The Elegant Universe (7 page)

The light clock discussion also makes clear that the precise time difference between stationary and moving clocks depends on how much farther the sliding clock’s photon must travel to complete each round-trip journey This in turn depends on how quickly the sliding clock is moving—from the viewpoint of a stationary observer, the faster the clock is sliding, the farther the photon must travel to the right. We conclude that in comparison to a stationary clock, the rate of ticking of the sliding clock becomes slower and slower as it moves faster and faster.2

To get a sense of scale, note that the photon traverses one round-trip in about a billionth of a second. For the clock to be able to travel an appreciable distance during the time for one tick it must therefore be traveling enormously quickly—that is, some significant fraction of the speed of light. If it is traveling at more commonplace speeds like 10 miles per hour, the distance it can move to the right before one tick is completed is minuscule—just about 15 billionths of a foot. The extra distance that the sliding photon must travel is tiny and it has a correspondingly tiny effect on the rate of ticking of the moving clock. And again, by the principle of relativity, this is true for all clocks—that is, for time itself. This is why beings such as ourselves who travel relative to one another at such slow speeds are generally unaware of the distortions in the passage of time. The effects, although present to be sure, are incredibly small. If, on the other hand, we were able to grab hold of the sliding clock and move with it at, say, three-quarters the speed of light, the equations of special relativity can be used to show that stationary observers would see our moving clock ticking at just about two-thirds the rate of their own. A significant effect, indeed.

Life on the Run

We have seen that the constancy of the speed of light implies that a moving light clock ticks more slowly than a stationary light clock. And by the principle of relativity, this must be true not only for light clocks but also for any clock—it must be true of time itself. Time elapses more slowly for an individual in motion than it does for a stationary individual. If the fairly simple reasoning that has led us to this conclusion is correct, then, for instance, shouldn’t one be able to live longer by being in motion rather than staying stationary? After all, if time elapses more slowly for an individual in motion than for an individual at rest, then this disparity should apply not just to time as measured by watches but also to time as measured by heartbeats and the decay of body parts. This is the case, as has been directly confirmed—not with the life expectancy of humans, but with certain particles from the microworld: muons. There is one important catch, however, that prevents us from proclaiming a newfound fountain of youth.

When sitting at rest in the laboratory, muons disintegrate by a process closely akin to radioactive decay, in an average of about two millionths of a second. This disintegration is an experimental fact supported by an enormous amount of evidence. It’s as if a muon lives its life with a gun to its head; when it reaches two millionths of a second in age, it pulls the trigger and explodes apart into electrons and neutrinos. But if these muons are not sitting at rest in the laboratory and instead are traveling through a piece of equipment known as a particle accelerator that boosts them to just shy of light-speed, their average life expectancy as measured by scientists in the laboratory increases dramatically. This really happens. At 667 million miles per hour (about 99.5 percent of light speed), the muon lifetime is seen to increase by a factor of about ten. The explanation, according to special relativity, is that “wristwatches” worn by the muons tick much more slowly than the clocks in the laboratory, so long after the laboratory clocks say that the muons should have pulled their triggers and exploded, the watches on the fast-moving muons have yet to reach doom time. This is a very direct and dramatic demonstration of the effect of motion on the passage of time. If people were to zip around as quickly as these muons, their life expectancy would also increase by the same factor. Rather than living seventy years, people would live 700 years.3

Now for the catch. Although laboratory observers see fast-moving muons living far longer than their stationary brethren, this is due to time elapsing more slowly for the muons in motion. This slowing of time applies not just to the watches worn by the muons but also to all activities they might undertake. For instance, if a stationary muon can read 100 books in its short lifetime, its fast-moving cousin will also be able to read the same 100 books, because although it appears to live longer than the stationary muon, its rate of reading—as well as everything else in its life—has slowed down as well. From the laboratory perspective, it’s as if the moving muon is living its life in slow motion; from this viewpoint the moving muon will live longer than a stationary one, but the “amount of life” the muon will experience is precisely the same. The same conclusion, of course, holds true for the fast-moving people with a life expectancy of centuries. From their perspective, it’s life as usual. From our perspective they are living life in hyper-slow motion and therefore one of their normal life cycles takes an enormous amount of our time.

Who Is Moving, Anyway?

The relativity of motion is both the key to understanding Einstein’s theory and a potential source of confusion. You may have noticed that a reversal of perspective interchanges the roles of the “moving” muons, whose watches we have argued run slowly, and their “stationary” counterparts. Just as both George and Gracie had an equal right to declare that they were stationary and that the other was moving, the muons we have described as being in motion are fully justified in proclaiming that, from their perspective, they are motionless and that it is the “stationary” muons that are moving, in the opposite direction. The arguments presented can be applied equally well from this perspective, leading to the seemingly opposite conclusion that watches worn by the muons we christened as stationary are running slow compared with those worn by the muons we described as moving.

We have already met a situation, the signing ceremony with the light bulb, in which different viewpoints lead to results that seem to be completely at odds. In that case we were forced by the basic reasoning of special relativity to give up the ingrained idea that everyone, regardless of state of motion, agrees about which events happen at the same time. The present incongruity, though, appears to be worse. How can two observers each claim that the other’s watch is running slower? More dramatically, the different but equally valid muon perspectives seem to lead us to the conclusion that each group will claim that it is the other group that dies first. We are learning that the world can have some unexpectedly strange features, but we would hope that it does not cross into the realm of logical absurdity. So what’s going on?

As with all apparent paradoxes arising from special relativity, under close examination these logical dilemmas resolve to reveal new insights into the workings of the universe. To avoid ever more severe anthropomorphizing, let’s switch from muons back to George and Gracie, who now, in addition to their flashing lights, have bright digital clocks on their spacesuits. From George’s perspective, he is stationary while Gracie with her flashing green light and large digital clock appears in the distance and then passes him in the blackness of empty space. He notices that Gracie’s clock is running slow in comparison to his (with the rate of slowdown depending on how fast they pass one another). Were he a bit more astute, he would also note that in addition to the passage of time on her clock, everything about Gracie—the way she waves as she passes, the speed with which she blinks her eyes, and so on—is occurring in slow motion. From Gracie’s perspective, exactly the same observations apply to George.

Although this seems paradoxical, let’s try to pinpoint a precise experiment that would reveal a logical absurdity. The simplest possibility is to arrange things so that when George and Gracie pass one another they both set their clocks to read 12:00. As they travel apart, each claims that the other’s clock is running slower. To confront this disagreement head on, George and Gracie must rejoin each other and directly compare the time elapsed on their clocks. But how can they do this? Well, George has a jetpack that he can use, from his perspective, to catch up with Gracie. But if he does this, the symmetry of their two perspectives, which is the cause of the apparent paradox, is broken since George will have undergone accelerated, non-force-free motion. When they rejoin in this manner, less time will indeed have elapsed on George’s clock as he can now definitively say that he was in motion, since he could feel it. No longer are George’s and Gracie’s perspectives on equal footing. By turning on the jet-pack, George relinquishes his claim to being at rest.

If George chases after Gracie in this manner, the time difference that their clocks will show depends on their relative velocity and the details of how George uses his jet-pack. As is by now familiar, if the speeds involved are small, the difference will be minuscule. But if substantial fractions of light speed are involved, the differences can be minutes, days, years, centuries, or more. As one concrete example, imagine that the relative speed of George and Gracie when they pass and are moving apart is 99.5 percent of light speed. Further, let’s say that George waits 3 years, according to his clock, before firing up his jet-pack for a momentary blast that sends him closing in on Gracie at the same speed that they were previously moving apart, 99.5 percent of light speed. When he reaches Gracie, 6 years will have elapsed on his clock since it will take him 3 years to catch her. However, the mathematics of special relativity shows that 60 years will have elapsed on her clock. This is no sleight of hand: Gracie will have to search her distant memory, some 60 years before, to recall passing George in space. For George, on the other hand, it was a mere 6 years ago. In a real sense, George’s motion has made him a time traveler, albeit in a very precise sense: He has traveled into Gracie’s future.

Getting the two clocks back together for direct comparison might seem to be merely a logistical nuisance, but it is really at the heart of the matter. We can imagine a variety of tricks to circumvent this chink in the paradox armor, but all ultimately fail. For instance, rather than bringing the clocks back together, what if George and Gracie compare their clocks by cellular telephone communication? If such communication were instantaneous, we would be faced with an insurmountable inconsistency: reasoning from Gracie’s perspective, George’s clock is running slow and hence he must communicate less elapsed time; reasoning from George’s perspective, Gracie’s clock is running slow and hence she must communicate less elapsed time. They both can’t be right, and we would be sunk. The key point of course is that cell phones, like all forms of communication, do not transmit their signals instantaneously. Cell phones operate with radio waves, a form of light, and the signal they transmit therefore travels at light speed. This means that it takes time for the signals to be received—just enough time delay, in fact, to make each perspective compatible with the other.

Let’s see this, first, from George’s perspective. Imagine that every hour, on the hour, George recites into his cell phone, “It’s twelve o’clock and all is well,” “It’s one o’clock and all is well,” and so forth. Since from his perspective Gracie’s clock runs slow, at first blush he thinks that Gracie will receive these messages prior to her clock’s reaching the appointed hour. In this way, he concludes, Gracie will have to agree that hers is the slow clock. But then he rethinks it: “Since Gracie is receding from me, the signal I send to her by cell phone must travel ever longer distances to reach her. Maybe this additional travel time compensates for the slowness of her clock.” George’s realization that there are competing effects—the slowness of Gracie’s clock vs. the travel time of his signal—inspires him to sit down and quantitatively work out their combined effect. The result he finds is that the travel time effect more than compensates for the slowness of Gracie’s clock. He comes to the surprising conclusion that Gracie will receive his signals proclaiming the passing of an hour on his clock after the appointed hour has passed on hers. In fact, since George is aware of Gracie’s expertise in physics, he knows that she will take the signal’s travel time into account when drawing conclusions about his clock based on his cell phone communications. A little more calculation quantitatively shows that even taking the travel time into account, Gracie’s analysis of his signals will lead her to the conclusion that George’s clock ticks more slowly than hers.

Exactly the same reasoning applies when we take Gracie’s perspective, with her sending out hourly signals to George. At first the slowness of George’s clock from her perspective leads her to think that he will receive her hourly messages prior to broadcasting his own. But when she takes into account the ever longer distances her signal must travel to catch George as he recedes into the darkness, she realizes that George will actually receive them after sending out his own. Once again, she realizes that even if George takes the travel time into account, he will conclude from Gracie’s cell phone communications that her clock is running slower than his.

So long as neither George nor Gracie accelerates, their perspectives are on precisely equal footing. Even though it seems paradoxical, in this way they both realize that it is perfectly consistent for each to think the other’s clock is running slow.

Motion’s Effect on Space

The preceding discussion reveals that observers see moving clocks ticking more slowly than their own-that is, time is affected by motion. It is a short step to see that motion has an equally dramatic effect on space. Let’s return to Slim and Jim on the drag strip. While in the showroom, as we mentioned, Slim had carefully measured the length of his new car with a tape measure. As Slim is speeding along the drag strip, Jim cannot apply this method to measure the length of the car, so he must proceed in an indirect manner. One such approach, as we indicated earlier, is this: Jim starts his stopwatch just when the front bumper of the car reaches him and stops it just as the rear bumper passes. By multiplying the elapsed time by the speed of the car, Jim can determine the car’s length.