15 The Speeial Theory of Relativity 15-1 The principle of relativity For over 200 years the equations of motion enunciated by Newton were be- 15-1 The principle of relativity lieved to describe nature correctly,and the first time that an error in these laws was discovered,the way to correct it was also discovered.Both the error and its 15-2 The Lorentz transformation correction were discovered by Einstein in 1905. 15-3 The Michelson-Morley Newton's Second Law,which we have expressed by the equation experiment F=d(mu)/dt, 15-4 Transformation of time 15-5 The Lorentz contraction was stated with the tacit assumption that m is a constant,but we now know that this is not true,and that the mass of a body increases with velocity.In Einstein's 15-6 Simultaneity corrected formula m has the value 15-7 Four-vectors mo (15.1) 15-8 Relativistic dynamics √1-02/c2 15-9 Equivalence of mass and energy where the "rest mass"mo represents the mass of a body that is not moving and c is the speed of light,which is about3X105km·sec-1 or about186,000 mi·sec-l. For those who want to learn just enough about it so they can solve problems, that is all there is to the theory of relativity-it just changes Newton's laws by introducing a correction factor to the mass.From the formula itself it is easy to see that this mass increase is very small in ordinary circumstances.If the velocity is even as great as that of a satellite,which goes around the earth at 5 mi/sec, then v/c 5/186,000:putting this value into the formula shows that the cor- rection to the mass is only one part in two to three billion,which is nearly impossible to observe.Actually,the correctness of the formula has been amply confirmed by the observation of many kinds of particles,moving at speeds ranging up to practi- cally the speed of light.However,because the effect is ordinarily so small,it seems remarkable that it was discovered theoretically before it was discovered experimentally.Empirically,at a sufficiently high velocity,the effect is very large, but it was not discovered that way.Therefore it is interesting to see how a law that involved so delicate a modification (at the time when it was first discovered) was brought to light by a combination of experiments and physical reasoning. Contributions to the discovery were made by a number of people,the final result of whose work was Einstein's discovery. There are really two Einstein theories of relativity.This chapter is concerned with the Special Theory of Relativity,which dates from 1905.In 1915 Einstein published an additional theory,called the General Theory of Relativity.This latter theory deals with the extension of the Special Theory to the case of the law of gravitation;we shall not discuss the General Theory here. The principle of relativity was first stated by Newton,in one of his corollaries to the laws of motion:"The motions of bodies included in a given space are the same among themselves,whether that space is at rest or moves uniformly forward in a straight line."This means,for example,that if a space ship is drifting along at a uniform speed,all experiments performed in the space ship and all the phenom- ena in the space ship will appear the same as if the ship were not moving,pro- vided,of course,that one does not look outside.That is the meaning of the princi- ple of relativity.This is a simple enough idea,and the only question is whether it is true that in all experiments performed inside a moving system the laws of physics 15-1
will appear the same as they would if the system were standing still. Let us first investigate whether Newtons laws appear the same in the moving sys Suppose that Moe is moving in the x-direction with a uniform velocity u, and he measures the position of a certain point, shown in Fig. 15-1. He designates the x-distance"of the point in his coordinate system as x'. Joe is at rest, and measures the position of the same point, designating its x-coordinate in his system as x The relationship of the coordinates in the two systems is clear from the diagram After time t Moe's origin has moved a distance ut, and if the two systems originally coincided y =y, (152) X,y12) x If we substitute this transformation of coordinates into Newton 's laws we find that these laws transform to the same laws in the primed system; that is, the laws Fig. 15-1. Two coordinate systems of Newton are of the same form in a moving system as in a stationary system, and in uniform relative motion along the therefore it is impossible to tell, by making mechanical experiments, whether the x-axes. system is moving or not. The principle of relativity has been used in mechanics for a long time. It was employed by various people, in particular Huygens, to obtain the rules for the collision of billiard balls, in much the same way as we used in Chapter 10 discuss the conservation of momentum. In the past century interest in it wa heightened as the result of investigations into the phenomena of electricity, mag netism, and light. A long series of careful studies of these phenomena by many people culminated in Maxwells equations of the electromagnetic field, which Maxwell equations did not seem to obey the principle of relativity. That ss/ the describe electricity, magnetism, and light in one uniform system. However, tI transform Maxwells equations by the substitution of equations 15.2, their form does not remain the same, therefore, in a moving space ship the electrical and optical phenomena should be different from those in a stationary ship Thus one could use these optical phenomena to determine the speed of the ship; in particular, one could determine the absolute speed of the ship by making suitab optical or electrical measurements. One of the consequences of Maxwells equa- tions is that if there is a disturbance in the field such that light is generated, these electromagnetic waves go out in all directions equally and at the same speed c, or 186,000 mi/sec. Another consequence of the equations is that If the source of the disturbance is moving, the light emitted goes through space at the same speed c. This is analogous to the case of sound, the speed of sound waves being likewise independent of the motion of the source This independence of the motion of the source, in the case of light, brings up an interesting problem Suppose we are riding in a car that is going at a speed u, and light from the rear is going past the car with speed c. Differentiating the first equation in (15.2) gives x'/dt= dx/dt -u, which means that according to the galilean transformation the apparent the passing light, as we measure it in the car, should not be c but should be For instance, if the car is going 100,000 mi /sec, and the light is going mi/sec, then apparently the light going past the car should go 86,000 mi/sec. In any case, by measuring the speed of the light going past the car(if the galilean transformation is correct for light), one could determine the speed of the car. A number of experiments based on this general idea were performed to determine the velocity of the earth, but they all failed-they gave no velocity at all. We shall discuss one of these experiments in detail, to show exactly what was done wrong with the equations of physics. What could it be?
15-2 The Lorentz transforrnation When the failure of the equations of physics in the above case came to the first thought that occurred was that the trouble must lie in the new maxwell equations of electrodynamics, which were only 20 years old at the time. It seemed almost obvious that these equations must be wrong, so the thing to do was to change them in such a way that under the Galilean transformation the principle of relativity would be satisfied. When this was tried, the new terms that had to be put into the equations led to predictions of new electrical phenomena that did not exist at all when tested experimentally, so this attempt had to be abandoned Then it gradually became apparent that Maxwell's laws of electrodynamics were correct, and the trouble must be sought elsewhere. In the meantime, H. A. Lorentz noticed a remarkable and curious thing when he made the following substitutions in the Maxwell equations namely, Maxwell's equations remain in the same form when this transformation is applied to them! Equations(15.3)are known as a Lorentz transformation Einstein, following a suggestion originally made by Poincare, then proposed that all the physical laws should be of such a kind that they remain unchanged under a Lorentz transformation. In other words, we should change not the laws of electro dynamics, but the laws of mechanics, How shall we change Newton,s laws so that they will remain unchanged by the Lorentz transformation? If this goal is set, we then have to rewrite Newtons equations in such a way that the conditions we have imposed are satisfied. As it turned out, the only requirement is that the mass m in Newton's equations must be replaced by the form shown in Eq.(15.1). When this change is made, Newton's laws and the laws of electrod- namics will harmonize. Then If we use the Lorentz transformation in comparing Moe's measurements with Joes, we shall never be able to detect whether either is moving, because the form of all the equations will be the same in both coordinate It is interesting to discuss what it means that we replace the old transformation between the coordinates and time with a new one, because the old one(Gallean) seems to be self-evident, and the new one(Lorentz) looks peculiar. We wish to know whether it is logically and experimentally possible that the new, and not the old, transformation can be correct. To find that out, it is not enough to study the laws of mechanics but, as Einstein did, we too must analyze our ideas of space and time in order to understand this transformation. We shall have to discuss these ideas and their implications for mechanics at some length, so we say in in phase advance that the effort will be justified, since the results agree with experiment 15-3 The Michelson-Morley experiment Fig. 15-2. Schematic diagram of the As mentioned above, attempts were made to determine the absolute velocity Michelson-Morley experiment of the earth through the hypothetical"ether"that was supposed to pervade all space. The most famous of these experiments is one performed by Michelson and Morley in 1887. It was 18 years later before the negative results of the experi ment were finally explained, by Einstein The Michelson-Morley experiment was performed with an apparatus like that shown schematically in Fig. 15-2. This apparatus is essentially comprised of a light source A, a partially silvered glass plate B, and two mirrors C and E, all mounted on a rigid base. The mirrors are placed at equal distances L from B The plate B splits an oncoming beam of light, and the two resulting beams con- 15-3
tinue in mutually perpendicular directions to the mirrors, where they are reflected back to B. On arriving back at B, the two beams are recombined as two superposed beams, D and F. If the time taken for the light to go from b to e and back is the phase and will reinforce each other, but if the two times differ slightly, the beap" same as the time from b to C and back, the emerging beams d and F will be will be slightly out of phase and interference will result. If the apparatus is"at rest"in the ether, the times should be precisely equal, but if it is moving toward the right with a velocity u, there should be a difference in the times. Let us see why First, let us calculate the time required for the light to go from b to e and pack. Let us say that the time for light to go from plate b to mirror E is 11, and the time for the return is t2. Now, while the light is on its way from b to the mirror the apparatus moves a distance ut1, so the light must traverse a distance L ut at the speed c. We can also express this distance as cti, so we have cII=L+ utI t1=L/(c-l) (This result is also obvious from the point of view that the velocity of light relative to the apparatus is c-u, so the time is the length L divided by c -u )In a like manner, the time t2 can be calculated. During this time the plate B advances a distance ut2, so the return distance of the light is L-ut2. Then we have ct2EL-ut2, or I2 =L/(c+ u Then the total time is 1+t2=2Lc/(2-u2) For convenience in later comparison of times we write this as t1+t2 (154) Our second calculation will be of the time fa for the light to go from b to the mirror C. As before, during time fa the mirror C moves to the right a distance ut 3 to the position C; in the same time, the light travels a distar hypotenuse of a triangle, which is BC. For this right triangle we have (ct3)2=L2+(ut3)2 23-u213=(c2-n2)r3, For the return trip from C the distance is the same, as can be seen from the symmetry of the figure; therefore the return time is also the same, and the total time is 2f 3. With a little rearrangement of the form we can write 2L We are now able to compare the times taken by the two beams of light. In expressions(15.4)and(15.5)the numerators are identical, and represent the time that would be taken if the apparatus were at rest. In the denominators, the term u"/c will be small, unless u is comparable in size to c. The denominators represent the modifications in the times caused by the motion of the apparatus. And behold these modifications are not the same-the time to go to C and back is a little les than the time to E and back, even though the mirrors are equidistant from B, and all we have to do is to measure that difference with precisio Here a minor technical point arises--suppose the two lengths L are not exactly equal? In fact, we surely cannot make them exactly equal. In that case re simply turn the apparatus 90 degrees, so that bC is in the line of motion and BE is perpendicular to the motion. Any small difference in length then becomes
unimportant, and what we look for is a shift in the interference fringes when we rotate the apparatus. In carrying out the experiment, Michelson and Morley oriented the apparatus so that the line be was nearly parallel to the earth's motion in its orbit (at certain times of the day and night). This orbital speed is about 18 miles per second, and any"ether drift "should be at least that much at some time of the day or night and at some time during the year. The apparatus was amply sensitive to observe such an effect, but no time difference was found-the velocity of the earth through the ether could not be detected. The result of the experiment was null The result of the Michelson-Morley experiment was very puzzling and most disturbing. The first fruitful idea for finding a way out of the impasse came from Lorentz. He suggested that material bodies contract when they are moving, and hat this foreshortening is only in the direction of the motion, and also, that if the length is Lo when a body is at rest, then when it moves with speed u parallel D), is given by (156) When this modification is applied to the Michelson-Morley interferometer appara tus the distance from B to C does not change, but the distance from b to E is hortened to Lyl-u2/c2. Therefore Eq. (15.5)is not changed but the L of Eq(15. 4)must be changed in accordance with Eq. ( 15.6). When this is done we (2L/c)√1-u2/c2 r1+t2= (157) Comparing this result with Eq.(15.5), we see that t1+ 12= 2t3. So if the ap- paratus shrinks in the manner just described, we have a way of understanding why the Michelson-Morley experiment gives no effect at all. Although the contraction hypothesis successfully accounted for the negative result of the experiment, it wa open to the objection that it was invented for the express purpose of explaining way the difficulty, and was too artificial. However, in many other experiments to discover an ether wind, similar difficulties arose, until it appeared that nature was in a"conspiracy"to thwart man by introducing some new phenomenon to undo every phenomenon that he thought would permit a measurement of u It was ultimately recognized, as Poincare pointed out, that a complete conspiracy is itself a law of nature! Poincare then proposed that there is such a law of nature, that it is not possible to discover an ether wind by any experiment; that is, there is no way to determine an absolute velocity. 15-4 Transformation of time Othe i n checking out whether the contraction idea is in harmony with the facts experiments, it turns out that everything is correct provided that the times are also modified, in the manner expressed in the fourth equation of the set (15.3) That is because the time t3, calculated for the trip from b to C and back, is not the same when calculated by a man performing the experiment in a moving space ship as when calculated by a stationary observer who is watching the space ship To the man in the ship the time is simply 2L/c, but to the other observer it is (2L/c)/v1-u2c2(Eq. 15.5). In other words, when the outsider sees the man in the space ship lighting a cigar, all the actions appear to be slower than normal. while to the ma normal rate. So not only must the lengths shorten, but also the time-measuring instruments ("clocks")must appa ently slow down. That is, when the clock in the space ship records 1 second elapsed, as seen by the man in the ship, it shows 1/v1-u2/c2 second to the man outside This slowing of the clocks in a moving system is a very peculiar phenomenon and is worth an explanation. In order to understand this, we have to watch the machinery of the clock and see what happens when it is moving. Since that is 15-5