The Elegant Universe (6 page)

The Effect on Time: Part I

With minimal effort, we can make use of the constancy of the speed of light to show that the familiar everyday conception of time is plain wrong. Imagine that the leaders of two warring nations, sitting at opposite ends of a long negotiating table, have just concluded an agreement for a cease-fire, but neither wants to sign the accord before the other. The secretary-general of the United Nations comes up with a brilliant resolution. A light bulb, initially turned off, will be placed midway between the two presidents. When it is turned on, the light it emits will reach each of the presidents simultaneously, since they are equidistant from the bulb. Each president agrees to sign a copy of the accord when he or she sees the light. The plan is carried out and the agreement is signed to the satisfaction of both sides.

Flushed with success, the secretary-general makes use of the same approach with two other embattled nations that have also reached a peace agreement. The only difference is that the presidents involved in this negotiation are sitting at opposite ends of a table inside a train traveling along at constant velocity. Fittingly, the president of Forwardland is facing in the direction of the train’s motion while the president of Backwardland is facing in the opposite direction. Familiar with the fact that the laws of physics take precisely the same form regardless of one’s state of motion so long as this motion is unchanging, the secretary-general takes no heed of this difference, and carries out the light bulb-initiated signing ceremony as before. Both presidents sign the agreement, and along with their entourage of advisers, celebrate the end of hostilities.

Just then, word arrives that fighting has broken out between people from each country who had been watching the signing ceremony from the platform outside the moving train. All those on the negotiation train are dismayed to hear that the reason for the renewed hostilities is the claim by people from Forwardland that they have been duped, as their president signed the agreement before the president of Backwardland. As everyone on the train—from both sides—agrees that the accord was signed simultaneously, how can it be that the outside observers watching the ceremony think otherwise?

Let’s consider in more detail the perspective of an observer on the platform. Initially the bulb on the train is dark, and then at a particular moment it illuminates, sending beams of light speeding toward both presidents. From the perspective of a person on the platform, the president of Forwardland is heading toward the emitted light while the president of Backwardland is retreating. This means, to the platform observers, that the light beam does not have to travel as far to reach the president of Forwardland, who moves toward the approaching light, as it does to reach the president of Backwardland, who moves away from it. This is not a statement about the speed of the light as it travels toward the two presidents—we have already noted that regardless of the state of motion of the source or the observer, the speed of light is always the same. Instead, we are describing only how far, from the vantage point of the platform observers, the initial flash of light must travel to reach each of the presidents. Since this distance is less for the president of Forwardland than it is for the president of Backwardland, and since the speed of light toward each is the same, the light will reach the president of Forwardland first. This is why the citizens of Forwardland claim to have been duped.

When CNN broadcasts the eyewitness account, the secretary-general, the two presidents, and all of their advisers can’t believe their ears. They all agree that the light bulb was secured firmly, exactly midway between the two presidents and that therefore, without further ado, the light it emitted traveled the same distance to reach each of them. Since the speed of the emitted light to the left and to the right is the same, they believe, and in fact observed, that the light clearly reached each president simultaneously.

Who is right, those on or off the train? The observations of each group and their supporting explanations are impeccable. The answer is that both are right. Like our two space inhabitants George and Gracie, each perspective has an equal claim on truth. The only subtlety here is that the respective truths seem to be contradictory. An important political issue is at stake: Did the presidents sign the agreement simultaneously? The observations and reasoning above ineluctably lead us to the conclusion that according to those on the train they did while according to those on the platform they did not. In other words, things that are simultaneous from the viewpoint of some observers will not be simultaneous from the viewpoint of others, if the two groups are in relative motion.

This is a startling conclusion. It is one of the deepest insights into the nature of reality ever discovered. Nevertheless, if long after you set down this book you remember nothing of this chapter except for the ill-fated attempt at détente, you will have retained the essence of Einstein’s discovery. Without highbrow mathematics or a convoluted chain of logic, this completely unexpected feature of time follows directly from the constancy of the speed of light, as the scenario illustrates. Notice that if the speed of light were not constant but behaved according to our intuition based on slow-moving baseballs and snowballs, the platform observers would agree with those on the train. A platform observer would still claim that the photons have to travel farther to reach the president of Backwardland than they do to reach the president of Forwardland. However, usual intuition implies that the light approaching the president of Backwardland would be moving more quickly, having received a “kick” from the forward-moving train. Similarly, these observers would see that the light approaching the president of Forwardland would be moving more slowly, being “dragged” back by the train’s motion. When these (erroneous) effects were considered, the observers on the platform would see that that the light beams reached each president simultaneously. However, in the real world light does not speed up or slow down, it cannot be kicked to a higher speed or dragged to a slower one. Platform observers will therefore justifiably claim that the light reached the president of Forwardland first.

The constancy of the speed of light requires that we give up the age-old notion that simultaneity is a universal concept that everyone, regardless of their state of motion, agrees upon. The universal clock previously envisioned to dispassionately tick off identical seconds here on earth and on Mars and on Jupiter and in the Andromeda galaxy and in each and every nook and cranny of the cosmos does not exist. On the contrary, observers in relative motion will not agree on which events occur at the same time. Once again, the reason that this conclusion—a bona fide characteristic of the world we inhabit—is so unfamiliar is that the effects are extremely small when the speeds involved are those commonly encountered in everyday experience. If the negotiating table were 100 feet long and the train were moving at 10 miles per hour, platform observers would “see” that the light reached the president of Forwardland about a millionth of a billionth of a second before it reached the president of Backwardland. Although this represents a genuine difference, it is so tiny that it cannot be detected directly by human senses. If the train were moving considerably faster, say at 600 million miles per hour, from the perspective of someone on the platform the light would take almost 20 times as long to reach the president of Backwardland compared with the time to reach the president of Forwardland. At high speeds, the startling effects of special relativity become increasingly pronounced.

The Effect on Time: Part II

It is difficult to give an abstract definition of time—attempts to do so often wind up invoking the word “time” itself, or else go through linguistic contortions simply to avoid doing so. Rather than proceeding down such a path, we can take a pragmatic viewpoint and define time to be that which is measured by clocks. Of course, this shifts the burden of definition to the word “clock”; here we can somewhat loosely think of a clock as a device that undergoes perfectly regular cycles of motion. We will measure time by counting the number of cycles our clock goes through. A familiar clock such as a wristwatch meets this definition; it has hands that move in regular cycles of motion and we do indeed measure elapsed time by counting the number of cycles (or fractions thereof) that the hands swing through between chosen events.

Of course, the meaning of “perfectly regular cycles of motion” implicitly involves a notion of time, since “regular” refers to equal time durations elapsing for each cycle. From a practical standpoint we address this by building clocks out of simple physical components that, on fundamental grounds, we expect to undergo repetitive cyclical evolutions that do not change in any manner from one cycle to the next. Grandfather clocks with pendulums that swing back and forth and atomic clocks based on repetitive atomic processes provide simple examples.

Our goal is to understand how motion affects the passage of time, and since we have defined time operationally in terms of clocks, we can translate our question into how motion affects the “ticking” of clocks. It is crucial to emphasize at the outset that our discussion is not concerned with how the mechanical elements of a particular clock happen to respond to shaking or jostling that might result from bumpy motion. In fact, we will consider only the simplest and most serene kind of motion—motion at absolutely constant velocity—and therefore there will not be any shaking or jostling at all. Rather, we are interested in the universal question of how motion affects the passage of time and therefore how it fundamentally affects the ticking of any and all clocks regardless of their particular design or construction.

For this purpose we introduce the world’s conceptually simplest (yet most impractical) clock. It is known as a “light clock” and consists of two small mirrors mounted on a bracket facing one another, with a single photon of light bouncing back and forth between them (see Figure 2.1). If the mirrors are about six inches apart, it will take the photon about a billionth of a second to complete one round-trip journey. “Ticks” on the light clock may be thought of as occurring every time the photon completes a round-trip—a billion ticks means that one second has elapsed.

We can use the light clock like a stopwatch to measure the time elapsed between events: We simply count how many ticks occur during the period of interest and multiply by the time corresponding to one tick. For instance, if we are timing a horse race and count that between the start and finish the number of round-trip photon journeys is 55 billion, we can conclude that the race took 55 seconds.

The reason we use the light clock in our discussion is that its mechanical simplicity pares away extraneous details and therefore provides us with the clearest insight into how motion affects the passage of time. To see this, imagine that we are idly watching the passage of time by looking at a ticking light clock placed on a nearby table. Then, all of sudden, a second light clock slides by on the table, moving at constant velocity (see Figure 2.2) The question we ask is whether the moving light clock will tick at the same rate as the stationary light clock?

To answer the question, let’s consider the path, from our perspective, that the photon in the sliding clock must take in order for it to result in a tick. The photon starts at the base of the sliding clock, as in Figure 2.2, and first travels to the upper mirror. Since, from our perspective, the clock is moving, the photon must travel at an angle, as shown in Figure 2.3. If the photon did not travel along this path, it would miss the upper mirror and fly off into space. As the sliding clock has every right to claim that it’s stationary and everything else is moving, we know that the photon will hit the upper mirror and hence the path we have drawn is correct. The photon bounces off the upper mirror and again travels a diagonal path to hit the lower mirror, and the sliding clock ticks. The simple but essential point is that the double diagonal path that we see the photon traverse is longer than the straight up-and-down path taken by the photon in the stationary clock; in addition to traversing the up-and-down distance, the photon in the sliding clock must also travel to the right, from our perspective. Moreover, the constancy of the speed of light tells us that the sliding clock’s photon travels at exactly the same speed as the stationary clock’s photon. But since it must travel farther to achieve one tick it will tick less frequently. This simple argument establishes that the moving light clock, from our perspective, ticks more slowly than the stationary light clock. And since we have agreed that the number of ticks directly reflects how much time has passed, we see that the passage of time has slowed down for the moving clock.

You might wonder whether this merely reflects some special feature of light clocks and would not apply to grandfather clocks or Rolex watches. Would time as measured by these more familiar clocks also slow down? The answer is a resounding yes, as can be seen by an application of the principle of relativity. Let’s attach a Rolex watch to the top of each of our light clocks, and rerun the preceding experiment. As discussed, a stationary light clock and its attached Rolex measure identical time durations, with a billion ticks on the light clock occurring for every one second of elapsed time on the Rolex. But what about the moving light clock and its attached Rolex? Does the rate of ticking on the moving Rolex slow down so that it stays synchronized with the light clock to which it is attached? Well, to make the point most forcefully, imagine that the light clock-Rolex watch combination is moving because it is bolted to the floor of a windowless train compartment gliding along perfectly straight and smooth tracks at constant speed. By the principle of relativity, there is no way for an observer on this train to detect any influence of the train’s motion. But if the light clock and Rolex were to fall out of synchronization, this would be a noticeable influence indeed. And so the moving light clock and its attached Rolex must still measure equal time durations; the Rolex must slow down in exactly the same way that the light clock does. Regardless of brand, type, or construction, clocks that are moving relative to one another record the passage of time at different rates.