Special Relativity Space and Time - Woodside High School

Space and time are related.

Special Relativity Space and Time

Everyone knows that we move in time. at the rate of 24 hours per day. And everyone knows that we can move through space. at

rates ranging from a snails pace to those of super-sonic aircraft and space shuttles. But relam eh few people know that motion through space is related to motion in time.

The first person to understand the relationship between space and time was Albert Einstein.* Einstein went beyond common sense when he stated in 1905 that in moving through space we also change our rate of proceeding into the future—time itself is altered This view was introduced to the world in his special theory of relativity. This theory describes how time is affected by motion in space at constant velocity, and how mass and energy are related. Ten years later Einstein announced a similar theory, called the general theory of relativity, which encompasses accelerated motion as well. These theories have enormously changed the way scientists view the work-ings of the universe. This book discusses only the special theory and leaves the general theory for follow-up study later in your education.

This chapter will serve merely to acquaint you with the basic ideas of special relativity as they relate to space and time. Chapter 16 will continue with the relationship between mass and energy. These ideas, for the most part, are not common to your everyday experi-ence. As a result, they don't agree with common sense. So please be patient -will: yoursolf At you fiod that you do nu i -lind,,,— ,d-Perhaps your children or grandchildren will find them very much a part of their everyday experience. If so, they should find an under-standing of relativity considerably less difficult.

" The concerns of Albert Einstein (1879-1955) were not limited to physics. As a German

citizen in Nazi Germany he spoke out against Hitler's racial and political policies, which

prompted his resignation from the University of Berlin. He fled Germany in 1933 and

became an American citizen in 1940.

212 Chapter 15 Special Relativity—Space and Time

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15.1 Space-Time

Newton and other investigators before Einstein thought of space as an infinite expanse in which all things exist. We are in space, and we move about in space. It was never clear whether the universe exists in space, or space exists within the universe. Is there space outside the universe? Or is space only within the universe? The same ques-tion could be raised for time. Does the universe exist in time, or does time exist only within the universe? Was there time before the ;Jr:lye:caeca= to be? Will-there Le rime if arid ;A:licit ullivcrac ceases to exist? Einstein's answer to these questions is that both space and time exist only within the universe. There is no time or space "outside."

Einstein reasoned that space and time are two parts of one whole called space-time. To begin to understand this, consider your present knowledge that you are moving through time at the rate of 24 hours per day. This is only half the story. To get the other half, convert your thinking from "moving through time" to "moving through space-time." From the viewpoint of special relativity, you travel through a combination of space and time. You travel through space-time. When you stand still, then all your traveling is through time. When you move a bit, then some of your travel is through spac-e and most of it is still through time. If you were somehow able to travel through space at the speed of light, what changes would

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Figure 15.1 The universe does not exist in a certain part of infinite space, nor does it exist during a certain era in time. It is the other way around: space and time exist within the universe.

Figure 15.2 A When you stand still, you are traveling at the maximum rate in time: 24 hours per day. If you traveled at the maximum rate through space (the speed of light), time would stand still.

Figure 15.3 A The bag of groceries has an appreciable speed in the frame of reference of the building, but in the frame of reference ot the freely falling elevator it has no speed at all.

you experience in time? The answer is that all your traveling would be through space, with no travel through time! You would be as age-less as light, for light travels through space only (not time) and is timeless. From the frame of reference of a photon traveling from one part of the universe to another, the journey takes no time at all!

Motion in space affects motion in time. Whenever we move through space, we to some degree alter our rate of moving into the future. This is time dilation, a stretching of time that occurs ever so slightly for everyday speeds, but significantly for speeds approaching the speed of light. If spacecraft of the future reach sufficient speed, people will be able to travel noticeably in time. They will be able to jump centuries ahead, just as today people can jump from Earth to the moon. To understand time dilation and how this can be, you first need to understand several ideas: the relativity of motion and the fundamental assumptions (postulates) of special relativity

15.2 Motion Is Relative

Recall from Chapter 2 that whenever we discuss motion, we must specify the position from which the motion is being observed and measured. For example, you may walk along the aisle of a moving bus at a speed of 1 km/h relative to your seat, but at 100 km/h rela-tive to the road outside. Speed is a relative quantity. Its value depends upon the place—the frame of reference—where it is observed and measured. An object may have different speeds rela-tive to different frames of reference.

Suppose your friend always pitches a baseball at the same speed of 60 km/h. Neglecting air resistance and other small effects, the ball is moving at 60 km/h when you catch it. Now suppose your friend pitches the ball to you from the flatbed of a truck that moves toward you at 40 km/h. How fast does the ball meet you? You'll have to be sure to wear a catcher's mitt, because the speed of the ball will be 100 km/h (the 60 km/h relative to the truck plus the 40 km/h relative to the ground). Speed is relative.

Figure 15.4 A

Your speed is 1 km/h relative to your seat, and 100 km/h relative to the road.

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Suppose the truck moves away from you at 4(1 km/h and your Mend again pitches the ball to you. This time you will need no glove at all, for the ball will reach you at a speed nt 3) km :h (since 60 km/h minus 40 km/h is 20 km/h,. This is not surprising, for you expect that the ball will be traveling faster when the truck approaches you and that the ball will be traveling more slowly when the truck recedes.

The idea that speed is a relative quantity goes back to Galileo and was known lOng before the time of Einstein. As you will learn in this chapter, Einstein expanded the relativity of speed to include the relativity of things that seem unchangeable.

15.3 The Speed of Light Is Constant

Suppose you actually caught baseballs thrown off a moving truck out in a parking lot and found that no matter what the speed or direc-tion Lilo 'truck, the Lmill 1/4,1,3 - - 60 km/h. That is to say, if the truck zoomed toward you at 50 km/h and your friend pitched the ball at his speed of 60 km/h, you would catch the ball with the same speed as if the truck were not moving at all. Furthermore, if the truck moved away from you at whatever speed, the ball would still get to you at 60 km/h. This all seems quite impossible, for it is contrary to common experience. And if you did experience this, you would have to reevaluate your whole notion of reality To put it mildly, you would be quite confused.

Baseballs do not behave this way. But it turns out that light does! Every measurement of the speed of light in empty space gives the

4 Figure 15.5 The ball is always pitched at 60 km/h relative to the truck. (a) When both you and the truck are at relative rest the ball is traveling at 60 km/h when you catch it. (b) When the truck moves toward you at 40 km/h, the ball is traveling at 100 km/h when you catch it. (c) When the truck moves away from you at the same speed, the ball is travel-ing at 20 km/h when you catch it.

215

Figure 15.6 • The speed of light is found to be the same in all frames of reference.

same value of 300 000 km/s, regardless of the speed of the smiitthe speed of the receiver.*We do not ordinarily notice this beea light travels so incredibly fast.

The fact that light has only one speed in empty space was covered at the end of the last century** Light from an approac source reaches an observer at the same speed as light from a rec ing source. And the speed of light is the same whether we move toward or away from a light source. How did the physics comm regard this finding? They were as perplexed as you would be if yo caught baseballs at only one speed no matter how they were thr Experiments were done and redone, and always the results were same. Nothing could vary the speed of light. Various interpretation were proposed, but none were satisfactory. The foundations of-physics were on shaky ground.

Albert Einstein looked at the speed of light in terms of the de non of speed. What is speed? It is the amount of space traveled co

pared to the time of travel. Einstein recognized that the classical ideas of space and time were suspect. He concluded that space and time were a part of a single entity—space-time. The constancy of speed of light, Einstein reasoned, unifies space and time.

The special theory of relativity that Einstein developed rests on two fundamental assumptions, or postulates.

15.4 The First Postulate of Special Relativity

Einstein reasoned that there is no stationary hitching post in the universe relative to which motion should be measured. Instead, all motion is relative and all frames of reference are arbitrary A space-ship cannot measure its speed relative to empty space, but only relative to other objects. If, for example, spaceship A drifts past spaceship B in empty space, spaceman A and spacewoman B will each observe only the relative motion. From this observation each, will be unable to determine who is moving and who is at rest, if either.

* The presently accepted value for the speed of light is 299 792 km/s, which we round ot

to 300 000 km/s. This corresponds to 186 000 mi/s.

** In 1887 two American physicists, A. A. Michelson and E. W. Morley, performed an experiment to determine differences in the speed of light in different directions. They thought that the motion of Earth in its orbit about the sun would cause shifts in the speed of light. They thought that the speed of fight should have been faster when it was

going in the direction Earth was moving and slower when it was going opposite to the direction Earth was moving. Using a device called an interferometer, they found that the

speed seemed to be the same in all directions. For Michelson's many experiments on the speed of light, he was the first American honored with a Nobel Prize.


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This is a familiar experience to a passenger in a car at rest wait-ing for the traffic light to change. If you look out the window and see the car in the next lane begin moving backward, you may be sur-prised to find that the car you're observing is really at rest—your car is moving forward. If you could not see out the windows, there would be no way to determine whether your car was moving with constant velocity or was at rest.

In a high-speed jetliner we flip a coin and catch it just as we would if the plane were at rest. Coffee pours from the flight atten-dam's coffee pot as it does when the plane is standing on the run-wa:, If We swing a pendulum, it will move no differently when the plane is moving uniformly (constant velocity) than when not moving a! all. There is no physical experiment we can perform to determine our state of uniform motion. Of course, we can look outside and see Earth whizzing by, or send a radar signal out. However, no experi-ment confined within the cabin itself can determine whether or not :ten- is uniform motion_ The laws of physics within the uniformly 71: alrig cabin are the same as those in a stationary laboratory

These examples illustrate one of the two building blocks of spe-cial relativity It is Einstein's first postulate of special relativity:

All the laws of nature are the same in all uniformly moving frames of reference.

Any number of experiments can be devised to detect accelerated motion, but none can be devised, according to Einstein,ro detect the state of uniform motion.

15.5 The Second Postulate of Special Relativity

One of the questions that Einstein as a youth asked himself was, "What would a light beam look like if you traveled along beside it?" According to classical physics, the beam would be at rest to such an observer. The more Einstein thought about this, the more

Figure 152 A A person playing pool on a smooth and fast-moving ocean liner does not have to make adjustments to compensate for the speed of the ship. The laws of physics are the same for the ship whether it is moving uni-formly or is at rest.

217

Figure 15.9 110. The speed of a light flash emit-ted by either the spaceship Of

the space station is measured as c by observers on the ship or the space station. Everyone who measures the speed of light will get the same value c.

SPACE SPACE _ cTIME — TIME -

Figure 15.10 A All space and time measure-ments of light are unified by c.

218 Chapter 15

convinced he became of its impossibility He came to the concha that (fan observer could travel close to the speed of light, he wo measure the light as moving away from him at 300 000 km/s.

This is the idea that makes up Einstein's second postulate of special relativity:

The speed of light in empty space will always have the same value regardless of the motion of the source or the motion of the observer.

The speed of light in all reference frames is always the same. - Consider, for example, a spaceship departing from the space static shown in Figure 15.9. A flash of light is emitted from the station at 300 000 km/s—a speed we'll simply call c. No matter what the spee of the spaceship relative to the space station is, an observer on the spaceship will measure the speed of the flash of light passing her a c. If she sends a flash of her own to the space station, observers o the station will measure the speed of these flashes as c. The speed o the flashes will be no different if the spaceship stops or turns aroun and approaches. All observers who measure the speed of light will find it has the same value, c.

The constancy of the speed of light is what unifies space and time. And for any observation of motion through space, there is a corresponding passage of time. The ratio of space to time for light is the same for all who measure it. The speed of light is a constant.

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Pretend you are in a spaceship at rest in a part of your "hometown" where a large public clock is displayed. Suppose the clock reads "12 2:-.Lmi." Ti say it reads "12 noon" is to say that light reflects from the clock and carries the information "12 noon" toward you in the direc-tion of your line of sight. If you suddenly move your head to the side, instead of meeting your eye, the light carrying the information will continue past, presumably out into space. Out there an observer who later receives the light could say, "Oh, it's 12 noon on Earth now." But from your point of view it isn't. You and the distant observer will see 12 noon at different times. Now suppose your spaceship is moving as fast as the speed of light (just pretending) Then you'd keep up with the clock's information that says "12 noon." Traveling at the speed of light, then, tells you its always 12 noon back home. Time at home is

Special Relativity—Space and Time

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frozen! So if your spaceship is not moving, you will see the home-tovvn clock move into the future at the rate of 60 seconds per minute; if you could move at the speed of light, you'd see seconds on the

• clock taking infinite time. These are the two extremes. What's in between? How would the time readings appear to you moving at less than the speed of light? A little thought will show that the clock will be seen to run somewhere between the rate 60 seconds per minute and 60 seconds per an infinity of time if your speed is between zero

• and the speed of light. From your high-speed (but less than c) mov-ing frame of reference, the clock and all events in the reference frame of the clock will be seen in slow motion. Time will be stretched. How much depends on speed. This is time dilation.

Special relativity turns around some of our conceptions about the world. We agree that speed is relative, that it depends on the speeds of the source and the observer. Yet, one speed, the speed of light, is absolute—independent of the speeds of the source or observer. Time, on the other hand, is usually thought of as absolute. It seems to pass at the same rate regardless of what is happening. Yet, our imaginary spaceship experience shows this is not true. Einstein proposed that time depends on the motion between the observer and the events being observed.

We measure time with a clock. A clock can be any device that measures periodic intervals, such as the swings of a pendulum, the oscillations of a balance wheel, or the vibrations of a quartz crystal. We are going to consider a "light clock," a rather impractical device. but one that will help to describe time dilation.

Imagine an empty tube with a mirror at each end (Figure 15 1 A flash of light bounces back and forth between the parallel mirrtxs The mirrors are perfect reflectors, so the flash bounces indefunteh• If the tube is 300 000 km in length, each bounce will take 1 s in the frame of reference of the light clock. If the tube is 3 km long, each bounce will take 0.00001 s.

Suppose we view the light clock as it whizzes past us in a high-speed spaceship (Figure 15.12). We see the light flash bouncing up and down along a longer diagonal path.

Figure 15.12 A (a) An observer moving with the spaceship observes the light flash moving

vertically between the mirrors of the light clock. (b) An observer who is Passed by the moving ship observes the flash moving along a diagonal path.

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Figure 15.11 • A statIonary light clock. Light bounces between parallel mir-r -..rs and -non off" equal inter-vals a t ;roe

219

Figure 15.13 The longer longer distance taken by the light flash in following the diag-onal path must be divided by a correspondingly longer time interval to yield an unvarying value for the speed of light.

But remember the second postulate of relativity: The speed will be measured by any observer as c. Since the speed of light will not increase, we must measure more time between bounces! For us, looking in from the outside, one tick of the light clock takes longer than it takes for occupants of the spaceship. The spaceship's clock, according to our observations, has slowed down—although, for occupants of the spaceship, it has not slowed at all!

DISTANCE _ DISTANCE =cTIME — TIME

The slowing of time is not peculiar to the light clock. It is time itself in the moving frame of reference, as viewed from our frame of reference, that slows. The heartbeats of the spaceship occupants will have a slower rhythm. All events on the moving ship will be observed by us as slower. We say that time is "dilated."

How do the occupants on the spaceship view their own time? Do they perceive themselves moving in slow motion? Do they experi-ence longer lives as a result of time dilation? As it turns out, they notice none of these things. Time for them is the same as when they do not appear to us to be moving at all. Recall Einstein's first postu-late: All laws of nature are the same in all uniformly moving frames of reference. There is no way the spaceship occupants can tell uni-form motion from rest. They have no clues that events on board are seen to be dilated when viewed from other frames of reference.

How do occupants on the spaceship view our time? Do they see our time as speeded up? The answer is no—motion is relative, and

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Figure 15.14 shows three successive positions of the light clock as it moves to the right at constant speed v. The

diagonal lines represent the path of the light flash as it starts from the lower minor at position 1, moves to the upper mirror at position 2, and then back to the lower mirror at position 3.

The symbol to represents the time it takes for the flash to move between the mirrors as measured from a frame of refer-ence fixed to the light clock. This is the time for straight up or down motion. Since the speed of light is always c, the light flash is observed to move a vertical distance cto in the frame of refer-ence of the light clock. This is the distance between mirrors and is at right angles to the horizontal motion of the light clock. This vertical distance is the same in both reference frames.

The symbol t represents the time it takes the flash to move from one mirror to the other as measured from a frame of refer-ence in which the light clock moves to the right with speed v. Since the speed of the flash is c and the time to go from position 1 to position 2 is t, the diagonal distance traveled is ct. During this time t, the clock (which travels horizontally at speed v) moves a horizontal distance vt from position 1 to position 2.

These three distances make up a right triangle in the figure. in which ct is the hypotenuse, and ct, and vt are legs. A well-known theorem of geometry (the Pythagorean theorem) states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. If we apply this to the figure, we obtain

(ct)2 = (ct)2 + (vt)2

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The mathematical derivation of this equation for time dilation is included here mainly to show that it involves only a bit of geometry and elementary algebra. It is not expected that you master RI (If you take a follow-up physics course, you can master it then.)

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Figure 15.15 • The graph shows how 1 second on a stationary clock is stretched out, as measured on a moving clock. Note that the stretching tz:H.T.e,mer: speeds near the speed of light.

• Questions

1. Does time dilation mean that time really passes more slowly in moving systems or that it only seems to pass more

2. If you were moving in a spaceship at a high speed relative to Earth, would you notice a difference in your pulse rate? In the pulse rate of the people back on Earth?

3. Will observers A and 8 agree on measurements of time if A moves at half the speed of light relative to B? If both A and B move together at 0.5c relative to Earth?

from their frame of reference it appears that we are the ones who are moving. They see our time running slow, just as we see their time running slow. Is there a contradiction here? Not at all. It is physically impossible for observers in different frames of reference to refer to one and the same realm of space-time. The measurements in one frame of reference need not agree with the measurements made in another reference frame. There is only one measurement they will always agree on: the speed of light.

15.7 The Twin Trip

A dramatic illustration of time dilation is afforded by identical twins, one an astronaut who takes a high-speed round-trip journey while the other stays home on Earth. When the traveling twin returns, he is younger than the stay-at-home twin. How much younger depends on the relative speeds involved. If the traveling twin maintains a speed of 50% the speed of light for one year (according to clocks aboard the spaceship), 1.15 years will have elapsed on Earth. If the traveling twin maintains a speed of 87% the speed of light for a year, then 2 years will have elapsed on Earth. At 99.5% the speed of light,

NI Answers

1. The slowing of time in moving systems is not nterely an illusion resulting from motion

Time really does pass more slowly in a moving system compared with one at relative rest, as we shall see in the next section. Read on!

2. There would be no relative speed between you and your own pulse, so no relativistic effects would be noticed. There would be a relativistic effect between you and people back on Earth. You would find their pulse rate slower than normal land they would find your pulse rate slower than normal). Relativity effects are always attributed to "the other guy."

3. When A and B have different motions relative to each other, each will observe a slow-ing of time in the frame of reference of the other. So they will not agree on measure-

ments of time. When they are moving in unison, they share the same frame of reference and will agree on measurements of time. They will see each other's time as

passing normally, and each one will see events on Earth in the same slow motion.

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10 Earth years would pass in one spaceship year. At this speed the traveling twin would age a single year while the stay-at-home twin ages 10 years.

The question arises, since motion is relative, why isn't it just as well the other way around—why wouldn't the traveling twin return to find his stay-at-home twin younger than himself? We will show that from the frames of reference of both the Earthbound twin and the traveling twin, it is the Earthbound twin who ages more.

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First, consider a spaceship hovering at rest relative to a distant planet. Suppose the spaceship sends regularly spaced brief flashes of light to the planet (Figure 15.17). Some time will elapse before the !lashes get to the planet, just as 8 minutes elapses before sunlight gets to Earth. The light flashes will encounter the receiver on the

Al Figure 15.16 The traveling twin does not age as fast as the stay-at-home twin.

2 Develop

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Figure 15.18 le. When the sender moves towaya the receiver, the flashes are seen more frequently.

planet at speed c. Since there is no relative motion between the sender and receiver, successive flashes will be received as frequently as they are sent. For example, if a flash is sent from the ship every 6 minutes, then after some initial delay, the receiver will receive a flash every 6 minutes. With no motion involved, there is nothing unusual about this.

When motion is involved, the situation is quite different. It is important to note that the speed of the flashes will still be c, no matte how the ship or receiver may move. How frequently the flashes are seen, however, very much depends on the relative motion involved. When the ship travels toward the receiver, the receiver sees the flashes more often—that is, more frequently. This happens not only because time is altered due to motion, but mainly because each suc-ceeding flash has less distance to travel as the ship gets closer to the receiver. If the spaceship emits a flash every 6 minutes, the flashes will be seen at intervals of less than 6 minutes. Suppose the ship is traveling fast enough for the flashes to be seen twice as frequently. Then they are seen at intervals of 3 minutes. Note in Figure 15.18 that the flashes for approach are closer together and equally spaced.

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If the ship recedes from the receiver at the same speed and still emits flashes at 6-mM intervals, these flashes will be seen half as fre-quently by the receiver, that is, at 12-min intervals (Figure 15.19). This is mainly because each succeeding flash has a longer distance to travel as the ship gets farther away from the receiver.

The effect of moving away is just the opposite of moving closer to the receiver. So if the flashes are received twice as frequently when the spaceship is approaching (6-min flash intervals are seen every 3 mm ). they are rprpiwpd hJfa crecprrIti.;,.±.— It is receding (6-ralli

flash intervals are seen every 12 min).*

* The frequencies for approach and for recession are reciprocals of each other. That's, flashes that are seen 2 times as frequently for approach are seen f as frequently for recession. If seen 3 times as frequently for approach, the flashes are seen as fre-quently for recession, and so on for higher speeds. This reciprocal relationship does not hold for waves that require a medium. In the case of sound waves, for example, a speed that results in a doubling of emitting frequency for approach produces the emitting frequency for recession.

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The light flashes make up a light clock, or timer. In the frame of reference of the receiver, something like taking a bath or cooking pancakes that takes 6 min in the spaceship, is seen to take 12 min when the spaceship recedes, and only 3 mm when the ship is approaching.

• Questions

1. Here's a simple arithmetic question: If a spaceship travels for one hour and emits a flash every 6 m m , how many flashes will be emitted?

2. A spaceship sends equally spaced flashes while approaching a receiver at constant speed. Will the flashes be equally spaced when they encounter the receiver?

3. A spaceship emits flashes every 6 min for one hour. If the receiver sees these flashes at 3-min intervals, how much time will occur between the first and the last flash (in the frame of reference of the receiver)?

Let's apply this doubling and halving of flash intervals to the twins. Suppose the traveling twin recedes from the Earthbound twin at the same high speed for one hour, and then quickly turns around and returns in one hour. The traveling twin takes a round trip of two hoorq, according to all clocks aboard the spaceship. This trip will not be seen to take two hours from Earths frame oi reference, however. We can see this with the help of the flashes from the ship's light clock.

• Answers

1. Ten flashes, since (60 min)/(6 min/flash) = 10 flashes.

2. Yes, as long as the ship moves at constant speed, the equally spaced flashes will be

seen equally spaced but more frequently. (If the ship accelerated while sending flashes,

then they would not be seen at equally spaced intervals.)

3. All 10 flashes will be seen in 30 min 1(10 flashes) x (3 min/flash) = 30 minl.

Figure 15.19 When the sender moves away from the receiver, the flashes are spaced farther apart and are seen less frequently.

225

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Figure 15.20 • The spaceship emits flashes each 6 minutes during a two-hour trip. During the first hour, it recedes from Earth. During the second hour, it approaches Earth.

As the ship recedes from Earth, it emits a flash of light every 6 min. These flashes are received on Earth every 12 min During the hour of going away from Earth, a total of 10 flashes are emitted. If the ship departs from Earth at noon, clocks aboard the ship will read 1 p.m. when the tenth flash is emitted. What time will it be on r-t-tta this mnth. finch rnarbac Ftrtt-t? The mcnier is 7 p.m why? Because the time it takes Earth to receive 10 flashes at 12-min intervals is (10 flashes) x (12 min/flash), or 120 min (.= 2 h).

Suppose the spaceship is somehow able to suddenly turn around in a negligibly short time and return at the same high speed. During the hour of return it emits 10 more flashes at 6-mM intervals. These flashes are received every 3 min on Earth, so all 10 come in 30 mm . A clock on Earth will read 2:30 p.m. when the spaceship com-pletes its two-hour trip. We see that the Earthbound twin has aged a half hour more than the twin aboard the spaceship!


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EARTH FRAME OF REFERENCE: 10 FLASHES @ 12 MIN = 120 MIN 10 FLASHES @ 3 MIN = 30 MIN

150 MIN

212 HOURS

SPACESHIP FRAME OF REFERENCE:

_ _ - 20 FLASHES® 6 MIN = 120 MINI

2 HOURS

The result is the same from either frame of reference. Consider the same trip again, only this time with flashes emitted from Earth at regularly spaced 6-min intervals in Earth time. From the frame of reference of the receding spaceship, these flashes are received at 12-min intervals (Figure 15.22a). This means that 5 flashes are seen by the spaceship during the hour of receding from Earth. During the spaceship's hour of approaching, the light flashes are seen at 3-min intervals (Figure 15.22b), so 20 flashes will be seen.

A

N ‘ ‘

\ \ ‘

e '. •- - SENDIS FLASH EVERY t / `, 6 MINUTES 1 i

f 24 ACI

i / i

/ / i

/ I

' SEES FLASH EVERY , 12 MINUTES

,

SENDS FLASH EVERY t\ 6 MINUTES

4( ;I -I* , day ,/

SEES FLASH EVERY 3 MINUTES /

1 Figure 15.21 The trip that takes 2 hours in the frame of reference of the space-ship takes 2.5 hours in Earth's frame of reference.

Figure 15.22 Flashes sent from Earth at 6-min intervals are seen at 12-min inter-vals by the ship when it recedes, and at 3-min intervals when it approaches.

227

cks ereicar4 rip taroundEarth

't 1 A4'4W4 .'7-4-11-..*‘'a "- 1:0:j9; ,,, planesUpon ‘ land -•.. • ingctne trayding CHM= were a few billionthaof .,•• t -- ' a second;yOUnger" than twin clocks that stayed behind. Atomic docks now cruise overhead at even greater speeds in the satellites that are part of the global positioning sys-tem (GPS). In designing this system, which can

• pinpoint positions on Earth to within meters, scientists and engineers

. had to accommodate for relativistic time dilation. If they didn't, GPS could not precisely locate positions on Earth. Time dilation is a fact of everyday life to scientists and engineers—especially those who design equipment for global navigation work.

EARTH FRAME OF REFERENCE • 25 FLASHES @ 6 MIN = 150 MI

HOURti at a

i I

,, SPACESHIP FRAME OF REFERENCE ' 5 FLASHES @ 12 MIN = 60 mini

20 FLASHES @ 3 MIN = 60 MIN 120 MIN

2. HOURS

Figure 15.23 • A time interval of 2.5 hours on Earth is seen to be 2 hours in the spaceship frame of reference

Hence. the' 'pact -ship receives a total of 25 flashes during its two-hour tnp .44-cording to clocks on Earth, however, the time it took to emit tne 25 flashes at 6-min intervals was (25 flashes) x (6 mini flash ; . or ISO min (= 2.5 h). This is shown in Figure 15.23.

So both twin1/4 agree on the same results, with no dispute as to who age" Trent- than the other. While the stay-at-home twin rem in a single reference frame, the traveling twin has experienced two different frames of reference, separated by the acceleration of the spaceship in turning around. The spaceship has in effect experi-enced two different realms of time, while Earth has experienced a still different but single realm of time. The twins can meet again at the same place in space only at the expense of time.

15.8 Space and Time Travel

Before the theory of special relativity was introduced, it was argued that humans would never be able to venture to the stars. It was thought that our life span is too short to cover such great distances—at least for ihe distant stars. Alpha Centauri is the nearest star to Earth, after the sun, and it is 4 light-years away.* It was therefore thought that a round-trip even at the speed of light would require 8 years. The center of our galaxy is some 30 000 light-years away, so it was reasoned that a person traveling even at the speed of light would have to survive for 30 000 years to make such a voyage! But these arguments fail to take into account time dilation. Time for a person on Earth and time for a person in a high-speed spaceship are not the same.

228 Chapter 15

* A light-year is the distance that light travels in one year (9.46 x1012 km).

Special Relativity—Space and Time

A person's heart beats to the rhythm of the realm of time it is in. One realm of lime seems the same as any other realm of time to the person, but not to an observer who is located outside the person's frame of reference—for she sees the difference. As an example, astro-nauts traveling at 99% the speed of light could go to the star Procyon (11.4 fight-years distant) and back in 23.0 years in Earth time. It would take light itself 22.8 years in Earth time to make the same round trip. Because of time dilation, it would seem that only 3 years had gone by for the astronauts. All their clocks would indicate this, and biologically they would be only 3 years older. It would be the space officials greet-ing them on their return who would be 23 years older.

At higher speeds the results are even more impressive. At a speed of 99.99% the speed of light, travelers could travel slightly more than 70 light-years in a single year of their own time. At 99.999% the speed of light, this distance would be pushed appreciably farther than 200 years. A 5-year trip for them would take them farther than light travels in 1000 Earth-time years.

Such journeys seem impossible to us today. The amounts of energy involved to propel spaceships to such relativistic speeds are billions of times the energy used to put the space shuttles into orbit. The problems of shielding radiation induced by these high speeds seems formidable. The practicalities of such space journeys are pro-hibitive, so fat

If and when these problems are overcome and space travel becomes routine, people might have the option of taking a trip and returning in future centuries of their choosing. For example, one might depart from Earth in a high-speed ship in the year 2150, travel for 5 years or so, and return in the year 2500. One might live among Earthlings of that period for a while and depart again to try out the year 3000 for style. People could keep jumping into the future with some expense of their own time, but they could not trip into the past. They could never return to the same era on Earth that they bid farewell to.

Time, as we know it, travels only one way—forward. Here on Earth we constantly move into the future at the steady rate of 24 hours per day. An astronaut leaving on a deep-space voyage must live with the fact that, upon her return, much more time will have elapsed on Earth than she has experienced on her voyage. Star travelers will not bid "so long, see you later" to those they leave behind but, rather, a permanent "good-bye."

We ran CPP into the nact but IMP Cannot ?(-) into the pact. When we look at stars in the night skies, the starlight we see left those stars dozens, hundreds, even millions of years ago. What we see is the stars as they were long ago. We are thus eyewitnesses to ancient his-tory We can only speculate about what may have happened to the stars since then.

When we think of time and the universe, we may wonder what went on before the universe began. We wonder what will happen if the universe ceases to exist in time. But the concept of time applies to events within the universe. Time is "in" the universe; the universe Is not "in" time. Without the universe, there is no time; no before, no after. Likewise, space is "in" the universe; the universe is not "in" a

, region of space. There is no space "outside" the universe. We see that space-time exists within the universe. Think about that!

Figure 15.24 A From Earth's frame of reference, light takes 30 000 years to travel from the center of the Milky Way galaxy to our solar system. Fromthe frame of reference of a high-speed spaceship, the trip takes less time. From the frame of ref-erence of light itself, the trip takes no time. There is no time in a speed-of-light frame of reference.

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Chapter Assessment

EGo Online ."‘ NISchools=

For: Study and Review Visit: PHSchool.com Web Code: csd -1150

Coiteepi According to Einstein's special theory of relativity time is affected by motion in space at constant velocity.

• Time appears to pass more slowly in a frame of reference that is moving relative to the observer.

All the laws of nature are the same in all uni-formly moving frames of reference.

• No experiment can be devised to detect whether the observer is moving at constant velocity

The speed of light in empty space is the same in all frames of reference.

• The speed of light has the same value regard-less of the motion of the source or the motion of the observer.

Key Terms first postulate of special relativity (15.4)

postulate (15.3)

second postulate of special relativity (15.5)

space-time (15.1)

special theory of relativity (15.0)

time dilation (15.1)

Review Questions Check Concepts

1. What is space-time? (15.1)

2. Can you travel while remaining in one place in space? Explain. (15.1)

230 Chapter 15 Special Relativity-Space and Time

3. Does light travel through space? Through time? Through both space and time? (15.1)

4. What is time dilation? (15.1)

5. What does it mean to say that motion is relative? (15.2)

6. The speed of a ball you catch that is thrown from a is-stAiLlg Ltuati,perids un tif.C. Speed and direction of the truck. Does the speed of light caught from a moving source similarly depend on the speed and direction of the source? Explain. (15.3)

7. What does it mean to say that the speed of light is a constant? (15.3)

8. What is the first postulate of special relativity? (15.4)

9. What is the second postulate of special relativity? (15.5)

10. The ratio of velocity gain to time for a freely falling body is g. Similarly, what is the ratio of distance to time for light waves? (15.5)

11. The path of light in a vertical "light clock" in a high-speed spaceship is seen to be longer when viewed from a stationary frame of refer-ence. Why, then, does the light not appear to be moving faster? (15.6)

12. If we view a passing spaceship and see that the inhabitants' time is running slow, how do they see our time running? (15.6)

13. When a flashing light source approaches you, does the speed of light, or the frequency of arrival of light flashes, or both, increase? (15.7)

14. a. How many frames of reference does the stay-at-home twin experience in the twin trip?

K How many frames of reference does the traveling twin experience? (15.7)

15. Is it possible for a person with a 70-year life span to travel farther than light travels in 70 years? Explain. (15.8)

Plug and Chug Use Equations

16. If a spaceship moves away from you at half the speed of light and fires a rocket away from you at half the speed of light relative to the space-ship, common sense may tell you the rocket moves at the speed of light relative to you. But it doesn't! The relativistic addition of velocities (not covered in the chapter) is given by

UI + 1)2

1 4_ 1111/2

C2

Substitute 0.5c for both v1 and v2 and show that the velocity Vof the rocket relative to you is 0.8c.

17. If the spaceship of the previous question somehow travels at c relative to you, and it somehow fires its rocket at c relative to itself, use the equation to show that the speed of the rocket relative to you is still c!

IS. Substitute small values of v1 and v2 in the preceding equation and show for everyday speeds that Vis practically equal to V1 + V2.

Think and Explain Think Critically

19. If you were in a smooth-riding train with no windows, could you sense the difference between uniform motion and rest? Between accelerated motion and rest? Explain how you could do this with a bowl filled with water,

20. Suppose you 're playing catch with a friend in a moving train. When you toss the ball in the direction the train is moving, how does the speed of the ball appear to an observer stand-ing at rest outside the train? (Does it incti=l-iso Or appt:cti the same as lithe observer were riding on the train?)

21. Suppose you're shining a light while riding on a train. When you shine the light in the direc-tion the train is moving, how would the speed of light appear to an observer standing at rest Outside the train? (Does it increase or appear the same as if the observer were riding with the train?)

22. Light travels a certain distance in, say, 10 000 years. Can an astronaut travel more slowly than the speed of light and yet travel the same distance in a 10-year trip? Explain.

23. Can you get younger by traveling at speeds near the speed of light? Explain.

24. Explain why it is that when we look out into the universe, we see into the past.

25. One of the fads of the future might be "cen-tury hopping," where occupants of high-speed spaceships would depart from Earth for a few years and return centuries later. What are the present-day obstacles to such a practice?

26. If you were in a high-speed spaceship travel-ing away from Earth at a speed close to that of light, would you measure your normal pulse to be slower, the same, or faster? How would your measurements of pulses of friends back on Earth be if you could monitor them from your ship? Explain.

27. Is it possible for a person to be biologically older than one's parents? Explain.

Think and Solve Develop Problem-Solving Skills

28. Jose and Josie are twins. Suppose Josie takes a ride in a spaceship to a distant star some 4 light-years distant, traveling at a constant speed of 0.8c. After arriving at the star Josie quickly turns around and immediately comes back at the same speed.

a. How much older is Jose when Josie returns?

mucii ()icier is Josie when she returns?

c. VVhy isn't the reverse true, that Josie is older than Jose?

29. Joe Burpy is 30 years old and has a daughter who is 6 years old. Joe leaves on a Greyhound space bus and takes a 5-year (space-bus time) round-trip at 0.99c. How old will he and his daughter be when he returns?

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Special Relativity Space and Time - Woodside High School

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