton, is a consequence of the existence of absolute space. And Newton argued that once one has any solid evidence for something’s existence, no matter how indirect, that ends the discussion. In one clever stroke, Newton shifted the debate about space from philosophical ponderings to scientifically verifiable data. The effect was palpable. In due course, Leibniz was forced to admit, “I grant there is a difference between absolute true motion of a body and a mere relative change of its situation with respect to another body.”8 This was not a capitulation to Newton’s absolute space, but it was a strong blow to the firm relationist position.

During the next two hundred years, the arguments of Leibniz and others against assigning space an independent reality generated hardly an echo in the scientific community.9 Instead, the pendulum had clearly swung to Newton’s view of space; his laws of motion, founded on his concept of absolute space, took center stage. Certainly, the success of these laws in describing observations was the essential reason for their acceptance. It’s striking to note, however, that Newton himself viewed all of his achievements in physics as merely forming the solid foundation to support what he considered his really important discovery: absolute space. For Newton, it was all about space.10

Mach and the Meaning of Space

When I was growing up, I used to play a game with my father as we walked down the streets of Manhattan. One of us would look around, secretly fix on something that was happening—a bus rushing by, a pigeon landing on a windowsill, a man accidentally dropping a coin—and describe how it would look from an unusual perspective such as the wheel of the bus, the pigeon in flight, or the quarter falling earthward. The challenge was to take an unfamiliar description like “I’m walking on a dark, cylindrical surface surrounded by low, textured walls, and an unruly bunch of thick white tendrils is descending from the sky,” and figure out that it was the view of an ant walking on a hot dog that a street vendor was garnishing with sauerkraut. Although we stopped playing years before I took my first physics course, the game is at least partly to blame for my having a fair amount of distress when I encountered Newton’s laws.

The game encouraged seeing the world from different vantage points and emphasized that each was as valid as any other. But according to Newton, while you are certainly free to contemplate the world from any perspective you choose, the different vantage points are by no means on an equal footing. From the viewpoint of an ant on an ice skater’s boot, it is the ice and the arena that are spinning; from the viewpoint of a spectator in the stands, it is the ice skater that is spinning. The two vantage points seem to be equally valid, they seem to be on an equal footing, they seem to stand in the symmetric relationship of each spinning with respect to the other. Yet, according to Newton, one of these perspectives is more right than the other since if it really is the ice skater that is spinning, his or her arms will splay outward, whereas if it really is the arena that is spinning, his or her arms will not. Accepting Newton’s absolute space meant accepting an absolute conception of acceleration, and, in particular, accepting an absolute answer regarding who or what is really spinning. I struggled to understand how this could possibly be true. Every source I consulted—textbooks and teachers alike—agreed that only relative motion had relevance when considering constant velocity motion, so why in the world, I endlessly puzzled, would accelerated motion be so different? Why wouldn’t relative acceleration, like relative velocity, be the only thing that’s relevant when considering motion at velocity that isn’t constant? The existence of absolute space decreed otherwise, but to me this seemed thoroughly peculiar.

Much later I learned that over the last few hundred years many physicists and philosophers—sometimes loudly, sometimes quietly—had struggled with the very same issue. Although Newton’s bucket seemed to show definitively that absolute space is what selects one perspective over another (if someone or something is spinning with respect to absolute space then they are really spinning; otherwise they are not), this resolution left many people who mull over these issues unsatisfied. Beyond the intuitive sense that no perspective should be “more right” than any other, and beyond the eminently reasonable proposal of Leibniz that only relative motion between material objects has meaning, the concept of absolute space left many wondering how absolute space can allow us to identify true accelerated motion, as with the bucket, while it cannot provide a way to identify true constant velocity motion. After all, if absolute space really exists, it should provide a benchmark for all motion, not just accelerated motion. If absolute space really exists, why doesn’t it provide a way of identifying where we are located in an absolute sense, one that need not use our position relative to other material objects as a reference point? And, if absolute space really exists, how come it can affect us (causing our arms to splay if we spin, for example) while we apparently have no way to affect it?

In the centuries since Newton’s work, these questions were sometimes debated, but it wasn’t until the mid-1800s, when the Austrian physicist and philosopher Ernst Mach came on the scene, that a bold, prescient, and extremely influential new view about space was suggested—a view that, among other things, would in due course have a deep impact on Albert Einstein.

To understand Mach’s insight—or, more precisely, one modern reading of ideas often attributed to Mach2—let’s go back to the bucket for a moment. There is something odd about Newton’s argument. The bucket experiment challenges us to explain why the surface of the water is flat in one situation and concave in another. In hunting for explanations, we examined the two situations and realized that the key difference between them was whether or not the water was spinning. Unsurprisingly, we tried to explain the shape of the water’s surface by appealing to its state of motion. But here’s the thing: before introducing absolute space, Newton focused solely on the bucket as the possible reference for determining the motion of the water and, as we saw, that approach fails. There are other references, however, that we could naturally use to gauge the water’s motion, such as the laboratory in which the experiment takes place—its floor, ceiling, and walls. Or if we happened to perform the experiment on a sunny day in an open field, the surrounding buildings or trees, or the ground under our feet, would provide the “stationary” reference to determine whether the water was spinning. And if we happened to perform this experiment while floating in outer space, we would invoke the distant stars as our stationary reference.

This leads to the following question. Might Newton have kicked the bucket aside with such ease that he skipped too quickly over the relative motion we are apt to invoke in real life, such as between the water and the laboratory, or the water and the earth, or the water and the fixed stars in the sky? Might it be that such relative motion can account for the shape of the water’s surface, eliminating the need to introduce the concept of absolute space? That was the line of questioning raised by Mach in the 1870s.

To understand Mach’s point more fully, imagine you’re floating in outer space, feeling calm, motionless, and weightless. You look out and you can see the distant stars, and they too appear to be perfectly stationary. (It’s a real Zen moment.) Just then, someone floats by, grabs hold of you, and sets you spinning around. You will notice two things. First, your arms and legs will feel pulled from your body and if you let them go they will splay outward. Second, as you gaze out toward the stars, they will no longer appear stationary. Instead, they will seem to be spinning in great circular arcs across the distant heavens. Your experience thus reveals a close association between feeling a force on your body and witnessing motion with respect to the distant stars. Hold this in mind as we try the experiment again but in a different environment.

Imagine now that you are immersed in the blackness of completely empty space: no stars, no galaxies, no planets, no air, nothing but total blackness. (A real existential moment.) This time, if you start spinning, will you feel it? Will your arms and legs feel pulled outward? Our experiences in day-to-day life lead us to answer yes: any time we change from not spinning (a state in which we feel nothing) to spinning, we feel the difference as our appendages are pulled outward. But the current example is unlike anything any of us has ever experienced. In the universe as we know it, there are always other material objects, either nearby or, at the very least, far away (such as the distant stars), that can serve as a reference for our various states of motion. In this example, however, there is absolutely no way for you to distinguish “not spinning” from “spinning” by comparisons with other material objects; there aren’tany other material objects. Mach took this observation to heart and extended it one giant step further. He suggested that in this case there might also be no way to feel a difference between various states of spinning. More precisely, Mach argued that in an otherwise empty universe there is no distinction between spinning and not spinning—there is no conception of motion or acceleration if there are no benchmarks for comparison—and so spinning and not spinning are the same. If Newton’s two rocks tied together by a rope were set spinning in an otherwise empty universe, Mach reasoned that the rope would remain slack. If you spun around in an otherwise empty universe, your arms and legs would not splay outward, and the fluid in your ears would be unaffected; you’d feel nothing.

This is a deep and subtle suggestion. To really absorb it, you need to put yourself into the example earnestly and fully imagine the black, uniform stillness of totally empty space. It’s not like a dark room in which you feel the floor under your feet or in which your eyes slowly adjust to the tiny amount of light seeping in from outside the door or window; instead, we are imagining that there are no things, so there is no floor and there is absolutely no light to adjust to. Regardless of where you reach or look, you feel and see absolutely nothing at all. You are engulfed in a cocoon of unvarying blackness, with no material benchmarks for comparison. And without such benchmarks, Mach argued, the very concepts of motion and acceleration cease to have meaning. It’s not just that you won’t feel anything if you spin; it’s more basic. In an otherwise empty universe, standing perfectly motionless and spinning uniformly are indistinguishable.3

Newton, of course, would have disagreed. He claimed that even completely empty space still has space. And, although space is not tangible or directly graspable, Newton argued that it still provides a something with respect to which material objects can be said to move. But remember how Newton came to this conclusion: He pondered rotating motion and assumed that the results familiar from the laboratory (the water’s surface becomes concave; Homer feels pressed against the bucket wall; your arms splay outward when you spin around; the rope tied between two spinning rocks becomes taut) would hold true if the experiment were carried out in empty space. This assumption led him to search for something in empty space relative to which the motion could be defined, and the something he came up with was space itself. Mach strongly challenged the key assumption: He argued that what happens in the laboratory is not what would happen in completely empty space.

Mach’s was the first significant challenge to Newton’s work in more than two centuries, and for years it sent shock waves through the physics community (and beyond: in 1909, while living in London, Vladimir Lenin wrote a philosophical pamphlet that, among other things, discussed aspects of Mach’s work11). But if Mach was right and there was no notion of spinning in an otherwise empty universe—a state of affairs that would eliminate Newton’s justification for absolute space—that still leaves the problem of explaining the terrestrial bucket experiment, in which the water certainly does take on a concave shape. Without invoking absolute space—if absolute space is not a something—how would Mach explain the water’s shape? The answer emerges from thinking about a simple objection to Mach’s reasoning.

Mach, Motion, and the Stars

Imagine a universe that is not completely empty, as Mach envisioned, but, instead, one that has just a handful of stars sprinkled across the sky. If you perform the outer-space-spinning experiment now, the stars—even if they appear as mere pinpricks of light coming from enormous distance— provide a means of gauging your state of motion. If you start to spin, the distant pinpoints of light will appear to circle around you. And since the stars provide a visual reference that allows you to distinguish spinning from not spinning, you would expect to be able to feel it, too. But how can a few distant stars make such a difference, their presence or absence somehow acting as a switch that turns on or off the sensation of spinning (or more generally, the sensation of accelerated motion)? If you can feel spinning motion in a universe with merely a few distant stars, perhaps that means Mach’s idea is just wrong—perhaps, as assumed by Newton, in an empty universe you would still feel the sensation of spinning.

Mach offered an answer to this objection. In an empty universe, according to Mach, you feel nothing if you spin (more precisely, there is not even a concept of spinning vs. nonspinning). At the other end of the spectrum, in a universe populated by all the stars and other material objects existing in our real universe, the splaying force on your arms and legs is what you experience when you actually spin. (Try it.) And—here is the point—in a universe that is not empty but that has less matter than ours, Mach suggested that the force you would feel from spinning would lie between nothing and what you would feel in our universe. That is, the force you feel is proportional to the amount of matter in the universe. In a universe with a single star, you would feel a minuscule force on your body if you started spinning. With two stars, the force would get a bit stronger, and so on and so on, until you got to a universe with the material content of our own, in which you feel the full familiar force of spinning. In this approach, the force you feel from acceleration arises as a collective effect, a collective influence of all the other matter in the universe.

Again, the proposal holds for all kinds of accelerated motion, not just spinning. When the airplane you are on is accelerating down the runway, when the car you are in screeches to a halt, when the elevator you are in starts to climb, Mach’s ideas imply that the force you feel represents the combined influence of all the other matter making up the universe. If there were more matter, you would feel greater force. If there were less matter, you would feel less force. And if there were no matter, you wouldn’t feel anything at all. So, in Mach’s way of thinking, only relative motion and relative acceleration matter. You feel acceleration only when you accelerate relative to the average distribution of other material inhabiting the cosmos. Without other material—without any benchmarks for comparison—Mach claimed there would be no way to experience acceleration.

For many physicists, this is one of the most seductive proposals about the cosmos put forward during the last century and a half. Generations of physicists have found it deeply unsettling to imagine that the untouchable, ungraspable, unclutchable fabric of space is really a something—a something substantial enough to provide the ultimate, absolute benchmark for motion. To many it has seemed absurd, or at least scientifically irresponsible, to base an understanding of motion on something so thoroughly imperceptible, so completely beyond our senses, that it borders on the mystical. Yet these same physicists were dogged by the question of how else to explain Newton’s bucket. Mach’s insights generated excitement because they raised the possibility of a new answer, one in which space is not a something, an answer that points back toward the relationist conception of space advocated by Leibniz. Space, in Mach’s view, is very much as Leibniz imagined—it’s the language for expressing the relationship between one object’s position and another’s. But, like an alphabet without letters, space does not enjoy an independent existence.

Mach vs. Newton

I learned of Mach’s ideas when I was an undergraduate, and they were a godsend. Here, finally, was a theory of space and motion that put all perspectives back on an equal footing, since only relative motion and relative acceleration had meaning. Rather than the Newtonian benchmark for motion—an invisible thing called absolute space—Mach’s proposed benchmark is out in the open for all to see—the matter that is distributed throughout the cosmos. I felt sure Mach’s had to be the answer. I also learned that I was not alone in having this reaction; I was following a long line of physicists, including Albert Einstein, who had been swept away when they first encountered Mach’s ideas.

Is Mach right? Did Newton get so caught up in the swirl of his bucket that he came to a wishy-washy conclusion regarding space? Does Newton’s absolute space exist, or had the pendulum firmly swung back to the relationist perspective? During the first few decades after Mach introduced his ideas, these questions couldn’t be answered. For the most part, the reason was that Mach’s suggestion was not a complete theory or description, since he never specified how the matter content of the universe would exert the proposed influence. If his ideas were right, how do the distant stars and the house next door contribute to your feeling that you are spinning when you spin around? Without specifying a physical mechanism to realize his proposal, it was hard to investigate Mach’s ideas with any precision.

From our modern vantage point, a reasonable guess is that gravity might have something to do with the influences involved in Mach’s suggestion. In the following decades, this possibility caught Einstein’s attention and he drew much inspiration from Mach’s proposal while developing his own theory of gravity, the general theory of relativity. When the dust of relativity had finally settled, the question of whether space is a something—of whether the absolutist or relationist view of space is correct—was transformed in a manner that shattered all previous ways of looking at the universe.