The fi rst postulate can be traced to the ideas of Galileo and Newton on relativ- ity, 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 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 travel- ing in a boat with velocity vSboat relative to still water. If there are waves traveling at velocity vSwave relative to still water and if the boat is traveling along the wave propagation direction, Newton’s mechanics (and Galilean relativity) predict that the velocity of the waves relative to this observer is Svwave Svboat. 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 fi elds, 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 diffi cult to recon- cile 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
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 “fi xed” 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 defi nition of what it means to be “inertial.” Nowadays, we defi ne an inertial reference frame as one in which Newton’s fi rst law holds. Recall that New- ton’s fi rst 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 ques- tion—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 New- ton’s fi rst law and the concept of inertia, so the defi nition may seem a bit circu- lar. However, the notion of Galilean relativity also asserts that Newton’s other laws of mechanics are valid in all inertial frames. Hence, Newton’s second law 1g FS5mSa2 and third law (the action–reaction principle) should apply in all iner- tial frames, too.
Alternate defi nition of an inertial reference frame Alternate defi nition of an inertial
reference frame
Figure 27.3 According to our intuition, water waves on a lake obey Galilean relativity. The veloc- ity of these waves relative to the water (the wave medium) is vSwave, whereas the velocity of a boat relative to the water is vSboat. In this example, the velocity of the waves relative to the boat is vSwave2Svboat.
vboat
S
Observer
Water waves on a lake
vwave
S
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 stand- ing 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
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 New- ton’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.
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 view- point—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.
27.3 | TIME DILATION 921 Figure 27.4 A light clock. Each round-trip motion of the light pulse between the two mirrors cor- responds to one tick of the clock.
2, c Mirror
Mirror
c , Round-trip time
v v
Ted
Light clock
v
c c
A B
Ted
Ted’s clock
Round-trip time measured by Ted:
Dt0
c 2,c
vDt 2
z z
vDt z
C D
Ted Ted’s clock
vt
2 ,
( (2
Alice and her clock
,
,2
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 Dt052,/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 z5!,211v Dt/222, which is longer than the round-trip distance 2, seen by Ted in part B.
If t0 is the time required for the clock to make one “tick” as measured by Ted, we have the result
Dt052,
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,
z25 ,21 av Dt
2 b2 (27.5) Since z is half the total round-trip distance, Alice fi nds
z5 c Dt 2 or
z25 c21Dt22
4 (27.6)
Combining Equations 27.5 and 27.6 gives c21Dt22
4 5 ,21 v21Dt22 4 We next solve for t:
1Dt225 4,2 c2 1v2
c2 1Dt22 1Dt22a12 v2
c2b 54,2 c2 1Dt225 4,2/c2
12v2/c2
Taking the square root of both sides and using Equation 27.4 fi nally leads to Dt5 Dt0
"12v2/c2 (27.7)
Recall that t and t0 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 t0 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
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 t0 (measured by Ted) is
Dt
Dt0 5 1
"12v2/c2 (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/t0 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/Dt051/!12v2/c2 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/t0 is very close to 1. For example, when v 50 m/s (about 100 mi/h), the ratio is
Dt
Dt0 5 1
"12v2/c25 1
"12 150 m/s22/13.003108 m/s22 Dt
Dt0 51.000000000000014 (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 t0 is completely negligible.
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 2c into Equation 27.8, we get
Dt
Dt05 1
"12 12c22/c2 5 1
"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.
Time dilation: moving clocks run slow.
Time dilation: moving clocks run slow.
27.3 | TIME DILATION 923 Figure 27.6 Time dilation fac- tor t/t0 as a function of v/c.
0 2 4 6 8
Dt0
c
0.2 0.4 0.6 0.8 1.0 For common terrestrial speeds, v/c is very small and Dt ⬇ Dt0.
Dt
v
Insight 27.1
RELATIVISTIC CALCULATIONS WHEN V IS SMALL
The factor !12v2/c2 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 your calculator may have trouble dealing with it. In such cases, the approximations
"12v2/c2<12 v2 2c2 and
1
"12v2/c2<11 v2 2c2 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 expres- sions like 1/111A2, where A is very small. In such cases, we can use the approximation
1
11A<12A
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 11,000 m/s. What is the ratio t/t0 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 t0, and an observer on the Earth measures a longer time interval t.
Figure 27.7 Example 27.1.
Moon Earth
v Alice
Astronauts
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.
IDENT IF Y T HE REL AT IONSHIPS
We can fi nd t/t0 using our analysis of Figure 27.5 (and Eq. 27.8), substituting v 1.1 104 m/s and the known speed of light.
SOLV E
Inserting the given values, we have Dt
Dt05 1
"12v2/c2
5 1
"12 11.13104 m/s22/13.003108 m/s22 5 1.00000000067 When v is such a small fraction of the speed of light, you may be limited by the num- ber of signifi cant fi gures your calculator can display. In such cases, you can use one of the approximations given in Insight 27.1. The second approximation gives
Dt
Dt0<11 v2
2c2 511 11.13104 m/s22
213.003108 m/s22< 116.7310210 (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 pre- dicted 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
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 t0 for a particular clock or process is measured by an observer at rest relative to the clock (Ted in Fig. 27.5). The quantity t0 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 mov- ing 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 mov- ing 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 Figure 27.8 Concept Check
27.1.
Alice Ted
Bird Astronaut
vastro v
v
v
Yo-yo
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 fl ying 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?
27.3 | TIME DILATION 925
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 0.99c, very close to that of light. Muons are unstable, with an average lifetime of about t 2.2 106 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 labora- tory (Fig. 27.9). What lifetime would physicist 2 measure?
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 mov- ing along with the muon measures the proper time t0, 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.
SK E TCH T HE PROBLEM
Figure 27.9 shows the problem schematically.
IDENT IF Y T HE REL AT IONSHIPS
Applying the time dilation result from Equation 27.7, we have Dt5 Dt0
"12v2/c2
The lifetime for the muon at rest (i.e., measured by a clock at rest relative to the muon) is t0 t. We are given v 0.99c.
SOLV E
The lifetime of the moving muon is Dt5 t
"12v2/c2 5 t
"12 10.99c22/c2 Dt< 7.13 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 labora- tory. This is another surprising and counterintuitive result of special relativity.
The Twin Paradox
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
Figure 27.9 Example 27.2.
Physicist 1
Physicist 2
Clock at rest on the Earth Muon
v
A clock moving with the muon measures the proper time for the muon’s lifetime.
1Sirius is actually a double star, but that does not affect the mission; Ted gets to visit both stars.