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SpringerBriefs in Physics For further volumes: http://www.springer.com/series/8902 Editorial Board Egor Babaev, University of Massachusetts, USA Malcolm Bremer, University of Bristol, UK Xavier Calmet, University of Sussex, UK Francesca Di Lodovico, Queen Mary University of London, London, UK Maarten Hoogerland, University of Auckland, New Zealand Eric Le Ru, Victoria University of Wellington, New Zealand James Overduin, Towson University, USA Vesselin Petkov, Concordia University, Canada Charles H.-T. Wang, The University of Aberdeen, UK Andrew Whitaker, Queen’s University Belfast, UK Péter Hraskó Basic Relativity An Introductory Essay 123 Emeritus Professor at University of Pe´cs, Hungary Péter Hraskó University of Pécs H-7633 Pécs Szántó Kovács János u. 1/b Hungary e-mail: peter@hrasko.com ISSN 2191-5423 e-ISSN 2191-5431 ISBN 978-3-642-17809-2 e-ISBN 978-3-642-17810-8 DOI 10.1007/978-3-642-17810-8 Springer Heidelberg Dordrecht London New York � Péter Hraskó 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcast- ing, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar, Berlin/Figueres Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Traditional presentations of relativity theory start with the introduction of Lorentz- transformations from which the peculiar phenomena of the theory (time dilation, Lorentz contraction, the velocity addition formula, etc.) follow. Though this is certainly the most logical approach, it seems rather unfortunate from a pedagogical point of view, since a convincing and conceptually transparent explanation of the Lorentz-transformation itself presents a task of considerable difficulty. Lorentz- transformation is based on both the constancy of the light speed and Einstein’s synchronization prescription, and the interrelation between these two constituents is open to the frequent misunderstanding that constancy of the light speed is enforced by the special synchronization of clocks rather than being the law of nature. In order to avoid this pitfall an ad hoc though rigorous presentation of the theory’s perplexing properties in Part 1 precedes the introduction of the Lorentz- transformation (and any synchronization procedure). After the introduction of these transformations in Part 2 those same relativistic effects are reconsidered this time in a systematic manner. Part 3 is devoted to the fundamentals of general relativity. The book is based on the lectures given at the post graduate course in physics education at the Eötvös Loránd University (Budapest). Budapest, December 2010 Péter Hraskó v Contents 1 From Time Dilation to E0 = mc 2. . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Reference Frames and Inertial Frames . . . . . . . . . . . . . . . . . . 1 1.2 The Optical Doppler-Effect and Time Dilation . . . . . . . . . . . . 4 1.3 The Relativity of Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 The Proper Time and the Twin Paradox . . . . . . . . . . . . . . . . . 11 1.5 The Lorentz Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 Velocity Addition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 The Equation of Motion of a Point Particle. . . . . . . . . . . . . . . 17 1.8 Does Mass Increase with Velocity? . . . . . . . . . . . . . . . . . . . . 20 1.9 The Kinetic Energy of a Point Mass. . . . . . . . . . . . . . . . . . . . 20 1.10 The Rest Energy: The E0 = mc 2 Formula . . . . . . . . . . . . . . . . 22 1.11 Is Mass Conserved? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.12 The Popular View on the Mass–Energy Relation . . . . . . . . . . . 26 2 The Lorentz-Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1 The Coordinate Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Independence of the Constancy of c from Synchronization . . . . 32 2.3 The Minkowski Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 The Lorentz-Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Classification of Spacetime Intervals . . . . . . . . . . . . . . . . . . . 38 2.6 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7 The Causality Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.8 Demonstration of Time Dilation on Spacetime Diagram . . . . . . 48 2.9 Doppler-Effect Revisited. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.10 The Connection of the Proper Time and Coordinate Time in Inertial Frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.11 The Magnitude of the Twin Paradox . . . . . . . . . . . . . . . . . . . 52 2.12 The Coordinate Time in Accelerating Frames: the Twin Paradox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.13 The Coordinate Time in Accelerating Frames: the Rotating Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 vii 2.14 Lorentz Contraction Revisited . . . . . . . . . . . . . . . . . . . . . . . . 60 2.15 Is the Perimeter of a Spinning Disc Contracted? . . . . . . . . . . . 62 2.16 Do Moving Bodies seem Shorter? . . . . . . . . . . . . . . . . . . . . . 63 2.17 Velocity Addition Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.18 Equation of Motion Revisited . . . . . . . . . . . . . . . . . . . . . . . . 64 2.19 The Energy–Momentum Four Vector . . . . . . . . . . . . . . . . . . . 66 2.20 Massless Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.21 The Transformation of the Electromagnetic Field . . . . . . . . . . 69 2.22 The Thomas-Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.23 The Sagnac Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.1 Gravitational and Inertial Mass . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 The Equivalence Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3 The Meaning of the Relation m* = m . . . . . . . . . . . . . . . . . . . 78 3.4 Locality of the Inertial Frames. . . . . . . . . . . . . . . . . . . . . . . . 79 3.5 The Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.6 The GP-B Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.7 Light Deflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.8 Perihelion Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.9 Gravitational Red Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5 Selected Problems to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Index . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 viii Contents Chapter 1 From Time Dilation to E0 5 mc 2 Abstract Time dilation and the relativity of simultaneity are deduced from the Doppler-effect. Lorentz contraction and the equation of motion are derived from time dilation. Mass-energy relation is proved and its popular interpretation is critically examined. Keywords Reference frames � Time � Simultaneity � Contraction �Mass � Energy 1.1 Reference Frames and Inertial Frames Physical phenomena are always described relative to some object (laboratory, the surface of the Earth, moving traincar, spacecraft, etc.). Objects of reference of this kind are called reference frames. Though reference frames and coordinate systems are two very different notions they are not always clearly distinguished from each other. When, in order to study a certain phenomenon, a measurement is performed the instruments (including clocks and measuring rods among them) are always at rest in the reference frame used but nothing like ‘coordinate system’ is found there. Coordinate systems serve to assign a triple of numbers to the points of space in order to make calculations possible, while the purpose of the reference frames is to accommodate measuring apparatuses and their personnel. Phenomena which we try to observe and predict are coincidences, i.e. encounters of bodies, whose coordinates are important but unobservable auxiliary quantities. A coordinate system requires more than just three mutually perpendicular axes through the origin: the set of coordinate lines must cover a whole domain of space. Such an infinitely dense set of coordinate lines exists only in our minds and a great many misunderstandings could be avoided if the really existing (or imagined as such) reference frames were never called coordinate systems (and vice versa). Reference frames with respect to which the laws of Nature take their simplest possible form are called inertial frames. This rather informal definition P. Hraskó, Basic Relativity, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-17810-8_1, � Péter Hraskó 2011 1 presupposes that when the basic laws of a new field of physical phenomena have been successfully developed the concept of the inertial frame must be suitably adapted. In the first period of the modern history of physics, before the advent of electrodynamics, it was mechanics that reached a sufficiently high level of sophistication to formulate a precise law, the Newtonian law of mass� acceleration ¼ force, on which the definition of the inertial frames could be based. It is this formula which in Newtonian physics permits us to select inertial frames from the multitude of reference frames by the absence of inertial forces, i.e. by the criterion that in these frames one needs to take into consideration only forces, originating from well identifiable physical objects (true forces). In the special case when sources of this kind are absent (or are very far away) an isolated body retains its rectilinear uniform motion or remains at rest (the law of inertia). This is a practically applicable criterion to decide whether the reference frame a body is referred to is an inertial frame or not. A laboratory on the surface of the Earth is not an inertial frame since the plane in which the Foucault pendulum swings rotates with respect to it. This rotation is caused by the Coriolis force which is an inertial force. When the effect of the Coriolis force is negligible such laboratories can be considered as approximately inertial frames. But no laboratory on the Earth can be assumed an isolated inertial frame since all bodies in it are subjected to the action of the gravitation which from the Newtonian point of view is a true force.1 Therefore, in the laboratories on the Earth the law of inertia must be formulated in a counterfactual form: were grav- itation switched off (or compensated) the velocities of isolated bodies would remain constant. Given an inertial frame all the other reference frames which move uniformly or remain at rest with respect to it are, according to Newtonian physics, also inertial frames. Since in all of the inertial frames the basic laws of mechanics are of the same form these frames are, within the range of Newtonian mechanics, equivalent to each other. On the other hand, owing to the great variety of the inertial forces, generic reference frames are endowed with individual properties which make all of them intrinsically distinguishable from the others. The fundamental laws of electrodynamics are expressed by the Maxwell equations, according to which light propagates with the same velocity in all directions (isotropy). Vacuum light velocity is denoted by c. Einstein assumed that in their original form Maxwell equations are valid in the inertial frames which means that their observable consequences can be proved true with respect to these frames. In particular, it is only in the (isolated) inertial frames that speed of light is 1 It is a remarkable fact that because of weightlessness in them satellites, orbiting freely around the Earth, have the properties of a truly isolated inertial frame. Nevertheless, in the Newtonian framework they cannot be qualified as such since their center of mass is accelerating and bodies within them are subjected to the action of the corresponding inertial force. However, this force is precisely compensated by the gravitational attraction of the Earth. This question will be taken up again in Sect. 3.2 in connection with general relativity. 2 1 From Time Dilation to E0 = mc 2 equal to the same c in any direction. In this respect propagation of light is fun- damentally different from that of sound which is isotropic only with respect to its medium at rest. More generally, inertial frames free from outside influences are, from the point of view of both mechanics and electrodynamics, equivalent to each other. Though the inclusion of electrodynamics does not invalidate the mechanical equivalence of the inertial frames and in particular the validity of the law of inertia in them it leads to a slight modification of the form of the Newtonian equation of motion which retains its original form mass� acceleration ¼ force only for velocities much smaller than c (see Sects. 1.7 and 2.18). As far as it is known today the equivalence of the inertial frames extends actually far beyond mechanics and electrodynamics into the realms of weak and strong interactions too. This assumption which is a far reaching generalization of the constancy of the light velocity constitutes the first of the two postulates of the special relativity theory. This theory preserves the important property of the inertial frames that their relative motion is uniform and rectilinear. These prop- erties are, however, lost in general relativity. As it can be guessed from its name this theory is the generalization of special relativity which emerged from Ein- stein’s attempts to extend this latter theory to the gravitation. In pursuing this aim Einstein realized that gravitation cannot be forced into the Procrustean bed of special relativity but special relativity can be extended so as to provide a surpri- zingly natural place to gravitation. This more general approach does not, of course, invalidate special relativity but, as can be expected, it recognizes the limits of its applicability. In what follows we will confine ourselves mostly to the special theory which in itself covers important areas of physics. The basic principles of general relativity will be outlined in Chap. 3. Returning to the electrodynamics let us notice that as far as the considerations are restricted to some given inertial frame the constancy of the light speed presents no problem.It can be experimentally verified by any method which has been accepted as legitimate procedure to measure light velocity as e.g. the rotating disc experiment of Fizeau or Foucault’s rotating mirror method.2 Either procedure is based on the path/time notion of velocity and they were performed as two-way experiments rather than unidirectional one’s with the only aim to improve accu- racy (see Sect. 2.2). But, as a matter of fact, it would be an extremely difficult task to measure light velocity in a number of inertial frames in relative motion with an accuracy sufficient to convince ourselves of its constancy. Instead, we may resort to an indirect reasoning. Should light speed not the same in the different inertial frames to a high degree of accuracy, this fact had already been come to light, owing to its numerous consequences. It is in fact the whole body of the twentieth century physics which testifies in favour of the relativistic postulate of light 2 Strictly speaking, it would be unreasonable to expect that speed of light should be constant in reference frames, resting on the Earth, since Coriolis force and gravitation do certainly influence the propagation of light. The influence of the rotation of the Earth manifests itself in the Sagnac effect (see Sect. 2.23), but the effect of the gravitation is extremely small (see Sect. 3.7). 1.1 Reference Frames and Inertial Frames 3 velocity. In what follows we will, therefore, consider the independence of the light velocity of the motion of the inertial frames a well established empirical fact. When, on the other hand, a given phenomenon is analysed simultaneously from the point of view of several inertial frames in different states of motion one, as a rule, runs into conflict with intuition. The essence of special relativity theory is to explicate these paradoxes and explain how to resolve them in a consistent manner. This chapter will be devoted to this theme. 1.2 The Optical Doppler-Effect and Time Dilation Imagine a light source which is continuously emitting sharp signals with a period of T0 (i.e. at a rate equal to m0 ¼ 1=T0) and a receiver which detects them. When the latter is at rest with respect to the emitter it will detect the signals with the same frequency. But when it is moving the observed frequency m (and the period T ¼ 1=m) will be different from m0 (and T0). This phenomenon is known as the Doppler-effect. Assume that the emitter and the receiver recede from each other with the constant velocity V (and both are inertial frames). Then the ratio m=m0 is smaller than 1 and, according to the equivalence of the inertial frames, its value is the same regardless of whether the emitter or the receiver is taken to be at rest.3 Fig. 1.1 Calculation in the rest frame of the receiver (RFR) 3 Note that in acoustics the propagation of sound is influenced, beside the motion of the emitter and the receiver, by the state of motion of the medium too. 4 1 From Time Dilation to E0 = mc 2 How does this ratio depend on V? Let us perform the calculation in the rest frame of the receiver (RFR). The trajectory of the emitter (X ¼ x0 þ Vt) and those of the light signals (X ¼ konst:� ct) are shown on Fig. 1.1 while the trajectory of the receiver is the t-axis itself. The intersections of the trajectories allow us to identify the time intervals T0 and T . In order to establish their connection the altitude h of the enlarged shaded triangle has to be expressed both through the slope of the emitter’s trajectory (h ¼ T0V ) and that of signals’ trajectories h ¼ ðT � T0Þcð Þ. The relation between the periods and frequencies follows from the equality of these two expressions: T ¼ T0 1 þ V=cð Þ; ð1:2:1Þ m ¼ m0 1 þ V=c : ð1:2:2Þ Using Fig. 1.2, a completely analogous calculation can be performed with respect to the emitter’s rest frame (RFE). In this case h ¼ TV ¼ ðT � T0Þc and hence T ¼ T0 1 � V=c ð1:2:3Þ m ¼ m0 1 � V=cð Þ: ð1:2:4Þ A glance at these formulae reveal that they are in plain contradiction with the assumed equivalence of RFR and RFE since the two ratios m=m0 differ from each other: in the first case (relative to RFR) m=m0 is equal to ð1 þ V=cÞ�1 while in the second (relative to RFE) it is given by ð1 � V=cÞ. In both cases the frequency m is Fig. 1.2 Calculation in the rest frame of the emitter (RFE) 1.2 The Optical Doppler-Effect and Time Dilation 5 smaller than m0 but in different proportion. The equality of the light velocity alone in RFR and RFE is, therefore, not sufficient to ensure their equivalence from the point of view of the Doppler effect. Something important must still be lacking. To reveal it the Doppler effect itself must be scrutinized in some more depth. The starting point of our calculation with respect to RFR was the seemingly obvious (but hidden) assumption that the frequency m0 of the emitter is not altered by its motion and the frequency m registered by the receiver differs from m0 solely because the subsequent signals are emitted farther and farther away from it. The merit of Fig. 1.1 is the graphical expression of this fact. But might it not be that the emitter’s frequency itself has also been changed due to its motion? An analogous question can be asked concerning the calculation with respect to RFE too. In this case we have started from the tacit assumption that the counter of the receiver clicks at a rate less than m0 only because between two subsequent clicks the receiver gets farther and farther away from the emitter. Figure 1.2 expresses this fact visually. But might it not be that the rate of the receiver’s clicks itself is somehow influenced by the receiver’s motion too? Let us weigh on this possibility. Assume that the duration of the time interval between two events on a moving object is influenced by the velocity V of the object itself and this influence can be accounted for multiplying the time interval between the events by some function cðVÞ of the velocity. Since in the receiver’s rest frame it is the emitter which is moving, in (1.2.1) T0 has to be replaced by cT0 while in (1.2.3), referring to the emitter’s rest frame, it is T which is to be replaced by cT . Having performed these substitutions we obtain the formulae T ¼ T0c 1 þ V=cð Þ; Tc ¼ T0 1 � V=c which replace (1.2.1) and (1.2.3) respectively. We can now try to choose c in such a way as to make these two formulae identical. This is very easy to do. If we assume c ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V2=c2p ; ð1:2:5Þ then both expressions reduce to the same one given by T ¼ T0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ V=c 1 � V=c s ; ð1:2:6Þ or expressed in terms of the frequencies m ¼ m0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V=c 1 þ V=c s : ð1:2:7Þ 6 1 From Time Dilation to E0 = mc 2 Since m is still smaller than m0 we obtained an acceptable formula but the decisive moment for its acceptance is that the experimental study of the light emitted by moving atoms speaks unequivocally in its favour. The correctness of our reasoning is strongly supported by the existence of the transverse Doppler-effect. So far we have dealt with the longitudinal effect when the motion takes place along the straight line through the emitter and the receiver. In the transversal case the motion is perpendicular to this direction. Assume that the receiver revolves on a circle around the emitter. According to the prerelativistic conception of the Doppler-effect, in this case no frequency shift is expected to occur since the distance between the emitter and the receiver is not changing: T ¼ T0, m ¼ m0. If, however, the motion of the receiver alone is suffi- cient to influence the rate at which the light signals are perceived by it, then T must be replaced in the first of the above formulae by cT and we will have cT ¼ T0 and m=c ¼ m0, i.e. T ¼ T0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V2=c2 p ;m ¼ m0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V2=c2p : The validity of this formula has been proved experimentally, using Mössbauer effect. Now it will be shown that the effect of the factor cðVÞ can be summarized in the following statement: on a moving object time is flowing slower than expected (time dilation). To see this we first observe that the T in (1.2.6) is equal to the geometric mean of the T -s in (1.2.1) and (1.2.3) (see Problem 1 for the significance of this) and the former is smaller than the latter: T0 1 þ V=cð Þ\T0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ V=c 1 � V=c s \ T0 1 � V=c : Expressed somewhat hazily this can be written also as (1.2.1) \ (1.2.6) \ (1.2.3). Therefore, if the emitter is in motion then, comparing (1.2.1) with (1.2.6), we see that the signals arrive at the receiver at rest with a longer time lag as expected. This continuous delay can obviously been attributed to the slowering of the flow of time in the emitter’s reference frame due to its motion which means that all the processes on it, including the speed of clocks, become slower. When, on the other hand, it is the receiver which moves then, according to (1.2.3) and (1.2.6), the time interval it observes between two subsequent signals is smaller than expected, considering the rate at which the emitter at rest works. The decrease of this time interval is naturally attributed to the slowering of the flow of time in the receiver’s rest frame due to its motion, in particular to the slowing down of the speed of the clocks in it. The transverse effect discussed above demonstrates this phenomenon in its purest form. 1.2 The Optical Doppler-Effect and Time Dilation 7 1.3 The Relativity of Simultaneity If the above explanation of the Doppler effect is correct then both the statement that in RFR time flows slower than in RFE and its opposite are true, i.e. time dilation is a symmetrical phenomenon: the time in either of the inertial frames flows slower than in all the others. Is this not a complete nonsense? Before facing this question it has to be stressed that as far as the experimental facts are concerned the symmetry of time dilation leads to nothing like contra- diction. Let us assume that, instead of a separate emitter and receiver, we have two equipments of similar construction, both containing a combined emitter-receiver setup. Let they move with the velocity V with respect to each other. The emitters in both of them send signals of the same frequency m0 and both receivers perceive these signals with the frequency m predicted by (1.2.6). The smooth operation of the apparatuses is by no means disturbed by our recognition that time in either of them flows slower than in the other. Our intuitive notion of time is, however, in sharp contradiction with this interpretation. Let both inertial frames, containing the combined setups, are also equipped with ideal clocks4 say O and O0. Assume that at the moment the two setups passed by each other both clocks showed 0 s. Since O0 goes slower than O, at the same moment when O shows say 5 s the hand of O0 points only at 4 s. But O is also slower than O0, hence at the same moment when O0 shows 4 s the hand of O points only to 3 s. But this is impossible, since our starting assumption was that at this same moment O shows 5 s. We have arrived at a logical contradiction because a clock cannot show two different times at the same moment. However, this seemingly impeccable argumentation has a weak point. It is the tacit assumption that simultaneity is an intrinsic property of a pair of events and, therefore, if two events are simultaneous with respect to the rest frame of O they remain simultaneous with respect to the rest frame of O0 too (the simultaneity is absolute). The most enlightening discovery of Einstein was that the constancy of the light velocity is incompatible with this assumption: the simultaneity of distant events taken in themselves is a meaningless statement. A simple example may be helpful to elucidate the content of an assertion of this kind. Imagine two bodies which move in the same direction along some straight line in an inertial frame I . Let their velocities be v1 and v2. Assume that the bodies are at the same time observed from another inertial frame I0 too whose velocity V with respect to I lies between v1 and v2. Now ask the question: do the bodies in themselves move in the same or in the opposite direction with respect to each other? Neither of these possibilities are more true (or false) than the other since I and I0 are equivalent inertial frames and with respect to the former the bodies move in the same direction while with respect 4 The most important steps in the improvement of clocks are: sand and water clocks, clocks with escapement mechanism, pendulum clocks, chronometers, quartz clocks, atomic clocks, etc. The miniaturited ideal clock is the extrapolated endpoint of this real development. 8 1 From Time Dilation to E0 = mc 2 to the latter they move in the opposite direction. Therefore, any statement, con- cerning the relative direction of the bodies’ motion in itself, is meaningless. As Einstein proved by his famous train (& platform) thought experiment, something very similar is true for the distant simultaneity of a pair of events also. Let a flash of light is given off at the center of a traincar which passes by the platform of a station. The light signal triggers an explosion on both ends of the car. Are the explosions simultaneous or not? Since light propagates with the same velocity with respect to both the train and the platform in any direction no unambiguous answer can be given to that question. In the rest frame of the train the explosions are simultaneous because the light flash was given off at center of the car. However, in the rest frame of the platform the rear of the traincar moves toward the point at which the flash was given off, its front moves away from this point, while the light signals propagate with the same velocity toward both ends of the car. Therefore, the explosion at the rear of the car takes place earlier than at its front (see Problem 2). We can now ask: are the explosions in themselves simultaneous or not? The answer is the same as above: neither of these possibilities are more true (or false) than the other since the rest frames of the railcar and that of the platform are both inertial frames, intrinsically equivalent to each other. In short, the conclusion is that no absolute simultaneity exists. How the recognition of this relativity of simultaneity eliminates the contra- dictory readings of the clock O found above? Let the encounter of the clocks be the event E0. At this moment they both show 0 s. With respect to the rest frame I of O the clock O0 is moving and keeps going slower than O; when e.g. O shows 5 s (let this be the event E5) the hand of the clock O0 points, at the same moment, to only the 4 s (call this the event E4). These two events are, therefore, simulta- neous in I . But since they take place at some distance from each other (the former on the clock O while the latter on the clock O0) their simultaneity is not absolute. In the rest frame I0 of O0 with respect to which O is going slower the moment simultaneous with E4 may differ from E5. It may happen that when O0 shows 4 s the pointer of O at the same moment stands only at 3 s and, owing to the lack of absolute simultaneity, this event E3 need not be the same as E5. It is this possi- bility which permits us to avoid the absurd conclusion that O should have two different time readings at the same moment.5 A further aspect of the relativity of simultaneity can be elucidated by the following example. Imagine a Mars rover (an automated motor vehicle which propels itself across the surface of the Mars) which is climbing up a hill and, having arrived at the top, stops and sends a radio signal to the mission headquarterson the Earth. At the moment of the arrival of the signal a response is immediately released which causes the rover to immediately start its descent. How long was the vehicle standing? If in the time of the operation the Mars was at a distant, say, 5 We will return to this reasoning once again in Sect. 2.8. 1.3 The Relativity of Simultaneity 9 L ¼ 105 million kilometers from the Earth than it was keeping at rest during 2 � L=c ¼ 700 s. Assume that the clock at the mission headquarters showed 0 s at the moment when the signal from the Mars arrived and the response was sent. Where was the rover found at just the same moment of time? Obviously, it was staying on the top of the hill. For, if it had been sending signals continuously during its climbing up, they would have arrived at the mission center before the zero moment. If, moreover, the transmitter on the Earth had been continued to work even after that moment, its signals would have been perceived by the rover when it was moving down the hill. The radiostation on the Earth would have been, therefore, operating continuously: signals would have been either observed or transmitted. But the operation of the rover’s radioapparatus would have been interrupted for a period of 700 s the whole of which corresponded to the single zero moment on the Earth. Therefore, the rover was indeed staying on the top when the mission center’s clock showed 0 s but its staying there lasted 700 s. Would it have been possible to specify the moment within this interval which corresponded exactly to the zero moment on the clock on the Earth? That could be done only by means of an instantaneous signal from the Earth to the rover which would have marked the corresponding moment on the rover’s time keeping device. If e.g. there existed a rocket capable of reaching a velocity, say, 10c than, using it, the 700 s long ‘interval of simultaneity’ could have been shortened to 70 s. In this case it would be natural to assume that, though truly instantaneous signals of infinite velocity are probably beyond our reach, there had to exist, nevertheless, a unique moment of time at the top of the hill where the rover was standing which was precisely simultaneous to the zero moment on the mission center’s clock on the Earth. Newtonian physics is based on precisely this conception of simultaneity. But the explanation of the Doppler effect found in the previous section undermines this possibility since the factor 1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V2=c2 p becomes imaginary when the velocity V in it exceeds c. Obviously, this explanation can be held true only if bodies (reference frames) cannot be accelerated up to the velocity of light. The empirical success of the formula (1.2.6) prompts us to accept the second postulate of the relativity theory, according to which no bodies or signals can exceed the velocity of light in vacuo.6 As it will become clear in Sects. 1.6 and 1.7, this postulate is sufficient to ensure, that bodies which can serve as reference frames cannot indeed be accelerated up to c. At a given point of space, therefore, moments within a whole time interval may in principle be simultaneous with an event E at some other point. This is an essential aspect of the relativity of simultaneity since all the moments within this interval are indeed simultaneous with E in some given inertial frame. The interval of simultaneity is the longer the farther the points are situated from each other in space. It would, however, be completely misleading to draw from this the 6 The problem of superluminal signal propagation will be discussed in some more detail later in Sect. 2.7. 10 1 From Time Dilation to E0 = mc 2 conclusion that we now know much less than before because now we are unable to say unambiguously wether two distant events are simultaneous or not while earlier we could give a definite answer to this question. The situation is just the opposite since our knowledge has been substantially increased by the recognition that distant simultaneity is not an intrinsic property of pairs of events. The idea of the relativity of simultaneity is psychologically difficult to accept because our practice with now-questions (what happens now somewhere else, what is doing now somebody who is away) proves that they can be given sound answers. This is indeed the case because our experience is limited to distances at which the ‘interval of simultaneity’ discussed above is very short and, moreover, we are never faced with situations in which simultaneity of given pairs of events are to be related to different frames of reference. Considerations of the latter type are, however, indispensable in physics whose aim is to discover the most basic laws of Nature. 1.4 The Proper Time and the Twin Paradox The proper time of an object is equal to the time read off from a (fictitious or really existing) clock attached to the object. Proper time and proper time interval are among the most important notions of relativistic physics. It is strongly recom- mended to use the symbols s and Ds to denote them which, if necessary, can be specified further by suitable indices, bars, etc. If we wanted to apply this con- vention in retrospect, we should replace the symbols T and T0 in Sect. 1.2 by Ds and Ds0, since they are readings on two clocks attached to the receiver and the emitter respectively and are, therefore, proper time intervals (and m and m0 are proper time frequencies). As we already know, the proper time on moving objects flows slower than the time t found in the formulas like X ¼ x0 � Vt and X ¼ konst:� ct which describe their trajectories. This phenomenon was called time dilation. The relation between the increments ds and dt is given by the formula ds ¼ dt � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V 2=c2 p ð1:4:1Þ which means that in the infinitesimal interval ðt; t þ dtÞ, when the velocity is equal to VðtÞ, the increment ds of the proper time is equal to dt � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � VðtÞ2=c2 q . When the velocity is constant, ds and dt can be replaced by finite increments. The validity of (1.4.1) can be deduced from our interpretation of the Doppler effect. When e.g. the emitter was moving, we had to replace T0 with cT0. This means that the segment Dt of the t-axis which corresponds to the proper time interval T0 is equal to cT0. Hence, Dt ¼ T0= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V2=c2p which is nothing but the formula (1.4.1) applied to this special case. 1.3 The Relativity of Simultaneity 11 In (1.4.1) the velocity V is equal to ds=dt since the trajectory itself contains t rather than s. The simultaneity of distant events is established also by the equality of their moments of time t. In fact, within a given inertial frame, time t can be identified with the Newtonian time we all accustomed to in secondary school physics. The whole of this chapter is based on this familiar notion of the New- tonian time7 and the concept of the proper time. The physical meaning of the proper time is best seen in the phenomenon of the twin paradox. If two objects meet at some moment of time ta and again at a later moment tb then the proper time intervals Ds1 s Ds2 elapsed on them between the two encounters will be, as a rule, different from each other, and neither of them will be, in general, equal to the difference Dt ¼ ðtb � taÞ. It is only in the special case, when the object 1 remains at rest, that Ds1 is equal to Dt. The value of Ds2 will then be smaller than Dt since on every infinitesimal part of its trajectory ds ¼ dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � VðtÞ2=c2 q is smaller then dt. This is the essence of the twin paradox which is often demonstrated with two twins one of which sets forth a long space trip while the other remains at home (in an inertial frame). As a consequence of the time dilation, after their subsequent encounter the formerwill be younger then the latter (Ds2\Ds1 ¼ Dt). But there is a problem here. As we know, time dilation is symmetric and one might suppose that the same is true for the twin paradox: from the point of view of the spacecraft as the frame of reference the twin at home becomes younger because he was making a ‘‘journey’’. We have, therefore two opposite conclusions whose contradiction cannot be dispensed with by alluding to the relativity of simultaneity since they refer to a single event (the encounter of the twins) rather than a pair of distant events. The contradiction is resolved by noting that (1) the rest frames of the twins are not equivalent since if one of them is inertial the other is necessarily accelerating (otherwise the twin who made the trip could not have been returned), and (2) (1.4.1) is applicable only in inertial frames. In order to save their physical content, even the Newtonian equations of motion change their mathematical form when they are applied for some reason or another in a non-inertial frame. The same is true for the equation (1.4.1). Since the readings on a given clock attached to a moving object cannot depend on the reference frame the motion is referred to (the proper time is invariant) the formula (1.4.1) when applied in a non-inertial frame has to be modified so as to save its original physical content. This can be done in much the same way as the Newton equations are transformed from an inertial frame to an accelerating one. Therefore, independently of the frame with respect to which the phenomenon is described, it is always the twin subjected to accelera- tions who turns out younger.8 7 A deeper insight into the notion of the Newtonian time will be required only in Chap. 2 (see Sect. 2.1). 8 A more detailed discussion of these questions is found in Sects. 2.11 and 2.12. 12 1 From Time Dilation to E0 = mc 2 1.5 The Lorentz Contraction The length Dl of a moving train is equal to the distance at which its endpoints are found from each other at the same moment of time. At the moment when the rear end of the train goes past a given point, its front end is at a distance V � D�s from it, where D�s is the time the train was passing by the point chosen. Hence, the length of the moving train is equal to this product. Measurement of Dl, therefore, requires the measurement of D�s and V . Assume, that we are waiting for the train to come at a point P of the embankment. The time D�s can be measured simply with a stopwatch but the determination of the velocity requires some preliminary preparations. In the direction of the train’s course, at a distance Ds0 from P, we arrange a relay on the rail and a light source beside, which gives off a flash of light at the moment the front of the train sets the relay to work. Having these preparations finished, we occupy our position of observation at P with a stopwatch in each of our hands; one which serves to measure the time D�s the train passes by, the other to determine its velocity. Both will be started at the moment when the front of the train reaches P but the first will stop when the rear end goes past and the other when the flash of light is observed. The reading Ds of this second stopwatch is equal to the sum of two time intervals: the time Ds0=V the train reaches the relay and the time Ds0=c the light flash reaches the point P of observation: Ds ¼ Ds0 V þ Ds0 c : ð1:5:1Þ Solving this equation, the velocity of the train can be obtained. As we saw, the length Dl of the train is equal to V � D�s. Is this the same length Dl0 which the passengers measure with their measuring rods (or by the time the light signals travel from one end of the car to the other and back)? The answer is no, the length Dl turns out to be shorter than Dl0. This can be proved by reflecting on what passengers see. They see a man with stopwatches in his/her hands who passes by their car of length Dl0 in a time interval Dt and so they find his/her velocity V with respect to the train equal to Dl0=Dt. They calculate the proper time D�s elapsed on the man’s stopwatch (the first one) to be equal to Dt � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � V2=c2p . But Dt is equal to Dl0=V , hence D�s ¼ Dl0 V ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V2=c2p and, therefore, Dl ¼ V � D�s ¼ Dl0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V2=c2 p : ð1:5:2Þ As we see, the length of the train undergoes contraction along the dimension of motion, its (relativistic) length Dl being smaller than its proper length Dl0 (Lorentz contraction). As we have stressed at the beginning of the present section, the notion of the length of an object in motion involves simultaneity of pairs of distant events, which consist in being of the two ends in some given points at the same moment of time. It is, therefore, the relativity of distant simultaneity which 1.5 The Lorentz Contraction 13 makes the length of objects along their direction of motion to depend on their velocity. In our derivation of (1.5.2) this is reflected in the fact that the difference between Dt and Ds is also the manifestation of this property of simultaneity. As a corollary, lengths in dimensions perpendicular to the direction of motion do not undergo any modification since, as it is easy to see, their measurement does not involve measurement of time intervals. Lorentz contraction is a perfectly real phenomenon: the length of a moving object is equal to its relativistic length (1.5.2) in any context the notion of length is legitimately used. Consider e.g. the following thought experiment due to J. S. Bell. Imagine a very long solid rod which is to be transported by train to some distant place. Being too long to be mounted on a single car, somebody suggests to place the two ends of the rod on two identical motor-cars which are provided with appropriate bumpers to prevent the rod from moving back and forth. The motion of the motor-cars are controlled by independent identical computers which execute the commands of identical programs. The programs in the two computers are started at the same moment of time in the rest frame of the Earth’s surface with the aid of two simultaneous radiosignals sent from a point, symmetric with respect to the cars at rest. Subsequent timing of the computers is provided by their own ideal clocks. As a result of this procedure, the cars remain during their motion at the same distance from each other in the reference frame of the Earth, since commands to accelerate and decelerate are given at the same moment on both cars. But, owing to Lorentz contraction, the rod may fall in between them. Why the length of the rod and the distance between the cars do behave so differently? It is because the different nature of the laws, determining them. The length of the rod is determined by the laws of Quantum Theory. Since, according to the equivalence of the inertial frames, these laws are the same in all of these frames, the proper length of the rod is also the same in all of them. If the acceleration is sufficiently smooth, we may assume that the system of the cars with the rod on them, is at any instant at rest with respect to the inertial frame, moving with the instantaneous velocity of the cars. It is this instantaneous rest frame with respect to which the length of the rod remains the same while contracted in any other. The distance between the cars is, on the other hand, determined by the condition that identical program steps are performed on both cars simultaneously in the rest frame of the Earth’s surface. As seen from the instantaneous rest frame, these pairs of events cease to remain simultaneous. Just as in the Einstein’s train thought experiment,9 the time sequence between these pairs of program steps is modified so that the step in the front car is performed first. Therefore, in the accelerating initial period of the motion the front car in the instantaneous rest framewill be accelerated with respect to the rear one and, since the length of the rod in this frame remains the same, it can fall in between the cars. 9 When the explosions take place simultaneously on the fringes of the platform rather than on the ends of the train, it is the explosion in the direction of the train’s course which occurs first. 14 1 From Time Dilation to E0 = mc 2 However, the length of the rod and the distance between the cars may behave differently from each other only when they are accelerated. When merely observed from another inertial frame they both undergo Lorentz contraction in the same proportion (see Sect. 2.14). 1.6 Velocity Addition Consider two bodies, moving with the velocities V and U in opposite direction. The distance between them is then changing at rate ðV þ UÞ in both Newtonian physics and relativity theory, because the notion of the rate of change of the distance is a direct consequence of the definition of the velocity as Ds=Dt. This rate, therefore, can be larger than c even in relativity theory (but it cannot exceed 2c). When, on the other hand, the bodies are moving in the same direction their distance is changing at a rate jV � Uj. The relative velocity of the first body with respect to the second one is equal to the velocity of the first body in the rest frame of the other. Therefore, the relative velocity depends on how the motion is seen from different reference frames and, as a consequence, it need not be the same in Newtonian and relativistic physics. For the sake of definiteness, let us consider the case of bodies, moving in opposite direction. Their relative velocity in the Newtonian physics is given by the same formula ðV þ UÞ as above but, as we will show now, this is no longer true in relativity theory. Let us return to the measurement of the train’s velocity in the last section and try to describe the same procedure as seen from an inertial frame I0 , moving with the velocity U with respect to the embarkment in the direction opposite to the train. The rest frame of the embarkment will be the unprimed one I . As seen from I0 , the time Dt between the moment the front of the train goes past P and the light flash is observed consists of the same two parts as in the previous section. The train reaches the relay at the time interval Ds=ðV 0 � UÞ, where Ds is the distance between the point P and the relay, V 0 is the train’s velocity, and ðV 0 � UÞ is the rate of change of the distance between the train and the relay. All quantities are, of course, related to I0 , where both the train and the relay are moving in the same direction. Similarly, the light signal covers the same distance Ds at a time Ds=ðU þ cÞ, since the train and the signal move toward each other. Therefore, Dt ¼ Ds V 0 � U þ Ds U þ c : But since the stopwatches and the embarkment (with the observer and the relay on it) move with velocity U in I0 , we have the relations Dt ¼ D�s= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � U2=c2p and Ds ¼ Ds0 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � U2=c2p . Therefore 1.5 The Lorentz Contraction 15 Ds Ds0 ¼ ð1 � U2=c2Þ 1 V 0 � U þ 1 U þ c � � : This same ratio can also be expressed from (1.5.1) too and , equating them, we obtain ð1 � U2=c2Þ 1 V 0 � U þ 1 U þ c � � ¼ 1 V þ 1 c : Solving this equation for V 0, we obtain the relative velocity of the two bodies, recessing from each other, as V 0 ¼ ðV þ UÞ=ð1 þ VU=c2Þ. A more general form can be obtained if we choose along the line of motion a positive direction and assign a plus or minus sign to the velocities with respect to it. If e.g. V is positive and the bodies are moving in opposite direction, then U must be negative. Since in this case we should recover the formula just obtained, the relativistic law of velocity addition must have the form V 0 ¼ V � U 1 � VU=c2 ð1:6:1Þ The physical meaning of (1.6.1) is this: When a body is moving with respect to I with the velocity V , the inertial frame I0 is moving with the velocity U with respect to I along the same line (in either positive or negative direction), then the body moves the velocity V 0, given by (1.6.1), with respect to I0 . This is illustrated on Fig. 1.3 which shows a pair of parallel railway tracks as seen from above. The car on the two lower track is at rest, defining thereby the inertial frame I . The car on the upper track is moving with the velocity U with respect to I ; its rest frame is the inertial frame I0 . The two sides if the figure refer to two different moments of time: the left side to the moment t1 when the cars are alongside each other and the right one to a later moment t2 (Dt ¼ t2 � t1 [ 0). Between the tracks a body is moving. At the two moments shown it is found in the points A and B respectively. The observers in the cars find its velocity equal to V and V 0. The velocity addition formula establishes the relation between U, V and V 0. Intuitively, the connection should be V 0 ¼ V � U. The figure suggests that Dl0 ¼ Dl� U � Dt and if to divide this relation by Dt and use the definitions V 0 ¼ Fig. 1.3 Meaning of the velocity addition formula 16 1 From Time Dilation to E0 = mc 2 Dl0 Dt and V ¼ DlDt, we indeed obtain this simple relation. But as seen from (1.6.1), the Newtonian form V 0 ¼ V � U of velocity addition is recovered only in the limiting case of slow motion when VU=c2 � 1. In the opposite extreme when, instead of a body, a light signal is propagating between the points A and B, (1.6.1) leads to the conclusion that its velocity is equal to the same c in both of the inertial frames I and I0 (V 0 ¼ V ¼ c). This consequence of the relativistic velocity addition for- mula is a necessary condition of its consistency with the constancy of the light velocity which has actually been employed in its derivation. 1.7 The Equation of Motion of a Point Particle The validity of the Newtonian equation of motion ma ¼ F is strongly supported by experience for motions much slower than light. However, from the standpoint of relativity theory, it can only be an approximation to some law of general validity since, in its present form, it allows acceleration of bodies up to arbitrarily high velocities. In order to derive the precise form of the equation, it will be assumed that, at any moment chosen, equation ma ¼ F remains exactly valid in the instantaneous rest frame I0 , i.e. as seen from the inertial frame with respect to which the particle is instantaneously at rest. Such a starting point ensures that (1) this equation remains, to a good approximation, applicable to slow motions and (2) its exact form can be established simply by expressing the equation in I0 through quantities, referring to an arbitrarily moving inertial frame I . Therefore, mass is the measure of inertia of a body at rest (and remains approximately so if the body moves with a velocity v � c). Imagine a rocket whose fuel dosage is kept at a constant level by means of an automatic controlling device and let us follow its motion during a time interval sufficiently small for the mass loss, due to fuel consumption, to be negligibly small. According to Newtonian mechanics, under these conditions the rocket will be accelerated uniformly. Indeed, the thrust F in the equation m � dv ¼ F � dt ð1:7:1Þ will be constant and, therefore, the acceleration will remain also unchanged: a ¼ dv dt ¼ F m ¼ konstans: ð1:7:2Þ According to relativity theory, however, as seen by an observer at rest on the Earth (in the inertial frame I ) the acceleration of the rocket will be continuously diminishing. The most obvious reason for this is that the uniformity of the fuel supply must be understood on the proper time scale since the regulating device, including its clock, is moving together with the rocket. Then, as a consequence of time dilation, the fuel supply will be the slower the larger isthe velocity of the rocket. The acceleration suppressing effect of time dilation is, moreover, 1.6 Velocity Addition 17 considerably amplified due to the fact that the rate of velocity increase per unit fuel consumption, as seen by an observer on the Earth, turns out also to be a decreasing function of velocity. As explained above, our starting point is that for small velocities (in the limit v ! 0) Newtonian mechanics is still applicable. Therefore, with respect to the rocket (in its instantaneous rest frame I 0) (1.7.1) remains valid provided the time in it means proper time and dv means the velocity increase dv0 in I0 : m � dv0 ¼ F � ds: ð1:7:3Þ As a result, the acceleration of the rocket with respect to itself (in I0) will con- tinuously remain equal to F=m. In order to calculate its acceleration in I , ds and dv0 have to be expressed in (1.7.3) through their counterparts in I . As a first step, formula dt ¼ cds allows us to express ds through the differential dt of the time t on the Earth (in I ). The coefficient c here contains the velocity of the instantaneous rest frame I0 of the rocket with respect to the Earth which is equal simply to the rocket’s velocity v itself. Therefore, m � dv0 ¼ F ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p � dt: ð1:7:4Þ In this equation dv0 is still to be expressed through quantities, belonging to I . At any given point of its trajectory, the rocket is at rest in I0 . But it is accelerating continuously, hence the state of its rest lasts but a mathematical instant of time and, therefore, in the subsequent infinitesimal proper time interval it makes a small amount of distant, say dl0, with respect to I0 . During this dis- placement its velocity in I0 increases from zero to dv0. How to obtain the increment dv in I which corresponds to dv0 in I 0 ? Naturally enough, the ratio dv : dv0 of these velocity increments is equal to dl dt : dl0 ds where dl and dt are measures of dl0 and ds as seen in I . Since ds ¼ dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2p and dl0 ¼ dl= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2p , this equality allows us to connect dv=dv0 with the velocity of the motion 10: dv ¼ ð1 � v2=c2Þdv0: ð1:7:5Þ Substituting dv0 from this relation into (1.7.4), we arrive at the equation of motion which is valid in the inertial frame, with respect to which the rocket moves with the velocity v parallel to the force field: 10 This connection follows from the velocity addition formula too, in which now V ¼ dv0, U ¼ �v (since I moves with the velocity ð�vÞ with respect to I 0) and V 0 ¼ vþ dv. Substituting these into (1.6.1) we are led to the equation vþ dv ¼ dv0 þ v 1 þ v � dv0=c2 : Multiplying this by the denominator and neglecting terms quadratic in the differentials we arrive at (1.7.5) again. 18 1 From Time Dilation to E0 = mc 2 a ¼ dv dt ¼ ð1 � v2=c2Þ3=2 F m : ð1:7:6Þ This relativistic equation of motion specifies to what extent the effectiveness of the thrust of the engine is diminished due to time dilation and Lorentz contraction. Surely, the rocket is not a realistic example of the relativistic motion. Its only service was to make derivation as transparent as possible. Experimental investi- gation of the acceleration at high velocities is possible only by means of charged particles moving in an electromagnetic field (Kaufmann experiments, 1901–1902). The question is whether (1.7.6) is applicable to this case also? Since this formula refers to rectilinear motion, its only chance for applicability is when the point charges move along the field direction in a homogeneous electric field. In that case F ¼ QE where Q is the particles’ charge and E is the field strength.11 In (1.7.3), however, we must substitute the electric field E0 as seen in the instantaneous rest frame I0 , rather then the laboratory field E. The problem then arises, how to express E0 through the field E. The answer is given by (2.21.1) whose first formula shows that in the special case under consideration the field in both I0 and I is the same: E0 ¼ E. Formula (1.7.6) remains, therefore, applicable with F ¼ QE. Equation 1.7.6 may be rearranged so as to resemble more closely the original Newtonian equation: m 1 � v2=c2ð Þ3=2 dv dt ¼ F ð1:7:7Þ This form shows with particular clarity that, for given F , the acceleration dv dt is the smaller the larger the velocity v is (sinceð1 � v2=c2Þ�3=2 is an increasing function of the velocity). This property will be referred to as acceleration suppression. It may be shown that the velocity of the motion can never reach the light velocity c. Using the chain rule of the calculus, the law (1.7.7) can be cast into the compact form dp dt ¼ F ; ð1:7:8Þ in which p ¼ mvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2p : ð1:7:9Þ As it is indicated by the notation, this formula serves in relativity theory as the definition of the momentum. In the nonrelativistic limit v � c the usual expression of the momentum p ¼ mv is recovered from it. 11 The general case will be discussed in Sect. 2.21. 1.7 The Equation of Motion of a Point Particle 19 1.8 Does Mass Increase with Velocity? One of the basic assumptions of relativity theory is that the velocity of objects (reference frames) is always smaller than light velocity. As we saw in Sect. 1.6, the notion of velocity addition is in harmony with this requirement. The relativistic equation of motion must, of course, also agree with this principle, i.e. to lead, one way or another, to the phenomenon of the acceleration suppression whenever the velocity is approaching c. As we have stressed, Eq. 1.7.7 does indeed possess this property. The derivation of it in the last section has revealed that the origin of such a behaviour is rooted, through time dilation, in the most fundamental principles of relativity theory indeed. However, this impeccable explanation of acceleration suppression in terms of time dilation is largely ignored in favour of another interpretation based on the conception of a mass, depending on velocity. According to this view, the mass of a moving object is equal to its relativistic mass mr ¼ m= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2p which is an increasing function of velocity. Accordingly, the constant m in this formula is called rest mass which is the familiar mass in the Newtonian equation of motion. This interpretation, however, cannot be accepted as a true explanation of the acceleration suppression because it leads to vicious circle. Indeed, if one looks for the explanation of the relativistic increase of mass itself, the only possibility is to refer back to acceleration suppression: nothing in the derivation of (1.7.7) hints at a changing mass. Relativistic mass is, therefore, an outstanding example of non sequitur. The explanation through time dilation is, on the contrary, not circular since time dilation itself is not contingent upon acceleration suppression. Sometimes it is argued that relativistic mass is nothing but a convenient term to denote the quantity m= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2p without any prejudice as to its physical meaning. This is, however, not the case. The term ‘relativistic mass’ is always taken at its face value as the true mass of a moving object. The relativistic mass fallacy leads to numerous misinterpretations but its most unwelcome effect concerns mass–energy relation: unintentionally trivializing it the misconception of relativistic mass hinders to grasp the originality and true depth of this law. This will be discussed in some detail later on in this chapter (see Sect. 1.12). It is, therefore, strongly suggested to abandon altogether the use the notion of the relativistic mass. Then there will be no need in the term ‘rest mass’ either and the term ‘mass’ without adjectives will suffice. 1.9 The Kinetic Energy of a Point Mass The kinetic energy of a point mass is increasedby the amount of work done on it and is equal to zero when the body is at rest. Mathematically: dK ¼ F � dx ¼ F � v � dt, therefore 20 1 From Time Dilation to E0 = mc 2 dK dt ¼ vF : ð1:9:1Þ In order to obtain the formula for K it is necessary to substitute into (1.9.1) the force from the equation of motion and to express the right hand side as a time derivative. In Newtonian physics the equation to be used is of course ma ¼ F . Then dK dt ¼ vF ¼ mva ¼ mv dv dt ¼ d dt mv2 2 � � ; ð1:9:2Þ and we have K ¼ 12 mv2 þ konst: But since K must be zero when the body is at rest, the constant is equal to zero. Hence K ¼ 12 mv2. In relativity theory the procedure is the same but the force is taken from (1.7.7): dK dt ¼ vF ¼ mva ð1 � v2=c2Þ3=2 : If we express mva from (1.9.2) and use the chain rule of calculus to write 1 ð1 � v2=c2Þ3=2 ¼ c 2 v d dv 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p ! ; then the above equation can be transformed into the form dK dt ¼ vF ¼ d dt mc2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V2=c2p ! ; and K is read off as K ¼ mc 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2p þ konst: Now the constant must obviously be ð�mc2Þ, so for the relativistic kinetic energy we obtain the expression K ¼ mc 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2p �mc 2: ð1:9:3Þ When v2=c2 � 1, the factor 1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � v2=c2p can be replaced by ð1 þ v2=2c2Þ and the nonrelativistic expression mv2=2 of the kinetic energy is recovered. As we see, in relativity theory the kinetic energy increases with the velocity faster than in Newtonian mechanics. When v approaches the velocity of light their proportion tends to infinity. This behaviour is the immediate consequence of acceleration suppression, since kinetic energy increases due to the work done by the accelerating force (dK ¼ Fdx), and this work is larger when the path, required to reach a given value of v, becomes longer because of acceleration suppression. 1.9 The Kinetic Energy of a Point Mass 21 1.10 The Rest Energy: The E0 5 mc 2 Formula The energy content E0 of a body at rest is called rest energy or internal energy. Another property of a body at rest is its mass. According to relativity theory, any change DE0 in rest energy is related to a corresponding change Dm in the mass of the body by the amazingly simple general formula DE0 ¼ Dm � c2. The validity of this law has been demonstrated by Einstein with the help of the following thought experiment. Consider a body of mass m, resting in the coordinate system XYZ attached to the inertial frame I (the rest frame of the body). Let observe the same body from another inertial frame I0 too, equipped with the coordinate system X0Y 0Z0, whose axes are parallel to those of XYZ. Assume that I0 moves in I with an arbitrarily small velocity v in negative direction along the axis X. With respect to I0 the body will then move with velocity v in the positive X0 direction (see Fig. 1.4.). Imagine now two completely identical electromagnetic wave packets, which arrive from the positive and negative direction of the Y axis, and are absorbed by the body. According to classical electrodynamics, the energy � and the momentum p of either of the packets is related to each other through the formula � ¼ cp. Since energy is conserved, the absorption of the packets leads to an increase 2� in the energy of the body. This increase contributes solely to its internal energy because the moments of the packets compensate each other and, therefore, the body remains at rest. Viewing from the primed system X0Y 0Z0, however, the packets no longer move in strictly opposite direction but decline toward the direction of motion of the body, their angle of incidence with respect to Y 0 being equal to some angle a different from zero (aberration). Since v � c by assumption than, as follows from the vector diagram of the velocities, this angle is equal to v=c radian in both Newtonian and relativistic physics. Therefore, the moments of the packets do not fully compensate each other and a momentum 2p sin a ¼ 2p sin v c � 2p v c is transferred to the body in the X direction. Nevertheless, the velocity of the body remains unchanged since in the unprimed system it is at rest throughout and, therefore, its velocity with respect to the primed Fig. 1.4 Proof of the mass- energy relation 22 1 From Time Dilation to E0 = mc 2 one remains also equal to the same v both before and after the absorption of the packets. But how can then it acquire a momentum? The only possibility is that the mass of it becomes larger by an amount Dm. The value of Dm is fixed by momentum conservation. Since v is chosen arbi- trarily small the momentum of the body (1.7.9) is equal to mv even in relativity theory. Its increase Dm � v must then be equal to the momentum 2p v c of the absorbed radiation. Hence, Dm ¼ 2p c . But, as it has already been mentioned, between the momentum and the energy of the packet the relation p ¼ �=c holds true, therefore, Dm ¼ 2�=c2. But 2� is equal to DE0, the increase of the body’s rest energy, and so we arrive at the formula DE0 ¼ Dm � c2. The conclusion is that whenever a body’s internal energy is changed by an amount DE0 its mass is altered proportionally by an amount DE0=c2. Since the modified internal state of the body is independent of how the energy was supplied to it the mass–energy relation DE0 ¼ Dm � c2 is independent of it either. Relativity theory is, however, built on the somewhat stronger assumption that the mass–energy relation actually holds true between the internal energy and mass themselves: E0 ¼ mc2. As it will be seen in Sect. 2.19, the four-vector character of energy and momentum which is of fundamental importance in virtually all applications of the theory is based partially on this assumption. Therefore, the success of the theory as a whole testifies in favour of this stronger form of the energy–mass relation. There exist, furthermore, microscopic objects, as e.g. pos- itronium and p0 meson, which are capable to fully annihilate into radiation. The energetics of this process is governed by the law E0 ¼ mc2 without D-s. A further reason to accept this form is that the significance of energy consists in its conservation and from this point of view it is only the change in its value which matters. In other words: energy is defined only up to a constant. From this per- spective, the form E0 ¼ mc2 is equivalent to DE0 ¼ Dm � c2, being the result of the natural choice of the arbitrary constant.12 At first sight, the thought experiment described does not involve relativity theory at all. But it is there in the assumption that Maxwell theory, leading to the equation � ¼ cp, is valid in the rest frame of the body which can be any of the inertial frames. The total energy E of a free particle is equal to the sum of its kinetic energy K and rest energy E0: E ¼ Kþ E0. Using (1.9.3) we then have E ¼ Kþmc2 ¼ mc 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p : ð1:10:1Þ Rest energy E0 is recovered as the v ¼ 0 limit of this quantity. 12 The only place in physics where the energy itself is of importance is gravitation since, according to general relativity, the source of gravitation is the energy content of heavenly bodies rather than their mass. But the validity of the law E0 ¼ mc2 has not so far been challenged even in this field. (This aspect of the theory will be touched upon in Sect. 3.2.) 1.10 The Rest Energy: The E0 = mc 2 Formula 23 In order to better appreciate the meaning of the mass–energy relation E0 ¼ mc2 a couple of imaginary experiments will now be described. On Fig. 1.5 the boxes play the role of bodies whose internal energy may be manipulated by means of devices within them and whose mass is measured by the stretch of the spring they are hooked upon. Inthe boxes on the upper line two particles of mass M are fastened to the end of a pivoted massless rod which is at rest in the box on the left, but set rotating in the right one.13 The mass of the latter box is, therefore, greater than that of the former by an amount K=c2, where K is the kinetic energy of the particles. So the difference in the weights of the boxes is equal to g 2Mc2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2p � 2Mc 2 ! ð1:10:2Þ where v is the velocity of the particles. This difference is compensated for by the weight placed at the top of the left box. Owing to the friction, the rotation eventually comes to a stop but if the walls are adiabatic the weight of the right box remains unchanged, since any decrease dK in kinetic energy is compensated by the corresponding increase in heat: dKþ dQ ¼ 0. In the box on the right of Fig. 1.5b a particle of mass M is oscillating along a frictionless surface under the influence of a pair of springs. In the box on the left the particle is at rest in its equilibrium position. Again, the right box which M M M M M M (a) (b) Fig. 1.5 Meaning of the mass-energy relation 13 Unwanted effects of angular momentum may be excluded by using a pair of rods, rotating in opposite sense. 24 1 From Time Dilation to E0 = mc 2 contains the oscillating particle is heavier than the other, its weight being greater by an amount of gðKþ U �Mc2Þ. Here U is the elastic energy of the springs. The total energy ðKþ UÞ of the oscillation is constant but its form is changing con- tinuously: in the turning points it is pure elastic energy, in equilibrium point it is pure kinetic energy. The ‘weights’ of both are the same. Einstein’s famous paper ‘‘On the Electrodynamics of Moving Bodies’’ appeared in July of 1905 and contained all the essential ingredients of relativity theory with a single exception: the formula (1.9.3) for the kinetic energy was there but the energy–mass relation E0 ¼ mc2 was absent. This last formula was published three months later in a short paper under the title ‘‘Does the Inertia of a Body Depend on Its Energy Content?’’ but Einstein returned twice more to the same theme in 1935 and again in 1946. Here we followed the unsurpassably transparent derivation of this last paper. 1.11 Is Mass Conserved? The thought experiment discussed in the preceding section shows that it is not. A real example is the a-decay of Po210: Po210 �! Pb206 þ a: When the decaying nucleus is at rest, energy conservation takes the form Mc2 ¼ mc 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � V 2=c2p þ lc2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2p ð1:11:1Þ where M is the mass of Po210 while the pairs m, V and l, v refer to the nucleus Pb206 and the alpha-particle respectively. The right hand side of (1.11.1) is obviously greater than ðmþ lÞc2, and so M [ mþ l. Mass is, therefore, not conserved. It should indeed not, because mass is proportional to the rest energy of the particles and rest energy alone need not remain unchanged: it is only the total energy which is conserved. The essence of the processes like alpha decay is the transformation of internal energy into kinetic one as seen from (1.11.1) expressed through the kinetic energies of the decay products: ðM �m� lÞc2 ¼ KPb þKa: In macrophysics the change in mass which accompanies changes in internal energy is always negligibly small with respect to the mass itself: Dm � m. Internal combustion engines utilize the rest energy of their fuel but, in spite of the con- siderable amount of energy they provide, the corresponding decrease in their mass is absolutely negligible. Since Newtonian physics is the generalization of mac- roscopic experience, mass conservation rightly became one of its fundamental 1.10 The Rest Energy: The E0 = mc 2 Formula 25 postulates. It is, nevertheless, only an approximate conservation law, in contrast to the conservation of e.g. the electric charge which is a law of universal validity. 1.12 The Popular View on the Mass–Energy Relation The common view on the mass–energy relation is quite different from that outlined in the last two sections. It boils essentially down to the assertion that the mass of a moving body is equal to its total energy E divided by c2. Using (1.10.1), we then have E=c2 ¼ mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p which is the relativistic mass mr we already met (and refuted) in Sect. 1.8 This interpretation has evidently nothing to do with the mass–energy relation E0 ¼ mc2 proved in Sect. 1.10 and one may wonder where this false doctrine came from. Its most obvious source should be ignorance: people have not been taught to learn, that it is the equivalence of the rest energy and the mass, that has been proved true and this equivalence cannot be extended by mere fiat to other forms of energy (kinetic, electromagnetic, etc.). The other is the belief that increase of mass with velocity is an observable phenomenon rather than empty verbalism. Finally, virtual possibility to restore mass conservation appears to be a surprizingly strong motivation too. The imaginary experiment with the rotating particles on Fig. 1.5a has originally been devised in the hope to prove the reality of velocity dependent mass. But the experiment demonstrates only that the rotation of the particles makes the weight of the box greater. Mass–energy relation attributes this growth in weight to the increase of the internal energy of the box, while its popular interpretation ascribes it to the increase of the masses of the particles. The difference is subtle but can be unambiguously settled if other versions of the experiment are also invoked. Consider the experiment on Fig. 1.5b. If relativistic mass was in charge for the changes in the weight of the box, then the box should be oscillating up and down, since its weight would be the larger the faster the particle is moving. If, on the other hand, it is the internal energy of the box which determines its weight then this weight will remain constant in time. The experiment of Fig. 1.5a with friction can be analyzed similarly and leads to analogous conclusion. Therefore, the relativistic mass, if existed, could indeed be observed through its weight in the experiment on Fig. 1.5a, but it would contradict the experiment on Fig. 1.5b: the popular view on the mass–energy relation is, therefore, demon- strably different from the mass–energy relation proved by Einstein. Relativistic mass makes it possible to formally express energy conservation in the guise of mass conservation. If, for example, (1.11.1) is divided by c2 and the fractions on its right hand side are written as relativistic masses of the lead nucleus 26 1 From Time Dilation to E0 = mc 2 and the alpha particle respectively, we obtain M ¼ mr þ lr. In spite of its being expressed in terms of mass conservation, the content of this relation is still energy conservation, but even the purely verbal preservation of one of the most basic laws of Newtonian physics seems psychologically rewarding. This drive is so strong that in the popular interpretation mass conservation is extended even to the emission and absorption of radiation, assigning mass m ¼ E=c2 to the radiation field of energy E. For example, wave packets of the thought experiment of Sect. 1.10 are endowed with mass �=c2 and so the total mass in the process of their absorption can be said to conserve. Attempts to justify the assignment of mass to free electromagnetic radiation embrace light deflection and red shift in gravita- tional field but they fail both numerically and in principle (see Sects. 3.7 and 3.9). Our conclusion is that Einstein’s and popular views on the mass–energy relation are fundamentally different from each other. In the former interpretation E is restricted to the rest energy E0 of a body of mass m, and the relation itself is a law of Nature whose validity can be proved
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