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1 Newton's Mechanics And Relativity

# 1 Newton's Mechanics And Relativity

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Newton’s second law is thus obeyed in both of these inertial reference frames. This

result is similar to what we found in our analysis of the airplane accelerometer in

Chapter 4; there again, observers in different inertial reference frames both found

that Newton’s second law is obeyed.

The principle of Galilean relativity is certainly reasonable. It agrees with our

everyday intuition and the experiment in Figure 27.1, and was believed to be an

exact law of physics for more than two hundred years. The fi rst hint of a problem

came from Maxwell’s work in electromagnetism. According to Maxwell’s equations, the speed of light is (Eq. 23.1)

c5

1

(27.2)

"e0m0

Maxwell also showed that this result is independent of the motion of both the source

of the light and the observer. To see why this fi nding is surprising, suppose Ted is

again moving with a speed v relative to Alice when he turns on a flashlight that generates a light pulse traveling to the right in Figure 27.2. This is a one-dimensional

situation, so we need consider only the component of velocity along the horizontal

(x) direction. The light wave in Figure 27.2 has a speed c relative to Ted and his

flashlight. Since Ted’s speed relative to Alice is v, Newton’s mechanics predicts that

the speed of the light wave relative to Alice should be c ϩ v. According to Maxwell’s

theory, however, Ted and Alice should both observe the light wave to move with

speed c. Maxwell’s theory of electromagnetism is thus not consistent with the predictions of Galilean relativity for observers in different inertial frames.

Which prediction is correct? Initially, Maxwell and other physicists thought that

there must be a problem with Maxwell’s theory, but an experimental test is difficult. For example, if Ted in Figure 27.2 is moving at 100 m/s (about 200 mi/h),

Galilean relativity predicts that Alice will observe a speed of

v 1 Alice 2 5 c 1 v 1 Ted 2 5 3 1 3.00 3 108 2 1 100 4 m/s

Ted

Alice

c

y

y

x

v

x

Figure 27.2 Ted moves at speed

v relative to Alice when his flashlight emits a light pulse that moves

at speed c relative to him. What is

the speed of the light pulse relative

to Alice?

(27.3)

This differs from c by less than one part in one million, and such a small difference would be very hard to measure using the technology available in Maxwell’s

time. Successful experiments were carried out only well after Maxwell’s death and

showed that Maxwell’s theory has it right: the speed of light in a vacuum is always

c, and the prediction of Galilean relativity for how the speed of light depends on

the motion of the source is wrong. In 1905, Einstein worked out the correct theory,

which had profound implications for all physics.

2 7. 2

|

T H E P O S T U L AT E S O F S P E C I A L R E L AT I V I T Y

The theory Einstein developed to analyze the predicament of Ted and Alice in Figure 27.2 is called special relativity. According to Einstein, his work on this theory

was not motivated by any particular experiment; indeed, at the time he developed

his theory (1905), there were not yet any clear experimental results to show that

Galilean relativity was wrong. Those experiments came only after Einstein’s theory.

Instead, Einstein suspected that Maxwell’s result—that the speed of light is the

same in all reference frames—is correct, and he then worked out what that implies

for the other laws of physics such as Newton’s laws. In a sense, Einstein’s work was

similar to that of Newton: both proposed basic laws of physics and then worked

out the consequences. For Newton, these basic laws were his three laws of motion.

For Einstein, the basic laws are now known as two “postulates” about the laws

of physics. We’ll fi rst state the postulates and then spend most of the rest of this

chapter working out the consequences. Experiments performed after 1905 showed

that Newton’s theory breaks down for fast-moving objects (such as light) and that

Einstein’s theory gives a correct description of motion in such a regime.

27.2 | THE POSTULATES OF SPECIAL RELATIVITY

919

Observer

S

vboat

Water waves

on a lake

S

vwave

Figure 27.3 According to our

intuition, water waves on a lake

obey Galilean relativity. The velocity of these waves relative to the

S

water (the wave medium) is v wave ,

whereas the velocity of a boat

S

relative to the water is v boat . In this

example, the velocity of the waves

S

S

relative to the boat is v wave 2 v boat .

Postulates of Special Relativity

1. All the laws of physics are the same in all inertial reference frames.

2. The speed of light in a vacuum is a constant, independent of the motion of the

light source and all observers.

The fi rst postulate can be traced to the ideas of Galileo and Newton on relativity, but this postulate goes further than Galileo because it applies to all physical

laws, not just mechanics. The second postulate of special relativity is motivated

by Maxwell’s theory of light, which we have seen is not consistent with Newton’s

mechanics. Newton would have predicted that the speed of the light pulse relative

to Alice in Figure 27.2 is c ϩ v (Eq. 27.3), whereas Ted (who is in a different inertial

reference frame) would measure a speed c. According to the second postulate of

special relativity, both observers will measure the same speed c for light.

The postulates of special relativity will lead us to a new theory of mechanics that

corrects and extends Newton’s laws. Postulate 2 also tells us something very special

and unique about light. Light is a wave, and all the other waves we have studied

travel in a mechanical medium. For example, Figure 27.3 shows an observer travelS

ing in a boat with velocity v boat relative to still water. If there are waves traveling

S

at velocity v wave relative to still water and if the boat is traveling along the wave

propagation direction, Newton’s mechanics (and Galilean relativity) predict that

S

S

the velocity of the waves relative to this observer is v wave Ϫ v boat. An observer who

is stationary relative to the water would thus measure a different wave speed than

the observer in the boat. This is not what the two observers in Figure 27.2 fi nd; they

measure the same speed for the light wave.

The conclusion from Figures 27.2 and 27.3 is that our everyday experience with

conventional waves cannot be applied to light. According to postulate 2 of special

relativity, the speed of a light wave is independent of the velocity of the observer.

What role does the medium have in this case? Light does not depend on having a

conventional material medium in which to travel. For a light wave, the role of the

medium is played by the electric and magnetic fields, so a light wave essentially

carries its medium with it as it propagates (which is why light can travel through

a vacuum). The lack of a conventional medium is surprising and difficult to reconcile with one’s intuition, and Maxwell’s results were therefore slow to be accepted.

Experiments, however, show that nature does work this way; the speed of light is

independent of the motion of the observer. This example is just one of many for

which conventional intuition fails. We’ll come to more such failures as we study

special relativity.

Finding an Inertial Reference Frame

Alternate deﬁnition of an inertial

reference frame

920

CHAPTER 27 | RELATIVITY

Inertial reference frames played a role in our work with Newton’s mechanics, and

they play a crucial role in special relativity. We have stated that an inertial frame is

one that moves with constant velocity, but relative to what? Newton believed that the

heavens and stars formed a “fixed” and absolute reference frame to which all other

reference frames could be compared. We now know that stars are also in motion, so

we need a better definition of what it means to be “inertial.” Nowadays, we define an

inertial reference frame as one in which Newton’s fi rst law holds. Recall that Newton’s first law states that if the total force on a particle is zero, the particle will move

with a constant velocity, in a straight line with constant speed. So, we can test for an

inertial reference frame by observing the motion of a particle for which the total force

is zero. If the particle moves with a constant velocity, the reference frame in question—the reference frame we use to make the observation—is an inertial frame.

This defi nition of how to fi nd an inertial reference frame is thus tied to Newton’s fi rst law and the concept of inertia, so the defi nition may seem a bit circular. However, the notion of Galilean relativity also asserts that Newton’s other

lawsS of mechanics are valid in all inertial frames. Hence, Newton’s second law

S

1 g F 5 ma

2 and third law (the action–reaction principle) should apply in all inertial frames, too.

The Earth as a Reference Frame

Inertial reference frames move with constant velocity; hence, their acceleration is

zero. Since the Earth spins about its axis as it orbits the Sun, all points on the

Earth’s surface have a nonzero acceleration. Strictly speaking, then, a person standing on the surface of the Earth is not in an inertial reference frame. However, the

Earth’s acceleration is small enough that it can be ignored in most cases, so in most

situations we can consider the Earth to be an inertial reference frame.

2 7. 3

|

T I M E D I L AT I O N

Mirror

Einstein’s two postulates seem quite “innocent.” The fi rst postulate—that the laws

of physics must be the same in all inertial reference frames—is in accord with Newton’s laws, so it does not seem that this postulate can lead to anything new for

mechanics. The second postulate concerns the speed of light, and it is not obvious

what it will mean for objects other than light. Einstein, however, showed that these

two postulates together lead to a surprising result concerning the very nature of

time. He did so by considering in a very careful way how time can be measured.

Let’s use the postulates of special relativity to analyze the operation of the simple

clock in Figure 27.4. This clock keeps time using a pulse of light that travels back

and forth between two mirrors. The mirrors are separated by a distance ,, and light

travels between them at speed c. The time required for a light pulse to make one

round trip through the clock is thus 2 ,/c. That is the time required for the clock to

“tick” once.

c

, Round-trip time ϭ 2,

c

Mirror

Figure 27.4 A light clock. Each

round-trip motion of the light

pulse between the two mirrors corresponds to one tick of the clock.

Analysis of a Moving Light Clock

We now place our light clock on Ted’s railroad car in Figure 27.5A, so the clock

is moving at constant velocity with speed v relative to the ground. How does that

affect the operation of the clock? Let’s fi rst analyze the clock from Ted’s viewpoint—that is, in Ted’s reference frame—as we ride along on the railroad car in

Figure 27.5B. In this reference frame, the operation of the clock is identical to that

shown in Figure 27.4; the light pulse simply travels up and down between the two

mirrors. The separation of the mirrors is still ,, so the round-trip time is still 2 ,/c.

Ted

Light clock

v

Round-trip

Ted

time

Ted’s clock measured

by Ted:

2,

c

, Dt0 ϭ c

v

Ted

Ted’s clock

v Dt

2

c

c

z

,

( (

,2 ϩ v Dt

2

z

2

v

v ⌬t

Alice and

her clock

A

B

C

D

Figure 27.5 A A light clock traveling with Ted on his railroad car. B According to Ted, light

pulses travel back and forth in the clock just as in Figure 27.4. Each tick of the clock takes a

time Dt0 5 2 ,/c. According to Ted, the operation of the clock is the same whether or not his

railroad car is moving. C Motion of the light pulses in Ted’s clock as viewed by Alice, who is

at rest on the ground. D According to Alice, the round-trip travel distance for a light pulse is

2z, where z 5 ! ,2 1 1 v Dt/2 2 2, which is longer than the round-trip distance 2, seen by Ted in

part B.

27.3 | TIME DILATION

921

If ⌬t 0 is the time required for the clock to make one “tick” as measured by Ted, we

have the result

Dt0 5

2,

c

(27.4)

A second observer, Alice, is standing on the ground watching Ted’s railroad car

travel by and sees things differently. In her reference frame, Ted’s clock is moving

horizontally, so from her point of view the light pulse does not simply travel up and

down between the mirrors, but must travel a longer distance as shown in Figure

27.5C. According to postulate 2 of special relativity, the speed of light is the same

for Alice as it is for Ted. Because light travels a longer distance in Figure 27.5C than

in Figure 27.5B, according to Alice the light will take longer to travel between the

mirrors. Let’s use ⌬t to denote the round-trip time as observed by Alice; that is the

time it takes for the clock to complete one “tick” in Alice’s reference frame. We can

fi nd ⌬t by using a little geometry. In a time ⌬t (one “tick”), Alice sees the clock move

a horizontal distance v ⌬t. The path of the light pulse forms each hypotenuse z of

the back-to-back right triangles in Figure 27.5D. Using the Pythagorean theorem,

z2 5 ,2 1 a

v Dt 2

b

2

(27.5)

Since z is half the total round-trip distance, Alice fi nds

c Dt

2

z5

or

z2 5

c 2 1 Dt 2 2

4

(27.6)

Combining Equations 27.5 and 27.6 gives

c 2 1 Dt 2 2

v 2 1 Dt 2 2

5 ,2 1

4

4

We next solve for ⌬t:

1 Dt 2 2 5

4 ,2

v2

1 Dt 2 2

1

c2

c2

1 Dt 2 2 a1 2

1 Dt 2 2 5

v2

4 ,2

5

b

c2

c2

4 ,2 /c 2

1 2 v 2 /c 2

Taking the square root of both sides and using Equation 27.4 fi nally leads to

Dt 5

Dt0

"1 2 v 2 /c 2

(27.7)

Recall that ⌬t and ⌬t 0 are the times required for one tick of the light clock as

observed by Alice and Ted, respectively. In words, Equation 27.7 thus says that

these times are different! The operation of this clock depends on the motion of the

observer. Let’s now consider the implications of Equation 27.7 in more detail.

Moving Clocks Run Slow

The clock in Figure 27.5 is at rest relative to Ted, and he measures a time ⌬t 0 for

each tick. The same clock is moving relative to Alice, and according to Equation

27.7 she measures a longer time ⌬t for each tick. This result is not limited to light

clocks. Postulate 1 of special relativity states that all the laws of physics must be the

922

CHAPTER 27 | RELATIVITY

same in all inertial reference frames. We could use a light clock to monitor or time

any process in any reference frame. Since Equation 27.7 holds for light clocks, it

must therefore apply to any type of clock or process, including biological ones.

According to Equation 27.7, the ratio of ⌬t (the time measured by Alice) to the

time ⌬t 0 (measured by Ted) is

Dt

1

5

Dt0

"1 2 v 2 /c 2

(27.8)

Assuming v is less than the speed of light c (discussed further below), the factor on

the right-hand side is always greater than 1. Hence, the ratio ⌬t/⌬t 0 is larger than

1, and Alice measures a longer time than Ted does. In words, a moving clock will,

according to Alice, take longer for each tick. Hence, special relativity predicts that

moving clocks run slow. This effect is called time dilation.

This result seems very strange; our everyday experience does not suggest that

a clock (such as your wristwatch) traveling in a car gives different results than an

identical clock at rest. If Equation 27.8 is true (and experiments defi nitely show that

it is correct), why haven’t you noticed it before now? Figure 27.6 shows a plot of the

ratio Dt/Dt0 5 1/ !1 2 v 2 /c 2 as a function of the speed v of the clock. At ordinary

terrestrial speeds, v is much smaller than the speed of light c and the ratio ⌬t/⌬t 0 is

very close to 1. For example, when v ϭ 50 m/s (about 100 mi/h), the ratio is

Time dilation: moving clocks

run slow.

For common terrestrial speeds,

v/c is very small and D t Ϸ D t0.

8

6

Dt

4

D t0

2

Dt

1

1

5

5

Dt0

"1 2 v 2 /c 2

"1 2 1 50 m/s 2 2 / 1 3.00 3 108 m/s 2 2

Dt

5 1.000000000000014

Dt0

0

0.2

0.4

0.6

0.8

1.0

v

c

(27.9)

The result in Equation 27.9 is extremely close to 1, so for all practical purposes the

times measured by Ted and by Alice are the same if Ted moves at 50 m/s relative to

Alice. For typical terrestrial speeds, the difference between ⌬t and ⌬t 0 is completely

negligible.

Figure 27.6 Time dilation factor ⌬t/⌬t 0 as a function of v/c.

Nature’s Speed Limit

A curious feature of the time-dilation factor in Equation 27.8 is that its value is

imaginary when v is greater than the speed of light. For example, if we insert v ϭ

2c into Equation 27.8, we get

Dt

1

1

5

5

2 2

Dt0

"1 2 1 2c 2 /c

"23

This result is an imaginary number! Does it mean that special relativity predicts

that some time intervals ⌬t can be imaginary numbers? No, it does not. Speeds

greater than c have never been observed in nature. We’ll come back to this issue in

Section 27.9 when we discuss work and energy in special relativity, and we’ll see

why it is not possible for an object to travel faster than the speed of light.

Insight 27.1

RELATIVISTIC CALCULATIONS

WHEN V IS SMALL

The factor !1 2 v 2/c 2 arises often

in special relativity. When v is small

compared with c, this factor is very

close to 1. In fact, the difference

between it and 1 can be so small that

dealing with it. In such cases, the

approximations

"1 2 v 2/c 2 < 1 2

and

E X A M P L E 2 7. 1

Time Dilation for an Astronaut

The astronauts who traveled to the Moon in the Apollo missions hold the record for

the highest speed traveled by people, with v ϭ 11,000 m/s. What is the ratio ⌬t/⌬t 0 for

the Apollo astronauts’ clock?

RECOGNIZE T HE PRINCIPLE

The Apollo astronauts have the role of Ted in Figure 27.5 because we are interested in

a clock that travels with them, while an observer on the Earth has the role of Alice.

The time measured by the astronaut’s clock thus reads the time interval ⌬t 0, and an

observer on the Earth measures a longer time interval ⌬t.

1

<11

v2

2c 2

v2

2c 2

"1 2 v 2/c 2

are very handy and are quite accurate

at terrestrial speeds. In practice, they

can be used whenever v is less than

about 0.1c. (See Figs. 27.6 and 27.15.)

We sometimes also have expressions like 1/ 1 1 1 A 2 , where A is very

small. In such cases, we can use the

approximation

1

<12A

11A

27.3 | TIME DILATION

923

Alice

SK E TCH T HE PROBLEM

This problem is described by Figure 27.7. We assume the astronauts are carrying a

light clock with them to the Moon, just as Ted carried a clock in his railroad car in

Figure 27.5.

Astronauts

v

Moon

IDENT IF Y T HE REL AT IONSHIPS

We can fi nd ⌬t/⌬t 0 using our analysis of Figure 27.5 (and Eq. 27.8), substituting v ϭ

1.1 ϫ 104 m/s and the known speed of light.

Earth

Figure 27.7 Example 27.1.

SOLV E

Inserting the given values, we have

Dt

1

5

Dt0

"1 2 v 2 /c 2

5

1

"1 2 1 1.1 3 10 m/s 2 2 / 1 3.00 3 108 m/s 2 2

4

5 1.00000000067

When v is such a small fraction of the speed of light, you may be limited by the number of significant figures your calculator can display. In such cases, you can use one of

the approximations given in Insight 27.1. The second approximation gives

1 1.1 3 104 m/s 2 2

v2

Dt

<11 2511

< 1 1 6.7 3 10210

Dt0

2c

2 1 3.00 3 108 m/s 2 2

(1)

What does it mean?

Time dilation is a very small effect, even at this (relatively) high speed, yet it is

possible to make clocks that are accurate enough to observe the small amount of

slowing down in Equation (1). Experiments have shown that the time dilation predicted by special relativity is indeed correct. This result for ⌬t applies to all clocks,

including the biological clocks of the Apollo astronauts. Hence, these astronauts

aged slightly less than other people who stayed behind!

Proper Time

Astronaut

vastro ϾϾ v

Bird

v

Ted

Yo-yo

v

Alice

Figure 27.8 Concept Check

27.1.

924

CHAPTER 27 | RELATIVITY

We derived Equation 27.8 from an analysis of a light clock, but the result applies

to all time intervals measured with any type of clock. The time interval ⌬t 0 for a

particular clock or process is measured by an observer at rest relative to the clock

(Ted in Fig. 27.5). The quantity ⌬t 0 is called the proper time. The proper time is

always measured by an observer at rest relative to the clock or process that is being

studied. So, while Ted is moving on his railroad car in Figure 27.5, the clock is moving along with Ted. Hence, Ted is at rest relative to this clock and he measures the

proper time. On the other hand, Alice in Figure 27.5 is moving relative to the clock,

so she does not measure the proper time. The time interval ⌬t measured by a moving observer (Alice) for the same process is always longer than the proper time.

When an observer is at rest relative to a clock or process, the start and end of the

process occur at the same location for this observer. For the light clock in Figure

27.5B, Ted might be standing next to the bottom mirror, so from his viewpoint the

light pulse starts and ends at the same location. By comparison, for Alice in Figure

27.5C, the light pulse begins at the bottom mirror when the clock is at the left; the

pulse returns to this mirror when the clock is in a different location (relative to

Alice), and Alice measures a longer time interval ⌬t. The proper time is always the

shortest possible time that can be measured for a process, by any observer.

CO N C E P T C H E C K 2 7.1 | Measuring Proper Time

Ted is traveling in his railroad car with speed v relative to Alice, who is standing

on the ground nearby (Fig. 27.8). Ted is playing with his yo-yo and uses a clock

on the railroad car to measure the time it takes for the yo-yo to complete one

up-and-down oscillation. The yo-yo is also observed by Alice, by a bird flying

nearby (also with speed v), and by an astronaut who is cruising by at a very high

speed vastro. Which observer measures the proper time for the yo-yo’s period,

(a) Ted, (b) Alice, (c) the bird, or (d) the astronaut?

E X A M P L E 2 7. 2

Time Dilation for a Muon

Subatomic particles called muons are created when cosmic rays collide with atoms in

the Earth’s atmosphere. Muons created in this way have a typical speed v ϭ 0.99c,

very close to that of light. Muons are unstable, with an average lifetime of about t ϭ

2.2 ϫ 10Ϫ6 s before they decay into other particles. That is, physicist 1 at rest relative

to the muon measures this lifetime t. Another (physicist 2) studies the decay of muons

that are moving through the atmosphere with a speed of 0.99c relative to her laboratory (Fig. 27.9). What lifetime would physicist 2 measure?

A clock moving with the muon

measures the proper time for

Physicist 1

Muon

RECOGNIZE T HE PRINCIPLE

The muon acts as a sort of “clock” in which its lifetime corresponds to one “tick.”

Our results for a light clock, including Equation 27.8, apply to this muon “clock”

since the results of special relativity apply to all physical processes. A clock moving along with the muon measures the proper time ⌬t 0, just as Ted in Figure 27.5B

measures the proper time of a clock that travels along with him in his railroad car.

The muon is moving relative to physicist 2, so that physicist is just like Alice in Figure

27.5C. Hence, that physicist measures a longer time ⌬t for the muon’s lifetime.

v

Physicist 2

Clock at rest

on the Earth

SK E TCH T HE PROBLEM

Figure 27.9 shows the problem schematically.

Figure 27.9 Example 27.2.

IDENT IF Y T HE REL AT IONSHIPS

Applying the time dilation result from Equation 27.7, we have

Dt 5

Dt0

"1 2 v 2 /c 2

The lifetime for the muon at rest (i.e., measured by a clock at rest relative to the muon)

is ⌬t 0 ϭ t. We are given v ϭ 0.99c.

SOLV E

The lifetime of the moving muon is

Dt 5

t

"1 2 v /c

2

2

5

t

"1 2 1 0.99c 2 2 /c 2

Dt < 7.1 3 t

What does it mean?

According to physicist 2, the moving muon exists for a much longer time than a

muon at rest. Experiments with muons show that this result is correct: moving

muons do indeed “live” longer before decaying than muons at rest in the laboratory. This is another surprising and counterintuitive result of special relativity.

Example 27.2 describes the effect of time dilation on the lifetime of a muon, but a

similar result applies in other cases, including the lifetime of a person. Consider an

astronaut (Ted) who is on a mission to travel to the nearby star named Sirius1 and

1Sirius

is actually a double star, but that does not affect the mission; Ted gets to visit both stars.

27.3 | TIME DILATION

925

Figure 27.10 The twin paradox. A Astronaut Ted travels to the

distant star Sirius and then returns

while his trip is monitored by his

twin, Alice. B As viewed by Ted

(in his reference frame), Alice and

the Earth take a journey in the

direction opposite to that in part

A. Not to scale!

AS VIEWED

BY TED

AS VIEWED

BY ALICE

Alice

Alice

Ted

Earth

Ted

v ϭ 0.90 c

v ϭ 0.90 c

Sirius

Earth

B

A

then return to the Earth (Fig. 27.10A). Sirius is 8.6 light-years (ly) from the Earth,

which means that light takes 8.6 years to travel from the Earth to Sirius. Ted must

therefore travel in a very fast spaceship to complete his mission before he is too old

to be an astronaut, so NASA has given him a ship that travels at a speed of 0.90c.

Ted’s mission is being tracked by a second astronaut, Alice, who stays behind on

Earth. Suppose Alice is Ted’s twin, and one of her jobs is to monitor Ted’s health

and study how he ages during the trip. According to Alice, the round-trip distance

is 2 ϫ 8.6 ϭ 17.2 ly, so the trip takes a time ⌬t with

Dt 5

17.2 ly

5 19 years

0.90c

(27.10)

Alice also knows about time dilation and realizes that Ted’s body acts as a clock

(just like the muon in Example 27.2). Ted’s body clock measures the proper time ⌬t 0

since it travels along with him. From our work on time dilation, we know that the

proper time is shorter than Alice’s time in Equation 27.10, with (from Eq. 27.7)

Dt0 5 1 Dt 2 "1 2 v 2 /c 2

(27.11)

Inserting v ϭ 0.90c and our value of ⌬t gives

Dt0 5 1 Dt 2 "1 2 v 2 /c 2 5 1 19 years 2 "1 2 1 0.90c 2 2 /c 2 5 8.3 years

which is the time the trip takes according to Ted’s body clock; in other words, Ted

ages only 8.3 years, whereas his twin, Alice, ages by 19 years during the trip. When

they compare notes at the end of the journey, Ted will be younger than his twin!

This result can be understood in simple terms from the basic statement about time

dilation: moving clocks (in this case, Ted himself) run slow.

It is interesting to ask now how Ted views the trip. According to Ted, Alice and

the Earth both travel at a speed 0.90c relative to his spaceship, returning to him

at the same speed at the end of his journey as sketched in Figure 27.10B. Ted then

reasons that he can apply the results for time dilation described above to calculate

that, according to his clock, Alice’s trip will take 19 years while her body clock will

age by the proper time ⌬t 0 ϭ 8.3 years. Ted thus concludes that when they get back

together, Alice will be younger than he is!

This problem is called the twin “paradox”, as it appears that time dilation has

led to contradictory results. Alice and Ted cannot both be right; only one can be

the younger twin at the end of the journey. Alice’s analysis is the correct one: Ted

ages less than she does during the trip. The mistake in Ted’s analysis is that special

relativity applies only to inertial reference frames. Alice stays on the Earth, so she

is always in an inertial reference frame and she can apply the results of special relativity. On the other hand, Ted spends some of his time in an accelerating reference

frame when his spaceship turns around at Sirius to return to the Earth, and during

this time he cannot use special relativity to analyze how Alice ages. That is why

Ted’s conclusion is wrong, and is the resolution of this apparent paradox.

How Do We Know That Time Dilation Really Happens?

Our applications of time dilation to analyze the decay of a muon in Example 27.2

and the aging of two twins in the twin paradox (Fig. 27.10) are good examples of

926

CHAPTER 27 | RELATIVITY

2 7. 4

|

S I M U LTA N E I T Y I S N O T A B S O L U T E

Reprinted with permission of The Aerospace Corporation

special relativity, but they don’t seem very relevant to everyday life. A similar application of time dilation, however, does have important practical applications. The

Global Positioning System (GPS) consists of about 30 satellites that orbit the Earth

twice each day (Fig. 27.11). These satellites send signals to receivers on the Earth,

and the receivers use the signals to “triangulate” their position with an accuracy

of about 10 m. Each satellite contains a very accurate clock, and the GPS receivers

compare their clocks with the time signal from each satellite to do this triangulation. The GPS satellite clocks are moving in orbit, so they run slow by the factor

!1 2 v 2 /c 2 according to Equation 27.11. The GPS satellites move at a speed of

about 4000 m/s, which is much less than the speed of light. Even so, because of time

dilation the GPS clocks run slow by about 7 ϫ 10Ϫ6 s per day. The satellite signals

travel at the speed of light, so the corresponding distance is more than 200 m, which

is much larger than the theoretical accuracy of the GPS system. Only by accounting

for the effect of time dilation on the satellite clocks can the GPS system successfully

determine a position with an uncertainty of only 10 m. (We’ll apply time dilation to

analyze the GPS performance in Problem 70 at the end of the chapter.)

Figure 27.11 The Global Positioning System (GPS). Approximately 30 GPS satellites orbit

the Earth, sending signals to the

Earth. These signals are used by

location with an accuracy of better

than 10 m.

Two events are simultaneous if they occur at the same time. Our everyday experiences and intuition suggest that the notion of simultaneity is “absolute”; that is,

two events are either simultaneous or they are not, for all observers. However, to

determine if two events are simultaneous (or not), involves the measurement of

time, and our studies of time dilation show that different observers do not always

agree on measurements involving clocks and time intervals. So, let’s examine what

special relativity implies for the notion of simultaneity. If two events are judged

simultaneous by one observer, will other observers also fi nd them to be simultaneous? We can answer this question by analyzing the situation in Figure 27.12; Ted is

standing at the middle of his railroad car, moving with a speed v relative to Alice,

when two lightning bolts strike the ends of the car. The lightning bolts leave burn

marks on the ground (points A and B), which indicate the locations of the two ends

of the car when the bolts struck. We now ask, “Did the two lightning bolts strike

simultaneously?”

We fi rst ask this question of Alice, who notices that she is midway between the

burn marks at A and B. Alice also observes that the light pulses from the lightning

bolts reach her at the same time (Fig. 27.12C). Since she is midway between points

A and B and the light pulses reach her at the same time, Alice concludes that the

lightning bolts struck Ted’s railroad car at the same time. As viewed by Alice, the

bolts are simultaneous.

Ted

v

A

B

Burn

mark

Light

from

bolt A

A

c

v

c

B

v

A

B

Light

from

bolt A

Burn

mark

Alice

A

B

C

Figure 27.12 An experiment to study simultaneity. A Two lightning bolts strike at the ends

of Ted’s moving railroad car, leaving burn marks on the ground. B According to Alice, the

lightning bolts are simultaneous. She comes to this conclusion because she is midway between

the two burn marks and C the light pulses from the two bolts reach her at the same time.

27.4 | SIMULTANEITY IS NOT ABSOLUTE

927

Insight 27.2

SOME THINGS ARE NOT

“RELATIVE” IN SPECIAL

RELATIVIT Y.

Observers in different reference

things, such as the length of a meterstick or if two events are simultaneous,

but there are certain things on which

they always agree. For example, two

observers will always agree on the

order of two events that occur at the

same place. Consider two light beams

that are traveling toward Ted (like the

two light pulses in Fig. 29.12B) and

suppose these two beams have different colors, red and blue. Also suppose

Ted has two clocks that are sensitive

to these colors. Each clock stops when

hit by the appropriately colored light

beam. So, one clock stops when the

red light beam arrives at Ted, and the

other clock stops when the blue beam

arrives. Ted can then tell which beam

arrived fi rst by simply comparing the

readings of the two stopped clocks.

Moreover, Alice or any other observer

could also use the two stopped clocks

to tell which beam arrived fi rst and

would see the same clock readings as

Ted. So, all observers will agree on

which light beam arrived fi rst, that is,

on the order of these events.

What does Ted have to say? Ted stands at the middle of his railroad car, so (like

Alice) he is also midway between the places where the bolts strike. Hence, if the

two events are simultaneous as viewed by Ted, the light pulses should reach him

at the same time. Do they? Alice can also answer this question! Although the light

flashes are traveling to Alice, Ted’s railroad car is moving to the right. Alice realizes

that since Ted is moving, the flash emitted from B will reach him before the flash

from point A in Figure 27.12B reaches him. Two observers must always agree on the

order of two events that occur at the same point in space. (See Insight 27.2.) Hence,

Ted will agree that the light pulse from B reaches him fi rst. Ted therefore concludes

that the lightning bolt at B struck before the bolt at A. In Ted’s reference frame, the

two lightning bolts are not simultaneous. The two lightning bolts in Figure 27.12

are therefore simultaneous for one observer (Alice) but not for another observer

(Ted). The question of simultaneity is thus “relative” and can be different in different reference frames.

Time dilation and the relative nature of simultaneity mean that special relativity conflicts with many of our intuitive notions about time. Measurements of time

intervals and judgments about simultaneity depend on the motion of the observer.

That is very different from Newton’s picture, in which “time” is an absolute, objective quantity, the same for all observers.

2 7. 5

|

L E N G T H CO N T R A C T I O N

In the past few sections, we have seen that special relativity forces us to give up the

notion of absolute time. Measurements of time and simultaneity are “relative”; that

is, they can be different for different observers. Time is just one aspect of a reference

frame; reference frames also involve measurements of position and length. What

does relativity have to say about these quantities?

Let’s consider how Ted and Alice might work together to measure a length or

distance. Suppose Alice marks two spots A and B on the ground and measures

these spots to be a distance L 0 apart on the x axis (Fig. 27.13). Ted travels along

the x direction at constant speed v, and as he passes point A he reads his clock. Ted

reads his clock again when he passes point B and calls the difference between the

two readings ⌬t 0. This is a proper time interval because Ted measures the start and

fi nish times at the same location (the center of his railroad car) with the same clock.

From Section 27.3 (and Eq. 27.7), we know that when Alice measures the time it

takes for Ted to travel from A to B with her clock, she will fi nd a value

time measured by Alice 5 Dt 5

Ted and his

clock

v

A

B

L0

x

Dt0

distance measured by Alice 5 L0 5 v Dt

Alice

the distance between points A and

B by using a clock on his railroad

car to measure the time ⌬t 0 it takes

for him to travel from A to B,

together with his known speed v.

928

CHAPTER 27 | RELATIVITY

(27.13)

Ted will calculate the distance from A to B in the same way, but he will use ⌬t 0, the

time interval measured with his clock, so

distance measured by Ted 5 L 5 v Dt0

Figure 27.13 Ted can measure

(27.12)

"1 2 v 2 /c 2

Since Ted is traveling relative to Alice at speed v and points A and B are a distance

L 0 apart, Alice concludes that

L0

v5

Dt

which can be rearranged as

(27.14)

Comparing Equations 27.13 and 27.14, we see that since ⌬t is different from ⌬t 0 due

to time dilation, the lengths measured by Alice and Ted will also be different. Using

Equation 27.11, we get

L 5 v Dt0 5 v 1Dt "1 2 v 2 /c 2 2

L 5 L0"1 2 v 2 /c 2

(27.15)

Figure 27.14 A hypothetical experiment in which

Alice

Ted is at rest and Alice travels on a meterstick with

speed v relative to Ted. Ted fi nds that the moving

meterstick is shorter than the length measured by

Alice. This situation is very similar to Figure 27.13

because the meterstick (whose ends represent the

points A and B) is at rest relative Alice, while it has

speed v relative to Ted.

Ted

A

B

v ϭ Velocity of

Alice relative

Meterstick

to Ted

S

For common terrestrial speeds,

v/c is very small and L Ϸ L0.

1.0

0.8

L 0.6

L0

0.4

0.2

vTed ϭ 0

0

Hence, the length L measured by Ted is shorter than Alice’s length L 0. This effect

is called length contraction.

Proper Length

0.2

0.4

v

c

0.6

0.8

1.0

Figure 27.15 Length contraction factor L/L0 5 !1 2 v 2 /c 2 as a

function of v/c.

Length contraction also plays a role in the situation in Figure 27.14. Points A and

B are now the two ends of a meterstick, and Ted is measuring the length of the

meterstick as it moves past him. The meterstick is at rest relative to Alice, who

might be standing on it, just as she is at rest relative to the marks A and B on the

ground in Figure 27.13; she measures the meterstick in Figure 27.14 to have a length

of exactly L 0 ϭ 1 m. Ted measures a length L (Eq. 27.15), which is shorter than L 0.

We can thus say that moving metersticks are shortened. The quantity L 0 is called

the proper length because it is measured by an observer (Alice) who is at rest relative to the meterstick.

Length contraction is described by the factor

L

5 "1 2 v 2 /c 2

L0

(27.16)

This factor is plotted in Figure 27.15; it is very close to 1 when the speed is small

and approaches zero as v approaches c.

Notice that we denote the proper time by ⌬t 0 and the proper length by L 0, in

both cases using a subscript zero. In both cases, they are measurements made by

an observer who is at rest relative to the “thing” being measured. Proper time is

measured by an observer (Ted in Fig. 27.5B) who is at rest relative to the clock used

for the measurement. Proper length is measured by an observer (Alice in Fig. 27.14)

who is at rest relative to the object whose length is being measured.

Length contraction: moving

metersticks are contracted.

Proper time and proper length

Alice

v ϭ 100 m/s

Gio 01

E X A M P L E 2 7. 3

Length Contraction of a Moving Car

Consider a race car measured by Ted to be 4.0 m long when moving past him at a

speed of 100 m/s (about 200 mi/h). How long is the car when the race is fi nished and

the car is stopped?

Ted

Ted measures the contracted

length ϭ L as the car moves by.

A

RECOGNIZE T HE PRINCIPLE

Alice

This problem is an example of length contraction, with the race car playing the role

of the meterstick in Figure 27.14. The length measured during the race (when v ϭ

100 m/s) by an observer (Ted) watching from trackside is the contracted length L ϭ

4.0 m (Fig. 27.16A). The length of the race car at the end of the race (when it is not

moving) is the proper length L 0; that is the length that would be measured by an

observer at rest relative to the car, such as Ted in Figure 27.16B.

SK E TCH T HE PROBLEM

Figure 27.16 describes the problem.

vϭ0

Gio 01

Ted

Ted measures the proper

length ϭ L0 when the

car is at rest.

B

Figure 27.16 Example 27.3.

27.5 | LENGTH CONTRACTION

929

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