by Brian Greene
Think about speed for a moment. Speed is measured by how far something goes divided by how long it takes to get there. It is a measure of space (the distance traveled) divided by a measure of time (the duration of the journey). Ever since Newton, space had been thought of as absolute, as being out there, as existing "without reference to anything external." Measurements of space and spatial separations must therefore also be absolute: regardless of who measures the distance between two things in space, if the measurements are done with adequate care, the answers will always agree. And although we have not yet discussed it directly, Newton declared the same to be true of time. His description of time in the Principia echoes the language he used for space: "Time exists in and of itself and flows equably without reference to anything external." In other words, according to Newton, there is a universal, absolute conception of time that applies everywhere and everywhen. In a Newtonian universe, regardless of who measures how much time it takes for something to happen, if the measurements are done accurately, the answers will always agree.
These assumptions about space and time comport with our daily experiences and for that reason are the basis of our commonsense conclusion that light should appear to travel more slowly if we run after it. To see this, imagine that Bart, who's just received a new nuclear-powered skateboard, decides to take on the ultimate challenge and race a beam of light. Although he is a bit disappointed to see that the skateboard's top speed is only 500 million miles per hour, he is determined to give it his best shot. His sister Lisa stands ready with a laser; she counts down from 11 (her hero Schopenhauer's favorite number) and when she reaches 0, Bart and the laser light streak off into the distance. What does Lisa see? Well, for every hour that passes, Lisa sees the light travel 670 million miles while Bart travels only 500 million miles, so Lisa rightly concludes that the light is speeding away from Bart at 170 million miles per hour. Now let's bring Newton into the story. His ideas dictate that Lisa's observations about space and time are absolute and universal in the sense that anyone else performing these measurements would get the same answers. To Newton, such facts about motion through space and time were as objective as two plus two equaling four. According to Newton, then, Bart will agree with Lisa and will report that the light beam was speeding away from him at 170 million miles per hour.
But when Bart returns, he doesn't agree at all. Instead, he dejectedly claims that no matter what he did—no matter how much he pushed the skateboard's limit—he saw the light speed away at 670 million miles per hour, not a bit less. 3 And if for some reason you don't trust Bart, bear in mind that thousands of meticulous experiments carried out during the last hundred years, which have measured the speed of light using moving sources and receivers, support his observations with precision.
How can this be?
Einstein figured it out, and the answer he found is a logical yet profound extension of our discussion so far. It must be that Bart's measurements of distances and durations, the input that he uses to figure out how fast the light is receding from him, are different from Lisa's measurements. Think about it. Since speed is nothing but distance divided by time, there is no other way for Bart to have found a different answer from Lisa's for how fast the light was outrunning him. So, Einstein concluded, Newton's ideas of absolute space and absolute time were wrong. Einstein realized that experimenters who are moving relative to each other, like Bart and Lisa, will not find identical values for measurements of distances and durations. The puzzling experimental data on the speed of light can be explained only if their perceptions of space and time are different.
Subtle but Not Malicious
The relativity of space and of time is a startling conclusion. I have known about it for more than twenty-five years, but even so, whenever I quietly sit and think it through, I am amazed. From the well-worn statement that the speed of light is constant, we conclude that space and time are in the eye of the beholder. Each of us carries our own clock, our own monitor of the passage of time. Each clock is equally precise, yet when we move relative to one another, these clocks do not agree. They fall out of synchronization; they measure different amounts of elapsed time between two chosen events. The same is true of distance. Each of us carries our own yardstick, our own monitor of distance in space. Each yardstick is equally precise, yet when we move relative to one another, these yardsticks do not agree; they measure different distances between the locations of two specified events. If space and time did not behave this way, the speed of light would not be constant and would depend on the observer's state of motion. But it is constant; space and time do behave this way. Space and time adjust themselves in an exactly compensating manner so that observations of light's speed yield the same result, regardless of the observer's velocity.
Getting the quantitative details of precisely how the measurements of space and time differ is more involved, but requires only high school algebra. It is not the depth of mathematics that makes Einstein's special relativity challenging. It is the degree to which the ideas are foreign and apparently inconsistent with our everyday experiences. But once Einstein had the key insight—the realization that he needed to break with the more than two-hundred-year-old Newtonian perspective on space and time—it was not hard to fill in the details. He was able to show precisely how one person's measurements of distances and durations must differ from those of another in order to ensure that each measures an identical value for the speed of light. 4
To get a fuller sense of what Einstein found, imagine that Bart, with heavy heart, has carried out the mandatory retrofitting of his skateboard, which now has a maximum speed of 65 miles per hour. If he heads due north at top speed—reading, whistling, yawning, and occasionally glancing at the road—and then merges onto a highway pointing in a northeasterly direction, his speed in the northward direction will be less than 65 miles per hour. The reason is clear. Initially, all his speed was devoted to northward motion, but when he shifted direction some of that speed was diverted into eastward motion, leaving a little less for heading north. This extremely simple idea actually allows us to capture the core insight of special relativity. Here's how:
We are used to the fact that objects can move through space, but there is another kind of motion that is equally important: objects also move through time. Right now, the watch on your wrist and the clock on the wall are ticking away, showing that you and everything around you are relentlessly moving through time, relentlessly moving from one second to the next and the next. Newton thought that motion through time was totally separate from motion through space—he thought these two kinds of motion had nothing to do with each other. But Einstein found that they are intimately linked. In fact, the revolutionary discovery of special relativity is this: When you look at something like a parked car, which from your viewpoint is stationary—not moving through space, that is —all of its motion is through time. The car, its driver, the street, you, your clothes are all moving through time in perfect synch: second followed by second, ticking away uniformly. But if the car speeds away, some of its motion through time is diverted into motion through space. And just as Bart's speed in the northward direction slowed down when he diverted some of his northward motion into eastward motion, the speed of the car through time slows down when it diverts some of its motion through time into motion through space. This means that the car's progress through time slows down and therefore time elapses more slowly for the moving car and its driver than it elapses for you and everything else that remains stationary.
That, in a nutshell, is special relativity. In fact, we can be a bit more precise and take the description one step further. Because of the retrofitting, Bart had no choice but to limit his top speed to 65 miles per hour. This is important to the story, because if he sped up enough when he angled northeast, he could have compensated for the speed diversion and thereby maintained the same net speed toward the north. But with the retrofitting, no matter how hard he revved the skateboard's engine, his total speed—the combination of
his speed toward the north and his speed toward the east—remained fixed at the maximum of 65 miles per hour. And so when he shifted his direction a bit toward the east, he necessarily caused a decreased northward speed.
Special relativity declares a similar law for all motion: the combined speed of any object's motion through space and its motion through time is always precisely equal to the speed of light. At first, you may instinctively recoil from this statement since we are all used to the idea that nothing but light can travel at light speed. But that familiar idea refers solely to motion through space. We are now talking about something related, yet richer: an object's combined motion through space and time. The key fact, Einstein discovered, is that these two kinds of motion are always complementary. When the parked car you were looking at speeds away, what really happens is that some of its light-speed motion is diverted from motion through time into motion through space, keeping their combined total unchanged. Such diversion unassailably means that the car's motion through time slows down.
As an example, if Lisa had been able to see Bart's watch as he sped along at 500 million miles per hour, she would have seen that it was ticking about two-thirds as fast as her own. For every three hours that passed on Lisa's watch, she would see that only two had passed on Bart's. His rapid motion through space would have proved a significant drain on his speed through time.
Moreover, the maximum speed through space is reached when all light-speed motion through time is fully diverted into light-speed motion through space—one way of understanding why it is impossible to go through space at greater than light speed. Light, which always travels at light speed through space, is special in that it always achieves such total diversion. And just as driving due east leaves no motion for traveling north, moving at light speed through space leaves no motion for traveling through time! Time stops when traveling at the speed of light through space. A watch worn by a particle of light would not tick at all. Light realizes the dreams of Ponce de León and the cosmetics industry: it doesn't age. 5
As this description makes clear, the effects of special relativity are most pronounced when speeds (through space) are a significant fraction of light speed. But the unfamiliar, complementary nature of motion through space and time always applies. The lesser the speed, the smaller the deviation from prerelativity physics—from common sense, that is— but the deviation is still there, to be sure.
Truly. This is not dexterous wordplay, sleight of hand, or psychological illusion. This is how the universe works.
In 1971, Joseph Hafele and Richard Keating flew state-of-the-art cesium-beam atomic clocks around the world on a commercial Pan Am jet. When they compared the clocks flown on the plane with identical clocks left stationary on the ground, they found that less time had elapsed on the moving clocks. The difference was tiny—a few hundred billionths of a second—but it was precisely in accord with Einstein's discoveries. You can't get much more nuts-and-bolts than that.
In 1908, word began to spread that newer, more refined experiments were finding evidence for the aether. 6 If that had been so, it would have meant that there was an absolute standard of rest and that Einstein's special relativity was wrong. On hearing this rumor, Einstein replied, "Subtle is the Lord, malicious He is not." Peering deeply into the workings of nature to tease out insights into space and time was a profound challenge, one that had gotten the better of everyone until Einstein. But to allow such a startling and beautiful theory to exist, and yet to make it irrelevant to the workings of the universe, that would be malicious. Einstein would have none of it; he dismissed the new experiments. His confidence was well placed. The experiments were ultimately shown to be wrong, and the luminiferous aether evaporated from scientific discourse.
But What About the Bucket?
This is certainly a tidy story for light. Theory and experiment agree that light needs no medium to carry its waves and that regardless of the motion of either the source of light or the person observing, its speed is fixed and unchanging. Every vantage point is on an equal footing with every other. There is no absolute or preferred standard of rest. Great. But what about the bucket?
Remember, while many viewed the luminiferous aether as the physical substance giving credibility to Newton's absolute space, it had nothing to do with why Newton introduced absolute space. Instead, after wrangling with accelerated motion such as the spinning bucket, Newton saw no option but to invoke some invisible background stuff with respect to which motion could be unambiguously defined. Doing away with the aether did not do away with the bucket, so how did Einstein and his special theory of relativity cope with the issue?
Well, truth be told, in special relativity, Einstein's main focus was on a special kind of motion: constant-velocity motion. It was not until 1915, some ten years later, that he fully came to grips with more general, accelerated motion, through his general theory of relativity. Even so, Einstein and others repeatedly considered the question of rotating motion using the insights of special relativity; they concluded, like Newton and unlike Mach, that even in an otherwise completely empty universe you would feel the outward pull from spinning—Homer would feel pressed against the inner wall of a spinning bucket; the rope between the two twirling rocks would pull taut. 7 Having dismantled Newton's absolute space and absolute time, how did Einstein explain this?
The answer is surprising. Its name notwithstanding, Einstein's theory does not proclaim that everything is relative. Special relativity does claim that some things are relative: velocities are relative; distances across space are relative; durations of elapsed time are relative. But the theory actually introduces a grand, new, sweepingly absolute concept: absolute spacetime. Absolute spacetime is as absolute for special relativity as absolute space and absolute time were for Newton, and partly for this reason Einstein did not suggest or particularly like the name "relativity theory." Instead, he and other physicists suggested invariance theory, stressing that the theory, at its core, involves something that everyone agrees on, something that is not relative. 8
Absolute spacetime is the vital next chapter in the story of the bucket, because, even if devoid of all material benchmarks for defining motion, the absolute spacetime of special relativity provides a something with respect to which objects can be said to accelerate.
Carving Space and Time
To see this, imagine that Marge and Lisa, seeking some quality together-time, enroll in a Burns Institute extension course on urban renewal. For their first assignment, they are asked to redesign the street and avenue layout of Springfield, subject to two requirements: first, the street/avenue grid must be configured so that the Soaring Nuclear Monument is located right at the grid's center, at 5th Street and 5th Avenue, and, second, the designs must use streets 100 meters long, and avenues, which run perpendicular to streets, that are also 100 meters long. Just before class, Marge and Lisa compare their designs and realize that something is terribly wrong. After appropriately configuring her grid so that the Monument lies in the center, Marge finds that Kwik-E-Mart is at 8th Street and 5th Avenue and the nuclear power plant is at 3rd Street and 5th Avenue, as shown in Figure 3.2a. But in Lisa's design, the addresses are completely different: the Kwik-E-Mart is near the corner of 7th Street and 3rd Avenue, while the power plant is at 4th Street and 7th Avenue, as in Figure 3.2b. Clearly, someone has made a mistake.
After a moment's thought, though, Lisa realizes what's going on. There are no mistakes. She and Marge are both right. They merely chose different orientations for their street and avenue grids. Marge's streets and avenues run at an angle relative to Lisa's; their grids are rotated relative to each other; they have sliced up Springfield into streets and avenues in two different ways (see Figure 3.2c). The lesson here is simple, yet important. There is freedom in how Springfield—a region of space—can be organized by streets and avenues. There are no "absolute" streets or "absolute" avenues. Marge's choice is as valid as Lisa's—or any other possible orientation, for that matter.
Ho
ld this idea in mind as we paint time into the picture. We are used to thinking about space as the arena of the universe, but physical processes occur in some region of space during some interval of time. As an example, imagine that Itchy and Scratchy are having a duel, as illustrated in Figure 3.3a, and the events are recorded moment by moment in the fashion of one of those old-time flip books. Each page is a "time slice"—like a still frame in a filmstrip—that shows what happened in a region of space at one moment of time. To see what happened at a different moment of time you flip to a different page. 4 (Of course, space is three-dimensional while the pages are two-dimensional, but let's make this simplification for ease of thinking and drawing figures. It won't compromise any of our conclusions.) By way of terminology, a region of space considered over an interval of time is called a region of spacetime; you can think of a region of spacetime as a record of all things that happen in some region of space during a particular span of time.
