11.1 The Special Theory of Relativity

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1 Figure 1 Albert Einstein s ideas in phsis hanged our pereption of spae and time The Speial Theor of Relativit At the turn of the twentieth entur, most of the phsis ommunit enjoed a sense of aomplishment. Newtonian mehanis (also alled lassial mehanis), based on Newton s laws of motion, provided the priniples b whih all matter behaved. Phsiists epanded these priniples to the atomi level and established the idea of energ onservation. Building on James Clerk Mawell s mathematial desription of eletri and magneti fields, phsiists suessfull unified the subjets of eletriit, magnetism, and optis. Mawell s ideas had an apparent answer to the true nature of light: light is a ombination of osillating eletri and magneti fields. At that time, onl a few questions remained. The nature of the atom was still unlear. Phsiists had observed that light emitted from perfet radiators (alled blakbodies, whih ou will learn more about in Chapter 12) did not behave as predited. This gave rise to the so-alled blakbod problem. The phenomenon b whih metals give up eletrons under ertain kinds of light also had phsiists mstified. Moreover, ontrar to Mawell s assumption that eletromagneti waves, like all waves, needed a medium, there was no evidene to support this. Phsiists epeted to solve these problems within the framework of familiar phsial models and priniples. However, new models and priniples needed to be developed in order to eplain these problems, and these new ideas radiall hanged the wa phsiists understood their one-familiar universe. One of the main ontributors to these new models was Albert Einstein ( ) (Figure 1). CAREER LINK The Priniple of Relativit Suppose ou are inside a room that has no windows. A billiard table stands near the entre of the room. If ou strike the ue ball with the ue stik, ou will notie that the ball moves in the diretion of the applied fore. This is eatl what ou would epet to happen, aording to Newton s seond law of motion (F > 5 ma > ). Similarl, ou observe that linear momentum is onserved when the ue ball ollides with another ball. You will also note that mehanial energ is onserved if the ollision is elasti. Everthing that ou observe is onsistent with the priniples and laws of Newtonian mehanis. What ou do not know is that the room is not simpl a windowless room in a building it is in a railwa ar that an move on a set of traks. Now suppose that the railwa ar starts to move, and for a brief time aelerates from zero to a final onstant veloit. You will notie during this time that the laws of motion will appear different in the room. The billiard balls on the table will roll bakward on their own, as if ated upon b an unseen fore. You ma even have trouble walking beause of the sensation of a fore ating on ou. All of this is ourring beause everthing in the room is aelerating along with the railwa ar. B the time the ar is moving along a straight, level streth of smooth trak at a onstant speed, the onditions in the ar are the same as when it was at rest. You annot feel or hear the motion of the ar. The railwa ar is no longer aelerating, and nothing inside it not ou, not the table, not the balls on the table is aelerating either. What happens, then, if ou hit the ue ball with the stik in eatl the same wa as before? The answer is that the ball moves in eatl the same wa. Nothing about the motion of the balls, or an objet inside the room, indiates that the room is moving. The reason is that all objets inside the railwa ar share the motion of the ar. The relative motion between ou and the ball is the same for all onstant veloities of the railwa ar. This inludes the speial ase of zero veloit when the railwa ar is at rest. In fat, ou annot tell from the motion of an objet inside the railwa ar, inluding ourself, that the railwa ar is moving, as long as the railwa ar does not aelerate. 574 Chapter 11 Relativit NEL

2 The railwa ar in this situation is an eample of a frame of referene. Reall from earlier hapters that a frame of referene is a oordinate sstem that ou an use to observe and desribe motion. In this ase, ou ould measure the motion of the ue ball with respet to the referene frame. In partiular, no eternal fore ats on the objets in the frame of referene beause the frame (the railwa ar) does not aelerate. This is the ondition desribed in Newton s first law of motion, or the law of inertia. An frame of referene that is at rest or moves with a onstant veloit is an inertial frame of referene. Within an inertial frame, moving with onstant veloit v > (v > 5 0 is a speial ase of onstant veloit), mehanial laws appl in the same wa that the would if there were no movement. Would ou epet these laws to remain unhanged for an observer in a different inertial frame? As it turns out, there is onl one differene between two inertial referene frames. For an observer in one frame, the veloities of objets in the other frame must be added, b vetor addition, to the veloit with whih that frame moves awa from the first observer. This hanges the wa in whih the motion of the objet appears to the first observer. An eample is illustrated in Figure 2. One orreted, however, b onsidering the veloities of the two referene frames, the phsial results beome the same for both referene frames. This situation provides the basis for the priniple of relativit: for all inertial frames of referene, the laws of Newtonian mehanis are the same. The main idea of the priniple of relativit has a long histor, going bak at least to the work of Galileo in the 1600s. frame of referene a oordinate sstem relative to whih motion is desribed or observed inertial frame of referene a frame of referene that moves at zero or a onstant veloit; a frame in whih the law of inertia holds priniple of relativit the laws of motion are the same in all inertial frames referene frame 1 referene frame 2 observer 1 v v observer 2 (a) Figure 2 (a) When observer 1 throws a ball upward, he observes that the ball s motion is purel along the vertial () diretion. (b) When the railwa ar for observer 1 has a speed v relative to observer 2, the ball appears to undergo projetile motion with a displaement along both and in referene frame 2. However, as viewed b observer 1 using his referene frame and oordinates and, the ball s motion is still purel vertial, as in (a). (b) An two ars moving at different speeds on a straight road an eah define a different inertial frame of referene. Aelerating objets, suh as planets in orbit about their star and satellites in Earth orbit, are not inertial referene frames, but the are referene frames that move with respet to eah other. Earth s surfae is a non-inertial (aelerating) referene frame beause it spins on its ais, rotates about the Sun, moves through the gala as a part of the solar sstem, and so on. However, beause the aelerations involved in these motions are small, Earth s surfae is ver lose to being an inertial referene frame. We will use Earth s surfae to define an inertial frame of referene throughout this hapter. NEL 11.1 The Speial Theor of Relativit 575

3 Mini Investigation Understanding Frames of Referene Mini Investigation Skills: Performing, Observing, Analzing, Communiating In this ativit, ou will observe ations at slow, everda speeds. You will then desribe how the laws of Newtonian mehanis hange for objets in different frames of referene that move with respet to eah other. You will need to work in groups for this ativit. Equipment and Materials: small rubber or plasti ball 1. Have one member of the group stand still and toss the ball forward to a seond group member. Observe how the speed of the ball depends on the speed with whih the ball is tossed. 2. The student throwing the ball now walks slowl forward while tossing the ball to the seond student, taking are to throw the ball with the same speed as in Step Repeat Step 2, but with the student walking quikl forward as the ball is tossed. 4. Repeat Step 2, but with the student walking slowl bakwards as the ball is tossed. SKILLS HANDBOOK A2.5 A. Based on our observations in Step 2, infer how the speed of the ball depends on the speed with whih the ball is tossed and the speed of the student tossing the ball. K/U T/I B. Compare what happened in Step 3 with what happened in Step 2. What an ou infer about how the speed of the ball depends on the speed with whih the ball is tossed and the speed of the student tossing the ball? K/U T/I C. Compare what happened in Step 4 with what happened in Step 2. What an ou infer about how speed and diretion affet the speed of the ball? K/U T/I D. If ou know the speed of an objet in one inertial referene frame, an ou determine its speed in another inertial referene frame? Eplain our answer. T/I A ether the proposed medium through whih eletromagneti waves were one believed to propagate Is the Priniple of Relativit Universal? Mawell s mathematial desription of eletri and magneti fields unified the subjets of eletriit and magnetism but worked under the assumption that eletromagneti waves needed a medium through whih to travel. The medium was alled the luminiferous ether, or just ether. At the time of Mawell s theor, phsiists thought that ether filled the vauum of spae, had no mass, and had no drag effet on the motions of the planets. For 200 ears, phsiists knew that Newton s laws of motion remained the same, or were invariant, in all inertial frames. However, eletromagneti waves seemed to be different. As it was then understood, Mawell s theor stated that the speed of light with respet to the ether was alwas m/s. This meant that if ou were at rest and observed a light soure move relative to the ether, ou would measure the speed of light to be different from, depending on where ou were standing (Figure 3). This is similar to the motion of waves in other media: an observer moving through the medium would measure the veloit of the waves to be different from what an observer at rest with respet to the medium would measure. WEB LINK v rear front rear front net speed of light net speed of light at rear of ar v at front of ar v (a) (b) Figure 3 (a) When the railwa ar is at rest, the speed of light from the fl ashlight in both diretions is the speed of light with respet to the ether. (b) When the railwa ar moves at speed v relative to the ether, we would epet a stationar observer in front of the ar to measure 1 v for the speed of light beause it takes less time for light to reah the observer. B similar reasoning, we would epet a stationar observer behind the ar to measure 2 v for the speed of light beause it takes longer for light to reah the observer. 576 Chapter 11 Relativit NEL

4 The Development of Einstein s Postulates Eperimental evidene did not support the idea that the speed of light varied with the speed of the inertial frame. Eperiments performed b Mihelson and Morle in 1887, and b Trouton and Noble between 1901 and 1903, to tr to detet the ether showed no hange in the speed of light with the motion of Earth. Ultimatel, the results of these and similar eperiments suggested that eletromagneti waves do not require a medium in whih to propagate, and that the eistene of ether ould not be proven eperimentall. What were the impliations of the absene of ether? For one thing, the motion of the referene frame in whih the speed of light was being measured did not seem to affet that measurement. What did these eperimental results sa about the laws of phsis that govern the motion of material objets suh as eletrons and baseballs? Einstein was not onvined that eletromagneti phenomena depended on the motion of the referene frame. He eamined Mawell s ideas as applied to a frame-ofreferene eperiment that required onl a magnet and a losed oil of wire. Einstein used a method alled a thought eperiment, whih is an eperiment arried out in the imagination but not atuall performed. A thought eperiment eamines the logi behind a hpothesis, theor, or priniple b analzing a virtual version of an eperiment. The goal of a thought eperiment is to illustrate or eplore the onsequenes of a theor. Thought eperiments are useful when the atual eperiment ma be diffiult or impossible to perform, or when the researher wants to emphasize ertain details or unusual preditions of a theor. WEB LINK In the magnet-and-oil thought eperiment, Mawell s theor predits that when a magnet moves toward a oil of wire, an eletri field forms near the moving magnet. This eletri field moves harges within the oil, thus induing an eletri urrent. However, Mawell s theor also predits that if the oil moves and the magnet remains at rest, a urrent eists in the oil, not beause an eletri field forms, but beause the magneti field eerts a fore on the harges in the moving oil. To Einstein, it seemed illogial that seleting a frame of referene in whih either the magnet or the oil is at rest would hange the wa we understand what is happening. Wh would moving the magnet near a oil be different from moving the oil near a magnet? Nothing else in the universe desribed b phsis behaved in this wa. Given this unsatisfator onsequene of lassial eletromagnetism, as well as the lak of evidene for the ether, Einstein deided to address the problem in the most fundamental and straightforward wa possible. He began with the basi assumption that eletromagnetism, like Newtonian mehanis, did not hange when transformed between inertial frames. That is, the laws were the same in one inertial frame as in another. This meant that basi onservation priniples energ and momentum held in all frames. Then he proposed two postulates, or onditions believed or known to be true at the time. From these, the speial theor of relativit was developed. thought eperiment a mental eerise used to investigate the potential onsequenes of a hpothesis or postulate postulate a statement assumed to be true from whih a theor is developed Postulate 1: The Priniple of Relativit The laws of phsis are the same in all inertial frames of referene. No phsis eperiment an ever determine whether ou are at rest or moving at a onstant veloit. Postulate 2: The Speed of Light Priniple There is at least one inertial frame of referene in whih, for an observer at rest in this frame of referene, the speed of light,, in a vauum is independent of the motion of the soure of the light. NEL 11.1 The Speial Theor of Relativit 577

5 Postulate 1 states that eah inertial observer an onsider himself or herself to be at rest. If nature behaves in a ertain wa in one inertial frame, it should behave in the same wa in all inertial frames. You should not be able to detet inertial motion inside a losed room using eperiments of an kind. While not proven diretl, this is a reasonable guess. Postulate 2 is also reasonable. The speed of a wave is alwas independent of the speed of the soure. Sound waves, water waves, and earthquake waves travel at a fied speed relative to the medium through whih the are travelling. In Postulate 2, Einstein assumed that there eists a frame of referene in whih light waves behave in the same wa as other waves, regardless of whether a medium to arr them atuall eists. Separatel, the postulates were not etraordinar. When onsidered together, however, the two postulates led to an entirel new understanding about our universe. Postulate 1 implies that if Postulate 2 is true in one inertial frame of referene, it must be true in all frames of referene. Essentiall, the propert of light that is true for all models of waves annot be true in one frame and false in another; otherwise, the laws of phsis of light waves would differ in the two referene frames. The onsequene of this is that the speed of light must be onstant and the same in all inertial frames of referene beause the laws of phsis do not prefer one frame of referene over another. Put another wa, the speed of light is the same even when an observer is moving relative to the soure of the light. If one observer sees a soure of light in motion and another observer sees the soure at rest, both measure the speed of the light as. If ou move ver rapidl toward or awa from a flashlight, the speed of light ou measure is, not 1 v or 2 v. This result seems absurd, et it is the logial result of Einstein s postulates. Einstein published his work in 1905 in a paper titled On the Eletrodnamis of Moving Bodies. This paper was the basis for the speial theor of relativit. Speial Theor of Relativit All phsial laws are the same in all inertial frames of referene, and the speed of light is independent of the motion of the light soure or its observer in all inertial frames of referene. The priniple that the speed of light in a vauum is the same in all frames of referene is an essential feature of the speial theor of relativit. In fat, the laws of eletromagnetism did not hange at all as a result of speial relativit, but sientists understanding of the laws had to hange. The laws of phsis no longer made sense, and the understanding of spae and time had to hange. Einstein realized that the simplest and most natural wa to develop relativisti phsial laws would require a new understanding of spae and time. This understanding was not speifi to light it would affet all natural phenomena. We will onsider a number of features of the speial theor of relativit. First, the speed of light is muh greater than the speeds of objets that ou observe from da to da. As a result, in our everda life, ou annot observe the unusual effets upon spae and time hinted at in this setion. Another important outome is that, while advanes in eperimental tehniques at the beginning of the twentieth entur made measurements of the speed of light more aurate, it was still diffiult to test the results of the speial theor of relativit. Einstein used some of these eperimental results to draw his onlusions and then relied on thought eperiments to test the postulates and omplete the theor. Throughout the twentieth entur, numerous eperiments have onfirmed speial relativit. For eample, global positioning satellite sstems would not work auratel if the tehnolog did not take speial relativit into aount. You will learn more about this in Setion Chapter 11 Relativit NEL

6 11.1 Review Summar A frame of referene is a oordinate sstem in whih we an observe and measure the motion of an objet. A frame of referene that moves with a onstant veloit in whih the law of inertia holds is an inertial frame of referene. The priniple of relativit states that the laws of phsis are the same in all inertial frames of referene. The ether is a hpothetial medium through whih eletromagneti waves were thought to propagate. Tests suh as the Mihelson Morle eperiment failed to verif that there is an ether. Einstein s speial theor of relativit is based on two postulates: The laws of phsis are the same in all inertial frames of referene, and in at least one inertial frame of referene the speed of light is independent of the motion of the soure of the light. One result of Einstein s postulates is that the speed of light is the same in all inertial frames of referene, regardless of their veloities. Questions NEL 1. A student on inline skates moves with a onstant speed along the dek of a ruise ship, whih is moving with onstant veloit parallel to the shoreline. T/I C (a) Identif three distint frames of referene from whih to view the skater. (b) Desribe the skater s motion in eah referene frame. 2. Consider the properties of inertial and non-inertial frames of referene. K/U C (a) How does an inertial frame of referene differ from a non-inertial frame of referene? (b) Give two eamples of eah tpe of frame of referene. 3. Suppose an astronaut in a roket moving at 0.5 along Earth s surfae shines a light forward from the roket. K/U T/I A (a) Calulate the speed of the light ompared to that of the astronaut. (b) Calulate the speed of the light ompared to that of Earth s surfae. 4. Gabor stands on a art that moves with a onstant veloit. In his frame of referene, he tosses a ball vertiall up and down. Lutaaq stands on the ground nearb. She also tosses a ball vertiall up and down with respet to her frame of referene. T/I C A (a) Eplain what the motion of Gabor s ball looks like from Lutaaq s frame of referene. (b) Eplain what the motion of Lutaaq s ball looks like from Gabor s frame of referene. 5. What feature of Einstein s oil and magnet thought eperiment did he find troubling? K/U 6. State the two postulates of the speial theor of relativit. K/U C 7. What onlusion results from the ombination of Einstein s two postulates? K/U C 8. Eplain what a thought eperiment is and give an eample. K/U C 9. You are travelling in a spaeraft without windows. You are also far from an planets or stars. Desribe an eperiment that ou ould perform to determine whether ou are in an inertial or a non-inertial frame of referene. K/U C 10. Suppose ou are inside a windowless railwa ar. A billiard table is at the entre of the ar. K/U C (a) While rolling a ue ball forward, ou notie that the ball slows down suddenl, even though ou have not applied a bakward fore on the ball. Eplain the motion of the ball in terms of inertial or non-inertial frames of referene. (b) While rolling a ue ball forward, ou notie that the ball rolls to the right (Figure 4), even though ou have not applied a sidewas fore on the ball. Eplain the motion of the ball in terms of inertial or non-inertial frames of referene. Figure The Speial Theor of Relativit 579