1. RELATIVISTIC KINEMATICS

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1 1. RELATIVISTIC KINEMATICS The one truth of whih the human mind an be ertain indeed, this is the meaning of onsiousness itself is the reognition of its own existene. That we may be seure in this truth is assured us by Desartes famous axiom even if everything else, inluding Desartes, is a figment of our imagination. Nothing else an be proved. Our most fundamental belief, then, is that the universe exists around us, onsisting of three spatial dimensions and time, and that while we an move about in the three spatial dimensions, time flows inexorably onward, everywhere the same. Newton himself said it: absolute, true, and mathematial time, of itself, and from its own nature, flows equably and without relation to anything external. But he was wrong. The theory of relativity shows us that time and spae do not have the meaning we thought they had. In the words of Weyl, we are to disard our belief in the objetive meaning of simultaneity; it was the great ahievement of Einstein in the field of the theory of knowledge that he banished this dogma from our minds, and this is what leads us to rank his name with that of Copernius (italis his). But the disovery of relativity by Einstein in 1905 was not a bolt from the blue. People had been onerned about the nature of spae and time for at least hundreds of years before that, beoming more and more disturbed by the inonsistenies in our understanding of the physial world toward the end of the 19 th entury. Nevertheless, even after the true nature of spae and time beame lear, the theory of relativity so ontradited our most fundamental belief that it was rejeted for years. Einstein himself never reeived the Nobel Prize for this work that was, in the words of Bertrand Russell, probably the greatest syntheti ahievement of the human intellet up to the present time. Some sixteen years afterward he was relutantly awarded the Nobel Prize for a lesser work beause the greatest physiist of the entury, known to more people than the President of the United States, ould not be ompletely ignored. Yet, the truth Einstein taught us displayed one again nature s tendeny to assume the most beautiful, symmetri form, in spite of our objetions. And while the truth is a merger of spae and time that prevents us from ordering events absolutely in time, it does not result in haos, but preserves those features that we annot logially be denied. The priniple of ausality is never violated, and as eah of us progresses through this four-dimensional spae-time, our individual pereption of time as moving always forward is not ontradited Einstein s Postulates Einstein's solution to the dilemma of the veloity of light was as beautiful as it was radial. He hose the most symmetri form for nature, stating that all inertial referene frames are equivalent. He embodied this onept in his two postulates of speial relativity, whih state: 1. The laws of nature are idential in all inertial frames of referene. That is, if we transform the mathematial equations of physis from one inertial referene frame to another they remain in the same form.. The speed of light is the same to all observers at rest in inertial frames of referene. A more general statement of this priniple might be that the influene of one partile is 1

2 not felt instantaneously by another. Instead, the influene propagates at some (maximum) veloity. These two postulates an be summed up in a single postulate that states that there is no experiment that an be done to distinguish the absolute veloity of any oordinate system. Einstein s postulates appear beautifully simple and symmetri. In fat they are deeptively simple, sine within them lie profound onsequenes and more than a few startling paradoxes. Fundamental to all the paradoxes is the fat that simultaneity is no longer an objetive reality, as Weyl points out, but rather a subjetive one that depends on the observer. A simple example, shown in Figure 1, illustrates this. Consider the following "gedanken experiment" (thought experiment; Einstein loved gedanken experiments): Figure 1 On the moon, Buffy and Bubba, representating the lunar olonies Alpha and Beta, are having a green-heese eating ontest. When the winner is delared, the news is radioed to the folks bak home in Alpha and Beta, whih are equidistant from the site of the ontest. Eah olony reeives the news, whih travels at the speed of light, at exatly the same time. Right? Of ourse. But Hilda and Wolfgang, who are passing the moon in their spae ship on their way bak to earth after a vaation, wath the events on the moon and ome to a different onlusion. Sine, as they view it, the moon is moving to the left, the news reahes Beta, whih is moving toward the ontest, before it reahes Alpha. Right? Of ourse. Who is right? Well, they both are. There is no absolute meaning to the onept of simultaneity. In fat, let s hek in with Edgar and Eloise, who are in a spae ship going the opposite diretion from Wolfie and Hildie, just starting their vaation. As viewed by E and E, the moon is going the opposite diretion and the news reahes Alpha before it reahes Beta. We an t even get agreement on whih of two events (the news arriving at Alpha and at Beta) ourred first. In fat, there is really no absolute meaning to simultaneity. Time just doesn t work that way, although the effets are usually so small that you never notied it. You may (you should!) wonder what has happened to ause and effet. For example, if event A auses event B, what happens if someone else observes events A and B to our in the reverse order. This is similar to the logial diffiulty that ours when people travel bak in

3 time. Can Dr. No travel bak in time and kill his mother so that he himself is never born? Well, of ourse not, regardless of whether you liked Bak to the Future or not. In fat, the theory of relativity does not violate ausality. Two events an be reordered in time by other observers only if the events happen so far apart in distane and so lose together in time that neither light nor anything else (whih must travel slower than light) an get from the first event to the seond. Therefore, event A annot have any influene on (or ause) event B, and the priniple of ausality is not violated. Clearly, the two events alled news reahing Alpha and news reahing Beta are too far apart in distane to be onneted by a single light pulse. It takes two light pulses to reah the two events, so it is OK that they an be reordered in time by different observers. One of these two events an never ause or influene the other. 1.. Time dilation Let s do another gedanken experiment. This time we put a simple (in onept, at least) lok on the spae ship with Wolfie and Hildie. The lok sends a short laser pulse up to a mirror, and when it strikes the mirror and returns the lok tiks one and sends out the next pulse. If the distane to the mirror is L, the round-trip distane traveled by the laser pulse is d = L. If is the (universal) veloity of light, the lok tiks one in the time Δ t = d / = L/. But what do Buffy and Bubba, standing on the moon, think of this? Figure As they see it, the light makes a triangular trip up and down as the laser and the mirror move to the right at the veloity v. The total distane the light travels in one tik is found from Pythagoras' theorem: ( ) d' = L + v Δ t' (1.1) But light travels at the veloity, so the time for the lok to tik one is Δ t' = d'/. The moving lok goes too slow (that is, it is observed by Buffy and Bubba to take too long to tik) by the fator Δ t ' = γ (1.) Δt where 3

4 γ = 1 1 β (1.3) β = v/ (1.4) are the relativisti parameters. But this isn t just a ase of a lok going too slow. The lok is just fine. In fat, everything on the moving spae ship is going too slow. Wolfie and Hildie s hearts beat too slowly, and they are aging too slowly, at least aording to Buffy and Bubba. Wolfie and Hildie don t see anything wrong. There is no experiment they an do to detet their motion, after all. Should Bubba and Buffy be jealous that Wolfie and Hildie are getting old slower? Not at all; Wolfie and Hilda are not enjoying the extra time. Their thoughts, their days, everything is going slower for them. They don t experiene any extra time. In fat, when you think of it, Wolfie and Hildie see a lok belonging to Buffy and Bubba going too slow ompared with their lok. After all, they see themselves as stationary and Buffy and Bubba moving past them on a (very large) spae ship. So they see Buffy and Bubba getting old slower than they are. Eah one sees the other s lok as moving slower than their own! This is known as time dilation; time on a moving spae ship is observed to be strethed out. One again, this is not a problem aused by bad loks. This is the nature of time itself! How an this happen? The paradox is resolved when we onsider how the omparison is done. When Buffy and Bubba wath Wolfie and Hildie s lok, they do it (or an do it, and all methods are equivalent) by wathing one lok on the spae ship as it passes lose to two loks in two separate plaes on the moon. When Wolfie and Hildie do the omparison, they wath one moon lok as it goes past two separate loks on their spae ship. It turns out that sine the two pairs of observers an t agree on simultaneity, they have set their loks inorretly (relative to one another, in some sense) when they moved them into plae for the omparisons. In any event, there is no paradox sine the same loks are not being used in the two measurements. There is one way to get around the problem of multiple loks. Let s do another gedanken experiment. This time we have two twins. One twin gets in a spae ship and flies to alpha entauri and bak at high speed. Her twin brother on earth knows that her lok and her life proesses move slower than his. When she returns to earth he is not surprised to see that she is younger than he is. But shouldn t she see the same thing? That is, sine he (on earth) was moving relative to her spae ship, shouldn t she see that he is younger? Well, the orret answer is that she is younger than he is. The symmetry of the situation is broken beause she had to aelerate to fly away, deelerate to turn around at alpha entauri, and then aelerate and deelerate again to return home. Therefore, her loks behave differently. After all, Einstein s postulates apply only to oordinate systems travelling at a onstant veloity. Atually, this gedanken experiment has already been done. In very areful experiments, two atomi loks were flown around the earth in opposite diretions. When they returned to the original laboratory and were ompared with "stay-at-home" loks, they were slower (younger) by about a tenth of a miroseond, just the amount Einstein would have predited (atually, the effets of gravity had to be taken into aount sine the planes were flying high above the earth)! 4

5 1.3. Length ontration Just as times viewed in the lab frame oordinate seem longer than those measured in the moving frame, lengths measured in the lab frame seem shorter than those measured in the moving frame of referene. Going bak to Figure 1, let L 0 be the distane from the transmitting tower to the olony Beta. Buffy and Bubba, on the moon, an measure this with a long tape measure. They also know that the time it takes for the spae ship to get from the tower to the olony Beta is Δ t L / v 0 = 0, where v is the veloity of the spae ship. Meanwhile, Hilda and Wolfgang, on the spae ship, ompute the distane by measuring the time Δ t ' to get from the tower to Beta and using the formula L' = vδ t'. But due to time dilation, they get a different answer for the length: L' vδt' 1 = = (1.5) L0 vδt0 γ The length they measure is shorter! That is, moving objets (as viewed in this ase from the spae ship) appear shorter in the diretion of motion. This is alled length ontration. Note that the lengths of objets in diretions transverse to the diretion of motion are unaffeted. If they are the same height and standing on the same level, Bubba and Wolfgang will eah look diretly into eah other s eyes as they pass by. Neither will appear taller than the other Doppler shift Assoiated with time dilation and length ontration is the Doppler shift of light. As in the ase of sound waves, the motion of a soure of waves auses a hange in the frequeny and wavelength of the waves, but the formulas are different for light waves. Figure 3. Consider the gedanken experiment pitured in Figure 3. A spae ship traveling at veloity v emits two laser pulses in the forward diretion separated by the time (in its oordinate system) Δ t 0. An observer standing in front of the onoming spae ship sees the pulses arrive with a time separation Δ t. To ompute the time interval observed, we use the Minkowski diagram in Figure 3. In this diagram, we plot the time t and distane x, both measured in the lab frame. If the 5

6 first pulse is emitted at time t 1 = 0 and position x 1 = 0, the trajetory x = t of the pulse is a straight line at 45 degrees in this diagram. The trajetory of the spae ship is x = vt = βt, as shown. Beause of time dilation, the lok on the spae ship runs slow, so the time at whih the seond pulse is emitted is t = γδ t 0, and the position is x = vt = γβδ t0. The time interval between the two parallel light lines representing the two pulses in Figure 3 is therefore x 1 β Δ t = t = γδt0 γβδ t0 = γ ( 1 β) Δ t0 = Δt0. (1.6) 1+ β The time interval seen by the stationary observer is smaller than that of the observer in the spae ship. If there is a series of pulses emitted at the frequeny f 0 in the spae ship frame, the frequeny measured by the stationary observer is Doppler shifted by the fator f 1+ β = f0 1 β Note that as the veloity of the emitter approahes, the Doppler shift approahes infinity. (1.7) One of the most interesting appliations of the Doppler shift was Hubble s disovery of the expanding universe. By looking at the frequeny (olor) of the emission from ertain atoms in distant galaxies, Hubble disovered that all distant galaxies are moving away from us, moving faster at larger distanes. To explain this, he postulated that we live in a universe that started as a point and has been expanding for about 13 billion years sine then. This has, of ourse, all sorts of ramifiations for siene and even religion! 1.5. Intervals To disuss the onepts of relativisti kinematis it is useful to introdue Minkowski t, r in this 4- spae. For two spatial dimensions and time this is shown in Figure 4. Eah point ( ) dimensional spae-time is alled an event, and the path of a partile, t( s), ( s) parameter s, is alled the world line of the partile. r for some We now onsider the relationship between two events viewed in the referene frames K and K ' when the referene frame K ' is moving at the onstant veloity v relative to K. We Figure 4 6

7 note at the beginning that observers in both referene frames agree on the relative veloity v. That is, they agree on the absolute magnitude v = v, sine by symmetry neither ould observe a greater veloity than the other, but of ourse they differ on the sign of the vetor v. By symmetry, again, it is lear that the transformation from K to K ' differs from the inverse transformation (from K ' to K ) only in the sign of v. Central to the disussion is what we all the interval between two events. In ordinary 3- dimensional Eulidian geometry the distane between two infinitesimally separated points is dl = dx + dy + dz > 0. (1.8) For two infinitesimally separated events in 4-dimensional Minkowski spae, however, we define the intervals and ds = dt dx dy dz (1.9) ds ' = dt ' dx ' dy ' dz ' (1.10) in K and K ', respetively. These expressions are alled the metri equations for the two referene frames, and the quantity ds represents some sort of distane between the two events in 4-dimensional spae-time. Beause of the minus signs in these expressions the geometry of Minkowski spae is not Eulidian, but is alled pseudo-eulidian. Sine the square of the interval an be positive or negative some authors treat time as an imaginary oordinate, but we use a different approah here. The rationale for defining the interval as we have done here learly omes from the fat that the speed of light is the same in all inertial referene frames. If the two events separated by ds orrespond to the passage of a signal traveling at the veloity of light, then the interval as we define it vanishes in all referene frames ds = ds ' = 0. (1.11) Put another way, ds = 0 in one frame stritly implies that ds ' = 0 in any other, and vie versa. More generally we note that sine uniform motion in one inertial referene frame implies uniform motion in another, the transformation between K and K ' must be linear, so it must be true that ds ' = ads, (1.1) where a is some onstant that depends, at most, on the relative motion of K and K '. But a annot depend on the 4-dimensional oordinates themselves sine spae-time is presumed to be homogeneous, so it an depend at most on the veloity, a= a( v ). Sine the veloity v introdues a speial diretion into the disussion we might expet the interval to transform differently depending on the orientation of the interval relative to v. But sine all the spatial omponents of the interval enter (1.1) quadratially, the interval does not hange if any their signs are reversed, whih would hange the orientation of the interval relative to v. Equivalently, the transformation of the interval must be indifferent to the diretion of v, so that a ( ) = a( v) v. (1.13) 7

8 But as noted earlier, the inverse transformation is obtained from the forward transformation merely by hanging the sign of v, whih does not affet a( v ). Therefore, we see that from (1.1). It follows that ds ds ' ( ) = a v ds ' =, (1.14) a v ( ) ( ) a v = 1, (1.15) or a =± 1. We an disard the negative sign sine two suessive transformations must give the same result as a single transformation with the same final veloity, so that a = a. Therefore, we onlude that a = 1 and the interval ds is an invariant of the transformation between inertial oordinate systems. As we see shortly, this is all we need to define the Lorentz transformation between them. It is onvenient to lassify intervals in the following way: 0 ds >, timelike interval, (1.16) 0 ds <, spaelike interval, (1.17) 0 ds =, lightlike (null) interval. (1.18) Sine the interval is invariant, the haraterization of an interval as timelike, spaelike, or lightlike is independent of the oordinate system in whih the events are viewed. For example, suppose that two events our in the same plae but at different times in the oordinate system K ', so that d r ' = 0. Then ds ' = dt ' > 0, and the interval is timelike. In another referene frame K, the interval is still timelike, even though the events our in different plaes at different times. Conversely, if the interval between two events is timelike in the referene frame K, then there exists a referene frame K ' in whih the events our at the same plae. Speifially, if in K, then in a oordinate system ds dt d = r > 0, (1.19) K ' moving at the veloity d v = r (1.0) dt relative to K the events our at the same plae. In the moving referene frame are separated by the time K ' the events ds ' ds dr dt ' = = = dt. (1.1) In the same way, if two events viewed in the referene frame K ' our simultaneously at two different points, then ds ' = dr < 0, so the interval is spaelike. Viewed in the referene frame K, the events appear at two different plaes and two different times but the interval is still spaelike. Conversely, if two events are separated by a spaelike interval in the referene frame 8

9 K, then in some other referene frame the separation of the events is K ' the events are simultaneous. In this referene frame, dr ' = ds ' = ds = dr dt. (1.) Based on the lassifiations (1.16) - (1.18), we an divide Minkowski spae into the regions shown in Figure 5. For an event anywhere loated inside the light one the interval s = t r > 0 is timelike. Thus, in any other referene frame K ' the event ours in the same time order relative to the event at the origin as it does in the frame K, and may be regarded as in the absolute past or absolute future relative to the event at the origin. On the other hand, events loated outside the light one are related to the origin by spaelike intervals. Consequently, there is some referene frame K ' in whih the events are simultaneous. A seond transformation to a frame K '' moving with respet to the frame K ' will separate the events in time. However, by symmetry we see that if a relative veloity in one diretion plaes event A before B in K '', then a relative veloity in the opposite diretion will plae event B first. Thus, events separated by spaelike intervals annot be absolutely time ordered. We say that events in the region outside the light one are elsewhere relative to the event at the origin. For example, in the parable of the heese-eating ontest disussed earlier the arrival of the news at Alpha and Beta represents two events separated by a spaelike interval. Thus, the news arrived simultaneously at Alpha and Beta as observed by the olonists on the moon, but was observed by the spae travelers to arrive first at Alpha or first at Beta depending on the relative veloity of their spae ship. For an objet suh as a lok at rest in an inertial referene frame K ', the distane dr ' between two events along its world line vanishes, so the invariant interval ds ' = dt ' is just the time between the events. Viewed from the laboratory frame K the interval is the same, so if the rest frame K ' moves the with the veloity v relative to the laboratory frame, then the interval in the laboratory frame is ( ) ds = dt dr = v dt = dt '. (1.3) Therefore, ompared with the time dt elapsed on a lok in the laboratory frame the time elapsed on the moving lok is less, amounting to where dt = =, (1.4) γ dt ' 1 β dt dt Figure 5 9

10 and γ = β = v, (1.5) 1. (1.6) 1 β This shows that a lok at rest in the moving oordinate system indiates less elapsed time than loks at rest in the laboratory referene frame to whih it is ompared. This phenomenon is alled time dilation. Ample experimental evidene now exists to onfirm this effet. It has, of ourse, nothing to do with the failure of moving loks to perform orretly. It has to do with the nature of time itself, or, more preisely, the subjetive nature of simultaneity. Many physial phenomena, suh as the deay rate of subatomi partiles, an be used as loks. For example, when osmi rays strike the upper atmosphere of the earth, they reate a variety of partiles inluding muons. Ordinarily, muons have a half-life of. μs, so at the veloity of light they would travel, on average, about 600 m. However, due to time dilation a muon traveling at β = 0.999, whih orresponds to an energy of.4 GeV, lives for 50 μs, and travels 15 km. This aounts for the fat that large numbers of muons reated in the upper atmosphere are observed at the earth s surfae. In the same way, the subatomi partiles reated by ollisions in highenergy physis experiments would not be observable exept that their brief lifetimes are extended by time dilation. We all the time elapsed on a lok moving with an objet the proper time dτ for the objet. This is the time atually experiened by the objet. For objets that are not in uniform motion the motion within any brief interval may be regarded as uniform, and the elapsed proper time between two points on the world line of the objet is t dt τ τ = t t. (1.7) () t 1 1 γ t1 Note arefully that in this expression the terms τ 1 and τ refer to the time elapsed in an aelerating oordinate system, but t is the time in an inertial referene frame. A few remarks are in order. In the first plae, the proper time, defined by the invariant interval in the rest frame of the moving objet, is an invariant. That is, all observers agree on the proper time elapsed along the world line of the objet. Physially, this orresponds simply to the fat that the lok moving with the objet has an indiator (hands, or even a digital readout) on it to indiate the time. All observers, regardless of their relative motion, agree on what the indiator shows. That is, all observers agree on the numbers showing on the digital readout. In the ase of subatomi partiles, the time is indiated by the number of partiles that have or have not deayed. The number of partiles is the same to all observers. In the seond plae, when viewed from the rest frame of the moving objet, loks in the laboratory frame are moving with the veloity v = v so they indiate an elapsed time that is less than that in the objet s rest frame. Thus, to an observer in the laboratory frame the lok in the objet rest frame appears to be slow, while viewed from the objet rest frame a lok in the laboratory frame appears to be slow. The paradox is resolved by reognizing that when the time 10

11 Figure 6 indiated on the lok in the rest frame is observed from the laboratory frame, the observations are made by two observers at different plaes in the laboratory frame. They observe the moving lok as it passes by and ompare the time indiated on the moving lok with that on their individual loks. This is shown in Figure 6. Conversely, when the elapsed time on a lok in the laboratory frame is measured by observers in the moving frame, the laboratory lok is ompared with two separate loks in the moving frame. Thus, the measurements are not idential, and for this reason they do not give the same results. To avoid the problem of measuring elapsed time on a moving lok by omparing a single moving lok with two stationary loks, we start with two loks that are initially at rest in the laboratory frame. We then aelerate one lok to a high veloity, bring it to rest again, and then return it to its original position in the laboratory next to the other lok. When the two loks are ompared, it is found that the lok that has been aelerated and deelerated indiates a smaller elapsed time than the stationary lok, in aordane with (1.7). This is alled the twin paradox sine it is often stated in the form of an allegory in whih two twins are ompared. One twin stays on earth and grows old, while the other beomes an astronaut, flies off at high speed to a nearby star, and when she returns she is younger than her twin sister. It is easy to understand that the earthbound twin observes the astronaut s lok as progressing too slowly, but why doesn t the astronaut observe her twin sister s lok as progressing too slowly? In this ase, the paradox is resolved by reognizing that the astronaut twin must aelerate to leave the earth, deelerate and turn around after she arrives at the star, and aelerate bak toward earth and then deelerate again upon reahing home. The other twin does not aelerate at all, and this is what breaks the symmetry. More to the point, (1.7) involves transformations from one inertial frame to a sequene of inertial frames, eah of whih desribes the moving objet for a brief period. Sine the earthbound twin remains in an inertial frame of referene, this equation provides a valid desription of her observations of her astronaut sister. On the other hand, the astronaut twin is not in an inertial frame, and she annot use (1.7) to ompute her sister s age. In atual fat, while the astronaut is traveling at onstant veloity she does see her sister aging more slowly than herself. However, when she aelerates to turn around at the outbound end of her trip she observes her sister aging at an aelerated rate. Although tehnology has not reahed the stage where the astronaut experiment an atually be tried, experimental onfirmation of the twin paradox does exist. In areful experiments using atomi loks it has been observed that a lok that is flown around the world in an airplane arrives bak at the laboratory younger than a lok that remains at home. However, the differene in this ase is only hundreds of nanoseonds, and the effets of gravity (aounted for in general relativity) are of the same order of magnitude as the time dilation disussed here. 11

12 One final remark: if we draw the twins world lines on a Minkowski diagram we get paths like those shown in Figure 7. For the astronaut the elapsed time is given by (1.7). The elapsed time for the lok at rest is larger than this. In terms of the intervals, t t Δ dt s ( earth ) = dt s ' astronaut γ =Δ t1 t1 () t ( ). (1.8) That is, the interval along the straight line is larger. We see, therefore, that pseudoeulidian geometry is different from what we are used to in Eulidian geometry. A straight line in 4- dimensional spae-time (alled a geodesi) is the longest interval between two events, rather than the shortest. Figure The Lorentz Transformation To find the Lorentz transformation that relates the oordinates in two inertial referene frames we look for the most general linear transformation that leaves the interval ds invariant. The transformation must be linear beause uniform motion in one frame must orrespond to uniform motion in the other, as noted earlier. To redue the algebra, we make two simplifiations. In the first plae we assume that the axes of the two frames are parallel at all times, and that the origins oinide at time t = t' = 0, as shown in Figure 8. In the seond plae we assume that the referene frame K ' moves at the veloity v in the ˆx (and x ˆ ') diretion relative to the frame K. More general transformations an be obtained by rotations of K or K ' and simple orretions to the origins of the times and distanes. Figure 8. 1

13 Before Einstein disovered relativity, everyone, inluding Newton and Galileo, thought that spae and time were independent, and the transformation from K to K ' was simply t' = t (1.9) x' = x (1.30) y' = y (1.31) z' = z (1.3) This is alled a Galilean transformation. Now we know that things are not so simple. On physial grounds we see that in the diretions transverse to the relative motion the oordinates y and z transform into themselves. Sine the transformation is linear we may write y' = ay, (1.33) z ' = bz, (1.34) for some onstants a and b. But from the symmetry of the forward and bakward transformations we see that a= b= 1, so that y' = y, (1.35) z' = z. (1.36) In the longitudinal diretion the most general linear transformation is t ' = mt + nx, (1.37) x ' = pt + qx, (1.38) for some onstants m, n, p and q. To preserve the interval between the origin and the point ( t, r ), we require that ( ) ( ) t x = t' x' = mt+ nx pt+ qx ( ) ( ) ( ) = m p t + n q x + mn pq xt. (1.39) From the first term on the right-hand side we see that so we an write m p = 1, (1.40) for some ζ, and from the seond term we see that so we an write m = oshζ, p = sinhζ, (1.41) n q = 1, (1.4) for some ψ. From the third term we find that n = sinhψ, q = oshψ, (1.43) 13

14 mn pq = oshζ sinhψ + sinhζ oshψ = 0, (1.44) so that ψ = ζ. The most general transformation that preserves the interval is therefore t ' = t oshζ x sinhζ (1.45) x' = xoshζ tsinhζ, (1.46) where ζ is alled the boost parameter, or the rapidity. The transformation (1.45) and (1.46) resembles a rotation of oordinates exept that the sin and os are replaed by sinh and osh. In fat, a rotation of oordinates is the most general linear homogeneous transformation that preserves the length dx + dy = dx ' + dy ' in Eulidian geometry. Thus, the Lorentz transformation has the form of a pseudo-rotation in pseudo-eulidian spae. To determine the boost parameter ζ we onsider the motion of the origin of the K ' frame, as viewed in the K frame. Sine the origin ( x ' = 0) of the moving frame is at the position x = vt in the laboratory frame, we see from (1.46) that x' = 0 = vtoshζ tsinhζ, (1.47) so v tanhζ = = β. (1.48) Therefore, the oeffiients in the transformation are 1 1 oshζ = = = γ, (1.49) 1 tanh ζ 1 β and the omplete Lorentz transformation is sinhζ = tanhζ oshζ = βγ, (1.50) ( β ) ( β ) t ' = γ t x, (1.51) x ' = γ x t, (1.5) y' = y, (1.53) z' = z. (1.54) For a boost to a oordinate system moving to the right, tanhζ = β > 0. The relation between the new and old oordinate axes is shown in Figure 9, where we see that the new axes x ' = 0 and t ' = 0 are tilted toward the light line in the upper right quadrant. For a boost to a frame moving to the left, the axes are tilted away from this same light line. From Figure 7 it is easy to see how events separated by a spaelike interval an be reordered in time. For example, the point B is elsewhere with respet to the origin of the stationary system K, and ours later ( t > 0 ) in that system. In the K ' system the point B lies below the axis t ' = 0 and therefore ours earlier ( t ' < 0). 14

15 Figure 9 The inverse transformation an be found by solving for x and t, but it is easier simply to use the symmetry of the forward and inverse transformations. If we just hange the diretion of motion to v we immediately get ( ' β ') ( ' β ') t = γ t + x, (1.55) x = γ x + t, (1.56) y = y', (1.57) z = z'. (1.58) In the limit we reover Galilean relativity. To see this we write the Lorentz transformation expliitly in terms of the veloities and get v t x t' = t, (1.59) v 1 and x vt x ' = x vt. (1.60) v 1 Finally, we note that Galilean transformations ommute but Lorentz transformations do not. That is, if we transform first into a frame K ' moving at the veloity v 1 with respet to K and then into a frame K '' moving at the veloity v with respet to K ', we get a Galilean transformation diretly into the frame moving at the veloity v 3 = v 1 + v. It doesn t matter whih transformation omes first. The same is not true for Lorentz transformations. As we saw earlier, time dilation is the phenomenon that makes a moving lok appear to go slow, and length ontration is the phenomenon that makes a moving objet shrink in the diretion of motion. To see how time dilation and length ontration arise, we examine the measurement proess by whih eah is determined. To observe time dilation we onsider the 15

16 progress of a moving lok, that is, a single lok at rest at the point x ' in the frame K '. Differentiating the inverse transformation (1.55) with respet to t keeping x ' = onstant we get dt ' 1 = < 1. (1.61) dt γ That is, the moving lok always appears to go slower than the stationary loks to whih it is ompared. Time dilation refers to the fat that the tiks of the moving lok appear farther apart as viewed from the stationary frame. To observe length ontration, we measure the length of a moving rod by determining the positions of the two ends at a single time t in the stationary frame K. For an infinitesimal rod the orresponding length in the stationary frame is found by differentiating the forward transformation (1.5) with respet to x, holding t = onstant, to get dx ' = γ > 1. (1.6) dx Length ontration refers to the fat that the tik marks on the moving length sale appear loser together than those on the stationary sale. That is, a rod of length dx ' appears to have a length dx = dx '/ γ < dx ' in the stationary frame. Note that the fator 1/γ appears in time dilation where the fator γ appears in length ontration. This is beause in the first ase a oordinate ( x ') was held fixed in the moving frame, while in the seond ase a oordinate ( t ) was held fixed in the stationary frame. We have already introdued the proper time dτ, whih in the present disussion orresponds to dt ' ( x ' = onstant). We similarly define the proper length as the length observed in the referene frame in whih the objet is at rest, dλ = dx' ( t' = onstant). We saw previously that the proper time is invariant, ds ' = dt ' dx ' = dτ (holding dx ' = 0). (1.63) In the same way we see that the proper length is an invariant, ds ' = dt ' dx ' = dλ (holding dt ' = 0). (1.64) Physially, the Lorentz invariane of proper time just means that all observers agree on what a lok at rest in the moving frame indiates. Likewise, the invariane of proper length just means that all observers agree on the numbers that appear on a ruler or dial gauge at rest in the moving frame. Clearly, relativisti veloities do not add in the same way that nonrelativisti veloities do, for if we are traveling at the veloity v= 0.75 through a railroad station and another train passes us at 0.75, the passengers waiting in the station do not see the faster train traveling at 1.5. To see how veloities add we onsider a referene frame K ' that is moving at the veloity v with respet to the frame K, and a partile moving in the frame K ' at the veloity v', as shown in Figure 10. The inverse transformation between K and K ' is given by (1.55)-(1.58). For an infinitesimal movement dr' = v' dt' (1.65) 16

17 Figure 10 in the moving frame the motion in the stationary frame is vv ' x dt = γ dt '1 +, (1.66) dx = γ dt ' v + v, (1.67) ( ) ' x dy = v ' dt ', (1.68) y dz = v ' dt '. (1.69) z Dividing these expressions we obtain the relations dx v+ v' x vx = =, (1.70) dt vv ' x 1+ v v y z = dy v ' y dt =, (1.71) vv ' x γ 1+ = dz v ' z dt =. (1.7) vv ' x γ 1+ These are alled the Einstein veloity-addition laws. Clearly, the addition of veloities transverse to one another is different from the addition of veloities that are parallel to one another. As an example we onsider a relativisti partile that in some deay proess emits another partile with veloity v ' at the angle θ ' in the x ' y ' plane. In the laboratory frame the veloity of the seondary partile is v+ v'os θ ' vx =, (1.73) vv 'os θ ' 1+ v y = v 'sin θ '. (1.74) vv 'os θ ' γ 1+ 17

18 Figure 11 In the laboratory frame the emission angle is vy v 'sin θ ' tanθ = = v γ v+ v'os θ ' x ( ) 18. (1.75) For highly relativisti motion of the primary partile, γ >> 1, we see that most of the emission appears in the forward diretion in the laboratory frame with angles θ O( 1/ γ ). When the emitted partiles are photons, so that v' =, all radiation that is emitted in the forward hemisphere ( θ ' π /) in the rest frame of the primary partile is emitted inside the one 1 tanθ (1.76) βγ in the laboratory frame, as illustrated in Figure 11. This effet is quite pronouned in the radiation from synhrotron radiation soures and high-energy physis experiments where the 3 partile energy orresponds to γ >> 1, and often γ > Transformation of eletromagneti fields In the nonrelativisti ase, we an find the transformation laws for eletri and magneti fields from the Lorentz fore equation: F= q( E+ v B ) (1.77) In a oordinate system moving at the veloity V, the new veloity is v' = v V and the fore is where ( ' ) q( ' ' ') F= q E+ V B+ v B = E + v B (1.78) E' = E+ V B (1.79) B' = B (1.80) In relativisti physis, things are more ompliated. The eletri and magneti fields are not 4-vetors, and they transform differently. In ontrast with the ase of 4-vetors, the longitudinal omponents of the eletri and magneti fields are unhanged by the boost while the transverse omponents are hanged and mixed. That the longitudinal eletri field should remain unhanged by the boost an be understood on physial grounds if we onsider the field due to a parallel-plate apaitor whose axis (normal to the plates) is aligned parallel to the boost, as

19 Figure 1. shown in Figure 1. Sine transverse lengths are unaffeted by the boost the harge density on the plates is unhanged, but the separation between the plates is redued by length ontration. However, the field is independent of the separation of the plates so the eletri field in the diretion of the boost is unhanged. Similarly, if we onsider the magneti field of a solenoid aligned along the diretion of the boost, as shown in Figure 13, we see that the winding density of the solenoid is inreased by the Lorentz ontration, while the urrent in the solenoid is dereased by time dilation. The effets anel and leave the longitudinal magneti field unhanged. Figure 13. Thus, we find that there is no hange in the longitudinal eletri field and magneti field. To see how the transverse eletri field hanges, we onsider the field of a parallel-plate apaitor aligned with its axis perpendiular to the boost, as shown in Figure 14. Due to the Lorentz ontration the harge density on the plates is inreased by the fator γ, whih inreases the eletri field aordingly. The magneti field is slightly more ompliated. The magneti field in the referene frame K ' has new terms that depend on the eletri field in K that appear beause the harges that give rise to the eletri field in K are moving in K ' and onstitute a urrent. In the parallel-plate apaitor shown in Figure 14, there is no magneti field in the rest frame of the apaitor, but in the laboratory frame the harged plates onstitute a surfae urrent density γσ v, where σ is the harge density of the plates in the rest frame. This is proportional Figure

20 to the transverse eletri field and is responsible for the new terms in the magneti fields. If we evaluate the rest of the omponents we get E = E, (1.81) ' x x ( ) ( ) E = E vb, (1.8) ' y γ y z E = E + vb, (1.83) ' z γ z y B ' x = B, (1.84) x v B ' y = γ B y + E z, (1.85) v B ' z = γ B z E y. (1.86) The inverse transformation is obtained by hanging v v in (1.81) - (1.86). In the nonrelativisti limit v/ << 1 we reover the transformations whih we derived from the Lorentz fore law. This does not mean that the Lorentz fore law is valid only in the nonrelativisti limit. Rather, the fator γ and the higher-order terms in B ' y and B ' z appear in the transformed fields beause the motions in the new oordinate system are altered by time dilation and length ontration, and the fores must hange in the new oordinate system to reflet this. As an example of the transformation of eletromagneti fields we onsider the field of a point harge in uniform motion. In the moving frame the eletri field lines are direted radially away from the harge, but as observed in the laboratory frame the field is altered. The form of the resulting field is suggested shematially in Figure 15. Due to Lorentz ontration, the eletri field is ompressed in the longitudinal diretion but gets stronger in the transverse diretion where the lines of fore are now loser together. The magneti field lines are irles about the diretion of motion of the harge. Figure 15. 0

21 . RELATIVISTIC MECHANICS Beauty, at least in theoretial physis, is pereived in the simpliity and ompatness of the equations that desribe the phenomena we observe about us. Dira has emphasized this point and said It is more important to have beauty in one s equations than to have them fit experiment. It seems that if one is working from the point of view of getting beauty in one s equations, and if one has really a sound insight, one is on a sure line of progress. In this sense the beauty of lassial physis lies in the fat that it an all be derived from the postulates of relativity together with just one hypothesis, whih we all Hamilton s priniple. This inludes all of lassial mehanis and all of eletriity and magnetism. In fat, if we postulate other interations, suh as the Yukawa potential, the mathematial form of these interations is very restrited. The flexibility in the hoie of natural laws is very limited. In the future, as so-alled grand unified theories are developed, it is expeted that even this limited flexibility will be removed. One of the remarkable developments of modern physis has been the growing pereption that the laws of physis are inevitable. Hawking may have gone beyond the realm of pure physis when he asked the question Did God have any hoie? in the way She wrote the laws of physis. However, it seems that if the universe onsists of three spatial dimensions and time, and we require ausality, then there is little hoie in the laws of physis. Hamilton s priniple states that as a system moves from its onfiguration at time t 1 to that at time t it does so along the trajetory for whih a quantity alled the ation is minimized. Unfortunately, we don t have time to pursue this line of thought here, but it will be suffiient for our purposes to onsider those quantities that must be onserved from t 1 to time t. If we apply Einstein s postulates to these onserved quantities, we an learn a great deal about the laws of physis vetors To illustrate the problem with Newtonian mehanis, onsider the elasti ollision of two partiles of equal mass m shown in Figure 17. Viewed in the lab frame K, the partiles are v = v 1. After the ollision, the y inident with equal and opposite veloities, so that ( ) ( ) omponents of the veloities are reversed, and by symmetry we see that ( ) = ( 1) v v, where a bar indiates the value after the ollision. In Newtonian mehanis we would explain this by the onservation of momentum and energy, ( 1) + ( ) = ( 1) + ( ) ( 1) + ( ) = ( 1) + ( ) p p p p (1.87) E E E E (1.88) where p = mv (1.89) 1

22 Figure 16 1 mv If now we view this ollision in a oordinate system moving in the x -diretion with the veloity E = (1.90) V = v x of partile 1, the ollision looks like the right-hand diagram in Figure 16. In this frame of referene partile 1 simply goes up and down. Momentum is still onserved in the x diretion sine the veloities in this diretion are the same before and after the ollision. But what about the y diretion? If we use the veloity transformation laws, we find that for partile 1 the y omponent of the veloity is ( ) y( ) ( v ) v' y 1 v' 1 vy () 1 = = Vv ' x () 1 γ x γ ( V ) 1+ sine v ' () 1 = 0, but for partile we find x (1.91) v ' y vy ( ) = (1.9) Vv ' x ( ) γ ( V ) 1+ But whereas v y () 1 and v y ( ) were equal and opposite in the lab frame, the orresponding quantities v ' y () 1 and v ' y ( ) are not the same in the moving frame. Thus, the total momentum in the y diretion is not zero ( ) ( ) p ( ) p' y 1 + ' y 0 (1.93) and sine all the momenta are reversed after the ollision, the total momentum is not onserved. What went wrong? The problem (as always) lies in the nature of time. In the veloityaddition laws we divided the distane inrements by the time inrement, and this isn t the same for all the partiles. Sine we are dealing with a 4-dimensional Minkowski spae-time, we need to generalize our vetors to four dimensions. We therefore define the position 4-vetor as the set of four quantities (,,, ) (, ) r α = t x y z = t r (1.94)

23 where α = 0 3 and r is the 3-dimensional vetor position. Under a Lorentz transformation, this 4-vetor position transforms in aordane with Einstein s postulates of relativity. In partiular, the length of the vetor s ( t) = r, whih we all the interval, is invariant. We all suh an invariant quantity a 4-salar. In three dimensions, any set of three quantities, suh as the veloity, that transforms the same as the position is alled a vetor. Extending this to Minkowski spae, we an define the 4- vetor veloity as follows. For an inrement of position (,,, ) (, ) dr α = dt dx dy dz = v dt (1.95) orresponding to the inrement of proper time we define the 4-vetor veloity dt dτ = (1.96) γ α α dr u = = γ (, v ) (1.97) dτ Clearly, this is a 4-vetor sine the numerator is a good 4-vetor and the denominator is invariant under a Lorentz transformation. We note that in the limit of low veloity, γ 1 and the spaelike part u v, the nonrelativisti veloity, as it should. Curiously, the timelike 0 omponent of the 4-vetor veloity p is always larger than. It s straightforward from here to define the 4-vetor momentum ( γ, γ ) α α p = mu = m mv (1.98) where m is alled the rest mass. It is invariant, the same in all oordinate systems. The spaelike part is learly the relativisti generalization of the ordinary momentum, for p = γ mv mv (1.99) γ 1 But what about that timelike omponent? Well, at low veloities we an expand m 1 v 1 1 γ m = m 1 m mv v<< + = + (1.100) v 1 The seond term in the parentheses is just the familiar kineti energy. The first term is alled the rest energy, E rest = m (1.101) arguably the most famous equation in all of physis. We see, therefore, that the timelike omponent of the 4-vetor is just the total energy E divided by the speed of light. The 4-vetor momentum is therefore 3

24 where p α E =, p (1.10) p = γ mv (1.103) E = γ m (1.104) Finally, we an make one more use of the powerful mahinery of the Lorentz transformation. Sine it leaves the length of a 4-vetor invariant, we an evaluate the length in any oordinate system. Applying this to the 4-vetor momentum, we get E p = γ m γ mv = m (1.105) where the last expression on the right is the value in the rest frame of the partile, the frame in whih v = 0. Rearranging this gives us the very useful formula whih expresses the energy in terms of the momentum. 4 E = p + m (1.106) In 4-vetor notation, then, the onservation of momentum for N partiles of arbitrary mass beomes N N α α pn = pn, 0 3 n= 1 n= 1 α = (1.107) This replaes the separate energy and momentum onservation laws of Newton, and even inludes the ase when the masses before and after the ollision or other event are not onserved. In this ase, some of the mass is exhanged for kineti energy. As an example of the onservation of 4-vetor momentum (what we used to all the onservation of momentum and energy), we examine the emission of a gamma ray photon by a * * nuleus of mass m. Before the emission, the energy is E = m, and the momentum is p = 0. After the photon is emitted, the total energy is E' = γ m + hν, where hν is the photon energy (the quantity h is alled Plank s onstant, and ν is the frequeny; we ll learn about this in quantum mehanis) and γ is the relativisti fator for the nuleus, whih now has mass m. Sine a photon has no rest mass, the energy relation (1.106) beomes E photon = pphoton = hν. The hν total momentum is now p' = γ mv. The onservation equations are therefore * E = hν + γm = m (1.108) hν p = γ mv= 0 (1.109) 4

25 We an eliminate hν from these equations, and realling that γ = 1/(1 v / ) we an solve for β = v/. The result is simply * m m = 1+ β 1 β (1.110) 5