28.3

The Relativity of Time: Time Dilation

| Time Dilation

Common experience indicates that time passes just as quickly for a person standing on the ground as it does for an astronaut in a spacecraft. In contrast, special relativity reveals that the person on the ground measures time passing more slowly for the astronaut than for herself. We can see how this curious effect arises with the help of the clock illustrated in Figure 28-4, which uses a pulse of light to mark time. A short pulse of light is emitted by a light source, reflects from a mirror, and then strikes a detector that is situated next to the source. Each time a pulse reaches the detector, a “tick” registers on the chart recorder, another short pulse of light is emitted, and the cycle repeats. Thus, the time interval between successive “ticks” is marked by a beginning event (the firing of the light source) and an ending event (the pulse striking the detector). The source and detector are so close to each other that the two events can be considered to occur at the same location.

Figure 28-4 A light clock.

Suppose two identical clocks are built. One is kept on earth, and the other is placed aboard a spacecraft that travels at a constant velocity relative to the earth. The astronaut is at rest with respect to the clock on the spacecraft and, therefore, sees the light pulse move along the up/down path shown in Figure 28-5a. According to the astronaut, the time interval required for the light to follow this path is the distance 2D divided by the speed of light c; . To the astronaut, is the time interval between the “ticks” of the spacecraft clock—that is, the time interval between the beginning and ending events of the clock. An earth-based observer, however, does not measure as the time interval between these two events. Since the spacecraft is moving, the earth-based observer sees the light pulse follow the diagonal path shown in red in part b of the drawing. This path is longer than the up/down path seen by the astronaut. But light travels at the same speed c for both observers, in accord with the speed of light postulate. Therefore, the earth-based observer measures a time interval between the two events that is greater than the time interval measured by the astronaut. In other words, the earth-based observer, using her own earth-based clock to measure the performance of the astronaut’s clock, finds that the astronaut’s clock runs slowly. This result of special relativity is known as time dilation. (To dilate means to expand, and the time interval is “expanded” relative to .)

Figure 28-5 (a) The astronaut measures the time interval between successive “ticks” of his light clock. (b) An observer on earth watches the astronaut’s clock and sees the light pulse travel a greater distance between “ticks” than it does in part a. Consequently, the earth-based observer measures a time interval between “ticks” that is greater than .

The time interval that the earth-based observer measures in Figure 28-5b can be determined as follows. While the light pulse travels from the source to the detector, the spacecraft moves a distance to the right, where v is the speed of the spacecraft relative to the earth. From the drawing it can be seen that the light pulse travels a total diagonal distance of 2s during the time interval . Applying the Pythagorean theorem, we find that

But the distance 2s is also equal to the speed of light times the time interval , so that . Therefore,

Squaring this result and solving for gives

But , the time interval between successive “ticks” of the spacecraft’s clock as measured by the astronaut. With this substitution, the equation for can be expressed as

| (28.1)

The symbols in this formula are defined as follows:

proper time interval, which is the interval between two events as measured by an observer who is at rest with respect to the events and who views them as occurring at the same place

dilated time interval, which is the interval measured by an observer who is in motion with respect to the events and who views them as occurring at different places

v

relative speed between the two observers

c

speed of light in a vacuum
For a speed v that is less than c, the term in Equation 28.1 is less than 1, and the dilated time interval is greater than . Example 1 shows this time dilation effect.

| Example 1 | | | Time Dilation

The spacecraft in Figure 28-5 is moving past the earth at a constant speed v that is 0.92 times the speed of light. Thus, , which is often written as v = 0.92c. The astronaut measures the time interval between successive “ticks” of the spacecraft clock to be . What is the time interval that an earth observer measures between “ticks” of the astronaut’s clock?
Reasoning Since the clock on the spacecraft is moving relative to the earth observer, the earth observer measures a greater time interval between “ticks” than does the astronaut, who is at rest relative to the clock. The dilated time interval can be determined from the time dilation relation, Equation 28.1.
Solution The dilated time interval is

From the point of view of the earth-based observer, the astronaut is using a clock that is running slowly, because the earth-based observer measures a time between “ticks” that is longer (2.6 s) than what the astronaut measures (1.0 s).

Present-day spacecrafts fly nowhere near as fast as the craft in Example 1. Yet circumstances exist in which time dilation can create appreciable errors if not accounted for. The Global Positioning System (GPS), for instance, uses highly accurate and stable atomic clocks on board each of 24 satellites orbiting the earth at speeds of about 4000 m/s. These clocks make it possible to measure the time it takes for electromagnetic waves to travel from a satellite to a ground-based GPS receiver. From the speed of light and the times measured for signals from three or more of the satellites, it is possible to locate the position of the receiver (see Section 5.5). The stability of the clocks must be better than one part in 1013 to ensure the positional accuracy demanded of the GPS. Using Equation 28.1 and the speed of the GPS satellites, we can calculate the difference between the dilated time interval and the proper time interval as a fraction of the proper time interval and compare the result to the stability of the GPS clocks:

This result is approximately one thousand times greater than the GPS-clock stability of one part in 1013. Thus, if not taken into account, time dilation would cause an error in the measured position of the earth-based GPS receiver roughly equivalent to that caused by a thousand-fold degradation in the stability of the atomic clocks.

The physics of the Global Positioning System and special relativity.

| Proper Time Interval

In Figure 28-5 both the astronaut and the person standing on the earth are measuring the time interval between a beginning event (the firing of the light source) and an ending event (the light pulse striking the detector). For the astronaut, who is at rest with respect to the light clock, the two events occur at the same location. (Remember, we are assuming that the light source and detector are so close together that they are considered to be at the same place.) Being at rest with respect to a clock is the usual or “proper” situation, so the time interval measured by the astronaut is called the proper time interval. In general, the proper time interval between two events is the time interval measured by an observer who is at rest relative to the events and sees them at the same location in space. On the other hand, the earth-based observer does not see the two events occurring at the same location in space, since the spacecraft is in motion. The time interval that the earth-based observer measures is, therefore, not a proper time interval in the sense that we have defined it.
To understand situations involving time dilation, it is essential to distinguish between and . It is helpful if one first identifies the two events that define the time interval. These may be something other than the firing of a light source and the light pulse striking a detector. Then determine the reference frame in which the two events occur at the same place. An observer at rest in this reference frame measures the proper time interval .

| Space Travel

One of the intriguing aspects of time dilation occurs in conjunction with space travel. Since enormous distances are involved, travel to even the closest star outside our solar system would take a long time. However, as the following example shows, the travel time can be considerably less for the passengers than one might guess.

Shown here in orbit is astronaut David A. Wolf as he works on the International Space Station during a session of extravehicular activity.

| Example 2 | | | Space Travel

Alpha Centauri, a nearby star in our galaxy, is 4.3 light-years away. This means that, as measured by a person on earth, it would take light 4.3 years to reach this star. If a rocket leaves for Alpha Centauri and travels at a speed of v = 0.95c relative to the earth, by how much will the passengers have aged, according to their own clock, when they reach their destination? Assume that the earth and Alpha Centauri are stationary with respect to one another.

The physics of space travel and special relativity.

Reasoning The two events in this problem are the departure from earth and the arrival at Alpha Centauri. At departure, earth is just outside the spaceship. Upon arrival at the destination, Alpha Centauri is just outside. Therefore, relative to the passengers, the two events occur at the same place—namely, just outside the spaceship. Thus, the passengers measure the proper time interval on their clock, and it is this interval that we must find. For a person left behind on earth, the events occur at different places, so such a person measures the dilated time interval rather than the proper time interval. To find we note that the time to travel a given distance is inversely proportional to the speed. Since it takes 4.3 years to traverse the distance between earth and Alpha Centauri at the speed of light, it would take even longer at the slower speed of v = 0.95c. Thus, a person on earth measures the dilated time interval to be . This value can be used with the time-dilation equation to find the proper time interval .

Problem solving insight In dealing with time dilation, decide which interval is the proper time interval as follows: (1) Identify the two events that define the interval. (2) Determine the reference frame in which the events occur at the same place; an observer at rest in this frame measures the proper time interval .

Solution Using the time-dilation equation, we find that the proper time interval by which the passengers judge their own aging is

Thus, the people aboard the rocket will have aged by only 1.4 years when they reach Alpha Centauri, and not the 4.5 years an earthbound observer has calculated.

| Verification of Time Dilation

A striking confirmation of time dilation was achieved in 1971 by an experiment carried out by J. C. Hafele and R. E. Keating.* They transported very precise cesium-beam atomic clocks around the world on commercial jets. Since the speed of a jet plane is considerably less than c, the time-dilation effect is extremely small. However, the atomic clocks were accurate to about , so the effect could be measured. The clocks were in the air for 45 hours, and their times were compared to reference atomic clocks kept on earth. The experimental results revealed that, within experimental error, the readings on the clocks on the planes were different from those on earth by an amount that agreed with the prediction of relativity.
The behavior of subatomic particles called muons provides additional confirmation of time dilation. These particles are created high in the atmosphere, at altitudes of about 10 000 m. When at rest, muons exist only for about before disintegrating. With such a short lifetime, these particles could never make it down to the earth’s surface, even traveling at nearly the speed of light. However, a large number ofmuons do reach the earth. The only way they can do so is to live longer because of time dilation, as Example 3 illustrates.

| Example 3 | | | The Lifetime of a Muon

The average lifetime of a muon at rest is . A muon created in the upper atmosphere, thousands of meters above sea level, travels toward the earth at a speed of v = 0.998c. Find, on the average, (a) how long a muon lives according to an observer on earth, and (b) how far the muon travels before disintegrating.
Reasoning The two events of interest are the generation and subsequent disintegration of the muon. When the muon is at rest, these events occur at the same place, so the muon’s average (at rest) lifetime of is a proper time interval . When the muon moves at a speed v = 0.998c relative to the earth, an observer on the earth measures a dilated lifetime that is given by Equation 28.1. The average distance x traveled by a muon, as measured by an earth observer, is equal to the muon’s speed times the dilated time interval.
Solution

a. | The observer on earth measures a dilated lifetime given by

| (28.1)

b. | The distance traveled by the muon before it disintegrates is

Thus, the dilated, or extended, lifetime provides sufficient time for the muon to reach the surface of the earth. If its lifetime were only , a muon would travel only 660 m before disintegrating and could never reach the earth.

Problem solving insight The proper time interval is always shorter than the dilated time interval .

Check Your Understanding 1

One of the following statements concerning the dilated time interval between two events is false. Which is it? (a)The dilated time interval is always greater than the proper time interval. (b)The dilated time interval depends on the relative speed between the observers who measure the proper and dilated time intervals. (c)The dilated time interval depends on the speed of light in a vacuum. (d)The dilated time interval is measured by an observer who is at rest with respect to the events. (The answer is given at the end of the book.)
Background: In the theory of special relativity the time interval between two events can be either a proper or a dilated time interval. Which type of interval it is depends on who measures it—an observer at rest with respect to the events or an observer moving with respect to the events.
For similar questions (including calculational counterparts), consult Self-Assessment Test 28.1, which is described at the end of Section 28.4.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.