by Brian Greene
Figure 3.2 ( a ) Marge's street design. ( b ) Lisa's street design.
Figure 3.2 ( c ) Overview of Marge's and Lisa's street/avenue designs. Their grids differ by a rotation.
Now, following the insight of Einstein's mathematics professor Hermann Minkowski (who once called his young student a lazy dog), consider the region of spacetime as an entity unto itself: consider the complete flip book as an object in its own right. To do so, imagine that, as in Figure 3.3b, we expand the binding of the flip-card book and then imagine that, as in Figure 3.3c, all the pages are completely transparent, so when you look at the book you see one continuous block containing all the events that happened during a given time interval. From this perspective, the pages should be thought of as simply providing a convenient way of organizing the content of the block—that is, of organizing the events of spacetime. Just as a street/avenue grid allows us to specify locations in a city easily, by giving their street and avenue address, the division of the spacetime block into pages allows us to easily specify an event (Itchy shooting his gun, Scratchy being hit, and so on) by giving the time when the event occurred—the page on which it appears—and the location within the region of space depicted on the pages.
Figure 3.3 ( a ) Flip book of duel. ( b ) Flip book with expanded binding.
Here is the key point: Just as Lisa realized that there are different, equally valid ways to slice up a region of space into streets and avenues,
Figure 3.3 ( c ) Block of spacetime containing the duel. Pages, or "time slices," organize the events in the block. The spaces between slices are for visual clarity only; they are not meant to suggest that time is discrete, a question we discuss later.
Einstein realized that there are different, equally valid ways to slice up a region of spacetime—a block like that in Figure 3.3c—into regions of space at moments of time. The pages in Figures 3.3a, b, and c— with, again, each page denoting one moment of time —provide but one of the many possible slicings. This may sound like only a minor extension of what we know intuitively about space, but it's the basis for overturning some of the most basic intuitions that we've held for thousands of years. Until 1905, it was thought that everyone experiences the passage of time identically, that everyone agrees on what events occur at a given moment of time, and hence, that everyone would concur on what belongs on a given page in the flip book of spacetime. But when Einstein realized that two observers in relative motion have clocks that tick off time differently, this all changed. Clocks that are moving relative to each other fall out of synchronization and therefore give different notions of simultaneity. Each page in Figure 3.3b is but one observer's view of the events in space taking place at a given moment of his or her time. Another observer, moving relative to the first, will declare that the events on a single one of these pages do not all happen at the same time.
This is known as the relativity of simultaneity, and we can see it directly. Imagine that Itchy and Scratchy, pistols in paws, are now facing each other on opposite ends of a long, moving railway car with one referee on the train and another officiating from the platform. To make the duel as fair as possible, all parties have agreed to forgo the three-step rule, and instead, the duelers will draw when a small pile of gunpowder, set midway between them, explodes. The first referee, Apu, lights the fuse, takes a sip of his refreshing Chutney Squishee, and steps back. The gunpowder flares, and both Itchy and Scratchy draw and fire. Since Itchy and Scratchy are the same distance from the gunpowder, Apu is certain that light from the flare reaches them simultaneously, so he raises the green flag and declares it a fair draw. But the second referee, Martin, who was watching from the platform, wildly squeals foul play, claiming that Itchy got the light signal from the explosion before Scratchy did. He explains that because the train was moving forward, Itchy was heading toward the light while Scratchy was moving away from it. This means that the light did not have to travel quite as far to reach Itchy, since he moved closer to it; moreover, the light had to travel farther to reach Scratchy, since he moved away from it. Since the speed of light, moving left or right from anyone's perspective, is constant, Martin claims that it took the light longer to reach Scratchy since it had to travel farther, rendering the duel unfair.
Who is right, Apu or Martin? Einstein's unexpected answer is that they both are. Although the conclusions of our two referees differ, the observations and the reasoning of each are flawless. Like the bat and the baseball, they simply have different perspectives on the same sequence of events. The shocking thing that Einstein revealed is that their different perspectives yield different but equally valid claims of what events happen at the same time. Of course, at everyday speeds like that of the train, the disparity is small—Martin claims that Scratchy got the light less than a trillionth of a second after Itchy—but were the train moving faster, near light speed, the time difference would be substantial.
Think about what this means for the flip-book pages slicing up a region of spacetime. Since observers moving relative to each other do not agree on what things happen simultaneously, the way each of them will slice a block of spacetime into pages—with each page containing all events that happen at a given moment from each observer's perspective— will not agree, either. Instead, observers moving relative to each other cut a block of spacetime up into pages, into time slices, in different but equally valid ways. What Lisa and Marge found for space, Einstein found for spacetime.
Angling the Slices
The analogy between street/avenue grids and time slicings can be taken even further. Just as Marge's and Lisa's designs differed by a rotation, Apu's and Martin's time slicings, their flip-book pages, also differ by a rotation, but one that involves both space and time. This is illustrated in Figures 3.4a and 3.4b, in which we see that Martin's slices are rotated relative to Apu's, leading him to conclude that the duel was unfair. A critical difference of detail, though, is that whereas the rotation angle between Marge's and Lisa's schemes was merely a design choice, the rotation angle between Apu's and Martin's slicings is determined by their relative speed. With minimal effort, we can see why.
Imagine that Itchy and Scratchy have reconciled. Instead of trying to shoot each other, they just want to ensure that clocks on the front and back of the train are perfectly synchronized. Since they are still equidistant from the gunpowder, they come up with the following plan. They agree to set their clocks to noon just as they see the light from the flaring gunpowder. From their perspective, the light has to travel the same distance to reach either of them, and since light's speed is constant, it will reach them simultaneously. But, by the same reasoning as before, Martin and anyone else viewing from the platform will say that Itchy is heading toward the emitted light while Scratchy is moving away from it, and so Itchy will receive the light signal a little before Scratchy does. Platform observers will therefore conclude that Itchy set his clock to 12:00 before Scratchy and will therefore claim that Itchy's clock is set a bit ahead of Scratchy's. For example, to a platform observer like Martin, when it's 12:06 on Itchy's clock, it may be only 12:04 on Scratchy's (the precise numbers depend on the length and the speed of the train; the longer and faster it is, the greater the discrepancy). Yet, from the viewpoint of Apu and everyone on the train, Itchy and Scratchy performed the synchronization perfectly. Again, although it's hard to accept at a gut level, there is no paradox here: observers in relative motion do not agree on simultaneity—they do not agree on what things happen at the same time.
Figure 3.4 Time slicings according to ( a ) Apu and ( b ) Martin, who are in relative motion. Their slices differ by a rotation through space and time. According to Apu, who is on the train, the duel is fair; according to Martin, who is on the platform, it isn't. Both views are equally valid. In ( b ), the different angle of their slices through spacetime is emphasized.
This means that one page in the flip book as seen from the perspective of those on the train, a page containing events they consider simultaneous—such as Itchy's and Scratchy
's setting their clocks— contains events that lie on different pages from the perspective of those observing from the platform (according to platform observers, Itchy set his clock before Scratchy, so these two events are on different pages from the platform observer's perspective). And there we have it. A single page from the perspective of those on the train contains events that lie on earlier and later pages of a platform observer. This is why Martin's and Apu's slices in Figure 3.4 are rotated relative to each other: what is a single time slice, from one perspective, cuts across many time slices, from the other perspective.
If Newton's conception of absolute space and absolute time were correct, everyone would agree on a single slicing of spacetime. Each slice would represent absolute space as viewed at a given moment of absolute time. This, however, is not how the world works, and the shift from rigid Newtonian time to the newfound Einsteinian flexibility inspires a shift in our metaphor. Rather than viewing spacetime as a rigid flip book, it will sometimes be useful to think of it as a huge, fresh loaf of bread. In place of the fixed pages that make up a book—the fixed Newtonian time slices— think of the variety of angles at which you can slice a loaf into parallel pieces of bread, as in Figure 3.5a. Each piece of bread represents space at one moment of time from one observer's perspective. But as illustrated in Figure 3.5b, another observer, moving relative to the first, will slice the spacetime loaf at a different angle. The greater the relative velocity of the two observers, the larger the angle between their respective parallel slices (as explained in the endnotes, the speed limit set by light translates into a maximum 45° rotation angle for these slicings 9 ) and the greater the discrepancy between what the observers will report as having happened at the same moment.
The Bucket, According to Special Relativity
The relativity of time and space requires a dramatic change in our thinking. Yet there is an important point, mentioned earlier and illustrated now by the loaf of bread, which often gets lost: not everything in relativity is relative. Even if you and I were to imagine slicing up a loaf of bread in two different ways, there is still something that we would fully agree upon: the totality of the loaf itself. Although our slices would differ, if I were to imagine putting all of my slices together and you were to imagine doing the same for all of your slices, we would reconstitute the same loaf of bread. How could it be otherwise? We both imagined cutting up the same loaf.
Similarly, the totality of all the slices of space at successive moments of time, from any single observer's perspective (see Figure 3.4), collectively yield the same region of spacetime. Different observers slice up a region of spacetime in different ways, but the region itself, like the loaf of bread, has an independent existence. Thus, although Newton definitely got it wrong, his intuition that there was something absolute, something that everyone would agree upon, was not fully debunked by special relativity. Absolute space does not exist. Absolute time does not exist. But according to special relativity, absolute spacetime does exist. With this observation, let's visit the bucket once again.
Figure 3.5 Just as one loaf of bread can be sliced at different angles, a block of spacetime is "time sliced" at different angles by observers in relative motion. The greater the relative speed, the greater the angle (with a maximum angle of 45 corresponding to the maximum speed set by light).
In an otherwise empty universe, with respect to what is the bucket spinning? According to Newton, the answer is absolute space. According to Mach, there is no sense in which the bucket can even be said to spin. According to Einstein's special relativity, the answer is absolute spacetime.
To understand this, let's look again at the proposed street and avenue layouts for Springfield. Remember that Marge and Lisa disagreed on the street and avenue address of the Kwik-E-Mart and the nuclear plant because their grids were rotated relative to each other. Even so, regardless of how each chose to lay out the grid, there are some things they definitely still agree on. For example, if in the interest of increasing worker efficiency during lunchtime, a trail is painted on the ground from the nuclear plant straight to the Kwik-E-Mart, Marge and Lisa will not agree on the streets and avenues through which the trail passes, as you can see in Figure 3.6. But they will certainly agree on the shape of the trail: they will agree that it is a straight line. The geometrical shape of the painted trail is independent of the particular street/avenue grid one happens to use.
Einstein realized that something similar holds for spacetime. Even though two observers in relative motion slice up spacetime in different ways, there are things they still agree on. As a prime example, consider a straight line not just through space, but through spacetime. Although the inclusion of time makes such a trajectory less familiar, a moment's thought reveals its meaning. For an object's trajectory through spacetime to be straight, the object must not only move in a straight line through space, but its motion must also be uniform through time; that is, both its speed and direction must be unchanging and hence it must be moving with constant velocity. Now, even though different observers slice up the spacetime loaf at different angles and thus will not agree on how much time has elapsed or how much distance is covered between various points on a trajectory, such observers will, like Marge and Lisa, still agree on whether a trajectory through spacetime is a straight line. Just as the geometrical shape of the painted trail to the Kwik-E-Mart is independent of the street/avenue slicing one uses, so the geometrical shapes of trajectories in spacetime are independent of the time slicing one uses. 10
This is a simple yet critical realization, because with it special relativity provided an absolute criterion—one that all observers, regardless of their constant relative velocities, would agree on—for deciding whether or not something is accelerating. If the trajectory an object follows through spacetime is a straight line, like that of the gently resting astronaut (a) in Figure 3.7, it is not accelerating. If the trajectory an object follows has any other shape but a straight line through spacetime, it is accelerating. For example, should the astronaut fire up her jetpack and fly around in a circle over and over again, like astronaut (b) in Figure 3.7, or should she zip out toward deep space at ever increasing speed, like astronaut (c), her trajectory through spacetime will be curved—the telltale sign of acceleration. And so, with these developments we learn that geometricalshapes of trajectories in spacetime provide the absolute standard that determines whether something is accelerating. Spacetime, not space alone, provides the benchmark.
Figure 3.6 Regardless of which street grid is used, everyone agrees on the shape of a trail, in this case, a straight line.
In this sense, then, special relativity tells us that spacetime itself is the ultimate arbiter of accelerated motion. Spacetime provides the backdrop with respect to which something, like a spinning bucket, can be said to accelerate even in an otherwise empty universe. With this insight, the pendulum swung back again: from Leibniz the relationist to Newton the absolutist to Mach the relationist, and now back to Einstein, whose special relativity showed once again that the arena of reality—viewed as spacetime, not as space —is enough of a something to provide the ultimate benchmark for motion. 11
Figure 3.7 The paths through spacetime followed by three astronauts. Astronaut (a) does not accelerate and so follows a straight line through spacetime. Astronaut (b) flies repeatedly in a circle, and so follows a spiral through spacetime. Astronaut (c) accelerates into deep space, and so follows another curved trajectory in spacetime.
Gravity and the Age-old Question
At this point you might think we've reached the end of the bucket story, with Mach's ideas having been discredited and Einstein's radical updating of Newton's absolute conceptions of space and time having won the day. The truth, though, is more subtle and more interesting. But if you're new to the ideas we've covered so far, you may need a break before pressing on to the last sections of this chapter. In Table 3.1 you'll find a summary to refresh your memory when you've geared up to reengage.
Okay. If you're re
ading these words, I gather you're ready for the next major step in spacetime's story, a step catalyzed in large part by none other than Ernst Mach. Although special relativity, unlike Mach's theory, concludes that even in an otherwise empty universe you would feel pressed against the inside wall of a spinning bucket and that the rope tied between two twirling rocks would pull taut, Einstein remained deeply fascinated by Mach's ideas. He realized, however, that serious consideration of these ideas required significantly extending them. Mach never really specified a mechanism whereby distant stars and other matter in the universe might play a role in how strongly your arms splay outward when you spin or how forcefully you feel pressed against the inner wall of a spinning bucket. Einstein began to suspect that if there were such a mechanism it might have something to do with gravity.
This realization had a particular allure for Einstein because in special relativity, to keep the analysis tractable, he had completely ignored gravity.
Maybe, he speculated, a more robust theory, which embraced both special relativity and gravity, would come to a different conclusion regarding Mach's ideas. Maybe, he surmised, a generalization of special relativity that incorporated gravity would show that matter, both near and far, determines the force we feel when we accelerate.
Einstein also had a second, somewhat more pressing, reason for turning his attention to gravity. He realized that special relativity, with its central dictum that the speed of light is the fastest that anything or any disturbance can travel, was in direct conflict with Newton's universal law of gravity, the monumental achievement that had for over two hundred years predicted with fantastic precision the motion of the moon, the planets, comets, and all things tossed skyward. The experimental success of Newton's law notwithstanding, Einstein realized that according to Newton, gravity exerts its influence from place to place, from the sun to the earth, from the earth to the moon, from any-here to any-there, instantaneously, in no time at all, much faster than light. And that directly contradicted special relativity.