The Basics of the Special Theory of Relativity

Transcription

1 17.2 The aic of the Special Theory of Relatiity Eintein pecial theory of relatiity changed our fundaental undertanding of ditance, tie, and a. He ued hi faou thought experient to illutrate thee new concept. Thi ection contain eeral thought experient iilar to the one Eintein ued. Thought Experient 1: Siultaneity Iagine that you are itting high on a hill on Canada Day and you can ee two different celebration going on in the ditance. You are tartled when two et of firework ignite at exactly the ae tie one off to your left and the other far to your right. bout 100 behind you, a car i traelling along a highway at 95 k/h. Do the paenger in the car ee the firework igniting iultaneouly or do they think that one et ignited before the other? Your iediate reaction i probably, Of coure they aw the firework igniting iultaneouly they were iultaneou! ccording to Eintein pecial theory of relatiity, howeer, the anwer i not quite o iple. To retate the quetion ore preciely, are two eent that are iultaneou for an oberer in one inertial reference frae iultaneou for oberer in all inertial reference frae? The anwer i no. The contancy of the peed of light create proble with the iultaneity of eent, a the ituation in Figure 17.9 illutrate. In Figure 17.9 ), oberer and are eated equiditant fro a light ource S). The light ource flahe. Since the light ut trael an equal ditance to both oberer, they would ay that they receied the flah at exactly the ae tie, that the arrial of the x flah wa iultaneou for both of the. Now iagine that thee two oberer are actually itting on a railway flatcar that i oing to the right with elocity relatie to the ground and to oberer C in Figure 17.9 ). Oberer C ake two oberation. 1. i oing away fro the point fro which the light wa eitted. 2. i oing toward the point fro which the light wa eitted. SECTION Decribe Eintein thought experient relating to the contancy of the peed of light in all inertial frae of reference, tie dilation, and length contraction. KEY TERMS iultaneity tie dilation proper tie dilated tie length contraction proper length relatiitic peed gaa S OUTCOMES Figure 17.9 Do eent that appear to be iultaneou to oberer and alo appear to be iultaneou to oberer C and D? S C x D 2 Chapter 17 Special Theory of Relatiity MHR 815

2 C D c t o rocket light pule Oberer C conclude that it take longer for light to reach than it doe to reach. Thu, according to oberer C, oberer receied the flah firt and receied it econd. The arrial are not iultaneou in C frae of reference, and yet it i an inertial reference jut a uch a i the frae of reference of the flatcar. In the frae of reference for oberer D, who i oing to the right with a elocity of 2, the flatcar i oing toward the left with a elocity of. Now, it i who i oing away fro the point fro which the flah wa eitted and i oing toward that eiion point. The light would take longer to reach, o the light would arrie at oberer firt. you can ee fro thi exaple, the whole concept of iultaneity, of pat, preent, and future, i fuzzy in relatiity. What i a future eent in one frae of reference becoe a pat eent in another. Thi i due entirely to the fact that the peed of light i the ae in all inertial frae of reference, regardle of their relatie elocitie. Thought Experient 2: Tie Dilation Figure If the peed of light i the ae to all oberer, then light take longer to trael fro to than it doe to trael fro to. light pule t c t 816 MHR Unit 7 Wae and Modern Phyic Iagine yourelf back on the hilltop, watching firework. You look at your watch at the oent that the firework ignite and it ay 11:23 P.M. What do the watche of the paenger in the car read? If they aw the firework ignite at different tie, their watche cannot poibly agree with your. The contancy of the peed of light create proble with tie interal. The ter tie dilation applie to ituation in which tie interal appear different to oberer in different inertial frae of reference. To undertand the iplication of thi contant peed of light for tie eaureent, aue that an experienter ha deied a light clock. In it, a pule of light reflect back and forth between two irror, and. The tie that it take for the pule to trael between the irror i the baic tick of thi clock. Figure ) how uch a tick. Now, picture thi clock in a pacecraft that i peeding pat Earth. n oberer in the pacecraft ee the light a reflecting back and forth a it wa before, o the baic tick of the clock ha not changed. Howeer, an oberer on Earth would ee that the irror oed

3 while the light pule traelled fro to, a hown in Figure ). Since the pule actually ha to trael fro to, it ut take longer, a indicated in Figure C). The tick of the clock therefore take longer to occur in the Earth frae of reference than in the pacecraft oberer frae of reference. In fact, if the pacecraft oberer wa wearing a watch, the Earth oberer would ay that the watch wa counting out the econd too lowly. The pacecraft oberer, howeer, would ay that the watch and the light clock were working properly. The relationhip between tie a eaured in the pacecraft and on Earth can be deduced fro Figure D). ue that c i the peed of light, which i the ae for all oberer t i the tie that the Earth oberer ay it take for the pule to trael between the irror t o i the tie that the pacecraft oberer ay it take for the pule to trael between the irror The ditance fro to would be c t o. The ditance traelled by the pacecraft would be t, ince thi inole a ditance, peed, and tie obered by the Earth oberer. The Earth oberer clai that the light pule actually traelled a ditance of c t. Thee ditance repreent the length of the ide of a right-angled triangle, a een in Figure D). Notice how iilar thi reult i to the arrial-tie equation in the boat X-boat Y cenario on page 743 and 744. pply the Pythagorean theore and expand. c t = c t o + t t 2 = t 2 o + 2 t 2 ELECTRONIC LERNING PRTNER Relatiity Experient with near-light peed and tie dilation by uing your Electronic Learning Partner. Sole for t 2 o. Factor out a t 2. t 2 o = c2 t 2 2 t 2 t 2 o = t2 2 ) Diide by. Siplify, then take the quare root of both ide of the equation. Sole for t. t 2 o = t2 2 ) ) to 2 = t2 t o = t t o In any quetion inoling relatiitic tie, it i iportant to carefully identify the tie. t o i the tie a eaured by a peron at ret relatie to the object or the eent. It i called the proper tie. You could think of it a the ret tie, although thi ter i not generally ued. nother way to picture it i a the one-point tie, the tie for an oberer who ee the clock a taying at only one point. Math Link Note that the negatie quare root ha no eaning in thi ituation. oth tie will be een a poitie. In addition, ut be le than c. If it wa greater than c, the denoinator would becoe the quare root of a negatie nuber. lthough uch a quare root can be expreed uing coplex nuber, it i not expected that a tie eaureent would inole anything other than the et of real nuber. Chapter 17 Special Theory of Relatiity MHR 817

4 t i the expanded or dilated tie. Since the denoinator i le than one, t i alway greater than t o. It can alo be thought of a the two-point tie, the tie a eaured by an oberer who ee the clock a oing between two point. DILTED TIME The dilated tie i the quotient of the proper tie and the expreion: quare root of one inu the elocity of the oing reference frae quared diided by the peed of light quared. Quantity Sybol SI unit dilated tie t econd) proper tie t o econd) elocity of the oing etre per econd) reference frae peed of light c etre per econd) Unit nalyi econd econd = etre econd etre econd t o = econd = = MODEL PROLEM Relatie Tie rocket peed pat an ateroid at c. If an oberer in the rocket ee 10.0 pa on her watch, how long would that tie interal be a een by an oberer on the ateroid? Frae the Proble Proper tie, t o, and dilated tie, t, are not the ae. Tie interal appear to be horter to the oberer who i oing at a elocity cloe to the peed of light. Proper tie, t o, and dilated tie, t, are related by the peed of light, c. PROLEM TIP Since 2 2 i a ratio, the peed can c hae any unit a long a they are the ae for both the nuerator and the denoinator. It i often ueful to expre in ter of c. 818 MHR Unit 7 Wae and Modern Phyic

5 Identify the Goal The aount of tie, t, that pae for the oberer on the ateroid while 10.0 pae for the oberer on the rocket Variable and Contant Known Iplied Unknown rocket = c c = t t o = 10.0 Strategy Select the equation that relate dilated tie to proper tie. Subtitute into the equation. Sole. Calculation t o c) t 16.7 The tie a een by an oberer on the ateroid would be Validate The dilated tie i expected to be longer than the proper tie, and it i. PRCTICE PROLEMS 1. tau τ) particle ha a lifetie eaured at ret in the laboratory of If it i accelerated to c, what will be it lifetie a eaured in a) the laboratory frae of reference, and b) the τ particle frae of reference? 2. rocket pae by Earth at a peed of c. If a peron on the rocket take 245 to drink a cup of coffee, according to hi watch, how long would that ae eent take according to an oberer on Earth? 3. kaon particle κ) ha a lifetie at ret in a laboratory of t what peed ut it trael to hae it lifetie eaured a ? Thought Experient 3: Length Contraction Iagine the following ituation. Captain Quick i a coic book hero who can run at nearly the peed of light. In her hand, he i carrying a flare with a lit fue et to explode in 1.50 µ ). The flare ut be placed into it bracket before thi happen. The ditance L) between the flare and the bracket i 402. Chapter 17 Special Theory of Relatiity MHR 819

6 flare bracket L oberer bracket L Figure race againt tie Captain Quick run at 2 3 c /) and arrie at the bracket in tie. ccording to claical echanic, thi would not be poible becaue it hould take 2.01 µ a hown on the right. Howeer, to an oberer in the tationary frae of reference, the tie for the fue to burn will be dilated in relation to hi own frae of reference. It will take 2.01 µ for the fue to burn and therefore, Captain Quick will reach the bracket in tie. Since Captain Quick and the fue are in the ae frae of reference, howeer, Captain Quick hould obere the fue burning in 1.50 µ. How did he ake it in tie? Then he realized that the only way he could hae arried in tie wa if the ditance to the bracket in her oing frae of reference wa le than the 402 in the tationary frae. The ditance ut hae been ultiplied by the ae factor by which the tie wa diided in the oberer frae of reference. L or 2.01 µ t o c L = L o L = 402 ) L = 402 )0.7454) L = c 820 MHR Unit 7 Wae and Modern Phyic

7 If the ditance wa aller, then Captain Quick could ake it to the bracket before the fue burned out. L or 1.50 µ Thi thought experient illutrate that two idea go hand in hand. If two oberer are oing relatie to each other, then a tie dilation fro one oberer point of iew will be balanced by a correponding length contraction fro the other oberer point of iew. In the box below, L o repreent the proper length, which i the length a eaured by an oberer at ret relatie to the object or eent and L i the contracted length een by the oing oberer. LENGTH CONTRCTION The contracted length i the product of the proper length and the expreion, quare root of one inu the elocity of the oing reference frae quared diided by the peed of light quared. L = L o Quantity Sybol SI unit contracted length L etre) proper length L o etre) elocity of the oing etre per econd) reference frae peed of light c etre per econd) Unit nalyi etre = etre = etre econd etre econd = etre = Note: Length contraction applie only to length eaured parallel to the direction of the elocity. Length eaured perpendicular to the elocity are not affected. Chapter 17 Special Theory of Relatiity MHR 821

8 Thi thought experient ee to yield trange reult that go againt coon experience. Howeer, the reult explain a phenoenon inoling a tiny particle called the u eon or uon). Thi particle ha a lifetie of and i fored about aboe the urface of Earth, peeding downward at about c. t that peed according to claical echanic), it hould trael only about 660 before decaying into other particle, but it i obered in great nuber at Earth urface. The relatiitic explanation i that the uon lifetie a eaured by Earth-baed oberer ha been dilated a follow. t o c) The ditance traelled becoe d = t d = 0.998) ) ) d = t that peed, the uon lifetie i o expanded according to the oberer on Earth) that the particle can reach the urface. On the other hand, the uon ee it own lifetie a unchanged, and fro it frae of reference, Earth urface i ruhing toward it at c. The ditance it ee to Earth urface i gien by L = L o L = c)2 ) L = )0.0632) L = 632 Thi reduced ditance would take a horter tie, gien by x ) ) The uon therefore can reach Earth urface before decaying. 822 MHR Unit 7 Wae and Modern Phyic

9 Which I Correct? The phyicit tanding on the urface of Earth clai that the lifetie of the uon i and it height aboe Earth urface i Fro the uon point of iew, howeer, it lifetie i and it height i 632. Which i correct? oth tateent are correct. The alue of any eaureent i tied to the frae of reference in which that eaureent i taken. Going fro one inertial frae of reference to another will inole difference in the eaureent of length and tie. Norally, thee difference are too all to be obered, but a relatie peed approach the peed of light, thee difference becoe quite apparent. Gaa Sae Tie When oling proble inoling relatiitic peed peed approaching the peed of light), you will often need to calculate 1 the alue of. Phyicit hae aigned the ybol 1 gaa γ ) to thi alue, or γ =. Uing the γ notation, the length and tie equation becoe γ t o and L = L o γ. MODEL PROLEM Relatiitic Length pacecraft pae Earth at a peed of /. If oberer on Earth eaure the length of the pacecraft to be 554, how long would it be according to it paenger? Frae the Proble Length appear to be horter, or contracted, to the oberer who i oing relatie to the object being eaured. The aount of length contraction that occur i deterined by the relatie peed of the reference frae of the two oberer. Identify the Goal The length of the pacecraft, L o, a een by it paenger Variable and Contant Known Iplied Unknown = c = L o L = 554 continued Chapter 17 Special Theory of Relatiity MHR 823

10 continued fro preiou page Strategy Calculate gaa. Calculation 1 γ = γ = ) )2 γ = Ue the equation that decribe length contraction. L = L o γ L o = Lγ Sole. L o = 554 )1.342) L o 743 The length of the pacecraft a een by it paenger i 743. Validate The proper length i expected to be longer than the contracted length, and it i. PRCTICE PROLEMS 4. n ateroid ha a long axi of 725 k. rocket pae by parallel to the long axi at a peed of c. What will be the length of the long axi a eaured by oberer in the rocket? 5. n electron i oing at 0.95 c parallel to a etre tick. How long will the etre tick be in the electron frae of reference? 6. pacecraft pae a pherical pace tation. Oberer in the pacecraft ee the tation iniu diaeter a 265 and the axiu diaeter a 325. a) How fat i the pacecraft traelling relatie to the pace tation? b) Why doe the tation not look like a phere to the oberer in the pacecraft? PHYSICS FILE Eintein equation allow a particle to trael fater than light if it wa already traelling fater than light when it wa created. For uch particle called tachyon ), the peed of light repreent the lowet peed liit. lthough the equation ay that tachyon can exit, there i no eidence that they do. In fact, no one know how they would interact with noral atter. The Unieral Speed Liit Calculation of expanded tie and contracted length inole the expreion 2. Since tie and length are eaureent, c they ut be repreented by real nuber, o the alue under the quare root ut be a poitie real nuber. For thi to be true, 2 < 1. Thi iplie that < c. If approache c, the alue of gaa approache infinity. Conider what happen to t when t approache c in o. The denoinator approache zero. 824 MHR Unit 7 Wae and Modern Phyic

11 Diiion of a non-zero real nuber by zero i undefined o an object peed ut be le than the peed of light. Thi peed liit applie only to aterial object. Obiouly, light can trael at the peed of light. lo, once a light pule ha been lowed down by paing into a ediu uch a water, object can trael fater through that ediu than can the pule. The blue glow called Cerenko radiation ) eanating fro water in which radioactie aterial i being tored i created by highpeed electron beta particle) that are traelling through the water fater than the peed of light through water. Thi phenoenon i oetie copared to onic boo, in which particle in the for of a jet airplane) are traelling fater than the peed of ound in air. Figure The blue glow fro thi torage pool in a nuclear generating tation coe fro particle that are traelling through the water fater than the peed of light through water Section Reiew 1. K/U a) Explain what i eant by an inertial frae of reference. b) Would a rotating erry-go-round be an inertial frae of reference? Gie reaon for your anwer. 2. K/U Explain the eaning of the ter proper length and proper tie. 3. I n arrow and a pipe hae exactly the ae length when lying ide by ide on a table. The arrow i then fired at a relatiitic peed through the pipe, which i till lying on the table. Deterine whether there i a frae of reference in which the arrow can a) be copletely inide the pipe with extra pipe at each end b) oerhang the pipe at each end Gie reaon for your anwer. 4. K/U Explain the eaning of the ter length contraction and tie dilation. 5. C Explain why the reult of the Michelon-Morley experient were o iportant. continued Chapter 17 Special Theory of Relatiity MHR 825

12 continued fro preiou page 6. a) In the diagra, two tar and ) are equiditant fro a planet P) and are at ret relatie to that planet. They both explode into noa at the ae tie, according to an oberer on the planet. Fro the point of iew of paenger in a rocket hip traelling pat at relatiitic peed, howeer, which tar went noa firt? Gie reaon for your anwer. What will be the peed of that ball relatie to the ground? K/U b) Where could the oberer tand on the b) If the elocity of the otorcycle relatie to the ground i g, the elocity of the ball relatie to the otorcycle i b, and the elocity of the ball relatie to the ground i bg, tate the ector equation for calculating the elocity of the ball relatie to the ground. c) pply thi forula to a ituation in planet in order to ee both tar at the ae tie? P which the otorcycle i traelling at 0.60 c and the ball i thrown forward with a peed of 0.80 c. What i the peed of the ball relatie to the ground? What i wrong with thi anwer? x x d) In the pecial theory of relatiity, the P forula for adding thee elocitie i + bg = b b g 1 + c2 g What doe thi forula predict for the anwer to c)? What doe thi forula predict for the anwer to a)? Iagine that you are traelling in your car at a peed of 0.60 c and you hine a light bea ahead of you that trael away fro you at a peed of c. ccording to thi forula, what would be the peed of that light bea relatie to the ground? 7. Part ) of the diagra how a tar S) located at the idpoint between two planet ) and ), which are at ret relatie to the tar. The tar explode into a upernoa. K/U a) In the frae of reference of the planet, which planet aw the upernoa firt? Gie reaon for your anwer. b) pacecraft i paing by a hown in part ). In it frae of reference, the tar and planet are oing a hown in part C). In the pacecraft frae of reference, which planet aw the upernoa firt? Gie reaon for your anwer. S x x UNIT PROJECT PREP How would the general public hae receied the new inforation in Eintein pecial theory of relatiity? Do you beliee that at the turn of the twentieth century ociety had ore or le faith in cience than people do today? Why or why not? Draatic eent often teer thinking into new direction. Do you beliee that Eintein wa affected by any one particular eent a he deeloped hi theorie? re you able to link recent ocietal eent with current change in the direction of cientific reearch? S C 8. I S a) Iagine that you are riding along on a otorcycle at 22 / and throw a ball ahead of you with a peed of 35 /. 826 MHR Unit 7 Wae and Modern Phyic

13 CREERS IN PHYSICS Not Een the Sky the Liit! When a ignal leae a atellite or interplanetary pace probe, a pecial code i ebedded in it to gie it a tie-tap. When the ignal i picked up on Earth, that tie-tap i copared to a terretrial clock. Subtracting the two gie the trael tie between the atellite or pace probe and the ground tation. Since the ignal trael at the peed of light, all you hould need to do then i ultiply the tie by c to deterine the ditance but it not that iple. Graity i part of the proble. ccording to Eintein general theory of relatiity, clock run ore lowly in a graitational field. The tronger the field i, the lower the clock run. Clock on board pacecraft and atellite run lightly fater in interplanetary pace than they do near Earth. Thee tiing difference reult in ditance eaureent difference between what i obered fro a ground tation and fro a pacecraft. The ituation becoe een ore coplicated a the pacecraft dip into and out of the graitational field of planet that it encounter on it oyage. econd proble reult fro the relatie elocity between the pacecraft and the ground tation. Eintein pecial theory of relatiity, dicued in detail in thi chapter, decribe how tie interal and ditance eaureent ary between inertial frae of reference that are in otion relatie to each other. Thi relatie elocity i continually changing a a reult of the graity of the Sun and planet and due to Earth orbital and rotational elocitie. Relatiitic correction nuerical adjutent baed on the theory of relatiity are an ongoing challenge in pacecraft intruent deign. There are a whole uite of career that utilize thee thing, Stee Lichten, anager of the Tracking Syte and pplication Section of NS Jet Propulion Laboratory, ay of relatiity. Einteinian phyic i no longer the ole property of unierity reearcher. Coercial atellite anufacturer ut hae an undertanding of relatiity in order for their product to work. Theoretician, engineer, and coputer cientit ut work together to help a pacecraft counicate with it ground tation, o the copanie that anufacture pacecraft and coercial atellite are alway on the lookout for people with the neceary knowledge. Generally, an adanced graduate degree in engineering, phyic, or atheatic i preferred, although a bachelor degree in cience with a deontrated undertanding of the concept and technique inoled will go a long way. So bruh up your ath kill and keep doing thoe thought experient. Soe day, they ight take you to the tar! Going Further 1. Decribe oe exaple of atellite that require extreely precie ditance and tie eaureent. Explain why uch preciion i neceary for thoe atellite. 2. Many copanie that anufacture atellite or equipent for ue on atellite including pace tation) offer uer internhip progra for intereted tudent. Find out if any of thee copanie are located near you and call the. You ight be able to get a head tart on a great career! 3. Reearch the pace probe, uch a the one hown in the photograph, that are currently actie. Explain why precie knowledge of tie interal and ditance i of extree iportance to the operation of pace probe. Web Link For inforation about the NS Jet Propulion Laboratory pat, current, and planned pace iion, go to the aboe Internet ite and click on Web Link. Chapter 17 Special Theory of Relatiity MHR 827