Relativity Kinematics Harvard University

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XI 2 CHAPTER 11 RELATIVITY KINEMATICS, two topics kinematics and dynamics Kinematics deals with lengths times speeds etc. It is concerned only with the space and time coordinates of an abstract particle and not. with masses forces energy momentum etc Dynamics on the other hand does deal with. these quantities This chapter covers kinematics Chapter 12 covers dynamics Most of the. fun paradoxes fall into the kinematics part so the present chapter is the longer of the two. In Chapter 13 we ll introduce the concept of 4 vectors which ties much of the material in. Chapters 11 and 12 together,11 1 Motivation, Although it was obviously a stroke of genius that led Einstein to his theory of relativity. it didn t just come out of the blue A number of things going on in 19th century physics. suggested that something was amiss There were many efforts made by many people to. explain away the troubles that were arising and at least a few steps had been taken toward. the correct theory But Einstein was the one who finally put everything together and he. did so in a way that had consequences far beyond the realm of the specific issues that people. were trying to understand Indeed his theory turned our idea of space and time on its head. But before we get into the heart of the theory let s look at two of the major problems in. y y late 19th century physics 2, x x 11 1 1 Galilean transformations Maxwell s equations. S S Imagine standing on the ground and watching a train travel by with constant speed v in the. x direction Let the train frame be S 0 and the ground frame be S as shown in Fig 11 1. Figure 11 1 Consider two events that happen on the train For example one person claps her hands. and another person stomps his feet If the space and time separations between these two. events in the frame of the train are x0 and t0 what are the space and time separations. x and t in the frame of the ground Ignoring what we ll be learning about relativity. in this chapter the answers are obvious well in that incorrectly obvious sort of way as. we ll see in Section 11 4 1 The time separation t is the same as on the train so we have. t t0 We know from everyday experience that nothing strange happens with time. When you see people exiting a train station they re not fiddling with their watches trying. to recalibrate them with a ground based clock, The spatial separation is a little more exciting but still nothing too complicated The. train is moving so everything in it in particular the second event gets carried along at. speed v during the time t0 between the two events So we have x x0 v t0 As. a special case if the two events happen at the same place on the train so that x0 0. then we have x v t0 This makes sense because the spot on the train where the events. occur simply travels a distance v t by the time the second event happens The Galilean. transformations are therefore, Also nothing interesting happens in the y and z directions so we have y y 0 and.
The principle of Galilean invariance says that the laws of physics are invariant under. the above Galilean transformations Alternatively it says that the laws of physics hold in. 2 If you can t wait to get to the postulates and results of Special Relativity you can go straight to Section. 11 2 The present section can be skipped on a first reading. 11 1 MOTIVATION XI 3, all inertial frames 3 This is quite believable For example Newton s second law holds in all. inertial frames because the constant relative velocity between any two frames implies that. the acceleration of a particle is the same in all frames. Remarks Note that the Galilean transformations aren t symmetric in x and t This isn t auto. matically a bad thing but it turns out that it will in fact be a problem in special relativity where. space and time are treated on a more equal footing We ll find in Section 11 4 1 that the Galilean. transformations are replaced by the Lorentz transformations at least in the world we live in and. the latter are indeed symmetric in x and t up to factors of the speed of light c. Note also that eq 11 1 deals only with the differences in x and t between two events and not. with the values of the coordinates themselves The values of the coordinates of a single event depend. on where you pick your origin which is an arbitrary choice The coordinate differences between two. events however are independent of this choice and this allows us to make the physically meaningful. statement in eq 11 1 It makes no sense for a physical result to depend on the arbitrary choice. of origin and so the Lorentz transformations we derive later on will also involve only differences in. coordinates, One of the great triumphs of 19th century physics was the theory of electromagnetism. In 1864 James Clerk Maxwell wrote down a set of equations that collectively described. everything that was known about the subject These equations involve the electric and. magnetic fields through their space and time derivatives We won t worry about the specific. form of the equations here 4 but it turns out that if you transform them from one frame. to another via the Galilean transformations they end up taking a different form That is. if you ve written down Maxwell s equations in one frame where they take their standard. nice looking form and if you then replace the coordinates in this frame by those in another. frame using eq 11 1 then the equations look different and not so nice This presents. a major problem If Maxwell s equations take a nice form in one frame and a not so nice. form in every other frame then why is one frame special Said in another way Maxwell s. equations predict that light moves with a certain speed c But which frame is this speed. measured with respect to The Galilean transformations imply that if the speed is c with. respect to a given frame then it is not c with respect to any other frame The proposed. special frame where Maxwell s equations are nice and the speed of light is c was called. the frame of the ether We ll talk in detail about the ether in the next section but what. experiments showed was that light surprisingly moved with speed c in every frame no matter. which way the frame was moving through the supposed ether. There were therefore two possibilities Either something was wrong with Maxwell s. equations or something was wrong with the Galilean transformations Considering how. obvious the latter are the natural assumption in the late 19th century was that something. was wrong with Maxwell s equations which were quite new after all However after a. good deal of effort by many people to make Maxwell s equations fit with the Galilean. transformations Einstein finally showed that the trouble was in fact with the latter More. precisely in 1905 he showed that the Galilean transformations are a special case of the. Lorentz transformations valid only when the speed involved is much less than the speed of. light 5 As we ll see in Section 11 4 1 the coefficients in the Lorentz transformations depend. on both v and the speed of light c where the c s appear in various denominators Since c is. 3 It was assumed prior to Einstein that these two statements say the same thing but we will soon see. that they do not The second statement is the one that remains valid in relativity. 4 Maxwell s original formulation involved a large number of equations but these were later written more. compactly using vectors as four equations, 5 It was well known then that Maxwell s equations were invariant under the Lorentz transformations in. contrast with their non invariance under the Galilean transformations but Einstein was the first to recognize. their full meaning Instead of being relevant only to electromagnetism the Lorentz transformations replaced. the Galilean transformations universally,XI 4 CHAPTER 11 RELATIVITY KINEMATICS. quite large about 3 108 m s compared with everyday speeds v the parts of the Lorentz. transformations involving c are negligible for any typical v This is why no one prior to. Einstein realized that the transformations had anything to do with the speed of light Only. the terms in eq 11 1 were noticeable,As he pondered the long futile fight.
To make Galileo s world right,In a new variation,On the old transformation. It was Einstein who first saw the light, In short the reasons why Maxwell s equations were in conflict with the Galilean trans. formations are 1 The speed of light is what determines the scale on which the Galilean. transformations break down 2 Maxwell s equations inherently involve the speed of light. because light is an electromagnetic wave,11 1 2 Michelson Morley experiment. As mentioned above it was known in the late 19th century after Maxwell wrote down his. equations that light is an electromagnetic wave and that it moves with a speed of about. 3 108 m s Now every other wave that people knew about at the time needed a medium. to propagate in Sound waves need air ocean waves need water waves on a string of course. need the string and so on It was therefore natural to assume that light also needed a. medium to propagate in This proposed medium was called the ether However if light. propagates in a given medium and if the speed in this medium is c then the speed in. a reference frame moving relative to the medium will be different from c Consider for. example sound waves in air If the speed of sound in air is vsound and if you run toward. a sound source with speed vyou then the speed of the sound waves with respect to you. assuming it s a windless day is vsound vyou Equivalently if you are standing downwind. and the speed of the wind is vwind then the speed of the sound waves with respect to you. is vsound vwind, If this ether really exists then a reasonable thing to do is to try to measure one s speed. with respect to it This can be done in the following way we ll work in terms of sound. waves in air here 6 Let vs be the speed of sound in air Imagine two people standing on the. ends of a long platform of length L which moves at speed vp with respect to the reference. frame in which the air is at rest One person claps the other person claps immediately. when he hears the first clap assume that the reaction time is negligible and then the first. person records the total time elapsed when she hears the second clap What is this total. time Well the answer is that we can t say without knowing which direction the platform. is moving Is it moving parallel to its length or transverse to it or somewhere in between. Let s look at these two basic cases For both of these we ll view the setup and do the. calculation in the frame in which the air is at rest. Consider first the case where the platform moves parallel to its length In the frame of. the air assume that the person at the rear is the one who claps first Then it takes a time. of L vs vp for the sound to reach the front person This is true because the sound must. close the initial gap of L at a relative speed of vs vp as viewed in the air frame 7 By. 6 As we ll soon see there is no ether and light travels at the same speed with respect to any frame This. is a rather bizarre fact and it takes some getting used to It s hard enough to get away from the old way of. thinking even without any further reminders so I can t bring myself to work through this method in terms. of light waves in an ether I ll therefore work in terms of sound waves in air. 7 Alternatively relative to the initial back of the platform the position of the sound wave is v t and the. position of the front person is L vp t Equating these gives t L vs vp. 11 1 MOTIVATION XI 5, similar reasoning the time for the sound to return to the rear person is L vs vp The.
total time is therefore,t1 2 11 2 vp,vs vp vs vp vs vp2. This correctly equals 2L vs when vp 0 and infinity when vp vs. Now consider the case where the platform moves perpendicularly to its length In the vs. frame of the air we have the situation shown in Fig 11 2 Since. q the sound moves diagonally, the vertical component is by the Pythagorean theorem vs2 vp2 This is the relevant platform. component as far as traveling the length of the platform goes so the total time is. vs2 vp2 Figure 11 2, Again this correctly equals 2L vs when vp 0 and infinity when vp vs. The times in eqs 11 2 and 11 3 are not equal As an exercise you can show that. of all the possible orientations of the platform relative to the direction of motion t1 is the. largest possible time and t2 is the smallest Therefore if you are on a large surface that is. moving with respect to the air and if you know the values of L and vs then if you want to. figure out what vp is assume that it doesn t occur to you to toss a little piece of paper to. at least find the direction of the wind all you have to do is repeat the above setup with. someone standing at various points along the circumference of a given circle around you If. you take the largest total time that occurs and equate it with t1 then eq 11 2 will give. you vp Alternatively you can equate the smallest time with t2 and eq 11 3 will yield. the same vp Note that if vp vs we can apply Taylor series approximations to the above. two times For future reference these approximations give the difference in times as. 2L 1 1 Lvp2,t t1 t2 q 11 4,vs 1 vp2 vs2 1 v 2 v 2 vs3. The above setup is the general idea that Michelson and Morley used in 1887 to measure. the speed of the earth through the supposed ether 9 There is however a major complication. RELATIVITY KINEMATICS two topics kinematics and dynamics Kinematics deals with lengths times speeds etc It is concerned only with the space and time coordinates of an abstract particle and not with masses forces energy momentum etc Dynamics on the other hand does deal with these quantities This chapter covers kinematics

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