by Brian Greene
Figure 3.11 The earth stays in orbit around the sun because it follows curves in the spacetime fabric caused by the sun's presence. ( a ) 2-d version. ( b ) 3-d version.
General Relativity and the Bucket
Beyond giving the world a mathematically elegant, conceptually powerful, and, for the first time, fully consistent theory of gravity, the general theory of relativity also thoroughly reshaped our view of space and time. In both Newton's conception and that of special relativity, space and time provided an unchanging stage for the events of the universe. Even though the slicing of the cosmos into space at successive moments has a flexibility in special relativity unfathomable in Newton's age, space and time do not respond to happenings in the universe. Spacetime—the loaf, as we've been calling it—is taken as a given, once and for all. In general relativity, all this changes. Space and time become players in the evolving cosmos. They come alive. Matter here causes space to warp there, which causes matter over there to move, which causes space way over there to warp even more, and so on. General relativity provides the choreography for an entwined cosmic dance of space, time, matter, and energy.
This is a stunning development. But we now come back to our central theme: What about the bucket? Does general relativity provide the physical basis for Mach's relationist ideas, as Einstein hoped it would?
Over the years, this question has generated much controversy. Initially, Einstein thought that general relativity fully incorporated Mach's perspective, a viewpoint he considered so important that he christened it Mach's principle. In fact, in 1913, as Einstein was furiously working to put the final pieces of general relativity in place, he wrote Mach an enthusiastic letter in which he described how general relativity would confirm Mach's analysis of Newton's bucket experiment. 17 And in 1918, when Einstein wrote an article enumerating the three essential ideas behind general relativity, the third point in his list was Mach's principle. But general relativity is subtle and it had features that took many years for physicists, including Einstein himself, to appreciate completely. As these aspects were better understood, Einstein found it increasingly difficult to fully incorporate Mach's principle into general relativity. Little by little, he grew disillusioned with Mach's ideas and by the later years of his life came to renounce them. 18
With an additional half century of research and hindsight, we can consider anew the extent to which general relativity conforms to Mach's reasoning. Although there is still some controversy, I think the most accurate statement is that in some respects general relativity has a distinctly Machian flavor, but it does not conform to the fully relationist perspective Mach advocated. Here's what I mean.
Mach argued 19 that when the spinning water's surface becomes concave, or when you feel your arms splay outward, or when the rope tied between the two rocks pulls taut, this has nothing to do with some hypothetical—and, in his view, thoroughly misguided—notion of absolute space (or absolute spacetime, in our more modern understanding). Instead, he argued that it's evidence of accelerated motion with respect to all the matter that's spread throughout the cosmos. Were there no matter, there'd be no notion of acceleration and none of the enumerated physical effects (concave water, splaying arms, rope pulling taut) would happen.
What does general relativity say?
According to general relativity, the benchmarks for all motion, and accelerated motion in particular, are freely falling observers—observers who have fully given in to gravity and are being acted on by no other forces. Now, a key point is that the gravitational force to which a freely falling observer acquiesces arises from all the matter (and energy) spread throughout the cosmos. The earth, the moon, the distant planets, stars, gas clouds, quasars, and galaxies all contribute to the gravitational field (in geometrical language, to the curvature of spacetime) right where you're now sitting. Things that are more massive and less distant exert a greater gravitational influence, but the gravitational field you feel represents the combined influence of the matter that's out there. 20 The path you'd take were you to give in to gravity fully and assume free-fall motion—the benchmark you'd become for judging whether some other object is accelerating —would be influenced by all matter in the cosmos, by the stars in the heavens and by the house next door. Thus, in general relativity, when an object is said to be accelerating, it means the object is accelerating with respect to a benchmark determined by matter spread throughout the universe. That's a conclusion which has the feel of what Mach advocated. So, in this sense, general relativity does incorporate some of Mach's thinking.
Nevertheless, general relativity does not confirm all of Mach's reasoning, as we can see directly by considering, once again, the spinning bucket in an otherwise empty universe. In an empty unchanging universe—no stars, no planets, no anything at all—there is no gravity. 21 And without gravity, spacetime is not warped—it takes the simple, uncurved shape shown in Figure 3.9b—and that means we are back in the simpler setting of special relativity. (Remember, Einstein ignored gravity while developing special relativity. General relativity made up for this deficiency by incorporating gravity, but when the universe is empty and unchanging there is no gravity, and so general relativity reduces to special relativity.) If we now introduce the bucket into this empty universe, it has such a tiny mass that its presence hardly affects the shape of space at all. And so the discussion we had earlier for the bucket in special relativity applies equally well to general relativity. In contradiction to what Mach would have predicted, general relativity comes to the same answer as special relativity, and proclaims that even in an otherwise empty universe, you will feel pressed against the inner wall of the spinning bucket; in an otherwise empty universe, your arms will feel pulled outward if you spin around; in an otherwise empty universe, the rope tied between two twirling rocks will become taut. The conclusion we draw is that even in general relativity, empty spacetime provides a benchmark for accelerated motion.
Hence, although general relativity incorporates some elements of Mach's thinking, it does not subscribe to the completely relative conception of motion Mach advocated. 22 Mach's principle is an example of a provocative idea that provided inspiration for a revolutionary discovery even though that discovery ultimately failed to fully embrace the idea that inspired it.
Spacetime in the Third Millennium
The spinning bucket has had a long run. From Newton's absolute space and absolute time, to Leibniz's and then Mach's relational conceptions, to Einstein's realization in special relativity that space and time are relative and yet in their union fill out absolute spacetime, to his subsequent discovery in general relativity that spacetime is a dynamic player in the unfolding cosmos, the bucket has always been there. Twirling in the back of the mind, it has provided a simple and quiet test for whether the invisible, the abstract, the untouchable stuff of space—and spacetime, more generally—is substantial enough to provide the ultimate reference for motion. The verdict? Although the issue is still debated, as we've now seen, the most straightforward reading of Einstein and his general relativity is that spacetime can provide such a benchmark: spacetime is a some thing. 23
Notice, though, that this conclusion is also cause for celebration among supporters of a more broadly defined relationist outlook. In Newton's view and subsequently that of special relativity, space and then spacetime were invoked as entities that provide the reference for defining accelerated motion. And since, according to these perspectives, space and spacetime are absolutely unchangeable, this notion of acceleration is absolute. In general relativity, though, the character of spacetime is completely different. Space and time are dynamic in general relativity: they are mutable; they respond to the presence of mass and energy; they are not absolute. Spacetime and, in particular, the way it warps and curves, is an embodiment of the gravitational field. Thus, in general relativity, acceleration relative to spacetime is a far cry from the absolute, staunchly un-relational conception invoked by previous theories. Instead, as Einstein argued eloquently a
few years before he died, 24 acceleration relative to general relativity's spacetime is relational. It is not acceleration relative to material objects like stones or stars, but it is acceleration relative to something just as real, tangible, and changeable: a field—the gravitational field. 6 In this sense, spacetime—by being the incarnation of gravity—is so real in general relativity that the benchmark it provides is one that many relationists can comfortably accept.
Debate on the issues discussed in this chapter will no doubt continue as we grope to understand what space, time, and spacetime actually are. With the development of quantum mechanics, the plot only thickens. The concepts of empty space and of nothingness take on a whole new meaning when quantum uncertainty takes the stage. Indeed, since 1905, when Einstein did away with the luminiferous aether, the idea that space is filled with invisible substances has waged a vigorous comeback. As we will see in later chapters, key developments in modern physics have reinstituted various forms of an aetherlike entity, none of which set an absolute standard for motion like the original luminiferous aether, but all of which thoroughly challenge the naïve conception of what it means for spacetime to be empty. Moreover, as we will now see, the most basic role that space plays in a classical universe—as the medium that separates one object from another, as the intervening stuff that allows us to declare definitively that one object is distinct and independent from another—is thoroughly challenged by startling quantum connections.
4 - Entangling Space
WHAT DOES IT MEAN TO BE SEPARATE
IN A QUANTUM UNIVERSE?
To accept special and general relativity is to abandon Newtonian absolute space and absolute time. While it's not easy, you can train your mind to do this. Whenever you move around, imagine your now shifting away from the nows experienced by all others not moving with you. While you are driving along a highway, imagine your watch ticking away at a different rate compared with timepieces in the homes you are speeding past. While you are gazing out from a mountaintop, imagine that because of the warping of spacetime, time passes more quickly for you than for those subject to stronger gravity on the ground far below. I say "imagine" because in ordinary circumstances such as these, the effects of relativity are so tiny that they go completely unnoticed. Everyday experience thus fails to reveal how the universe really works, and that's why a hundred years after Einstein, almost no one, not even professional physicists, feels relativity in their bones. This isn't surprising; one is hard pressed to find the survival advantage offered by a solid grasp of relativity. Newton's flawed conceptions of absolute space and absolute time work wonderfully well at the slow speeds and moderate gravity we encounter in daily life, so our senses are under no evolutionary pressure to develop relativistic acumen. Deep awareness and true understanding therefore require that we diligently use our intellect to fill in the gaps left by our senses.
While relativity represented a monumental break with traditional ideas about the universe, between 1900 and 1930 another revolution was also turning physics upside down. It started at the turn of the twentieth century with a couple of papers on properties of radiation, one by Max Planck and the other by Einstein; these, after three decades of intense research, led to the formulation of quantum mechanics. As with relativity, whose effects become significant under extremes of speed or gravity, the new physics of quantum mechanics reveals itself abundantly only in another extreme situation: the realm of the extremely tiny. But there is a sharp distinction between the upheavals of relativity and those of quantum mechanics. The weirdness of relativity arises because our personal experience of space and time differs from the experience of others. It is a weirdness born of comparison. We are forced to concede that our view of reality is but one among many—an infinite number, in fact—which all fit together within the seamless whole of spacetime.
Quantum mechanics is different. Its weirdness is evident without comparison. It is harder to train your mind to have quantum mechanical intuition, because quantum mechanics shatters our own personal, individual conception of reality.
The World According to the Quantum
Every age develops its stories or metaphors for how the universe was conceived and structured. According to an ancient Indian creation myth, the universe was created when the gods dismembered the primordial giant Purusa, whose head became the sky, whose feet became the earth, and whose breath became the wind. To Aristotle, the universe was a collection of fifty-five concentric crystalline spheres, the outermost being heaven, surrounding those of the planets, earth and its elements, and finally the seven circles of hell. 1 With Newton and his precise, deterministic mathematical formulation of motion, the description changed again. The universe was likened to an enormous, grand clockwork: after being wound and set into its initial state, the clockwork universe ticks from one moment to the next with complete regularity and predictability.
Special and general relativity pointed out important subtleties of the clockwork metaphor: there is no single, preferred, universal clock; there is no consensus on what constitutes a moment, what constitutes a now. Even so, you can still tell a clockworklike story about the evolving universe. The clock is your clock. The story is your story. But the universe unfolds with the same regularity and predictability as in the Newtonian framework. If by some means you know the state of the universe right now—if you know where every particle is and how fast and in what direction each is moving—then, Newton and Einstein agree, you can, in principle, use the laws of physics to predict everything about the universe arbitrarily far into the future or to figure out what it was like arbitrarily far into the past. 2
Quantum mechanics breaks with this tradition. We can't ever know the exact location and exact velocity of even a single particle. We can't predict with total certainty the outcome of even the simplest of experiments, let alone the evolution of the entire cosmos. Quantum mechanics shows that the best we can ever do is predict the probability that an experiment will turn out this way or that. And as quantum mechanics has been verified through decades of fantastically accurate experiments, the Newtonian cosmic clock, even with its Einsteinian updating, is an untenable metaphor; it is demonstrably not how the world works.
But the break with the past is yet more complete. Even though Newton's and Einstein's theories differ sharply on the nature of space and time, they do agree on certain basic facts, certain truths that appear to be self-evident. If there is space between two objects—if there are two birds in the sky and one is way off to your right and the other is way off to your left—we can and do consider the two objects to be independent. We regard them as separate and distinct entities. Space, whatever it is fundamentally, provides the medium that separates and distinguishes one object from another. That is what space does. Things occupying different locations in space are different things. Moreover, in order for one object to influence another, it must in some way negotiate the space that separates them. One bird can fly to the other, traversing the space between them, and then peck or nudge its companion. One person can influence another by shooting a slingshot, causing a pebble to traverse the space between them, or by yelling, causing a domino effect of bouncing air molecules, one jostling the next until some bang into the recipient's eardrum. Being yet more sophisticated, one can exert influence on another by firing a laser, causing an electromagnetic wave—a beam of light—to traverse the intervening space; or, being more ambitious (like the extraterrestrial pranksters of last chapter) one can shake or move a massive body (like the moon) sending a gravitational disturbance speeding from one location to another. To be sure, if we are over here we can influence someone over there, but no matter how we do it, the procedure always involves someone or something traveling from here to there, and only when the someone or something gets there can the influence be exerted.
Physicists call this feature of the universe locality, emphasizing the point that you can directly affect only things that are next to you, that are local. Voodoo contravenes locality, since it involves doing
something over here and affecting something over there without the need for anything to travel from here to there, but common experience leads us to think that verifiable, repeatable experiments would confirm locality. 3 And most do.
But a class of experiments performed during the last couple of decades has shown that something we do over here (such as measuring certain properties of a particle) can be subtly entwined with something that happens over there (such as the outcome of measuring certain properties of another distant particle), without anything being sent from here to there. While intuitively baffling, this phenomenon fully conforms to the laws of quantum mechanics, and was predicted using quantum mechanics long before the technology existed to do the experiment and observe, remarkably, that the prediction is correct. This sounds like voodoo; Einstein, who was among the first physicists to recognize—and sharply criticize—this possible feature of quantum mechanics, called it "spooky." But as we shall see, the long-distance links these experiments confirm are extremely delicate and are, in a precise sense, fundamentally beyond our ability to control.
