Objectives• Introduce space-time and time dilation. (15.1) 15 THE BIG • Give examples of relative motion. (15.2) • Discuss the constancy of the speed of light. (15.3) • Describe time dilation. (15.4) • Describe why space travel at relativistic speeds seems impossible. (15.5) • Describe the conditions under which length contracts. (15.6) discover! MATERIALS soda straws, scissors Students will create a model for length contraction. EXPECTED OUTCOME ANALYZE AND CONCLUDE 1. Students will relate relativistic speed to length contraction for a straw. 2. You would see approximately half the soda straw’s length. 3. Relativistic effects become noticeable at speeds approaching the speed of light and hence are not observed under normal circumstances. Relativistic effects, such as length contraction, must be taken into account in the design of particle accelerators where particles routinely travel at extremely high speeds. 282 SPECIAL RELATIVITY— SPACE AND TIME ........ SPECIAL RELATIVITY— SPACE AND TIME IDEA Motion through space is related to motion in time. E veryone 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 snail’s pace to those of supersonic aircraft and space shuttles. But relatively 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.15.0 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. Ten years later Einstein announced a similar theory, called the general theory of relativity (discussed in the next chapter), that shows how gravity is related to space and time. These theories have enormously changed the way scientists view the workings of the universe. discover! How are Speed and Length Contraction Related? 1. Obtain six soda straws and determine the length of a single soda straw in centimeters. 2. Multiply the length of a soda straw by the following factors: 1, 0.9999999999999978, 0.999999944, 0.995, 0.5, and 0.045. 3. Use scissors to cut soda straw segments to the lengths determined in Step 2. If you find it impossible to cut the straws to the required lengths, simply leave them uncut. 4. Compare the lengths of the soda straws by placing them one above the other. 282 Analyze and Conclude 1. Observing The cut lengths represent the effects of length contraction you would observe if a soda straw were moving past you at the following speeds (c represents the speed of light, or roughly 3 ⫻ 108 m/s): 0, 20 m/s, 100,000 m/s, 0.1c, 0.87c, and 0.999c. 2. Predicting What fraction of a soda straw’s length would you see if you were moving past a soda straw at a speed of 0.87c? 3. Making Generalizations When do length contraction effects become noticeable? If you must omit some of the chapters of the text, this and the following chapter can be skipped without consequence to the chapters that follow. 15.1 Space-Time Newton and other investigators before Einstein thought of space as an infinite expanse in which all things exist. 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 question could be raised for time. Does the universe exist in time, or does time exist only within the universe? 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. 15.1 Space-Time Key Terms space-time, special theory of relativity, postulate Teaching Tip Explain that though time is one of those concepts with which we are all familiar, it is difficult to define. We can say it is what’s measured by a clock or that it is nature’s way of seeing that everything does not happen at once. Ask students for their thoughts on this. 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. Einstein’s special theory of relativity describes how time is affected by motion in space at constant velocity, and how mass and energy are related. From the viewpoint of special relativity, you travel through a combination of space and time. You travel through space-time. The colorful cloud of gas and dust particles in Figure 15.1 moves through 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.” When you stand still, like the girl in Figure 15.2, then all your traveling is through time. When you move a bit, then some of your travel is through space and most of it is still through time. If you were somehow able to travel through space at the speed of light, all your traveling would be through space, with no travel through time!15.1 You would be as ageless as light, for light travels through space only 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! CHAPTER 15 Teaching Tip After discussing Einstein and a broad overview of what special relativity is and is not, point out that the theory of relativity is grounded in experiment, and in its development it explained some very perplexing experimental facts (constancy of the speed of light, muon decay, solar energy, the nature of mass, etc.). It is not, as some people think, only the speculations of one man. FIGURE 15.2 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. SPECIAL RELATIVITY—SPACE AND TIME 283 Teaching Tip Challenge your students to think of space and time completely inside the universe as opposed to the universe being located in the vastness of space and time. No universe, no space; no universe, no time! Point out that the universe doesn’t occupy space; space (and time) are in the universe. Said another way, the expanding universe doesn’t spread out to fill a larger emptiness. The only space that exists is the space it creates as it goes. 283 for although in 1905 there was only one person to understand it. you travel though a combination of space and time. CONCEPT Einstein reasoned that there is no stationary hitching post in the universe relative to which motion should be measured. Newspapers during the early part of the twentieth century used to report that there were only 12 people in the world who understood special relativity.. or the stretching of time. but only relative to other objects. CONCEPT How can you describe a person’s travel from ... You travel through space-time. CHECK the viewpoint of special relativity? 15. if either. If you could not see out the windows. or postulates.3 Spaceman A considers himself at rest and sees spacewoman B pass by. If you look out the window and see the car in the next lane begin moving backward. all motion is relative and all frames of reference are arbitrary. But spacewoman B considers herself at rest and sees spaceman A pass by. you may be surprised to find that the car you’re observing is really at rest—your car is moving forward. From this observation each will be unable to determine who is moving and who is at rest. From the viewpoint CHECK of special relativity. people will be able to travel noticeably in time. A spaceship. This is inaccurate. Look at Figure 15. there would be no way to determine whether your car was moving with constant velocity or was at rest... Spaceman A and spacewoman B will both observe only the relative motion. spaceman A and spacewoman B will each observe only the relative motion... This is a familiar experience to a passenger in a car at rest waiting for the traffic light to change. Motion in space affects motion in time.. The special theory of relativity that Einstein developed rests on two fundamental assumptions. They will be able to jump centuries ahead.3.2 The First Postulate of Special Relativity . Whenever we move through space.. we to some degree alter our rate of moving into the future. Instead. cannot measure its speed relative to empty space. This is known as time dilation.2 The First Postulate of Special Relativity Key Term first postulate of special relativity FIGURE 15. It’s filled with electromagnetic radiation and streams of subatomic particles. Einstein himself. Teaching Resources • Reading and Study Workbook • PresentationEXPRESS • Interactive Textbook • Conceptual Physics Alive! DVDs Special Relativity I 15.Even empty space isn’t really empty. after Einstein published his paper and explained it.. If spaceship A drifts past spaceship B in empty space. 284 284 . large numbers of people in the community of physicists understood it. for example. just as today people can jump from Earth to the moon. If spacecraft of the future reach sufficient speed. The fact that physical experiments produce the same results in all uniformly moving frames leads to one of the fundamentals of special relativity: The speed of light is seen to be the same by all observers. Of course.. the beam would be at rest to such an observer. FIGURE 15. he would measure the light as moving away from him at 300. and a flight attendant need make no adjustments in pouring tea because of the plane’s high speed. no experiment confined within the cabin itself can determine whether or not there is uniform motion. The more Einstein thought about this. or send a radar signal out. we can look outside and see Earth whizzing by. Many experiments can detect accelerated motion. Teaching Tip Explain that the laws of physics are the same in all uniformly moving reference frames: A bee inside a fast-moving jet plane executes the same flying maneuvers regardless of the speed of the plane. There is no physical experiment we can perform to determine our state of uniform motion. The laws of physics within the uniformly moving cabin are the same as those in a stationary laboratory. “What would a light beam look like if you traveled along beside it?” According to classical physics... Teaching Resources • Reading and Study Workbook • PresentationEXPRESS • Interactive Textbook . the more convinced he became of its impossibility.. The first postulate of special relativity states that all the laws of nature are the same in all uniformly moving frames of reference.. The laws of physics are the same for the ship whether it is moving uniformly or is at rest. the test of their validity is that they lead to predictions that we can test. If we swing a pendulum. according to Einstein...4 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.000 km/s. Teaching Tip Emphasize that uniform motion is motion with no change in either speed or direction.15.3 CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 285 285 .. CONCEPT What does the first postulate of special CHECK relativity state? 15. but none can. However. The postulates themselves don’t have to make “common” sense. a coin dropped to the floor of the moving plane will fall as if the plane is at rest.4 does not have to make adjustments to his game as long as the ship moves at a constant velocity. detect the state of uniform motion.3 The Second Postulate of Special Relativity Key Term second postulate of special relativity One of the questions that Einstein as a youth asked himself was. In the cabin of a high-speed jetliner.. The person playing pool in Figure 15. As with all postulates in science. it will move no differently when the plane is moving uniformly (constant velocity) than when not moving at all. He came to the conclusion that if an observer could travel close to the speed of light. CONCEPT CHECK Einstein’s first postulate of special relativity assumes our inability to detect a state of uniform motion.. we flip a coin and catch it just as we would if the plane were at rest. The first postulate of special relativity states that all the laws of nature are the same in all uniformly moving frames of reference. . No experiment can be performed that will determine whether a closed cabin is at rest or moving at constant velocity.3 The Second Postulate of Special Relativity 15. Underline it. so write the word SPACE in correspondingly larger letters. c5 FIGURE 15. If she sends a flash of her own to the space station. The speed of light is a constant. . All observers measure the same ratio of space to time for light waves in free space.. The ratio of space to time for light is the same for all who measure it. The speed of light in CHECK empty space will always have the same value. observers on the station will measure the speed of these flashes as c.6. CHECK relativity state? .Demonstration As you stand still. The speed of the flashes will be no different if the spaceship stops or turns around and approaches. toss a piece of chalk in the air and catch it.5 shows. a spaceship departing from the space station shown in Figure 15. And for any observation of motion through space.. the constancy of the speed of light is what unifies space and time. A flash of light is emitted from the station at 300. Everyone who measures the speed of light will get the same value c. for example. Consider..6 The speed of a light flash emitted by either the spaceship or the space station is measured as c by observers on the ship or the space station. All observers who measure the speed of light will find it has the same value. c.. the space covered by the tossed chalk was seen to be greater. Suppose instead that their measurement of speed was the same.. Write the enlarged word TIME beneath the underline. there is a corresponding passage of time. then the greater space can be accounted for if the measured time is also greater. No matter what the speed of the spaceship relative to the space station is. the speed of light is constant regardless of the speed of the flashlight or the source. State that if they measure the same ratio of space to time.. State that from your frame of reference the measured speed is again the same. The second postulate of special relativity states that 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..7 As Figure 15. They should respond that the chalk was moving faster this time. Ask the class to suppose that all measurements show the chalk to have a constant average speed.7 shows. Einstein’s second postulate of special relativity assumes that the speed of light is constant. Write on the board. ‘ The speed of light in all reference frames is always the same. CONCEPT 286 All space and time measurements of light are unified by c. Proceed to walk at a brisk pace across the room and toss the chalk again.. SPACE TIME This represents speed as seen by you in your frame of reference. As Figure 15. State that from the frame of reference of the class.5 The speed of light is constant regardless of the speed of the flashlight or observer. Call this constant speed c. FIGURE 15.000 km/s—a speed we’ll simply call c. Ask if the speed looked different to them... 286 CONCEPT What does the second postulate of special . FIGURE 15. with uniformly sized letters. equating it to c. an observer on the spaceship will measure the speed of the flash of light passing her as c. each bounce will take 0. it has not slowed at all! CHAPTER 15 Does time dilation mean that time really passes more slowly in moving systems or that it only seems to pass more slowly? Explain. and a means of shielding astronauts from the radiation that would be produced by impact with interstellar matter.00001 s. but one that will help to describe time dilation. We see the light flash bouncing up and down along a longer diagonal path.11.4 Time Dilation 15. b.4. An observer moving with the spaceship observes the light flash moving vertically. slightly for everyday speeds. according to our observations. think! But remember the second postulate of relativity: The speed will be measured by any observer as c. explain that the time for light to go from one mirror to another is shorter for the person on the spaceship than for the person watching light move along the longer diagonal path from the rest frame.15. If the tube is 3 km long. Answer: 15. each bounce will take 1 s in the frame of reference of the light clock.9 Teaching Tip Discuss the prospects of “century hopping. for occupants of the spaceship. we must measure more time between bounces! For us.8 A stationary light clock is shown here. A flash of light bounces back and forth between the parallel mirrors.9. a FIGURE 15. and also to the material on page 289. but significantly for speeds approaching the speed of light. or even thousands of years.000 km in length. Light bounces between parallel mirrors and “ticks off” equal intervals of time. of course. or the vibrations of a quartz crystal. b The moving ship contains a light clock.” a scenario in which future space travelers may take relatively short trips of a few years or so. If the tube is 300. A Moving Light Clock Imagine an empty tube with a mirror at each end as shown in Figure 15. Suppose we view the light clock as it whizzes past us in a highspeed spaceship as shown in Figure 15. 15. A clock can be any device that measures periodic intervals. and 15. We measure time with a clock. one tick of the light clock takes longer than it takes for occupants of the spaceship. and return in decades. An observer who is passed by the moving ship observes the flash moving along a diagonal path. This is. pending the solution to two major problems: durable rocket engines and sufficient fuel supplies for prolonged voyages.” a rather impractical device. centuries. Teaching Tip While discussing Figure 15. The spaceship’s clock. The mirrors are perfect reflectors. looking in from the outside.9. such as the swings of a pendulum. The Time dilation occurs ever so stretching of time is time dilation. a. FIGURE 15. has slowed down—although.4 Time Dilation Key Term time dilation Einstein proposed that time can be stretched depending on the motion between the observer and the events being observed.1 SPECIAL RELATIVITY—SPACE AND TIME 287 287 . so the flash bounces indefinitely. the oscillations of a balance wheel. We are going to consider a “light clock. Since the speed of light will not increase.22.214.171.124. Teaching Tip Relate the analogy of your chalk-tossing sequence to the light clock discussed in the text and in Figures 15. 15. the pitter patter of a slanting rain when you run into the rain vs. would you notice a difference in your pulse rate? In the pulse rate of the people back on Earth? Explain.10 The longer distance taken by the light flash in following the diagonal path must be divided by a correspondingly longer time interval to yield an unvarying value for the speed of light. All events on the moving ship will be observed by us as slower.4.2 FIGURE 15. Teaching Tip Sketch a simple version of the art below on the board. Explain that for a ship at rest relative to the two observers on the distant planets. . How do the occupants on the spaceship view their own time? Time for them is the same as when they do not appear to us to be moving at all. There is no way the spaceship occupants can tell uniform motion from rest.10 show. v. But with motion. FIGURE 15. when you run away from it. light flashes emitted at 6-min intervals would be seen by both to also be at 6-min intervals. Recall Einstein’s first postulate: All laws of nature are the same in all uniformly moving frames of reference. We say that time is stretched—it is dilated. Give examples of the Doppler effect: the changing pitch of a car horn when it approaches and when it recedes. Answer: 15. the speed of the light clock has no effect on the speed of light. Einstein showed that the relation between the time t0 (proper time) in the observer’s own frame of reference and the relative time t measured in another frame of reference is t think! If you were moving in a spaceship at a high speed relative to Earth. They have no clues that events on board are seen to be dilated when viewed from other frames of reference. as viewed from our frame of reference. As the equation for time and Figure 15. The slowing of time is not peculiar to the light clock. The heartbeats of the spaceship occupants will have a slower rhythm.11 A light clock moves to the right at a constant speed. that slows. 288 288 t0 v 2 1( c ) where v represents the relative velocity between the observer and the observed and c is the speed of light. It is time itself in the moving frame of reference. the situation is different. It is physically impossible for observers in different frames of reference to refer to one and the same realm of space-time. and cto and vt are legs. Since the speed of light is always c. they’ll be stretched to 12-min intervals for recession? Once this is acceptable to your class.4 Figure 15. go on to discuss the Twin Trip. There is only one measurement they will always agree on: the speed of light.12 do the math! How Can You Derive the Time Dilation Equation? 15. If we apply this to the figure. These three distances make up a right triangle in the figure. 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. the light flash is observed to move a vertical distance cto in the frame of reference of the light clock. we obtain (ct)2 (ct )2 (vt)2 (ct)2 (vt)2 (ct )2 t 2 [1 (v 2/c 2)] t 2 t t 2 t2 1 (v 2/c 2) t Physicist Ken Ford emphasizes the meaning of the time dilation equation with his ninth-grade high school students. the flashes are spread apart twice as much so that the time between flashes is doubled? That if 6-min flash intervals are cut to 3-min for approach. The symbol t represents the time it takes the flash to move from one mirror to the other as measured from a frame of reference in which the light clock moves to the right with speed v. The time between flashes is cut in half. Is it reasonable to say the opposite occurs? That instead of being squeezed together. During this time t. They see our time running slow. The symbol to represents the time it takes for the flash to move between the mirrors as measured from a frame of reference fixed to the light clock. This vertical distance is the same in both reference frames. The measurements in one frame of reference need not agree with the measurements made in another reference frame. in which ct is the hypotenuse. 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. This is the distance between mirrors. Since the speed of the flash is c and the time to go from position 1 to position 2 is t. moves to the upper mirror at position 2. There is no contradiction here. Ask how the time between flashes would be seen by the observer on the left—who sees the source receding. FIGURE 15. the clock moves a horizontal distance vt from position 1 to position 2. just as we see their time running slow.How do occupants on the spaceship view our time? From their frame of reference. it appears that we are the ones who are moving. It is not expected that you master it! 1 (v 2/c 2) CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 289 289 . the diagonal distance traveled is ct. The diagonal lines represent the path of the light flash as it starts from the lower mirror at position 1. and then back to the lower mirror at position 3.11 shows three successive positions of the light clock as it moves to the right at constant speed v. Teaching Tip Ask your class to suppose the ship in your sketch moves so fast toward the right observer that the flashes reach the observer at twice the frequency—with flashes closer together so they appear at 3-min intervals. 13 The traveling twin does not age as fast as the stay-at-home twin. At this speed the traveling twin would age a single year while the stay-at-home twin ages 10 years. The question arises. when the traveling twin returns. If the traveling twin maintains a speed of 87% the speed of light for a year. 290 290 . 1. on the other hand. He has been in two realms of space-time. experiences a single frame of reference—one realm of space-time. he is younger than the stayat-home twin.15 years will have elapsed on Earth. As Figure 15. there’s a fundamental difference here. How much younger depends on the relative speeds involved. and due to time dilation is today twohundredths of a second younger than he would be if he’d never been in space! The Twin Trip A dramatic illustration of time dilation is afforded by identical twins. It may take more than a class period to develop these ideas.13 shows. If the traveling twin maintains a speed of 50% the speed of light for one year (according to clocks aboard the spaceship). The space-traveling twin experiences two frames of reference in his round trip—one receding from Earth. Cosmonaut Sergei Avdeyev spent more than two years aboard the orbiting Mir space station. The stay-athome twin.5% the speed of light. At 99. Please do the practice pages on The Twin Trip in the Concept Development Practice Book. since motion is relative. one an astronaut who takes a high-speed roundtrip journey while the other stays home on Earth. You’ll see that the twins can meet again at the same place in space only at the expense of time. then 2 years will have elapsed on Earth. 10 Earth years would pass in one spaceship year. 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? Aha.org Web Code: csn – 1504 FIGURE 15. separated by the event of turning around.SciLinks. and the other approaching Earth.Be patient with your students as they ponder the Twin Trip. For: Links on relativity of time Visit: www. As Figure 15.. will the effects of going and coming compensate each other? Amazingly. if you move at the speed of light. the light carrying the information continues past. Answer: 15.” Traveling at the speed of light. CONCEPT CHECK Teaching Resources • Reading and Study Workbook How does time dilation at everyday speeds compare with time dilation at light speed? • Concept-Development Practice Book 15-1. When you return and are once again sitting in the square. These are the two extremes. you see all events taking place in the reference frame of the clock on Earth as happening in slow motion.15 shows. From your highspeed (but less than c) moving frame of reference. like the one shown in Figure 15.. tells the time is always 12 noon at the village square. you’ll see all events occurring in the clock’s frame of reference as being speeded up.. CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 291 • Problem-Solving Exercises in Physics 9-1 • PresentationEXPRESS • Interactive Textbook • Next-Time Questions 15-1 291 .. you see the village-square clock move into the future at the rate of 60 seconds per minute. but significantly for speeds approaching the speed of light. is moving in a direction away from a huge clock displayed in a village square. then it would keep up with the information that says “12 noon.. then. “light left the clock at 12 noon on Earth”). The wristwatch you were wearing the whole time and the village clock will disagree.4. Out there an observer who later receives the light says. If you suddenly move your head to the side. Time dilation occurs ever so slightly for everyday speeds..3 The graph shows how 1 second on a stationary clock is stretched out. 1 second on a stationary clock is stretched out. Time at the village square is frozen! If the trolley car is not moving.” depending on what your speed is between the extremes of zero and the speed of light. as measured on a moving clock. Clockwatching on a Trolley-Car Ride Pretend you are Einstein in a trolley car that provided the high-speed travel back then. 15-2 think! FIGURE 15. You wonder more about this idea. The clock reads 12 noon...15 Will observers A and B 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. CONCEPT CHECK FIGURE 15.. as measured on a moving clock. it’s 12 noon on Earth now” (or more correctly. If the trolley car traveled as fast as the light. no! Time will be stretched. . “Oh. To say it reads 12 noon is to say that light carrying the information “12 noon” is reflected by the clock and travels toward you along your line of sight. If you reverse direction and travel at high-speed back toward the clock.. you see seconds on the clock taking infinite time.14 Light that carries the information “12 noon” is reflected by the clock and travels toward the trolley. instead of meeting your eye. What’s in between? What happens for speeds less than the speed of light? A little thought will show that you will receive the message “1 o’clock” anywhere from 60 minutes to an infinity of time after you receive the message “12 noon.. This is time dilation.5c relative to Earth? Explain.14. You and the distant observer will see 12 noon at different times. presumably out into space. Suppose the trolley car. wouldn’t we have tourists from the future? 15.5 Space and Time Travel The topic of space and time travel often leads to good class discussions (and topics for papers!).99% the speed of light. but not to an observer who is located outside the person’s frame of reference—for she sees the difference. At 99. so it was reasoned that a person traveling even at the speed of light would have to survive for 30. this distance would be pushed appreciably farther than 200 years. Link to TECHNOLOGY Relativistic Clocks In 1971 atomic clocks were carried around Earth in jet planes. it would seem that only 3 years had gone by for the astronauts. GPS could not precisely locate positions on Earth.000 light-years away. it was argued that humans would never be able to venture to the stars. The center of our galaxy is some 30.0 years in Earth time.15. Time for a person on Earth and time for a person in a high-speed spaceship are not the same. astronauts traveling at 99% the speed of light could go to the star Procyon (11. A person’s heart beats to the rhythm of the realm of time it is in.15.4 light-years distant) and back in 23.000 years to make such a voyage! But these arguments fail to take into account time dilation. Atomic clocks now cruise overhead at even greater speeds in the satellites that are part of the global positioning system (GPS). Because of time dilation. One realm of time seems the same as any other realm of time to the person. It would be the space officials greeting them on their return who would be 23 years older. Alpha Centauri is the nearest star to Earth. A 5-year trip for them would take them farther than light travels in 1000 Earth-time years.5 Space and Time Travel Before the theory of special relativity was introduced. scientists and engineers had to accommodate for relativistic time dilation.999% the speed of light. It was thought that our life span is too short to cover such great distances— at least for the distant stars. In designing this system. which can pinpoint positions on Earth to 292 292 within meters. It would take light itself 22. Time dilation is a fact of everyday life to scientists and engineers—especially those who design equipment for global navigation work. If they didn’t.5 It was therefore thought that a round-trip even at the speed of light would require 8 years. and biologically they would be only 3 years older. travelers could travel slightly more than 70 light-years in a single year of their own time. As an example. If traveling backward in time were possible. and it is 4 light-years away. All their clocks would indicate this. the traveling clocks were a few billionths of a second “younger” than twin clocks that stayed behind. after the sun. Upon landing. . At higher speeds the results are even more impressive.8 years in Earth time to make the same round trip. At a speed of 99. The amounts of energy required to propel spaceships to relativistic speeds are billions of times the energy used to put the space shuttles into orbit. but simply because of the impracticality of space travel. travels only one way—forward. Traveling close to the speed of light in order to take advantage of time dilation is completely consistent with the laws of physics. rather. CONCEPT CHECK Teaching Resources speeds seem impossible? • Reading and Study Workbook • PresentationEXPRESS • Interactive Textbook CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 293 293 . peer in. FIGURE 15. (Of course. and return in the year 2500. Star travelers will not bid “so long. An astronaut leaving on a deep-space voyage must live with the fact that. People could keep jumping into the future with some expense of their own time. the occupant decides that 10 minutes is enough so he or she presses the button.. interstellar space travel must be relegated to science fiction.000 years to travel from the center of the Milky Way galaxy to our solar system. They could never return to the same era on Earth that they bid farewell to. “Good gosh. You can only see the universe as it was in the past. in reality. is the key factor.. much more time will have elapsed on Earth than she has experienced on her voyage. Explain how one taking a “ride” in such a centrifuge might be strapped in a seat and told to press a button on the seat when he or she wishes the ride terminated. Such journeys seem impossible to us today.. Motion in space.From Earth’s frame of reference. see you later” to those they leave behind but. as we know it. those outside open the door. For the present. one would be crushed to death in such a case. but pretend that somehow one is physically unaffected by the crushing centripetal forces—the fictitiousness of this example). and ask. One might live among Earthlings of that period for a while and depart again to try out the year 3000 for style. signaling those outside to bring the machine to a halt. rather than space itself. The practicalities of such space journeys are prohibitive.” You can see into the past. even millions of years. but they could not travel into the past. Here on Earth we constantly move into the future at the steady rate of 24 hours per day. what have you been doing in there for the past 3 weeks!” In the laboratory frame of reference. And suppose that after being whirled about at rim speeds near the speed of light. so far. hundreds. one might depart from Earth in a high-speed ship in the year 2150. a permanent “good-bye. CONCEPT Why does space travel at relativistic CHECK The amounts of energy required to propel spaceships to relativistic speeds are billions of times the energy used to put the space shuttles into orbit. but you cannot go into the past. If these problems are ever overcome and space travel becomes routine. This is not because of scientific fantasy. people might have the option of taking a trip and returning in future centuries of their choosing. After the machine is halted. light takes 30. 3 weeks would have elapsed during a 10-min interval in the rotating centrifuge. Time.. The point: One does not have to necessarily travel through wide expanses of space for time dilation to be significant. The problems of shielding radiation induced by these high speeds seems formidable.. For example.. . upon her return...16 Teaching Tip Present this interesting but fictitious example of time dilation: Suppose that one could be whirled in a giant centrifuge up to relativistic speeds without physical injury. you’re looking at light that’s been on its way to you for dozens.. .. When you look at stars or galaxies at night. travel for 5 years or so. At greater speeds. length contraction dramatically shortens their distance to Earth. 0.18 shows. If it whizzed past at 99. seemingly too brief to reach the ground below before decaying. and state that if your students made accurate measurements of its length. so they will see our metersticks contracted—and us as well. 294 For moving objects. a meterstick aboard a spaceship whizzing past you at 87% the speed of light. FACT Teaching Tip Hold up a meter stick horizontally. Besides. Teaching Tip Point out that as far as people aboard a spaceship are concerned.18 In the frame of reference of the meterstick on the spaceship. Some are muons. Do people aboard the spaceship also see their metersticks—and everything else in their environment—contracted? The answer is no.5% the speed of light. State that your measurements and those of your students would now differ. relative to the class. At 99. When an object moves at a very high speed relative to an observer. new particles are made. As Figure 15. it would violate the first postulate of relativity. causes biological mutations.6 Length Contraction Key Term length contraction Common Misconception Objects can go faster than the speed of light from some frames of reference. But because muons move at nearly the speed of light. As relative speed gets closer and closer to the speed of light. it would appear to you to be contracted to one tenth its original length.15. there is a relative speed between themselves and our frame of reference. the measured lengths of objects contract closer and closer to zero. If they did. like that of all high-speed elementary particles. doesn’t change.17 shows. The effects of relativity are always attributed to “the other guy. space as well as time undergoes changes. For everyday speeds. Link to BIOLOGY Muons and Mutations When cosmic rays bombard atoms at the top of the atmosphere.” . its length is 1 meter. its measured length in the direction of motion is contracted. the contraction would be noticeable. If an object could reach the speed of light. State that if you were to travel at 87% the speed of light.5% the speed of light. Teaching Tip Walk across the room holding the meter stick like a spear. People in the spaceship see nothing at all unusual about the lengths of things in their own reference frame. would appear to you to be only 0. However. radioactive particles that streak downward toward Earth’s surface.6 Length Contraction 15. So we see a link between the effects of relativity and the evolution of living creatures on Earth. The amount of contraction is related to the amount of time dilation. People at the back of the room would have to compensate for its shorter appearance due to distance.17 A meterstick traveling at 87% the speed of light relative to an observer would be measured as only half as long as normal.5 meter long. for example. they would see it as only 10 cm long. perpendicular to the direction of travel. Observers from this frame see our metersticks contracted. Everyone would measure it as 1 m long. Recall that all the laws of physics are the same in all uniformly moving reference frames. The observable shortening of objects moving at speeds approaching the speed of light is length contraction. they would agree on its 1-m length. there is no relative speed between the people on the spaceship and the things they observe in their own reference frame. A muon’s average lifetime is only two millionths of a second. but nevertheless.5 m. the length of the stick would contract to zero from the point of view of the students. it would be even shorter. 294 FIGURE 15. As Figure 15. You are hit by hundreds of muons every second! Muon impact. they would measure the stick to be half as long. their measurements would agree with yours. the amount of contraction is much too small to be measured.” FIGURE 15. a rule of relativity is that changes due to alterations of space-time are always seen in the frame of reference of the “other guy. For relativistic speeds. The width of the stick. If you were traveling at the speed of light. they are at rest and other things move away from or toward them. its length would contract to zero which means that such an apparent speed is impossible. The stick moving in spear fashion appears shorter but it doesn’t appear thinner. So if an object. From your frame of reference. so that v ⫽ 0.19. It was stated earlier that if an object were moving at 87% the speed of light.87c is substituted for v in the equation. State that contraction takes place only in the direction of motion. the v in the equation is zero. its measured length in the direction of motion is contracted. Now consider a photon traveling from one star across the entire universe to the other. its length would contract to zero. This is one of the reasons that the speed of light is the upper limit for the speed of any object. When 0... as stated earlier.1L0. we find L ⫽ L0. If a distance of 20 light-years separates a pair of stars in our frame of reference. CONCEPT CHECK • Next-Time Question 15-2 • Problem-Solving Exercises in Physics 9-2 moving at a very high speed relative to an observer? CHAPTER 15 Ask Consider a pair of stars. whatever the speed. What. Or when 0. exactly. is time? Can we say it is nature’s way of seeing to it that everything does not all happen at once? And why does time seem to move in one direction? Has it always moved forward? Are there other parts of the universe where time moves backward? Perhaps these unanswered questions will be answered by the physicists of tomorrow. How exciting! A spacewoman travels by a spherical planet so fast that it appears to her to be an ellipsoid (egg shaped). Length contraction: moving objects are shorter (in the direction of motion). we find L ⫽ 0. Since you move with the stick you see no contraction. and L0 is the measured length of the object at rest. one on each “edge” of the universe. L is the length of the moving object as measured by the observer.. Teaching Tip Write the length-contraction formula on the board... From the frame of reference of the photon.5L0. no contraction takes place vertically. contraction in the direction of motion increases. the direction of motion. Einstein’s theory of relativity has raised many philosophical questions.. When 0 is substituted for v in the equation. SPECIAL RELATIVITY—SPACE AND TIME .87c between them would see them as only 10 lightyears apart. Teaching Resources CONCEPT How does the length of an object change when it is CHECK Teaching Tip With space travel between stars. is moving horizontally. If the object could reach the speed c. like the baseball in Figure 15.6 Suppose that an object is at rest. .. we find L ⫽ 0. v is the speed of the object relative to the observer..995c is substituted for v.19 As relative speed increases. Lengths along the direction perpendicular to this motion are the same in the two frames of reference. c is the speed of light. as we would expect. Lengths in the perpendicular direction do not change. the length contraction reaches zero. If she sees the short diameter as half the long diameter. it would contract to half its length. The contraction of speeding objects is the contraction of space itself. and so L 5 Lo. Contrast the students’ view of the stick with yours.6 In summary—Time dilation: moving clocks run slowly. a spaceship traveling at 0. think! Relativistic length contraction is stated mathematically: v2 c2 ( ) L L0 1 In this equation. 295 295 .. When an object moves at a very high speed relative to an observer. Contraction depends on the frame of reference. Space contracts in only one direction. FIGURE 15.15.. the distance as seen from our rest frame of reference is quite different than it is as seen from the frame of reference of a moving spaceship. what is the distance of separation between stars? Zero! From a frame of reference traveling at c. what is her speed relative to the planet? Answer: 15. and each one will see events on Earth in the same slow motion.com Web Code: csa – 1500 think! Answers 15. When an object moves at a very high speed relative to an observer. so no relativistic effects would be noticed.2 There would be no relative speed between you and your own pulse. . 284) first postulate of special relativity (p. but significantly for speeds approaching the speed of light.REVIEW Teaching Resources • TeacherEXPRESS • Conceptual Physics Alive! DVDs Special Relativity I 15 REVIEW Concept Summary • • • • • • From the viewpoint of special relativity. 15. you travel through a combination of space and time. 283) postulate (p. The second postulate of special relativity states that 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. So they will not agree on measurements of time. 287) length contraction (p. Relativity effects are always attributed to “the other guy. You travel through space-time. Key Terms •••••• space-time (p. 15. 294) For: Self-Assessment Visit: PHSchool. 286) time dilation (p.4. When they are moving in unison. 283) special theory of relativity (p. 285) 296 •••••• 296 second postulate of special relativity (p. They will see each other’s time as passing normally. they share the same frame of reference and will agree on measurements of time. Time dilation occurs ever so slightly for everyday speeds.3 When A and B have different motions relative to each other.6 The spacewoman passes the spherical planet at 87% the speed of light.1 The slowing of time in moving systems is not merely an illusion resulting from motion. There would be a relativistic effect between you and people back on Earth. You would find their pulse rate slower than normal (and they would find your pulse rate slower than normal). Time really does pass more slowly in a moving system compared with one at relative rest.” 15. its measured length in the direction of motion is contracted.4. The amounts of energy required to propel spaceships to relativistic speeds are billions of times the energy used to put the space shuttles into orbit. The first postulate of special relativity states that all the laws of nature are the same in all uniformly moving frames of reference. each will observe a slowing of time in the frame of reference of the other.4. What is the second postulate of special relativity? 7. 10. What is time dilation? 4.2 5.4 8. No change. Section 15. Does light travel through space? Through time? Through both space and time? 11. Stretching of time due to motion in space 12.3 6. but with its length perpendicular to its direction of motion? (Why are your answers to this question and the last question different?) 5. How long would a meterstick appear if it were traveling at 99. Section 15.5 4. Section 15. It takes correspondingly more time. Similarly. yes. 7. there is no time dilation. a person moving at speeds close to c lives longer. no. In your frame. not perpendicular. only change is parallel to motion. the receiver. Why.1 1. c is always the same whether the source. No. What are the present-day obstacles to interstellar space travel? 13. The ratio of velocity gain to time for a freely falling body is g.5% the speed of light. Slow also. the speed of light 8. 9.ASSESS ASSESS 15 ASSESS Check Concepts 1. 3. or both move. Yes. Yes. then. c. no Check Concepts 9. 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 reference. The laws of nature are the same in all uniformly moving frames. would clocks on board appear to you to be running slow? Defend your answer. How long would a meterstick appear if it were thrown like a spear at 99. 12. Relative speed of you and clocks is zero. What is space-time? 2. 10. What is the first postulate of special relativity? 6. Space and time are two parts of one whole.5% the speed of light? Section 15. One-tenth as long 14. Can you travel while remaining in one place in space? Explain. If we view a passing spaceship and see that the inhabitants’ time is running slow. what is the ratio of distance to time for light waves? Section 15. each sees the same effect in the other. how do they see our time running? •••••• Section 15. does the light not appear to be moving faster? CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 297 297 . Is it possible for a person with a 70-year life span to travel farther than light travels in 70 years? Explain. 2. If you were traveling in a high-speed rocket ship. in time 3. 14. Fuel energy and radiation 13.6 11. of the rocket relative to you is 0. A.8c Think and Rank •••••• Rank each of the following sets of scenarios in order of the quantity or property involved. B A C A B C Rank the following quantities from greatest to least. Think and Rank 15 ASSESS For: Study and Review Visit: PHSchool.com Web Code: csd – 4270 (continued) 16. a. No. C. then separate them with an equal sign. how frequently the flashes reach the same observer c. a. v. If scenarios have equal rankings. v 5 (0. the speeds of the probes as seen by an Earth observer b. A ⫽ B) 16.5c for both v1 and v2 and show that the velocity. Three spaceships shoot space probes at the speeds shown. common sense may tell you the rocket moves at the speed of light relative to you.5c) 4 [1 1 (0. 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 spaceship. A 5 B 5 C b.25 5 0. But it doesn’t! The relativistic addition of velocities (not covered in the chapter) is given by v v1 v2 vv c 1 2 1 2 Substitute 0. the speeds at which the flashes reach an observer to the right. A spaceship emits brief flashes of light at 1-second intervals. If you were Summary traveling in a high-speed spaceConcept •••••• ship. in front of the approaching spaceship b. B b.g. A.. (e.5c)2/c2] 5 c/1.5c 1 0. the speed of light reflected from the departing probes as seen by the spaceship Plug and Chug Rank the following quantities from greatest to least. B 17. A 5 B 5 C 15. an Earth observer •••••• 18. A.15. would metersticks on board appear contracted to you? Defend your answer. In your frame. C. The relative speed of you and sticks is zero.8c. 17. 298 298 . a. The circles represent light already emitted by the spaceship. the speeds of the spaceship as seen by you. there is no length contraction. Plug and Chug 18. C. B c. List them from left to right. a. If you were in a smooth-riding train with no windows. we see into the past. No. 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. When you shine the light in the direction the train is moving.” 27. Substitute small values of v1 and v2 in the preceding equation and show for everyday speeds that v is practically equal to v1 + v2. For small speeds. Time 25. Suppose you’re shining a light while riding on a train. Can an astronaut travel more slowly than the speed of light and yet travel the same distance in a 10-year trip? Explain. No. Can you get younger by traveling at speeds near the speed of light? Explain. Think and Explain •••••• 21. the speed of light is invariant. yes. 22. People who ride in a bus all know they’re moving through space. Yes. 23. 10. how does the speed of the ball appear to an observer standing at rest outside the train? (Does it increase or appear the same as if the observer were riding on the train?) CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 299 299 . Same. the effects of relativity always are attributed to the “other guy. 23. and so we always see distant places as they were when the light left—in the past. say. because of time dilation 26. v 5 (c 1 c)/[1 1 (c2/c2)] 5 2c/(1 1 1) 5 c ASSESS 15 ASSESS 20. Light travels a certain distance in. so v 5 (v1 1 v2)/(1 1 0) 5 v1 1 v2. What else are they moving through? 25. It takes time for light to travel from one place to another. 21. When you toss the ball in the direction the train is moving. 24. Explain why it is that when we look out into the universe. Suppose you’re playing catch with a friend in a moving train. 27. 26. If the spaceship in question 18 somehow travels at c relative to you. and it somehow fires its rocket at c relative to itself.19. But you know that they’re also moving through something else. by observing that the surface of water in a bowl is not horizontal 22. Think and Explain 19. 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?) 24.000 years. It appears to travel faster. v1v2/c2 < 0. use the equation to show that the speed of the rocket relative to you is still c! 20. 4 yr.00 3 108 m/s)] 5 6. Explain. 31.” where occupants of high-speed spaceships would depart from Earth for a few years and return centuries later.6c)2 / c2 5 1. Think and Solve •••••• 33. If you were in a high-speed spaceship traveling away from Earth at a speed close to that of light. Joe leaves on a space bus and takes a 5-year (space-bus time) round-trip at 0. a. you make similar measurements on Earthlings who measure their own heartbeats to be 1 s apart.5 m b. Joe Burpy is 30 years old and has a daughter who is 6 years old. no way of effectively shielding the occupants from the radiation resulting from the high-speed collisions with interstellar matter. t 5 t0 /√1 2 v2 / c2 5 1 s/ √1 2 (0. theirs would appear to be slower. his stay-athome daughter experiences t 5 (5 yr)/√1 2(0.99c)2/c2 5 35.5 meters long when at rest. then what is the shape according to observers on the rocket ship? 32.25 s c. length contraction shortens their journey. t 5 Lyou measure/v 5 1. What time do you measure between your own heartbeats? b. Her age will be 6 1 35.4 5 41. is 2. The shape is elliptical. 29. No means of propelling such a massive body to such speeds. t 5 1. a. 35. Same as above. Yes. How old will he and his daughter be when he returns? 300 300 For: Study and Review Visit: PHSchool. No relativistic changes occur in your own frame of reference. Show that you would measure a time of 6. Think and Solve 33. L 5 L0 √1 2 (v/c)2 5 2.3 ns for the length of his body to pass you.99c. and that you are in a spaceship that moves past Earth at 0.28. One of the fads of the future might be Concept Summary •••••• “century hopping. If stationary observers measure the shape of an emblem on fast-moving rocket ship as exactly circular.5 m/[0. Is it possible for a person to be biologically older than his or her parents? Explain. 32. What are the present-day obstacles to such a practice? 29.8c)2 / c2 5 1.25 s.4 yr. Assume that your heart normally beats once every second. because you’re at rest in your reference frame. To the electrons. c.com Web Code: csd – 4270 34.5 m 3 √12 (0.8 3 (3. As you and the spaceship whiz past Earth. 31. 30. . Due to time dilation. with the long axis in the direction of motion. From v 5 d/t. 30. if the person stays behind while the parents take a relativistic trip. The two-mile-long linear accelerator at Stanford University in California is less than a meter long to the electrons that travel in it.25 s apart. would you measure your normal pulse to be slower. How long will you measure him to be when he’s running by at 0. b. Same. You measure 1 s. 34. or faster? How would your measurements of pulses of friends back on Earth be if you could monitor them from your ship? Explain. Thomas.80c? b.3 3 1029 s 5 6. a rhino.3 ns 15 ASSESS (continued) 28. the same. a. How much time do you measure between their heartbeats? 35. Joe will be 30 1 5 5 35 years old when he returns. Show that your heartbeats are measured by someone on Earth to be 1.6c. a. 8 5 3. What cooking time will you measure for the egg? b. Yes. a. so no relativistic changes. you pack a meterstick in your luggage.50c. So Pinocchio decides to run past Gepetto fast enough that his 10-inch long nose will be seen by Gepetto to be only 2 inches long. Lizzie is scooting down the Interstate at 17 percent of the speed of light and measures the distance between mileposts to be less than 5. Pinocchio is concerned that Gepetto will see his long nose and realize that he has been lying. Your feet and shoes are in the same frame. If the meterstick is moving parallel to an observer resting on Hislaurels. What distance does she measure? b. L 5 L0 √12 (v/c)2 5 5280 ft 3 √12 (0. which is moving at 0. a. What distance would she measure at 32 percent the speed of light? 37. You are standing facing forward on the floor of your starship. so length measures L 5 L0√1 2 (v / c)2 5 1. both see the other’s nose shortened. Before leaving the planet Hislaurels for a starship voyage.00 m 3 √12 (0. will Pinocchio see Gepetto’s nose shortened? 40. The cook and the egg both age 3 min in their own frames of reference. Earth observers say feet and shoes contract by same amount so shoes should still fit—moving or not.60c. rather than overcooked? 37. Why should you not be surprised when the egg turns out to be perfectly cooked. so v 5 c √1 2 (2 / 10)2 5 0. you measured your feet to be 25 cm long. b.6c)2 / c2 5 3 min/0. you take the meterstick out of your bag. When running past Gepetto. tyou 5 tship/ √1 2 v2 / c2 5 3 min/ √1 2 (0. a. Do you need to be concerned now that the shoes that you packed for the trip will be too big? Teaching Resources • Computer Test Bank • Chapter and Unit Tests CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 301 301 . a. 40.60) 5 15 cm b. a. How fast must Pinocchio run? b. a.17c / c)2 5 5200 ft b.5c)2 / c2 5 0. √12 (0.280 feet. From L 5 L0 √1 2 (v / c)2. No relative motion between you and stick.8 min b. How long will you measure the meterstick to be? b. Someone aboard is making a 3-minute egg for breakfast. L 5 L0 √12 (v/c)2 5 5280 ft 3 38. An observer at rest sees the meterstick at v 5 0. a. v 5 c √1 2 (L / L0)2. how long will the observer measure the meterstick? 39. so you measure it to be 1 m long. Before you left Earth.50c.36. 39. ASSESS 15 ASSESS 36.87 m 38. A ship whizzes by you at 0. After the ship has settled down to a steady speed of 0.98c b. a.32c / c)2 5 5000 ft a.8c)2 / c2 5 25 cm(0. People on Earth will now measure your feet to be how long? b. a.80c relative to Earth. L 5 L0 √1 2 (v / c)2 5 25 cm √12 (0. Times differ in differently moving frames of reference.