Basic Relativity

SpringerBriefs in Physics
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

For further volumes: http://www.springer.com/series/8902

Péter Hraskó

Basic Relativity
An Introductory Essay

´ Emeritus Professor at University of Pecs, Hungary

123

Péter Hraskó University of Pécs H-7633 Pécs Szántó Kovács János u. 1/b Hungary e-mail: peter@hrasko.com

ISSN 2191-5423 ISBN 978-3-642-17809-2 DOI 10.1007/978-3-642-17810-8
Springer Heidelberg Dordrecht London New York Ó Péter Hraskó 2011

e-ISSN 2191-5431 e-ISBN 978-3-642-17810-8

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, broadcasting, 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 Lorentztransformations 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. Lorentztransformation 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 Lorentztransformation (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

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Contents

1

From 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12

Time Dilation to E0 = mc2 . . . . . . . . . . . . . . . . Reference Frames and Inertial Frames . . . . . . . . The Optical Doppler-Effect and Time Dilation . . The Relativity of Simultaneity . . . . . . . . . . . . . . The Proper Time and the Twin Paradox . . . . . . . The Lorentz Contraction . . . . . . . . . . . . . . . . . . Velocity Addition. . . . . . . . . . . . . . . . . . . . . . . The Equation of Motion of a Point Particle. . . . . Does Mass Increase with Velocity? . . . . . . . . . . The Kinetic Energy of a Point Mass. . . . . . . . . . The Rest Energy: The E0 = mc2 Formula . . . . . . Is Mass Conserved? . . . . . . . . . . . . . . . . . . . . . The Popular View on the Mass–Energy Relation .

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1 1 4 8 11 13 15 17 20 20 22 25 26 29 29 32 33 34 38 40 45 48 50 51 52 53 59 vii 2

The Lorentz-Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Coordinate Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Independence of the Constancy of c from Synchronization . 2.3 The Minkowski Coordinates . . . . . . . . . . . . . . . . . . . . . . 2.4 The Lorentz-Transformation . . . . . . . . . . . . . . . . . . . . . . 2.5 Classification of Spacetime Intervals . . . . . . . . . . . . . . . . 2.6 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Causality Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Demonstration of Time Dilation on Spacetime Diagram . . . 2.9 Doppler-Effect Revisited. . . . . . . . . . . . . . . . . . . . . . . . . 2.10 The Connection of the Proper Time and Coordinate Time in Inertial Frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 The Magnitude of the Twin Paradox . . . . . . . . . . . . . . . . 2.12 The Coordinate Time in Accelerating Frames: the Twin Paradox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 The Coordinate Time in Accelerating Frames: the Rotating Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 3

Lorentz Contraction Revisited . . . . . . . . . . . . . . Is the Perimeter of a Spinning Disc Contracted? . Do Moving Bodies seem Shorter? . . . . . . . . . . . Velocity Addition Revisited . . . . . . . . . . . . . . . Equation of Motion Revisited . . . . . . . . . . . . . . The Energy–Momentum Four Vector . . . . . . . . . Massless Particles . . . . . . . . . . . . . . . . . . . . . . The Transformation of the Electromagnetic Field The Thomas-Precession . . . . . . . . . . . . . . . . . . The Sagnac Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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60 62 63 64 64 66 68 69 71 72 75 75 77 78 79 81 81 83 84 85 89 93 101

General Relativity . . . . . . . . . . . . . . . . . . 3.1 Gravitational and Inertial Mass . . . . . 3.2 The Equivalence Principle. . . . . . . . . 3.3 The Meaning of the Relation m* = m . 3.4 Locality of the Inertial Frames. . . . . . 3.5 The Weight . . . . . . . . . . . . . . . . . . . 3.6 The GP-B Experiment . . . . . . . . . . . 3.7 Light Deflection. . . . . . . . . . . . . . . . 3.8 Perihelion Precession . . . . . . . . . . . . 3.9 Gravitational Red Shift . . . . . . . . . . .

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Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selected Problems to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . .

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1

From Time Dilation to E0 5 mc2

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

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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, Ó Péter Hraskó 2011 1

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1 From Time Dilation to E0 = mc2

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 massÂ 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 gravitation 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.

1.1 Reference Frames and Inertial Frames

3

equal to the same c in any direction. In this respect propagation of light is fundamentally 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 Â 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 properties 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 Einstein’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 surprizingly 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 accuracy (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

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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).

4 Fig. 1.1 Calculation in the rest frame of the receiver (RFR)

1 From Time Dilation to E0 = mc2

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

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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.

1.2 The Optical Doppler-Effect and Time Dilation Fig. 1.2 Calculation in the rest frame of the emitter (RFE)

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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: À 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 T0 V ) and that of signals’ trajectories ðh ¼ ðT À T0 ÞcÞ. The relation between the periods and frequencies follows from the equality of these two expressions: T ¼ T0 ð1 þ V =cÞ; m¼ m0 : 1 þ V =c ð1:2:1Þ 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 À T0 Þc and hence T ¼ T0 1 À V =c ð1:2:3Þ ð1:2:4Þ

m ¼ m0 ð1 À V =cÞ:

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 =cÞÀ1 while in the second (relative to RFE) it is given by ð1 À V =cÞ. In both cases the frequency m is

6

1 From Time Dilation to E0 = mc2

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 cðV Þ 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 T0 cð1 þ V =cÞ; Tc ¼ T0 1 À 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 1 c ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 1 À V 2 =c2 then both expressions reduce to the same one given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ V =c ; T ¼ T0 1 À V =c or expressed in terms of the frequencies sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V =c m ¼ m0 : 1 þ V =c ð1:2:5Þ

ð1:2:6Þ

ð1:2:7Þ

1.2 The Optical Doppler-Effect and Time Dilation

7

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 sufﬁcient 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 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V 2 =c2 ; m0 m ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 1 À V 2 =c2

The validity of this formula has been proved experimentally, using Mössbauer effect. Now it will be shown that the effect of the factor cðV Þ 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: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ V =c T0 \ : T0 ð1 þ V =cÞ\T0 1 À V =c 1 À 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.

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1 From Time Dilation to E0 = mc2

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 contradiction. 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 I 0 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 I 0 are equivalent inertial frames and with respect to the former the bodies move in the same direction while with respect
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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.

1.3 The Relativity of Simultaneity

9

to the latter they move in the opposite direction. Therefore, any statement, concerning 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 contradictory 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, simultaneous 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 I 0 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 possibility 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 headquarters on 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.

10

1 From Time Dilation to E0 = mc2

L ¼ 105 million kilometers from the Earth than it was keeping at rest during 2 Â 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 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ undermines this possibility since the factor 1= 1 À V 2 =c2 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.

1.3 The Relativity of Simultaneity

11

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 recommended 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 convention 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 Æ Vt and X ¼ konst: À 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 Â pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V 2 =c2 ð1:4:1Þ

which means that in the inﬁnitesimal interval ðt; t þ dtÞ, when the velocity is equal qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ to V ðtÞ, the increment ds of the proper time is equal to dt Â 1 À V ðtÞ2 =c2 . 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 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ interval T0 is equal to cT0 . Hence, Dt ¼ T0 = 1 À V 2 =c2 which is nothing but the formula (1.4.1) applied to this special case.

12

1 From Time Dilation to E0 = mc2

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 Newtonian 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 À ta Þ. 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 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ds ¼ dt 1 À V ðtÞ2 =c2 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 former will 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 accelerations 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.

1.5 The Lorentz Contraction

13

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 Á D" from it, s where D" is the time the train was passing by the point chosen. Hence, the length s of the moving train is equal to this product. Measurement of Dl, therefore, requires the measurement of D" and V . Assume, s that we are waiting for the train to come at a point P of the embankment. The time D" can be measured simply with a stopwatch but the determination of the velocity s 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 Ds0 þ : V 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 Á D". Is this the same length s 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" elapsed on the man’s stopwatch (the s pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬁrst one) to be equal to Dt Â 1 À V 2 =c2 . But Dt is equal to Dl0 =V , hence pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D" ¼ Dl0 1 À V 2 =c2 and, therefore, s V pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1:5:2Þ Dl ¼ V Á D" ¼ Dl0 1 À V 2 =c2 : s 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

14

1 From Time Dilation to E0 = mc2

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 frame will 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.

1.5 The Lorentz Contraction

15

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 À 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 I 0 , 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 I 0 , 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 À UÞ, where Ds is the distance between the point P and the relay, V 0 is the train’s velocity, and ðV 0 À UÞ is the rate of change of the distance between the train and the relay. All quantities are, of course, related to I 0 , 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 Ds : þ V0 À U U þ c

But since the stopwatches and the embarkment (with the observer and the relay on pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ it) move with velocity U in I 0 , we have the relations Dt ¼ D"= 1 À U 2 =c2 and s pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ds ¼ Ds0 Á 1 À U 2 =c2 . Therefore

16

1 From Time Dilation to E0 = mc2

Ds 1 1 þ ¼ ð1 À U 2 =c2 Þ 0 : Ds0 V ÀU Uþc This same ratio can also be expressed from (1.5.1) too and , equating them, we obtain 1 1 1 1 þ ð1 À U 2 =c2 Þ 0 ¼ þ : V ÀU Uþc V 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=c2 Þ. 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 V0 ¼ V ÀU 1 À 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 I 0 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 I 0 . 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 I 0 . 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 À 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 À U. The ﬁgure suggests that 0 Dl ¼ Dl À U Á 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

1.6 Velocity Addition
Dl0 Dt

17

Dl and V ¼ Dt, we indeed obtain this simple relation. But as seen from (1.6.1), the Newtonian form V 0 ¼ V À 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 I 0 (V 0 ¼ V ¼ c). This consequence of the relativistic velocity addition formula 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 I 0 , 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 I 0 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 Á dv ¼ F Á dt ð1:7:1Þ

will be constant and, therefore, the acceleration will remain also unchanged: a¼ dv F ¼ ¼ konstans: dt m 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 is the velocity of the rocket. The acceleration suppressing effect of time dilation is, moreover,

18

1 From Time Dilation to E0 = mc2

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 I 0 : m Á dv0 ¼ F Á ds: ð1:7:3Þ

As a result, the acceleration of the rocket with respect to itself (in I 0 ) will continuously 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 I 0 of the rocket with respect to the Earth which is equal simply to the rocket’s velocity v itself. Therefore, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m Á dv0 ¼ F 1 À v2 =c2 Á 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 I 0 . 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 I 0 . During this displacement its velocity in I 0 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 dl0 : ds where dl and dt are measures of dl0 and ds as seen in I . Since ds ¼ dt pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dt 1 À v2 =c2 and dl0 ¼ dl= 1 À v2 =c2 , this equality allows us to connect dv=dv0 with the velocity of the motion10: dv ¼ ð1 À v2 =c2 Þdv0 : 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 Àv (since I moves with the velocity ðÀvÞ with respect to I 0 ) and V 0 ¼ v þ dv. Substituting these into (1.6.1) we are led to the equation

v þ dv ¼

dv0 þ v : 1 þ v Á dv0 =c2

Multiplying this by the denominator and neglecting terms quadratic in the differentials we arrive at (1.7.5) again.

1.7 The Equation of Motion of a Point Particle

19

a¼

dv F ¼ ð1 À v2 =c2 Þ3=2 : dt 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 investigation 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 I 0 , 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 I 0 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 ð1 À v2 =c2 Þ3=2 dv ¼F dt ð1:7:7Þ

This form shows with particular clarity that, for given F, the acceleration dv is the dt smaller the larger the velocity v is (sinceð1 À v2 =c2 ÞÀ3=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 ¼ F; dt in which mv p ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 1 À v2 =c2 ð1:7:9Þ ð1:7:8Þ

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.

20

1 From Time Dilation to E0 = mc2

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 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ moving object is equal to its relativistic mass mr ¼ m= 1 À v2 =c2 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 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ denote the quantity m= 1 À v2 =c2 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 increased by the amount of work done on it and is equal to zero when the body is at rest. Mathematically: dK ¼ F Á dx ¼ F Á v Á dt, therefore

1.9 The Kinetic Energy of a Point Mass

21

dK ¼ vF: dt

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 dv d mv2 ¼ vF ¼ mva ¼ mv ¼ ; ð1:9:2Þ dt dt dt 2 and we have K ¼ 1 mv2 þ konst: But since K must be zero when the body is at 2 rest, the constant is equal to zero. Hence K ¼ 1 mv2 . 2 In relativity theory the procedure is the same but the force is taken from (1.7.7): dK mva ¼ vF ¼ : dt ð1 À v2 =c2 Þ3=2 If we express mva from (1.9.2) and use the chain rule of calculus to write ! 1 c2 d 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ¼ v dv 1 À v2 =c2 ð1 À v2 =c2 Þ3=2 then the above equation can be transformed into the form ! dK d mc2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ¼ vF ¼ dt dt 1 À V 2 =c2 and K is read off as mc2 K ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ konst: 1 À v2 =c2 Now the constant must obviously be ðÀmc2 Þ, so for the relativistic kinetic energy we obtain the expression mc2 ð1:9:3Þ K ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ À mc2 : 1 À v2 =c2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ When v2 =c2 ( 1, the factor 1= 1 À v2 =c2 can be replaced by ð1 þ v2 =2c2 Þ 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.

22

1 From Time Dilation to E0 = mc2

1.10 The Rest Energy: The E0 5 mc2 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 Á 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 I 0 too, equipped with the coordinate system X0 Y 0 Z0 , whose axes are parallel to those of XYZ. Assume that I 0 moves in I with an arbitrarily small velocity v in negative direction along the axis X. With respect to I 0 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 X0 Y 0 Z0 , 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 % 2p v is c c 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 massenergy relation

1.10

The Rest Energy: The E0 = mc2 Formula

23

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 arbitrarily small the momentum of the body (1.7.9) is equal to mv even in relativity theory. Its increase Dm Á v must then be equal to the momentum 2p v of the c absorbed radiation. Hence, Dm ¼ 2p. But, as it has already been mentioned, c 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 Á 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 Á 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. positronium 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 perspective, the form E0 ¼ mc2 is equivalent to DE0 ¼ Dm Á 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 K þ E0 . Using (1.9.3) we then have mc2 E ¼ K þ mc2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 1 À v2 =c2 Rest energy E0 is recovered as the v ¼ 0 limit of this quantity. ð1:10:1Þ

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.)

12

24 Fig. 1.5 Meaning of the mass-energy relation

1 From Time Dilation to E0 = mc2

(a)

M

M

M

M

(b)

M

M

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. In the 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 ! 2Mc2 2 ð1:10:2Þ g pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ À 2Mc 1 À v2 =c2 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: dK þ 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
Unwanted effects of angular momentum may be excluded by using a pair of rods, rotating in opposite sense.
13

1.10

The Rest Energy: The E0 = mc2 Formula

25

contains the oscillating particle is heavier than the other, its weight being greater by an amount of gðK þ U À Mc2 Þ. Here U is the elastic energy of the springs. The total energy ðK þ UÞ of the oscillation is constant but its form is changing continuously: 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 À Pb206 þ a: ! When the decaying nucleus is at rest, energy conservation takes the form mc2 lc2 Mc2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V 2 =c2 1 À v2 =c2 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 ðm þ lÞc2 , and so M [ m þ 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 À m À lÞc2 ¼ KPb þ Ka : 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 considerable amount of energy they provide, the corresponding decrease in their mass is absolutely negligible. Since Newtonian physics is the generalization of macroscopic experience, mass conservation rightly became one of its fundamental

26

1 From Time Dilation to E0 = mc2

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 m E=c2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À v2 =c2 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, demonstrably 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

1.12

The Popular View on the Mass–Energy Relation

27

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 gravitational 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 or disproved by measuring all the quantities in the equation DE0 ¼ Dm Á c2 independently of each other. According to the popular view formula E ¼ mc2 serves to deﬁne mass as m ¼ E=c2 for any form of the energy. This deﬁnition is, however, either useless or contradictory, since the masses of bodies have already been deﬁned by their role in the equation of motion or simply through their weights. This degradation of one of the most beautiful laws of Nature to mere deﬁnition was characterized above as its trivialization (Sect. 1.8) The main purpose of this chapter has been to introduce basic concepts of relativity theory without the preliminary knowledge of Lorentz-transformations. Though these transformations add nothing to the physical meaning of these concepts, they are, nevertheless, indispensable. Their function is at least threefold: to provide solid basis for the concepts introduced, to make feasible the analysis of phenomena beyond the simplest one’s, and to make the inner consistency of the theory transparent. Lorentz-transformations logically precede the effects treated in this chapter, but since they are based on subtle premises, it seemed pedagogically justiﬁed to give a sound preliminary survey of their perplexing physical consequences. Familiarity with these phenomena, from time dilation to mass–energy relation, may assist appreciation of the efforts to give them solid foundation through Lorentz-transformations.

Chapter 2

The Lorentz-Transformation

Abstract The notion of the coordinate time is introduced. The interrelation between constancy of c and synchronization is analyzed. Lorentz-transformations are derived and using them the relativistic effects are reconsidered. The causality paradox is discussed and the twin paradox is described from the point of view of both siblings. Keywords Time paradox

Á Synchronization Á Lorentz-transformation Á Causality Á Twin

2.1 The Coordinate Time
Physical processes take place in space and time. Their ultimate constituents are the pointlike instantaneous events characterized by four spacetime coordinates, i.e. by three space coordinates and a moment of time. Spacetime itself is the fourdimensional continuum of these points. Intuition is a much less reliable guide, concerning coordinate systems, in spacetime than in three-dimensional space alone. Therefore, even the simplest spacetime coordinate systems require careful description. As it has already been stressed in Sect. 1.1, the importance of coordinate systems (K) lies in the fundamental role they play in theoretical calculations. In sharp contrast to reference frames (R) and inertial frames (I ), coordinate systems exist only in our minds: think of the description of planets’ motion in Newtonian physics. When working in a given reference frame, the most convenient coordinate system is that in which our reference frame is at rest. Such a coordinate system is called attached to the reference frame: R !K; R 0 !K0 or I !K; I 0 !K0 : When, for example, a railcar is chosen to serve as a reference frame, one can imagine Cartesian coordinates attached to it whose origin is at the center of the car, x-axis is parallel to the rails and z-axis points upwards perpendicular to Earth’s
P. Hraskó, Basic Relativity, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-17810-8_2, Ó Péter Hraskó 2011 29

30

2 The Lorentz-Transformation

surface. Then the location at the moment t of a point mass is determined by the triplet xðtÞ; yðtÞ; zðtÞ of functions of time. Here t is the same time that in Sect. 2.1 has also been denoted by this letter and was advised to understand and use in the same way as learned in secondary school physics. Now ‘it is time’ to analyse it in some more detail. Let us ﬁrst give a name to it: it will be called coordinate time. This is a very appropriate term since on graphical representations of the function s ¼ f ðtÞ; which describes the motion, one of the coordinate axes is t-axis. It was, perhaps, Galileo who, in his famous experiments with balls rolling down a slope, used for the ﬁrst time the notion of time in the same sense we give now to coordinate time. Galileo measured the time it took to reach marked distances on the slope and established that this time is proportional to the square root of the corresponding distance. To measure the time, he made a hole in the bottom of a bucket, suspended a jar immediately below it and ﬁlled the bucket with water, keeping the hole closed. He let water to ﬂow from the bucket into the jar only during the time the ball was rolling from its initial position on the top of the slope up to one of the marks. The weight of the water in the jar was then proportional to the time during which the ball travelled the distance chosen. The experiment was naturally a very inaccurate one but as we will show now its idealized version permits us to formulate precisely what coordinate time means. Galileo had to roll the ball many times before he could draw the conclusion which is now written in the form s ¼ 1 gt2 : In a modern version of the experiment a 2 single rolling down would sufﬁce. Imagine that the slope is bordered with photoelectric cells each of which is provided by appropriate electric circuitry to generate a sharp signal at the moment when the ball passes by. The signals are collected in a time delay analyzer controlled by an ideal clock. Then the output of the analyzer will be the collection of points of the path versus time curve s ¼ f ðtÞ: Since the number of the photoelectric cells may in principle be arbitrarily large the curve can be determined very accurately. This accuracy, however, cannot be made arbitrarily high since the method has an inbuilt drawback: the necessity to transmit signals from one place to another. The time interval required for the signals to travel from the photoelectric cells to the analyzer must obviously be the same for all of the cells since otherwise the curve at the output of the analyzer would be distorted. Therefore, all the cables must be of equal length. But this is not enough because the speed of the signals, though largely determined by the construction of the cables, is slightly inﬂuenced also on how they are mounted, i.e. on their shape, environment and mutual arrangement. Since it is certainly impossible to ensure completely identical conditions for all the cables, methods, requiring signal transmittion, must be abandoned. This difﬁculty, however, can be easily circumvented if every photoelectric cell is provided by its own clock and camera. In this case, the signal of any particular cell may be used to take a snapshot of the face of its clock at the moment when the ball rolls by. These data can also be used to reconstruct the law of motion s ¼ f ðtÞ; provided the clocks are synchronized correctly.

2.1 The Coordinate Time

31

The most natural method of synchronisation consists in collecting all the clocks together in a suitable place where they can be easily surveyed and synchronized by bringing their hands simultaneously into the same position. Since they are of ideal construction by assumption, the rhythm of all of them is exactly the same and they will retain their synchronism forever. Therefore, they can be transported safely to their original places without the risk of being desynchronized (Newtonian synchronization). The persistence of syncronization can be veriﬁed by means of the following procedure. After synchronization having been performed but clocks still being together, one of them is transported to a distant place and back again. If in the course of transportation its synchronism with the rest of clocks is not spoiled, Newtonian method of synchronization may be accepted as a correct one. Now we are at a position to formulate the general deﬁnition of the coordinate time: coordinate time would be measured by a set Sof uniformly scattered correctly synchronized ideal clocks at rest if they were indeed there. As we could observe in Chap. 1, the meaning of basic physical quantities, such as velocity, momentum, energy, etc., is the same in both Newtonian and relativistic physics; it is their properties which are different, sometimes dramatically, in the two theories. The same is true for coordinate time. Its deﬁnition as given above in the context of Newtonian physics remains true in relativity theory as well. It is only the method of synchronization which is to be modiﬁed since, owing to twin paradox (or time dilation), the Newtonian synchronization would not, as a matter of fact, pass the test procedure described above.1 But relativity theory offers an alternative based on the constancy of light speed in inertial frames (Einstein synchronization). Consider a pair of clocks A and B; resting in an inertial frame at some distance from each other. A light signal sent from clock A at the moment of time tA1 is immediately reﬂected by a mirror from clock B and arrives back to clock A at the moment tA2 : Both tA1 and tA2 are, of course, read off from the clock A: Let the moment of reﬂection be tB as seen on clock B: Now, if the two clocks were correctly synchronized, the equality of light velocity in both direction could be expressed by the equation tB À tA1 ¼ tA2 À tB from which tB ¼ 1 0 2 ðtA1 þ tA2 Þ: If at the moment of reﬂexion the actual time on clock B was tB then it 1 0 0 should be set ahead by an amount tB À tB ¼ 2 ðtA1 þ tA2 Þ À tB : A and B may be either two particular real clocks or two arbitrary members of the virtual multitude S of clocks which deﬁne coordinate time in the inertial frame I chosen. Then constancy of light speed in I ensures that the distance Dl between two arbitrary points, travelled by light in time Dt; is given by the equation Dl2 ¼ Dx2 þ Dy2 þ Dz2 ¼ c2 Dt2 : ð2:1:1Þ

Since inertial frames are moving with respect to each other, all of them are provided by its own set of virtual clocks at rest, which show coordinate time in it.

1

Twin paradox vanishes only if transportation velocity is equal to zero. But then the procedure would last inﬁnitely long.

32

2 The Lorentz-Transformation

Individual clocks in each particular set are assumed to be correctly synchronized but, owing to time dilation, no synchronism can exist between a pair of clocks, belonging to different inertial frames. The only possibility to relate coordinate times of two different inertial frames consists in selecting a particular clock in both of them which are found at the same point of space and to assign on them the same time to their moment of encounter. The standard procedure is this: let the inertial frames I and I 0 be equipped with Cartesian coordinates K and K0 , the corresponding axes of which are parallel to each other and, moreover, their x-axes are in common. Their relative motion is along this common axis. Both I and I 0 have their own sets S and S 0 of virtual clocks which deﬁne coordinate time in them. Let us denote the particular members of S and S 0 at the origins of K and K0 by O and O 0 : The coordinate times in the two frames are then related by the assumption that O and O 0 show zero time at the moment of their encounter (standard setting). Remember now the deﬁnition of the proper time of a body: it would be measured by an ideal clock if it was attached to the body. Evidently, proper time and coordinate time are logically independent notions and, therefore, they need not coincide numerically. In Newtonian physics they actually do, in complete agreement with the Newtonian sychronization procedure. The splitting of the unique notion of time into two was recognized only in relativity theory.

2.2 Independence of the Constancy of c from Synchronization
It is sometimes claimed that Einstein synchronization of distant clocks A and B is circular. The argument is very simple: Einstein synchronization is based on the equality of light velocity on the path from A to B and back from B to A; but measurement of light velocity in one direction between two distant points is impossible unless the clocks at these points have already been synchronized. This argument is, however, fallacious. It is true that one-way measurement of light velocity can be performed only if clocks at the endpoints are synchronized correctly. But since they need not show the correct coordinate time, they can be synchronized without light signals by transporting them from a common site in a symmetrical manner. The procedure consists of the following steps: 1. Let our laboratory be an inertial frame. To ascertain this no measurement of time is required: isolated bodies at rest must remain so and axes of gyroscopes must be pointing continuously to the same piece of the wall. 2. Two ideal clocks are synchronized at a common site and transported in a symmetrical way into the points A and B: The symmetry of transportation is ensured by mounting them on two completely identical ‘land rovers’ which start to move at the same moment of time in exactly opposite direction. Their motion is controlled by the clocks themselves by means of identical computers, executing identical programs.

2.2 Independence of the Constancy of c from Synchronization

33

3. At the points A and B the clocks are either gaining or losing with respect to the coordinate time (they do not belong to the set S) but since their synchronism is not destroyed they make it possible to compare light velocities in the direction A À B and B À A: ! ! 4. The procedure is suitable to measure even the value of light velocity, since the distance between the points can be calculated from the number of turns and perimeter of the ‘land rover’s wheels. Repeating it many times, the isotropy of light velocity can be veriﬁed. As we see, the thought experiment described is capable to prove constancy of light speed if it is true, or to disprove it if it is false. It provides, therefore, solid logical foundation for Einstein’s synchronization prescription.

2.3 The Minkowski Coordinates
If an inertial frame is endowed with both Cartesian coordinates and a set S of correctly synchronized virtual clocks then it is called that a spacetime coordinate system ðK; SÞ is attached to it (Minkowski coordinates). Instead of these pair, Minkowski coordinate systems will mostly be denoted by the single symbol M: As coordinate systems generally do, Minkowski coordinate system too exisits only in our imagination. It serves to assign four well-deﬁned spacetime coordinates to any pointlike instantaneous event: three space coordinates x, y, z in K and a moment of time shown on the virtual clock at this point. Consider two inertial frames I and I 0 with Minkowski coordinates ðK; SÞ M and ðK0 ; S 0 Þ M0 attached to them. Assume that a ﬂash of light is given off at some point at a given moment of time (event A) and observed at a distant point at a later moment of time (event B). Either of these events have both primed and unprimed Minkowski coordinates whose differences are Dx ¼ xB À xA Dy ¼ yB À yA Dz ¼ zB À zA Dt ¼ tB À tA Dx0 ¼ x0B À x0A Dy0 ¼ y0B À y0A Dz0 ¼ z0B À z0A 0 0 Dt0 ¼ tB À tA

The equality of light speed in both I and I 0 is then expressed by the relation qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 Dx2 þ Dy2 þ Dz2 Dx0 2 þ Dy0 2 þ Dz0 2 Dl Dl ¼c( ) 0 ¼ c; Dt Dt Dt0 Dt which can also be expressed in the form ) Dx2 þ Dy2 þ Dz2 ¼ c2 Dt2 ( Dx0 þ Dy0 þ Dz0 ¼ c2 Dt0 ; where ( means that either part of the relation follows from the other. )
2 2 2 2

ð2:3:1Þ

34

2 The Lorentz-Transformation

Let us now suppose that the two Minkowski coordinate systems constitute a standard setting as deﬁned at the end of Sect. 2.1. Then the origin of K and K0 are moving on their common x-axis and the virtual clocks at the origin of both of them show zero time at the moment of their encounter. Assume that the event A (the light ﬂash) takes place at just this moment of time at the common origin of K and K0 . Then all the eight A-coordinates are equal to zero and (2.3.1) is reduced to x2 þ y2 þ z 2 ¼ c 2 t 2 ( x0 þ y0 þ z 0 ¼ c 2 t 0 )
2 2 2 2

ð2:3:2Þ

(subscript B has been omitted). The meaning of this relation is this: if E is any event with spacetime coordinates x, y, z, t in M and x0 , y0 , z0 , t0 in M0 and x2 þ y2 þ z2 ¼ c2 t2 then x0 2 þ y0 2 þ z0 2 ¼ c2 t0 2 is also true and vice versa. In Sect. 1.1 it has been stated that in their original form Maxwell equations are valid in the inertial frames. But Maxwell equations can be written down only if the spacetime coordinate system has already been speciﬁed. The most natural assumption is that these equations are valid in Minkowski coordinates, the decisive argument being, that the constancy of light speed which is the consequence of Maxwell equations, is the basic property of Minkowski coordinates attached to any inertial frame. The question now arises how to assign spacetime coordinates to an accelerating (non-inertial) reference frame R. The problem is that in such frames light velocity is not isotropic and is, in general, even time dependent. Einstein synchronization is, therefore, out of the question. If Newtonian synchronization was a valid procedure it could be applied in any reference frame but Einstein synchronization is restricted to inertial frames. But this is not a drawback since accelerating frames are always related in some deﬁnite way to the inertial ones. Newtonian equations of motion, for example, are originally given in the form suitable only in inertial frames. When they are applied in an accelerating frame R, their mathematical form can be computed from the known relation of R to some inertial frame I . Similarly, when in relativity theory a calculation is performed, for some reason or another, in a non-inertial frame R, the coordinate time is always chosen in relation to the Minkowski coordinates of an inertial frame (see Sect. 2.12 for an example of such a procedure).

2.4 The Lorentz-Transformation
Let us take up the discussion of the previous section which led to (2.3.1) and (2.3.2) and ask: How to calculate in a standard setting the primed spacetime coordinates of an event E if its unprimed coordinates are given? Formulae, answering this question, are known as the LorentzÀtransformation: The analogous question in Newtonian physics can be answered immediately: it is the GalileiÀtransformation t0 ¼ t; x0 ¼ x À Vt; y0 ¼ y; z0 ¼ z ð2:4:1Þ

2.4 The Lorentz-Transformation

35

which connects the primed and unprimed spacetime coordinates of an event in a standard setting of K and K0 attached to I and I 0 . In (2.4.1) V is the constant velocity of I 0 with respect to I along the common x-axis in positive direction if V [ 0 and in negative direction if V \0: For V ¼ 0 (2.4.1) reduces to the identity transformation as should be. Galilei-transformations are linear transformations, hence they have the same form when applied to coordinate differences of a pair of events: Dt0 ¼ Dt; Dx0 ¼ Dx À V Dt; Dy0 ¼ Dy; Dz0 ¼ Dz:

This property is an essential one. It is the consequence of the fact that the origin of a coordinate system attached to a given reference frame can be freely chosen. Its choice, therefore, must not lead to observable consequences. The coordinates themselves depend on the position of the origin but the coordinate differences are independent of it. The form containing D-s is, therefore, the more fundamental one. Formulae (2.4.1) are meaningful only with respect to the standard setting in which the origin of the Cartesian coordinates and the zero moment of time (i.e. the origin of the spacetime coordinate systems attached to the inertial frames) have already been chosen. The transformations (2.4.1) are, moreover, homogeneous since no constant term independent of the unprimed coordinates is found on their right hand side. This is the direct consequence of the fact that in a standard setting the primed coordinates of the event at x ¼ y ¼ z ¼ t ¼ 0 are equal also to zero: x0 ¼ y0 ¼ z0 ¼ t0 ¼ 0: Considerations concerning homogeneous linearity evidently apply to Lorentztransformations too. In order to write (2.3.1) as (2.3.2) we had to take event A of the light ﬂash for the origin of the coordinate system. By this choice we tacitly assumed that the origin of an attached coordinate system may be anywhere in spacetime. As a matter of fact, (2.3.1) and (2.3.2) can be both valid only for linear transformations. As for homogeniety, it remains valid for the same reason as above. The peculiarity of Lorentz-transformations lies in the requirement (2.3.2) which they have to obey. Galilei transformations evidently violate it. Our task is, therefore, to ﬁnd those homogeneous linear transformations with V -dependent coefﬁcients which obey (2.3.2) and for V ¼ 0 reduce to the identity transformation. As a ﬁrst step suppose that we already know the formulae of Lorentz-transformations and substitute them into the expression c2 t0 2 À x0 2 À y0 2 À z0 2 : We then obtain a quadratic expression of unprimed coordinates which, according to (2.3.2), must vanish when c2 t2 À x2 À y2 À z2 ¼ 0: This can only happen if it is proportional to this expression: À Á 2 2 2 2 ð2:4:2Þ c2 t0 À x0 À y0 À z0 ¼ FðV Þ Á c2 t2 À x2 À y2 À z2 : The constant of proportionality can, of course, depend on V : We show now that it is an even function of V : We have already made use of the fact that the origin of the attached coordinates may be freely chosen. This playes the role of a constraint on the possible form of Lorentz-transformations, enforcing them to be linear. FðV Þ is constrained to be an even function due to another freedom in the choice of the standard setting. Let us

36

2 The Lorentz-Transformation

rotate the Cartesian coordinate systems K and K0 in a standard setting by 180 around their respective z-axes. We arrive at another standard setting attached to the same bodies in which, however, the velocity of I 0 with respect to I has opposite sign. Since both settings are standard ones, neither V nor ÀV is the ‘true’ relative velocity and if FðV Þ and FðÀV Þ were different from each other it would be impossible to make a choice between them. We are, therefore, compelled to assume that FðÀV Þ ¼ FðV Þ: The inverse of a Lorentz-transformation is also a Lorentz-transformation with V replaced by ÀV : Applying (2.4.2) to it we will have 2 2 2 2 c2 t2 À x2 À y2 À z2 ¼ FðÀV Þ Á c2 t0 À x0 À y0 À z0 : Substituting this into (2.4.2), the relation À Á c2 t2 À x2 À y2 À z2 ¼ FðV Þ Á FðÀV Þ Á c2 t2 À x2 À y2 À z2 is obtained, from which FðV Þ Á FðÀV Þ ¼ 1 follows. Since FðÀV Þ ¼ FðV Þ we ﬁnd that FðV Þ ¼ Æ1: But Lorentz-transformation for V ¼ 0 reduces to the identity transformation. Hence (2.4.2) leads to Fð0Þ ¼ þ1 and so FðV Þ ¼ 1: We have, therefore, 2 2 2 2 c 2 t 0 À x0 À y0 À z 0 ¼ c 2 t 2 À x2 À y2 À z 2 ð2:4:3Þ which is a constraint much stronger than (2.3.2). Since the transformation is linear (2.4.3) is applicable also to coordinate differences. In order to constraint further the possible form of the Lorentz-transformations, we will consider a version of Einstein’s train thought experiment in which the explosives are above and below the source of light at equal distances from it. In this setting constancy of light velocity leads to no unexpected consequences: The explosions are simultaneous with respect to both the car and the platform. In the corresponding standard setting x-axis is parallel to the rails. If the vertical direction is taken for the z-axis, we will have the following relations between the coordinate differences of explosions: Dt0 ¼ Dt ¼ 0; Dy0 ¼ Dy ¼ 0; Dx0 ¼ Dx ¼ 0 jDz0 j ¼ jDzj 6¼ 0:

Analogous relations (with the role of y and z directions exchanged) are obtained when the explosives are installed in the y direction with respect to the light source. The corresponding relations for the original version of the experiment are Dt0 6¼ 0;
0

Dt ¼ 0;
0

Dx0 6¼ 0;

Dx 6¼ 0

Dy ¼ Dy ¼ 0 ¼ Dz ¼ Dz ¼ 0; the unprimed frame being the rest frame of the train. To meet these requirements Lorentz-transformations should be of the form 9 ct0 ¼ AðV Þ Á ct þ BðV Þ Á x = x0 ¼ CðV Þ Á ct þ DðV Þ Á x ð2:4:4Þ ; y0 ¼ y; z0 ¼ z:

2.4 The Lorentz-Transformation

37

The use of ct instead of t is expedient because the former has the same dimension of length as space coordinates do and all the coefﬁcients will be dimensionless. Due to the special form of (2.4.4) relation (2.4.3) reduces to c 2 t 0 À x0 ¼ c 2 t 2 À x2 :
2 2

ð2:4:5Þ

The coefﬁcients AðV Þ and BðV Þ can be determined by their physical meaning. Consider, for example, a clock at the origin of K and the events there when the readings on the clock are t and t þ Dt: For this pair of events we have Dx ¼ 0 and, according to the ﬁrst of the equations (2.4.4), the time Dt0 which elapsed betwen them in K0 is equal to AðV ÞDt: But Dt is evidenly equal to the proper time Ds between the events and, since the clock moves with the velocity -V in K0 ; Dt0 must qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ be equal to Ds= 1 À ðÀV Þ2 =c2 ¼ Ds= 1 À V 2 =c2 : Hence 1 AðV Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cV : 1 À V 2 =c2 Return now to the original setting of Einstein’s train thought experiment again as discussed in Sect. 1.3. There V denoted the velocity of the train and we retain this meaning of it. If I is the rest frame of the train then for the explosions we have Dt ¼ 0 and Dx ¼ l0 where l0 is the proper length of the traincar. In the rest frame I 0 of the platform, which moves with the velocity -V with respect to the train, the time Dt0 elapsed between the explosions is equal to BðÀV Þl0 : But we know from Problem 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ that Dt0 ¼ V Á l0 =ðc2 1 À V 2 =c2 Þ; therefore BðÀV Þ ¼ V =ðc 1 À V 2 =c2 Þ and so V =c BðV Þ ¼ À pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 1 À V 2 =c2 The rest of the coefﬁcients follows from (2.4.5). Substituting (2.4.4) into this relation and equating the coefﬁcient of c2 t2 ; x2 and ctx; we have A2 À C2 ¼ 1; B2 À D2 ¼ À1; AB À CD ¼ 0: ð2:4:6Þ

Substituting A and B; we obtain CðV Þ ¼ þBðV Þ and DðV Þ ¼ þAðV Þ or CðV Þ ¼ ÀBðV Þ and DðV Þ ¼ ÀAðV Þ: The correct solution is the ﬁrst one, since at V ¼ 0 (2.4.4) must reduce to the identity transformation, and so we must have Dð0Þ ¼ þAð0Þ ¼ 1 [and Cð0Þ ¼ Bð0Þ ¼ 0]. The Lorentz-transformations are, therefore, given by the formulae 9 ct À V x > 0 c > ct ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > > 1 À V 2 =c2 > > = x À Vt ð2:4:7Þ 0 x ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > 1 À V 2 =c2 > > > > > 0 0 y ¼ y; z ¼ z ; In the c À 1 limit they become identical to Galilei-transformations (2.4.1). !

38

2 The Lorentz-Transformation

Lorentz-transformations are often called pseudorotations of the Minkowski coordinate system around the origin. The term is based on the similarity of (2.4.3) to the equation x0 þ y0 þ z0 ¼ x2 þ y2 þ z2 ; which holds true for rotations of Cartesian coordinates in 3D (three dimensional) geometric space. The transformations (2.4.7) are, therefore, pseudorotations in the ðct; xÞ plane of spacetime. To stress that they are not the most general case of Lorentz-transformations, the name boost is sometimes used to denote them, but in what follows we will continue to call them Lorentz-transformation in conformity with general usage. In the last step of derivation of (2.4.7) we made use of our prior knowledge of time dilation and train thought experiment. Either of them would sufﬁce. If no such prior information is available at all we may start from the equation x ¼ Vt of the origin of K0 in K which in K0 is simply x0 ¼ 0: Using (2.4.4), this last equation can C be expressed through unprimed coordinates as C:ðctÞ þ Dx ¼ 0 or x ¼ À D ðctÞ: Since this is just another form of the equation x ¼ Vt, we have C=D ¼ ÀV =c: This formula together with (2.4.6) is sufﬁcient to derive (2.4.7). The material of Chap. 1 is not, therefore, a necessary prerequisite to derive Lorentz-transformations. On the contrary, most of phenomena discussed in Chap. 1 can be obtained in a systematic way from Lorentz-transformations. Nevertheless, Chap. 1 does not seem at all superﬂuous. First, Lorentz-transformation rests on Einstein synchronization whose proper understanding is greatly facilitated by prior knowledge of time dilation. Second, mass–energy relation is a piece of knowledge independent of Lorentz-transformation. Relativistic dynamics requires both of them for its foundation (see Sect. 2.19).
2 2 2

2.5 Classiﬁcation of Spacetime Intervals
Both three dimensional geometric space and spacetime of special relativity are manifolds equipped with metric: Points of geometric space and spacetime are in one-to-one correspondence with triples or quadruples of real numbers respectively; the former is, therefore, a three dimensional manifold, the latter is a four dimensional one. The correspondence can be accomplished in inﬁnitely many different ways, all of them being a possible coordinate system in the corresponding manifold. The possession of a metric is the expression of the fact that any two points of the manifold are at a well-deﬁned ‘squared distance’ from each other. The squared distance is, in principle, a measurable quantity and, therefore, independent of the coordinate system chosen: it is invariant: In 3D geometric space the formula for the squared distace has its simplest mathematical form in Cartesian coordinates while in spacetime it is most simple in Minkowski coordinates:

2.5 Classiﬁcation of Spacetime Intervals

39

Dl2 ¼ Dx2 þ Dy2 þ Dz2 ; Ds2 ¼ c2 Dt2 À Dx2 À Dy2 À Dz2

ð2:5:1Þ ð2:5:2Þ

(for inﬁnitesimal intervals D À d). Formula (2.5.1) refers to 3D euclidean space. ! The meaning of the adjective ‘euclidean’ is that Dl2 is positive deﬁnite, since it contains only positive terms. On the other hand, spacetime is a pseudoÀeuclidean manifold since the expression of Ds2 contains both positive and negative terms. As a consequence, it may be equal to zero or may even take on negative values. It is, therefore, quite misleading to write it as square of a quantity but, since it is the spacetime counterpart of Dl2 ; the notation Ds2 is an appropriate one. An important aspect of (2.5.1) and (2.5.2) is expressed in the form of negation. The meaning of (2.5.1), for example, is not fully exhausted by the statement that the sum on the right hand side has the same value in all of the coordinate systems which are rotated with respect to each other; its full content consists in the assertion that it is the sum only which remains invariant. Its separate terms depend on the coordinate system chosen and so they are of no intrinsic geometric signiﬁcance. The same is true for (2.5.2). A pair of events deﬁnes a unique Ds2 between them, but their time difference and spatial distance depend on the coordinate system chosen. Since different inertial frames involve different Minkowski coordinate systems, this property of spacetime is an expression of the relativity of simultaneity and of the Lorentz contraction on geometric language. Depending on the sign of Ds2 ; pairs of events (or spacetime intervals between them) are of three fundamentally different types. Since Ds2 is invariant, the type of a given pair is the same with respect to all the inertial frames. Let the events be denoted by Ea and Eb and assume that in the inertial frame I chosen Ea precedes Eb : Dt ¼ tb À ta [ 0 (and Dx ¼ xb À xa ). For the sake of simplicity we assume that Dy ¼ Dz ¼ 0: 1. When Ds2 ¼ 0 the pair is called lightlike, because Ea and Eb can be mediated by light pulse. Indeed, since Ds2 ¼ c2 Dt2 À Dl2 ; for Ds2 ¼ 0 we have Dt ¼ Dl=c: 2. If Ds2 [ 0 the events are timelike. For timelike pairs (and only for them) an inertial frame I 0 is always found in which the events happen at the same place. The time Dt0 ¼ t0b À t0a elapsed between the events in I 0 is proportional to Ds 2 since, owing to Dx0 ¼ x0b À x0a ¼ 0; Dt0 ¼ Ds2 =c2 : Two consecutive events on a pointlike body always constitute a timelike pair. If in I the body moves with constant velocity then I 0 is its rest frame and Dt0 is identical to the proper time interval between the events on it: Ds2 ¼ Ds2 =c2 . 3. When, ﬁnally, Ds2 \0 the pair is spacelike. According to the ﬁrst equation of (2.4.7), in I 0 the time elapsed between them is equal to Dt À ðV =c2 ÞDx 1 À V =VD Dt0 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ Dt Â pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 1 À V 2 =c2 1 À V 2 =c2 ð2:5:3Þ

40

2 The Lorentz-Transformation

where VD ¼ c2 ðDt=DxÞ is a quantity of the dimension of velocity. For a spacelike pair VD \c: The proof is simple: for Ds2 \0 we have
2 VD ¼ c 2

c2 Dt2 Dx2 þ Ds2 ¼ c2 \c2 : 2 Dx Dx2

Therefore, an inertial system I D exists which in I moves with velocity VD : As it can be seen from (2.5.3), in this frame the events Ea and Eb are simultaneous. In 0 0 the frames I 0 for which 1\V =VD ; Eb precedes Ea ðtb \ta Þ while in those with 1 [ V =VD the event Ea takes place earlier. The conclusion is that time order within a pair of spacelike events depends on the inertial frame chosen.2

2.6 Spacetime Diagrams
Spacetime diagrams (or Minkowski diagrams) help in visualisation of the content of Lorentz-transformations. Representation of Galilei-transformations by analogous diagrams is also possible but, owing to the simplicity of these transformations, they are never used. Yet we are going to start with this latter type of diagrams because they may be instrumental in proper understanding of how and why diagrams of this kind may be useful. Both (2.4.1) and (2.4.7) refer to standard setting in which coordinates y and z remain intact. Let us denote the residual ðt; xÞ part of the coordinate system, attached to I , by M2 : On a sheet of paper it can be represented as a rectilinear coordinate system ðt; xÞ whose vertical axis is, by tradition, the t-axis. The points of the ðt; xÞ coordinate plane represent events. For any point, the position and the moment of time of the corresponding event is obtained by parallel projection along the axes. The motion of a point mass is a continuous array of events which on the diagram is represented by a continuous curve called world line. In Newtonian physics world lines can be of any form with proviso that to any t there corresponds a unique value of x since no body can be found in several different places at the same moment of time. To a given x, on the contrary, there belong as many different moments of time as many times the body returns to the same place x. The world line of a uniformly moving body is a straight line x ¼ vt þ x0 : The angle b made by this line with the t-axis is connected to the velocity trough the formula v ¼ tan b: For a point mass at rest b = 0: the world line of a body at rest is parallel to the t-axis. In particular, the world line of a point mass, resting at the origin x = 0, is the t-axis itself, and it is often convenient to look at the t-axis as the x = 0 axis.

2

In the train thought experiment, for example, explosions are simultaneous in the train’s rest frame, the explosion at the rear end is the ﬁrst one in the platform’s rest frame and this time order is reversed as seen from another train, moving faster in the same direction

2.6 Spacetime Diagrams

41

Straight lines parallel to the x-axis cannot be world lines of point masses since they belong to inﬁnite velocity: b ¼ 90 and so v ¼ tan 90 ¼ 1: The events they contain are in M2 simultaneous to each other (the same t belongs to all of them). In particular, all the events on the x-axis take place at the moment t = 0 and the xaxis is, therefore, often called the t ¼ 0 axis. What has so far been said refers to both Newtonian and relativistic physics (in the latter case the velocity on the world lines must be smaller than c). They are summarized on Fig. 2.1. Let us now conﬁne ourselves to Galilean-transformations. When, using (2.4.1), we change our unprimed coordinates M2 to primed ones, attached to another inertial frame, nothing happens to the event E: it remains the same event though referred to two different coordinate systems. The correct representation of this process is, therefore, to leave the point which represents E at its original place, but to replace the unprimed rectilinear coordinate system M2 with an oblique M02 one in such a way that the coordinates ðt0 ; x0 Þ of E; obtained by parallel projection along the new axes, be equal to those given by (2.4.1). This is very easy to do. According to Galilei-transformations (2.4.1), equation x0 = 0 of the origin of M02 is described in M2 by the equation x0 ¼ x À Vt ¼ 0: The straight line x ¼ Vt passes through the origin and makes an angle b ¼ arctan V with the t-axis. But we have seen that the straight line x0 ¼ 0 is identical to the t0 -axis of M02 (see Fig. 2.2). The t- and t0 -axes are, therefore, different from each other. The x0 -axis, however, must be the same as the x-axis. This last axis contains the events which in M take place at the moment t ¼ 0: Since time in Newtonian physics is absolute ðt0 ¼ tÞ; x0 -axis consists of these same events (points), and the t0 ¼ constant coordinate lines are all of horizontal direction. As a conseqence, the scales on the t- and t0 -axes are different from each are, since the

Fig. 2.1 World lines on the Minkowski diagram

42 Fig. 2.2 Representation of Galilei-transformation on the Minkowski diagram

2 The Lorentz-Transformation

geometric distance between the origin and the points, belonging to 1 s on the t- and t0 -axes is in the former case smaller than in the latter. As seen on Fig. 2.2, the coordinates ðt; xÞ and ðt0 ; x0 Þ of the point E are indeed related to each other by the transformation (2.4.1). The equality t0 ¼ t is obviously fulﬁlled and x0 Ox0 ¼ Ox À x0 x ¼ x À tan b Á t ¼ x À Vt is also true. When I 0 is moving in the negative direction (V \0; b\0), the t0 -axis lies to the left of the t-axis. On spacetime diagrams rectilinear coordinates are by no means distinguished with respect to the oblique ones: it is the relative direction of the time axes which matters. When, for example, the t-axis of M is taken so as to make the angle b ¼ À tan V with respect to the vertical direction (and the former coordinates t and x of E are measured along these oblique axes) then the t0 -axis will be the vertical one. While Galilean spacetime diagrams have in fact no useful application, visualisation of the more complicated Lorentz-transformations by means of spacetime diagrams may often be helpful. The basic principle is the same in this more difﬁcult case too: the equation of the t0 -axis with respect to M2 is obtained from the condition x0 ¼ 0; and that of the x0 -axis is determined by the equation t0 ¼ 0; provided t0 and x0 are expressed in them, with the help of the Lorentz-transformation (2.4.7), through the unprimed coordinates. For the equation of the t0 -axis we obtain again x ¼ Vt so this axis makes on relativistic spacetime diagrams the same angle b ¼ arctan V with the t-axis as it

2.6 Spacetime Diagrams

43

does on Galilean diagrams. From now on, however, it will be more convenient to work with the variable ct instead of t: It is often denoted by x0 which is very convenient from a mathematical point of view, but we will continue to use the explicite form ct: Since x ¼ Vt ¼ ðV =cÞ Á ct; the ct0 axis makes an angle b ¼ arctanðV =cÞ with the upward ct-axis in clockwise direction (for b [ 0). The essential novelty of relativistic diagrams with respect to the Galilean ones consists in the direction of the x0 -axis. According to relativity theory events, which are simultaneous with each other in different inertial frames, belong to different domains of spacetime. Since x-axis (the line t ¼ 0) and the x0 -axis (t0 ¼ 0) are built up of simultaneous events in M2 and M02 respectively, they cannot coincide with each other. If we substitute t0 ¼ 0 into the ﬁrst line of (2.4.7) we obtain that the x0 axis is indeed different from the x-axis, its equation in unprimed coordinates being ct ¼ ðV =cÞ Á x: This line passes through the origin and, as it can be easily shown, makes the same angle b with the x-axis (in counterclockwise direction) which is made by the two time axes with each other. Let us discuss now the scales on the primed and unprimed axes. As we have already observed, on Galilean diagrams the scales on the t- and the t0 -axes are different: pieces of equal lengths on them correspont to different intervals of time. On relativistic diagrams this distortion, analogous to the distortions on geographical maps, is ampliﬁed and extended to the x- and x0 -axes too. The measure of distortion can be assessed by means of the hyperbolas c2 t2 À x2 ¼ constant 6¼ 0 which, according to (2.4.5), have the same mathematical form in all Minkowski coordinates. Take the points ct0 ¼ Æj on the t0 -axis for which c2 t0 2 À x0 2 ¼ j2 : The content of (2.4.5) is that the unprimed coordinates of this point satisfy the equation c2 t2 À x2 ¼ j2 : Hence, the points on all the possible time axes, possessing the same coordinate, lie on hyperbolas c2 t2 À x2 ¼ constant [ 0: The analogous statement for points on the x-axes contains hyperbolas c2 t2 À x2 ¼ constant\0: The common asymptotes of the two hyperbolas are the bisectrices x = ± ct of the angle between the coordinate axes. As it can be established by inspection of the relativistic diagrams (or proved with the help of Lorentz-transformation) the equation of them is of the same form in all Minkowski coordinates. Since their points satisfy equation c2 t2 À x2 ¼ 0 they do not indeed have common points with the hyperbolas c2 t2 À x2 ¼ constant 6¼ 0 though are approaching them inﬁnitely close. The points on relativistic diagrams can be classiﬁed according to their types with respect to the event E0 at the origin. Since any point of spacetime may be chosen for the origin this classiﬁcation refers in fact to two arbitrary events (or spacetime intervals between them). The interval between the points of the asymptotes x ¼ Æct and the origin is lightlike. They constitute the light cone of E0 : The upper half (t [ 0) of the light cone is the path a ﬂash of light at the origin takes through spacetime. It is the lightlike future of E0 : The lower part (t \ 0) consists of those events the light signals from which can reach the origin. They are, therefore, the lightlike past of the event at the origin.

44

2 The Lorentz-Transformation

The light cone splits spacetime into several domains. Points outside the light cone are those which are spacelike with respect to E0 ðc2 t2 À x2 \0Þ: As we have seen in the previous section, for any event E in this domain an inertial frame can be found in which E and E0 are simultaneous. The points of the outer domain constitute the ‘interval of simultaneity’ with respect to E0 which was discussed in Sect. 1.3 in connection with the Mars rover. On the spacetime diagram of M2 the outside domain appears divided into two disjuct parts but this is true only for the two dimensional sections of it. If Minkowski coordinates are imagined in three dimensional space where y-direction can also be visualized the outside domain is seen to be a connected part of spacetime. The inside domain of light cone contains events which are timelike with respect to E0 : Like the light cone itself, this domain splits also into two disconnected part according to the sign of t. Events with t [ 0 constitute the timelike future of the origin while points with t \ 0 constitute its timelike past. To any event E in these domains an inertial frame can be found in which E and E0 happen at the same place. E precedes or follows E0 ; depending on whether it belongs to the timelike past or future of it. This order of events cannot be altered by any choice of the coordinate system. The properties of the relativistic spacetime diagrams discussed so far are summarized on Fig. 2.3. Remember, how Lorentz-transformations are represented on them. Primed and unprimed coordinates of an event E are obtained by means of parallel projection along the corresponding axis, if only the unity is taken on them properly (Fig. 2.4). Primed axes on Fig. 2.4 correspond to motion of I 0 with positive velocity (V [ 0; b [ 0). In the opposite case (V \0; b\0) the time axis is rotated in counterclockwise, the x-axis in clockwise direction. The primed axes
Fig. 2.3 Structure of the relativistic spacetime diagram

2.6 Spacetime Diagrams Fig. 2.4 Representation of Lorentz-transformation on the Minkowski diagram

45

will then make an obtuse angle with each other. Since primed and unprimed frames are equivalent, rectilinear coordinates are by no means distinguished with respect to the oblique ones. If, for example, the primed coordinates are taken in the vertical and horizontal direction then the unprimed coordinates will make an obtuse angle between them.

2.7 The Causality Paradox
If superluminal signals existed, sending messages into past would be possible due to the relativity of simultaneity. In a ﬁt of despair, for example, one could hire a killer in the past to put one’s father to death when he is still in his infancy. Suppose the murder did his job. On who’s behalf was he acting? Inﬂuences annulling their own cause would beset with perplexing consequences if subjects were able to choose and act as their own will dictated (as in the story above). Let us illustrate with an example how superluminal velocity could in relativity theory lead to inﬂuence of this kind. Consider two inertial frames I and I 0 in standard setting equipped at their origins O and O0 with identical cannons C and C0 , aiming at each other. Examine the following scenario, described in terms of the coordinate time of I : at the moment t1 cannon C ﬁres a projectile toward I 0 which hits it at the moment t2. The shot, however, misses cannon C0 which retaliates immediately. The projectile

46

2 The Lorentz-Transformation

ﬁred by C0 hits cannon C at the moment t3 and completely destroys it. Since we are investigating consequences of superluminal signals, the muzzle velocity v of the cannons will be allowed to be either smaller or greater than light velocity (but it must be greater than the relative velocity V of the inertial frames in order to make the scenario possible). The problem is to calculate t3 as a function of t1, V and v. In a standard setting the trajectory of O0 in K is x = Vt. The trajectory of the projectile in K shot by C is x = v(t - t1). The moment t2 of its arrival to O0 is determined by the equation Vt2 ¼ vðt2 À t1 Þ; from which we obtain t2 ¼ v t1 : vÀV Vv t1 [ 0 vÀV ð2:7:1Þ

At this moment O0 is found at the point x2 ¼ Vt2 ¼ ð2:7:2Þ

of K. Let us denote by u the velocity with respect to K of the projectile ﬁred by C0 . According to the rule of relativistic velocity addition we ﬁnd that u¼ V Àv ; 1 À Vv c2 ð2:7:3Þ

since we have now in (1.6.1) V0 = u, V =-v and U =-V. The trajectory of this second projectile in K is x ¼ x2 þ uðt À t2 Þ: At the moment t3 of its hit it is found at x = 0, hence x2 t3 ¼ t2 À : u

ð2:7:4Þ

Substituting (2.7.1), (2.7.2) and (2.7.3) into (2.7.4) we obtain, after some rearrangements, the ﬁnal formula t3 ¼ À 1 À V2 c Á2 Á t1 : 1ÀV v
2

ð2:7:5Þ

Remember that v need not be smaller than c but it cannot be less than V. For v [ V the derivative of t3 with respect to v is negative, therefore, t3 is a decreasing function of v as it should be. It is very large when v is only slightly greater than V: as v À V ; t3 À 1: In the opposite limit, when v tends to inﬁnity, the denom! ! inator of (2.7.5) is practically equal to unity and t3 % ð1 À V 2 =c2 Þ Á t1 : This value is smaller than t1! When, therefore, v is sufﬁciently great, cannon C is destroyed before it could begin the duel. But then no retaliation by C0 is followed, cannon C remains ready to shoote at t1 which it indeed does, its projectile hits O0 , etc., etc.

2.7 The Causality Paradox

47

We arrived at a logical contradiction because we are both afﬁrming and negating the same facts. In order to analyze the sequence of the events in some more detail, denote the consecutive events of ﬁring by C, C0 and the destruction of the cannon C at O by E1 ; E2 and E3 whose coordinates in M are ð0; t1 Þ; ðx2 ; t2 Þ and (0, t3), respec0 0 tively. In I E2 happens later than E1 ðt2 [ t1 Þ and in I 0 E3 follows E2 (t3 [ t2 ). When v \ c nothing paradoxical happens. The pairs E1 ; E2 and E2 ; E3 are timelike and their time order is the same in any inertial frame (it is invariant). E3 is, therefore, subsequent to E2 in the frame I too. As a consequence, in this frame E3 follows E1 : t3 [ t1 : But E1 and E3 are timelike with respect to each other since they both happen at the same place and so their time order is also invariant. The time sequence of the events is, therefore, E1 ! E2 ! E3 as expected, independently of the coordinate system chosen. This order, however, can be changed dramatically when the projectiles move with faster-than-light velocity. In this case the pairs E1 ; E2 and E2 ; E3 are spacelike and E3 may precede E2 in I in spite of the fact that in I 0 the former is subsequent to the latter. When t3 becomes sufﬁciently smaller than t2 the causality paradox arises. If, for example, t1 = 75 s, V = c/2 and v = 8c then t3 ¼ 64 s\t1 : Let us denote by vk the critical value of v, above which the paradox t3 [ t1 occurs. When v is equal to vk ; t3 is equal to t1. From this condition, using (2.7.5), we obtain for the critical velocity the following expression: vk ¼ V pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ [ c: 1 À 1 À V 2 =c2 ð2:7:6Þ

When v [ vk formula (2.7.3) leads to positive velocity u of the response projectile3 shot by C0 ; in the numerical example above u = + 5c/2. Does this mean the projectile ﬂies away from O instead of approaching it? No, it ﬂies toward O since the sign of u coincides with the direction of motion (i.e. with the sign of dx) only if dt [ 0. But when the return projectile travels in I backward in time (dt \ 0) then a positive u is needed to ensure that dx ¼ u Á dt be negative. On Fig. 2.5 the spacetime diagram of the scenario is shown for two special values of v: on Fig. 2.5a v = c and on Fig. 2.5b v ¼ 1: When v = c the trajectories of the projectiles make an angle 45° with the ct-axis and the time sequence of the events is the natural one. It is the more so the smaller the velocity v is. On Fig. 2.5b both the pairs E1 ; E2 and E2 ; E3 are simultaneous but with respect to different inertial frames: the trajectory E1 À E2 is parallel to the x-axis, while ! the trajectory E2 À E3 is parallel to the x0 -axis. This is the reason why t3 turns ! out smaller than t1. The coordinate time t is obviously identical to the proper time of the cannon C, resting in the origin of K, and so the moments t1 and t3 are independent of the coordinate system chosen.

The velocity u is positive already for v [ c2/V According to (2.7.4), in this domain t3 \t2 : Since vk [ c2/V it is true a fortiori in the domain v [ vk.

3

48

2 The Lorentz-Transformation

(a)

(b)

Fig. 2.5 Illustration of the causality paradox

As we see, superluminal signals are a potential source of the causality paradox. The paradox itself consists in reversal of the time order within a causally related timelike pair of events (E1 and E3 in the example). This reversal is made possible by the noninvariance of time order within spacelike pairs. The problem cannot be disposed of by saying that the cause is always the event wich took place ﬁrst because cause and effect (the shot and the explosion) are events intrinsically different from each other. It has been shown in Sect. 1.7 that bodies cannot be accelerated up to the velocity of light. Why then bother about superluminal velocities and causal paradox? The answer is that signals are not necessarily moving bodies, the obvious example being a light pulse. There may also exist elementary particles which are born in a radioactive process as faster-than-light objects. No such exotic particle has so far been observed but a name has already been given to them: They are called tachyons. The mathematical formalism of relativity theory does not forbid tachyons but their real existence would certainly undermine causality. Since causaliy plays an outstanding role in physics, we close this section by stating its content. In physics causality means that (1) cause and its effect are different in nature and (2) the effect always follows in time its cause.

2.8 Demonstration of Time Dilation on Spacetime Diagram
In Sect. 1.3 we outlined an argument to the effect that the symmetry of time dilation is incompatible with the Newtonian notion of time. At the same time it has been argued that contradictions can be removed if simultaneity of distant events is allowed to depend on the inertial frame in which they are observed. Spacetime diagram provides an ideal tool for making this argument transparent.

2.8 Demonstration of Time Dilation on Spacetime Diagram Fig. 2.6 Explanation of the time dilation

49

Let the clocks O and O 0 are resting at the origins O and O0 of K and K0 in a standard setting. The ct- and ct0 -axes on Fig. 2.6 are the world lines of them. The origin is the event of their encounter when both clocks show zero second. In the rest frame of O clock O 0 is moving. When O shows t1 s (event ct1 on the ct-axis) the hand of O 0 points to t2 (event ct2 on the ct0 -axis). These two points are found on the same (horizontal) x-coordinate line. Owing to the different scales on these axes, t2 is smaller than t1. This is time dilation from the ‘unprimed point of view’. Analogously, in the rest frame of O 0 the clock O is moving. When the former shows t2 s the reading on the latter is t3 s. Again, the difference of the scales leads to t3 \t2 which is time dilation from the ‘primed point of view’. In this latter case simultaneity is determined by the (oblique) x0 - coordinate lines. Notice that the x-coordinate line which connects the points ct1 and ct2 on the 2 corresponding time axes is tangent to the hyperbola c2 t2 À x2 ¼ c2 t1 : Similarly, 0 the x -coordinate line which passes through the points ct2 and ct3 on them is 2 tangent to the hyperbola c2 t2 À x2 ¼ c2 t2 : The ﬁrst of these statement is obvious but the second one needs veriﬁcation. As a matter of fact, the second statement would be evident too if the primed coordinate system had been drawn as the rectilinear one. This would be a permitted choice since primed and unprimed frames are equivalent to each other. The property to be proved is expressed by the following theorem: the tangent to the hyperbolas c2 t2 À x2 ¼ constant [ 0 at the point of their intersection with the ct0 -axis is parallel to the x0 -axis, and the tangent to the hyperbolas c2 t2 À x2 ¼ constant\0 at the point of their intersection with the x0 -axis is parallel to the ct0 -axis.

50

2 The Lorentz-Transformation

The proof is very simple. The differential of the equation c2 t2 À x2 ¼ constant of the hyperbola is 2ct Á dðctÞ ¼ 2x Á dx: Its slope is, therefore, given by the equation dðctÞ x ¼ : dx ct But on the ct0 -axis (i.e. on the x0 = 0 axis) x = Vt, therefore, the slope of the hyperbola at its point of intersection with this axis is dðctÞ V ¼ : dx c On the other hand, the equation of the x0 -axis in the unprimed coordinates is ct À V x ¼ 0 and its slope d(ct)/dx is also equal to V/c. This proves the ﬁrst part of c the theorem and the second part can be proved in the same way. Notice that Fig. 2.6 is identical to Fig. 2.5b, because the moments of the shoot and the hit are simultaneous events in the inertial frame where muzzle velocity is inﬁnitely large.

2.9 Doppler-Effect Revisited
Consider a monochromatic electromagnetic plane wave whose phase in I is given by the expression x 2p mðx À ctÞ: 2p À mt ¼ k c The second form has been obtained using the relation mk ¼ c: Let us express in this formula the unprimed coordinates ðct; xÞ through the primed ones. To this end we have to reverse in (2.4.7) the role of primed and unprimed coordinates and, since K is moving with respect to K0 with the velocity ÀV ; the sign of V must also be changed to the opposite. Then ! 2p 2pm V mðx À ctÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðx0 þ VtÞ À ct0 þ x0 ¼ c c c 1 À V 2 =c2 ! 2pm V ðx0 À ct0 Þ : ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À 2 2 c c 1 À V =c Comparing this expression with the phase 2p m0 ðx0 À ct0 Þ of the same light wave in c I 0 we obtain for the frequency m0 the expression sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V =c 0 m ¼m 1 þ V =c

2.9 Doppler-Effect Revisited

51

in agreement with the formula (1.2.7). We see that the same formula is applied to both the rate of light signals and the frequency of light waves. In Sect. 1.2 we have emphasized that m and m0 are frequencies measured in the proper time of the apparatuses which are at rest in I and I 0 ; respectively, while the present derivation concerns frequencies in the coordinate times t and t0 : But frequencies are measured by clocks at rest for which dt ¼ ds so from the point of view of frequencies the two different kinds of time are equivalent to each other.

2.10 The Connection of the Proper Time and Coordinate Time in Inertial Frames
A clock ‘laid down’ in an inertial frame goes at the rhythm of the coordinate time. Therefore, the coordinate time and the proper time of a clock at rest is related to each other by the equation ds ¼ dt independently of their synchronisation. But clocks moving in I go slower. How many times slower? The trajectory of a pointlike clock in K attached to I is described by equations of the form x ¼ f ðtÞ; y ¼ gðtÞ; z ¼ hðtÞ

in which t is the coordinate time. The square of the instantaneous velocity vðtÞ is then 2 2 2 df dg dh _ _ v2 ¼ þ þ x2 þ y 2 þ z 2 : _ ð2:10:1Þ dt dt dt The functions f ðtÞ; gðtÞ and hðtÞ are such that v2 \c2 : At the moment t the clock is at rest with respect to the instantaneous rest frame I 0 , moving with uniform velocity vðtÞ: At this moment its proper time is ﬂowing with the same speed as the coordinate time t0 of I 0 : ds ¼ dt0 : Since at this same moment the velocity of the clock in I 0 is zero, this equation can also be written as c2 ds2 ¼ c2 dt 2 ¼ c2 dt 2 À dx 2 À dy 2 À dz 2 :
0 0 0 0 0

ð2:10:2Þ

But the expression on the right hand side is the relativistic squared distance ds2 whose value can be calculated by means of this same formula in any inertial frame: c2 dt 2 À dx 2 À dy 2 À dz 2 ¼ c2 dt2 À dx2 À dy2 À dz2 :
0 0 0 0

ð2:10:3Þ

Substituting this into the right hand side of (2.10.2), factoring out c2 dt2 and using (2.10.1), we obtain c2 ds2 ¼ c2 dt2 ð1 À v2 =c2 Þ: Therefore, at the point of the trajectory where the velocity is equal to v the proper time ﬂows

52

2 The Lorentz-Transformation

ds pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1 À v2 =c2 1=cðvÞ dt

ð2:10:4Þ

times slower than the coordinate time. This formula has already been obtained in Sect. 1.4 when descriptions of the Doppler-effect from the point of view of two different inerial frames were compared with each other. That was a convincing heuristic argument in which the time t within an inertial frame was taken identical to the familiar Newtonian time. That was a natural simplifying assumption but not a misleading one since in the intuitive notion of the Newtonian time subtlities of synchronization play no role.

2.11 The Magnitude of the Twin Paradox
The twin paradox had already been discussed in Sect. 1.4 but the calculation of its magnitude requires Minkowski coordinates introduced only in the present section. The most straightforward formulation of it uses two ideal clocks. The paradox consists in the statement that if they move on different trajectories then they in general measure two different values of the proper time between subsequent encounters. Let the trajectories of the clocks in M attached to I is given by the functions x ¼ fi ðtÞ; y ¼ gi ðtÞ; z ¼ hi ðtÞ ði ¼ 1; 2Þ:

The proper time measured by them between encounters at the moments ta and tb is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Rt equal to Dsi ¼ tab dsi : Since dsi ¼ dt 1 À vi 2 =c2 ; we have vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ " # Ztb Ztb u u dfi 2 dgi 2 dhi 2 t1 À 1 dt: ð2:11:1Þ dsi ¼ þ þ Dsi ¼ c2 dt dt dt ta ta

The magnitude of the paradox is the difference between Ds1 and Ds2 : In the special case when clock 1 remains at rest we obtain Ds1 ¼ tb À ta Ds2 \ðtb À ta Þ ¼ Ds1 ;

which indicates that the shorter elapsed time is shown by that clock which was subjected to larger acceleration. R The integral ds by which the elapsed time is calculated is mathematically akin R to the path integral dl in analytic geometry which is used to calculate the arc length of a curve. The distinctive property of this type of integrals is that they are independent of the coordinate system chosen in conformity with the fact that both proper time and arc length are invariant quantities. The magnitude of the twin paradox was experimentally veriﬁed in the Gravity Probe A (GP-A) experiment of NASA in 1976. The probe was launched nearly

2.11

The Magnitude of the Twin Paradox

53

vertically upward, reaching a height of 10,000 km. It housed a hydrogen maser, a highly accurate frequency standard, to compare the time elapsed on the spacecraft and on the Earth. The experiment conﬁrmed the prediction of relativity theory to an accuracy of about 70 ppm. In this experiment, however, gravitation played an important role and, according to the general relativity theory, gravitation inﬂuences the proper time of bodies. It can be shown that in the vicinity of the Earth (2.10.4) is modiﬁed to pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ds ¼ dt 1 À v2 =c2 þ 2U=c2 ; ð2:11:2Þ in which U ¼ ÀGM=r is the Earth’s gravitational potential which tends to zero at inﬁnity. In the GP-A experiment the validity of this formula was veriﬁed. We notice ﬁnally that twin paradox (time dilation in general) has found already its way to the engineering application since the GPS navigation system would not function correctly if the internal clocks were not adjusted for relativity.

2.12 The Coordinate Time in Accelerating Frames: the Twin Paradox
In this section our task will be to compare the detailed description of the twin paradox as experienced by either of the twins, say Alice and Bob, when Alice remains at rest in an inertial frame, and her twin-brother Bob makes a trip. Let the place they live in call Blackwood and assume that Bob decides to travel by train from Blackwood to Whitewood, which is at some distance from it, and then back to Blackwood. The area, surrounding Blackwood and extending beyond Whitewood, constitutes the rest frame of Alice who will be awaiting Bob’s return patiently at the railway station at Blackwood. Since we will forget Earth’s rotation, her rest frame can be taken for an inertial one. The clocks at the railway stations within this area will all be assumed to show the correct Minkowski coordinate time. Early morning of the day of departure Bob and Alice walk to the railway station at Blackwood. They shoot a quick look at the station’s clock and ascertain that their ideal wrest-watches are keeping good time. On the way to Whitewood Bob compares the reading sB on his watch (proper time!) with the time t on the clocks at the stations the train passes by (coordinate time!) and observes that it is the more behind the more time has passed since the departure. This tendency continues to hold on the way back too. When he ﬁnally gets off the train at Blackwood and compares his watch with that of Alice, they ﬁnd that while he has spent a time TB on the train she was waiting for him for the longer time TA: the difference TA À TB is ‘the magnitude of the twin paradox’. It could have been approximately calculated in advance by means of (2.11.1), using the time table of Bob’s train. When the train moves with the same constant velocity in both directions and spends no pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ TB time at Whitewood then sB ¼ TA t with TB ¼ TA 1 À V 2 =c2 :

54

2 The Lorentz-Transformation

Is it possible to give a similarly unambiguous account of how Alice’s watch gets more and more ahead with respect to the coordinate time of the train’s rest frame? As we have already stressed, the magnitude of the effect is an invariant quantity which can be unambigouosly calculated in any reference frame. This follows from the invariance of the integral (2.11.1) whose value is independent of whether the paths of Bob and Alice are referred to the coordinates t, x in which Alice rests at x = 0 or to h , n attached to the train where Bob is resting at, say, n = 0 (the notation of the coordinates attached to the train by greek characters serves to remind us that they do not constitute a Minkowski coordinate system). But this is the global effect only which in itself tells nothing on how the time lag is building up gradually, station by station. When the train is declared to be at rest it is Alice who, together with the whole area of Blackwood and Whitewood, sets to depart in the direction opposite to the original motion of the train. In this alternative description she is expected to compare her watch with the clocks which are at rest on the train but this is clearly impossible unless the train is at least as long as the distance between Blackwood and Whitewood and appropriate clocks are indeed installed on it at more or less regular distances from each other. Though a train of such an enormous length would be a rather monstrous construction, it does not seem to contradict any physical principle and so we may be, perhaps, reconciled to its existence. But the proper synchronization of the clocks mentioned above does constitute a principal problem. Accelerations, decelerations and changes in the direction of motion of the area with respect to the train at rest are felt as inertial forces in the train and make the speed of light to depend in an irregular way on both of its direction of propagation and time. As a consequence, no universal method of synchronization of clocks on the train exists which would correspond to the synchronization of clocks at the stations, resting in an inertial frame. As a result, no universal scenario exists of how Alice’s watch is gaining gradually more and more. The behaviour of any particular member of the train’s clocks can be followed up unambiguously, provided its reading is prescribed at some point of its trajectory. As noted above, this is ensured by the possibility to relate the integral in (2.11.1) to Minkowski coordinates in an inertial frame. Since synchronization consists precisely of such kind of prescriptions detailed analysis of the twin paradox is possible also in coordinate systems attached to accelerating reference frames like our train once a coordinate time is ﬁxed in them in some particular way. But the claim to give a universal account of it prior to chosing a particular kind of synchronization is unfounded since any prediction of this kind could be veriﬁed only using real clocks synchronized in some deﬁnite manner. No sensible physical theory should give answer to such an ill posed problem. In this respect accelerating reference frames differ signiﬁcantly from the inertial ones in which Einstein synchronization and Minkowski coordinates are always a possible preferred choice. What has been said above will now be illustrated by two particular examples of synchronization in the train’s rest frame. Let us assume ﬁrst that, while staying at Blackwood, the clocks on the train are adjusted to show the coordinate time t in the Blackwood–Whitewood area (remember that the train is now longer than the distance

2.12

The Coordinate Time in Accelerating Frames: the Twin Paradox

55

between them). From the point of view of the train (and of Bob on it) at the moment of departure Alice (and the whole area) begins to move toward the rear end of the train. Passing by the ‘public clocks’ on the train Alice compares the reading on her watch sA (proper time!) with the time h (coordinate time!) shown on these clocks. Assume that the point n = 0 of the train where Bob is sitting reaches Whitewood at the moment t = 0 and at this moment the velocity of the whole train suddenly changes sign. Such a sudden synchronous velocity reversal can be accomplished only if all the cars are motor-cars and the motormen on them are aware of the time t due to e.g. the public clocks along the railroad. Our object of study is the dependence of sA on h i.e. the function f1(h) in the relation sA ¼ f1 ðhÞ (the index 1 refers to the ﬁrst method of synchronization). If the train’s motion is to a good approximation uniform with the same velocity V both to Whitewood and backward from it we have f1 ðhÞ ¼ TA h where TB ¼ TB pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 TA 1 À V =c : Indeed, looking at the series of encounters of Alice with the clocks on the train from the point of view of her rest frame, we observe that the later a train clock passes by Alice the longer path it had already travelled and, therefore, the more it is losing compared to her watch. This result is the strict analogue of Bob’s expeTB rience which has been summed up in the formula sB ¼ TA t Consider now the second method of synchronization. Assume that the coordinate time h on the train has been chosen according to Einstein’s synchronization procedure when the train was moving with the constant velocity V toward Blackwood. Suppose that at that time Bob has already been sitting on the train and compares his clock with that of Alice later while in motion through Blackwood. The velocity reversal takes place at the moment h = 0, when Bob arrives at Whitewood. The motormen now perform this operation according to their own correctly synchronized clocks. Since the rest frame of Alice is an inertial one, formula (2.11.1) is applicable in pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ it, and leads immediately to the relation TB ¼ TA 1 À V 2 =c2 : But there is a problem here. Now the rest frame of Bob is also an inertial one both on the way toward Whitewood and back from it. Therefore, (2.11.1) is applicable in them also pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ and leads obviously to the opposite result TA ¼ TB 1 À V 2 =c2 : This last formula is certainly wrong but it may not be immediately clear how to rectify it. To solve the puzzle let us attach a coordinate system to the train. Consider the auxiliary inertial frame I which moves with velocity +V with respect to the ground and attach Minkowski coordinates t, x to it. The trajectory of any given point P of the train is given in it by the equation & n if t 0; x¼ ð2:12:1Þ n þ Ut if t ! 0; where n is a constant and U¼ À2V 1 þ V 2 =c2 ð2:12:2Þ

56

2 The Lorentz-Transformation

is equal to the velocity of the train on its way back from Whitewood with respect to I . It has been assumed that the direction of motion of the train has changed at t = 0. Each piece of the train has its own constant n which can, therefore, be taken as space coordinate of the coordinate system attached to the train. Bob will be assumed to be sitting at n = 0. The most natural choice for the coordinate time h of the attached system is to assume that h is equal to the proper time of the correctly synchronized clocks (virtual or real, including Bob’s watch) which are ﬁxed on the train and show zero at the moment of the velocity reversal: & tpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ if t 0; h¼ ð2:12:3Þ t 1 À U 2 =c2 if t ! 0: Then the relation between the coordinates t, x and h, n can be summarized as ( ( n if h 0 h if h 0 Uh h x ¼ n þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ if h ! 0 t ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ if h ! 0 ð2:12:4Þ 2 2 2 2
1ÀU =c 1ÀU =c

and & n¼ x x À Ut if t 0 if t ! 0 & h¼ tpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t 1 À U 2 =c2 if t 0 if t ! 0: ð2:12:5Þ

In these formulae U must be expressed through V using (2.12.2). If in ds2 ¼ c2 dt2 À dx2 we express the differentials by means of (2.12.4) through dh and dn we obtain the expression ( if h\0 c2 dh2 À dn2 2 ds ¼ ð2:12:6Þ 4V =c c2 dh2 þ 1ÀV 2 =c2 cdh Á dn À dn2 if h [ 0 where U has been expressed through V. The construction of the attached coordinate system has now been completed. The next task is to write down the trajectory of Alice in it. It is obvious that at t \ 0 (which is the same as h \ 0) this trajectory is n = - V(h ? he) where he is a constant. Since Bob is sitting at the point n = 0 their ﬁrst encounter takes place at the moment h = - he \ 0. At that moment Bob’s watch shows -he s. Since the train at h [ 0 moves with the same speed as before, the second encounter will take place when the reading on Bob’s watch is +he (i.e. TB ¼ 2he ). The trajectory of Alice is, therefore, determined by the equations4

4

The velocity of Alice at h [ 0 can also be obtained directly, using (2.12.5) and (2.12.2): dn dx À U Á dt ÀV À U ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ þV ; dh dt 1 À U 2 =c2 1 À U 2 =c2

since the velocity

dx dt

of Alice with respect to the auxiliary frame I is equal to -V all the time.

2.12

The Coordinate Time in Accelerating Frames: the Twin Paradox

57

& n¼

ÀV ðh þ he Þ þV ðh À he Þ

if h 0 if h ! 0:

ð2:12:7Þ

Now we can ﬁnd the functional form of the relation sA ¼ f2 ðhÞ by integration: s ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Zh qﬃﬃﬃﬃﬃﬃﬃ Zh Zh 1 1 ds 2 Á dh: ds2 ¼ ds ¼ sA ¼ c c dh
Àhe Àhe Àhe

Using (2.12.6) we obtain for the integrand the expression qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 s À Á2 1 À c12 dn if h\0 1 ds 2 < dh qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ : 1 þ 4V =c Á 1 dn À 1 ÀdnÁ2 if h [ 0: c dh
1ÀV 2 =c2 c dh c2 dh

ð2:12:8Þ

According to (2.12.7) dn ¼ ÇV in the ﬁrst and second line, respectively. The dh integrand is, therefore, constant in both domains of h which makes the integration trivial. After a little algebra we obtain 8 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ < 1 À V 2 =c2 Á ðh þ he Þ if h 0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sA ¼ f2 ðhÞ ¼ ð2:12:9Þ 1þV 2 =c2 : pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h þ 1 À V 2 =c2 Á he if h ! 0: 2 2
1ÀV =c

This is our ﬁnal formula for the second method of synchronization which replaces sA ¼ f1 ðhÞ ¼ ðTA =TB Þh of the ﬁrst method in which TA =TB ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1= 1 À V 2 =c2 : For this ratio (2.12.9) leads to this same expression since pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ TA ¼ f2 ðhe Þ ¼ 2he = 1 À V 2 =c2 and, as we have already seen, 2he ¼ TB : But the functions f1(h) and f2(h) differ from each other (see Fig. 2.7).
Fig. 2.7 Time dependence of the twin paradox

58

2 The Lorentz-Transformation

The puzzle, found in the naiv consideration of the second method, has been thereby solved. The mistake was committed in the treatment of the second half of the motion. It is true that the train is an inertial frame in this phase too but the coordinates attached to it, which are in conformity with the continuous operation of clocks ﬁxed on the train, are not Minkowski coordinates since for h [ 0 ds2 6¼ c2 dh2 À dn2 : Therefore, (2.11.1) is not valid in these coordinates. The change in the form of ds2 is the lasting effect of the instantaneous acceleration at h = 0. One of its consequences is that a free body which is originally at rest acquires at the moment h = 0 the velocity -U = 2V/(1 ? V2/c2) which, in the absence of friction, is a lasting effect. A less obvious persistent effect is desynchronization of correctly synchronized pairs of ideal clocks attached to the train. Assume that the light velocity measured by a pair of clocks ﬁxed on the train is originally the same in both directions along the coordinate n. This will no longer be the case at h [ 0. The light velocity dn can be obtaind from the condition dh ÀdnÁ 2 ds2 2 ¼ 0; using (2.12.6) divided by dh . For h \ 0 we obtain dh Æ ¼ Æc; but for dh h [ 0 the two solutions of the equation 2 dn 4V dn À À c2 ¼ 0 dh 1 À V 2 =c2 dh are of different magnitude in the positive and negative direction: dn V =c Æ ð1 þ V 2 =c2 Þ ¼c : dh Æ 1 À V 2 =c2 This difference can be ascribed only to clocks’ desynchronization since, in spite of the fact that in h [ 0 the form of ds2 is different from the Minkowskian form c2 dh2 À dn2 ; the train is, none the less, an inertial frame in this domain of time too. This can be formally proven by demonstrating that h, n can be replaced by a new attached coordinate system t0 , x0 in which ds2 ¼ c2 dt0 2 À dx0 2 holds true. The ﬁrst step is to replace h by t0 : ch ¼ ct0 À 2 Then we have after some algebra ds2 ¼ c2 dt0 À Rescaling n according to the relation n¼ 1 À V 2 =c2 0 x 1 þ V 2 =c2 ð2:12:10Þ
2

V =c n: 1 À V 2 =c2

2 1 þ V 2 =c2 dn2 : 1 À V 2 =c2

2.12

The Coordinate Time in Accelerating Frames: the Twin Paradox

59

leads to the desired Minkowski form of ds2. The crucial moment here is that objects with ﬁxed n will have ﬁxed x0 coordinate too. The primed coordinates are, therefore, indeed attached to the train. Consider two ﬁxed points at n and n þ Dn: Since at negative times h; n are Minkowski coordinates, the distance Dl between them is equal to jDnj: For the same reason, at positive times their distance is changed to Dl0 ¼ jDx0 j: Using (2.12.10) we then obtain Dl0 ¼ 1 þ V 2 =c2 Dl: 1 À V 2 =c2

This is another persistent consequence of the velocity reversal: increase of the distance between given points of the train. Therefore, the material composition of our ‘model train’ which serves to modellize the coordinate transformation (2.12.4), (2.12.5) must be such as to make the necessary deformation possible. Closing this section the main message of it is emphazised again: detailed behaviour of the twin paradox from the point of view of the accelerating twin depends on how the coordinate time is speciﬁed in his/her rest frame.

2.13 The Coordinate Time in Accelerating Frames: the Rotating Earth
Rotating Earth is an accelerating reference frame considerably more relevant than the train commuting between Blackwood and Whitewood. But consider ﬁrst the case of the rotating disc. The coordinate time on it may be deﬁned by the readings of (virtual) clocks rotating together with the disc. It may be assumed that, before the disc starts rotating, these clocks keep in synchronism with those (virtual) clocks that are at rest on the ground and show the Minkowski time in the inertial frame I attached to the latter. As the disc sets rotating time dilation causes the clocks attached to it to lose. This slowering of them can in principle be easily established since in the course of their rotation either of these clocks periodically passes by the clock on the ground in the neighbourhood of which it had rested before the rotation began. The time lag is the larger the farther the clock is located from the axis of rotation but this fact by no means forbids to base coordinate time on their readings. Spacetime coordinate system based on this deﬁnition of coordinate time and on a cylindrical coordinate system whose z-axis coincides with the axis of rotation will be denoted by K01 : As a matter of fact this coordinate system is never used in practice because calculations usually become much simpler if coordinate time is chosen instead in the following manner: at the point P of the rotating disc the coordinate time is deﬁned by the reading on that clock ﬁxed to the ground which P is just passing by. Coordinate system based on this notion of coordinate time will be called K02 :

60

2 The Lorentz-Transformation

The coordinate system attached to the ground is the unprimed K. Its spatial part is a cylindrical system of the same orientation as described above. Then transformation from K to K02 is given by the formulas t0 ¼ t; r0 ¼ r; u0 ¼ u À xt; z0 ¼ z: ð2:13:1Þ

Clocks attached to the disc do not show the coordinate time, they are losing more and more with respect to it. Much as unnatural this choice may seem it is usually the most convenient one. The coordinate time on the Earth should conform with the practice that the duration of the solar day must be the same at every point of the Earth i.e. it must be independent of the latitude. This requirement excludes the choice K01 since, owing to time dilation, the time elapsed between two successive returns of the Sun to the local meridian would be on smaller latitudes shorter than on higher ones. Since the speed of the Earth’s rotation is very slow with respect to light velocity this problem is not a practical one. Let us now suppose, however, that Earth rotates much faster than it actually does and, as a consequence, the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ value of 1 À V 2 =c2 on the Equator is considerably smaller than unity but for the sake of simpliﬁcity forget the revolution of the Earth around the Sun. Then the solar day which determines the rhythm of everyday life will coincide with the sidereal day equal to the period of rotation of the Earth with respect to the ﬁxed stars. By this assumption the case of the Earth becomes indistinguishable from that of the rotating disc provided I is identiﬁed with the inertial frame of the ﬁxed stars. The coordinate system K02 adapted to the Sun’s path as seen from the Earth is then deﬁned by the correctly synchronized virtual clocks, resting in that frame. But on the hypothetical Earth, rotating with considerable angular velocity, this natural choice of coordinate time would require certain developments in the construction of time keeping devices since on different latitudes they should be calibrated to different speeds. Clocks, therefore, should provide an option to set the precise value of the latitude at which they are to work. We are fortunately not obliged to deal with problems of this kind till time dilation due to the rotation of our real Earth can be considered negligibly small.

2.14 Lorentz Contraction Revisited
Lorentz contraction is illustrated on the spacetime diagram of Fig. 2.8. Consider two points ﬁxed in the inertial frame I 0 whose distance is equal to l0 (proper distance). In Minkowski coordinates their world lines are a pair of vertical lines at, say, x0 = 0 and x0 ¼ l0 : Lorentz contraction is the consequence of the geometrical fact that segments of different length are cut out by these world lines from the axes 2 x0 and x. The position of the hyperbola c2 t0 À x2 ¼ Àl2 on the ﬁgure shows that 0 0 the distance l cut out from the x axis is smaller than l0.

2.14

Lorentz Contraction Revisited

61

Fig. 2.8 To the derivation of the Lorentz contaction formula

Let us express the magnitude of the Lorentz contracted length between the points by means of Lorentz transformation as x0 À Vt0 l ¼ x ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 1 À V 2 =c2 where ðx0 ; ct0 Þ are the coordinates in K0 of the point of intersection of the world line x0 ¼ l0 with the x-axis. According to Sect. 2.6, the equation of the x-axis in K0 is ct0 ¼ V x0 : Hence ðx0 ; ct0 Þ ¼ ðl0 ; Vl0 =cÞ: Using these values in the above c formula we obtain qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x0 À Vt0 1 À V 2 =c2 l ¼ x ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ l0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ l0 1 À V 2 =c2 : 1 À V 2 =c2 1 À V 2 =c2 Earlier, in Sect. 1.5, this formula has already been derived from time dilation which is the consequence of the relativity of simultaneity. As Fig. 2.8 shows clearly, contraction owes its existence to the nonzero angle between the x-axes, belonging to different inertial frames. As we have explained in Sect. 2.6, this property of Minkowski diagrams is the manifestation of the absence of absolute simultaneity in relativity theory. The derivation of Lorentz contraction in the present section leads, therefore, to the same conclusion as the derivation in Sect. 1.5: contraction is the consequence of the relativity of simultaneity. When, instead of K0 ; the points are at rest in K then their distance becomes shorter with respect to K0 : This conclusion can also be easily drawn from Fig. 2.8, since the world lines of the points are now the ct axis and the tangent to the

62

2 The Lorentz-Transformation

2 hyperbola c2 t0 À x2 ¼ Àl2 at its point of intersection with the x-axis (see the 0 0 theorem proved in Sect. 2.8). The point of intersection of this tangent with the x0axis is obviously nearer to the origin than the proper distance l0 of the points.

2.15 Is the Perimeter of a Spinning Disc Contracted?
Before tackling this problem let us return to the Bell’s thought experiment with the rod transported by a pair of railcars (Sect. 1.5). There we arrived at the conclusion that Lorentz contraction may cause the rod to slip between the cars. But what would happen if the rod was tightly bolted to the cars? If the cars’ motion were still kept in rigorous synchronicity the length of the rod would obviously remain unchanged but only at the cost of being streched. As a result, the work done by the engines of the motor-cars would be increased by the amount of the elastic energy of the rod and their fuel consumption might, therefore, be raised considerably. We are now ready to examine the case of the disc. Let us ﬁrst assume that it rests in the inertial frame I in a plane perpendicular to the vertical z-axis. Divide its perimeter into n arcs of equal length by means of the set of points A1 ; A2 ; . . .; An ; mark the corresponding points on the ground as B1 ; B2 ; . . .; Bn and then make the disc to rotate around the z-axis with a uniform angular velocity. The question is whether its dimensions will show up modiﬁcations correponding to Lorentz contraction. Centrifugal force will tempt the radius to increase. But this is a ‘case speciﬁc’ phenomenon because the magnitude of the radial deformation is contingent on the peculiar elastic properties of the disc: under identical circumstances the deformation is the less the more rigid material the disc is composed of. Lorentz contraction, on the contrary, is independent of material composition and even ideally rigid discs are subjected to it. Moreover, since radial direction is perpendicular to the velocity of rotation, no Lorentz contraction is expected to occur along it. We can, therefore, safely assume that for an ideally rigid disc no deformations take place along the radius. Along the circumference, however, Lorentz contraction may be in operation. It should manifest itself in the incongruence of the point set A1 ; A2 ; . . .; An on the disc with the corresponding set B1 ; B2 ; . . .; Bn on the ground. But an incongruousness of this kind is obviously impossible unless axial symmetry of the disc is destroyed by contraction which would be equivalent to its destruction. If this does not happen, all points Ai must obviously pass by the corresponding Bis simultaneously which is the same as to say that neither segment of the disc is contracted as compared to its counterpart on the ground. Does not this conclusion plainly contradict Lorentz contraction of a single rod? The example of the rod transported by traincars reveals that contraction takes place only when the rod can change its size freely with respect to the cars. The rotating disc is, however, analogous to the case when the rod is ﬁxed to the cars since adjecent segments of the disc mutually prohibit each other’s deformation in tangential

2.15

Is the Perimeter of a Spinning Disc Contracted?

63

direction. Stresses may arise but as long as they do not lead to instability and destruction of axial symmetry they prevent Lorentz contraction to manifest itself. Let us place now measuring rods (rulers) along the rim of the disc at rest without fastening them to each other or to the disc. For a sufﬁciently large disc the length of its circumference is equal to the number of such rods provided they ﬁll it without gaps. When, however, the disc is made to rotate the rods suffer contraction, they will cease to ﬁll the perimeter densely and gaps between them arise. What is the message to be drawn from this fact? When a ruler is employed to measure the distance between a pair of points of a body it must be at rest with respect to the body. Therefore, the contraction of the measuring rods on the rotating disc does not prompt us to make any conclusion concerning its perimeter as seen from the frame in which it is rotating. As elucidated above, the circumference understood in this sense is independent of whether the disc is spinning or not. But the gaps between the rulers along its perimeter are an incontestable fact which proves unambiguously that with respect to the rest frame of the rotating disc, in which the rulers are resting, the length of the perimeter did increase. The gaps cannot be attributed to the change in the length of the rulers because no length etalon except ideally rigid measuring rods are assumed to exist. Since rulers along the radius do not contract, the radius of the rotating disc is of the same length as that of the disc at rest. Therefore, the ratio of the cirmumference to the radius of a rotating disc measured in its rest frame is larger than 2p.

2.16 Do Moving Bodies seem Shorter?
George Gamow, the famous russian-american physicist, told in his book Mr. Tompkins in Wonderland the story of a certain bank clerk Mr. Tompkins who had once dropped in a popular lecture on relativity theory but in the middle of it had fallen asleep and had found himself in a wonderland where light velocity had had the value of only 15 km/h.
The hands of the big clock on the tower down the street were pointing to ﬁve o’clock and the streets were nearly empty. A single cyclist was coming slowly down the street and, as he approached, Mr Tompkins’s eyes opened wide with astonishment. For the bicycle and the young man on it were unbelievably shortened in the direction of the motion, as if seen through a cylindrical lens. The clock on the tower struck ﬁve, and the cyclist, evidently in a hurry, stepped harder on the pedals. Mr Tompkins did not notice that he gained much in speed, but, as the result of his effort, he shortened still more and went down the street looking exactly like a picture cut out of cardboard. Then Mr Tompkins felt very proud because he could understand what was happening to the cyclist—it was simply the contraction of moving bodies, about which he had just heard.

This vivid description is based on the conviction that Lorentz contraction may be observed by seeing it. In books on relativity theory the statement is often found that a moving sphere is seen as an oblate ellipsoid ﬂattened along the direction of its motion. But it is by no means obvious that when a moving body is seen the

64

2 The Lorentz-Transformation

same property of it is observed which determines its length, i.e. the distance between its endpoints at a given moment of time in the inertial frame of the observer. By seeing we perceive the angle of sight rather than a length and the light fronts which reach our eyes at a given moment had generally emerged from the endpoints of the body at different moments of time. The distortion that a passing sphere would appear to undergo if it were travelling a signiﬁcant fraction of the speed of light, was described independently by James Terrell and Roger Penrose in 1959. They found that the sphere does not appear oblate but seems to undergo a peculiar rotation (Penrose–Terrell effect).

2.17 Velocity Addition Revisited
In Sect. 1.6 the law of velocity addition has been derived for the special case when the velocity V of the moving body is parallel to the relative velocity U of the inertial frames. Lorentz-transformation permits us to obtain the corresponding formula for the more general case in a single step. In standard setting the x-axis can always be chosen parallel to the relative velocity: ðUx ; Uy ; Uz Þ ¼ ðU; 0; 0Þ; but the velocity V must be allowed to be of any direction. Then Eqs. (2.4.7) written for coordinate and time differentials lead at once to the required formulae:
0 Vx ¼

dx0 dx À Udt Vx À U ¼ ; ¼ U dt0 dt À c2 dx 1 À Vx U=c2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dy0 dy 1 À U 2 =c2 Vy 1 À U 2 =c2 0 Vy ¼ 0 ¼ ; ¼ U 1 À Vx U=c2 dt dt À c2 dx pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dz0 dz 1 À U 2 =c2 Vz 1 À U 2 =c2 0 Vz ¼ 0 ¼ : ¼ U 1 À Vx U=c2 dt dt À c2 dx

ð2:17:1Þ

As we have seen in Sect. 1.6, in relativity theory the relative velocity of two bodies differs from the rate of change of the distance between them. This is the direct consequence of the relativistic notion of the coordinate time. The rate of change of the distance refers to the coordinate time of a common K for both bodies, while, in calculating relative velocity, the velocity U of one of the bodies is understood in terms of the coordinate time in some K, while the velocity V of the other is expressed in the coordinate time of the rest frame of the former.

2.18 Equation of Motion Revisited
In Sect. 1.7 the basic principles, underlying equations of motion of point particles in relativity theory, have been clariﬁed, but the equations themselves have been

2.18

Equation of Motion Revisited

65

derived only for the special case when the force is parallel to the velocity. Below the equations in the general case will be derived with the help of the Lorentztransformations. The key point of the derivation is the transformation rule of the acceleration which in I and I 0 is deﬁned by the formulae dvx dvy dvz ; ; a ¼ ðax ; ay ; az Þ ¼ dt dt dt 0 dvx dv0y dv0z a0 ¼ ða0x ; a0y ; a0z Þ ¼ ; 0; 0 ; dt0 dt dt where v and v0 are the particle’s velocity in I and I 0 respectively. Since dx = vxdt, the differentials dt and dt0 are related to each other by Lorentz-transformation as dt À ðV =c2 Þ dx 1 À vx V =c2 dt0 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dt 1 À V 2 =c2 1 À V 2 =c2 ð2:18:1Þ

(as usual, V is the relative velocity of I 0 with respect to I in a standard setting). According to the ﬁrst of the equations (2.17.1) v0x ¼ ðvx À V Þ=ð1 À vx V =c2 Þ; hence d vx À V 1 À V 2 =c2 dvx : dv0x ¼ dvx ¼ 2 dvx 1 À vx V =c ð1 À vx V =c2 Þ2 We then have dv0x ð1 À V 2 =c2 Þ3=2 dvx ; ¼ dt0 ð1 À vx V =c2 Þ3 dt and ﬁnally a0x ¼ ð1 À V 2 =c2 Þ3=2 ð1 À vx V =c2 Þ3 ax :

The differential of the second equation of (2.17.1) is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! o vy 1 À V 2 =c2 o vy 1 À V 2 =c2 0 dvy ¼ dvx þ dvy : 1 À vx V =c2 1 À vx V =c2 ovx ovy Having performed the differentiations we obtain dv0y qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ vy 1 À V 2 =c2 Á pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V 2 =c2 dvx þ dvy : 1 À vx V =c2 ð1 À vx V =c2 Þ2 ðÀV =c2 Þ

Again, dividing by (2.18.1), we have for the primed y-component of the acceleration the expression a0y ¼ À vy V 1 À V 2 =c2 1 À V 2 =c2 Á ax þ ay : c2 ð1 À vx V =c2 Þ3 ð1 À vx V =c2 Þ2

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2 The Lorentz-Transformation

Assume that the primed frame is just the instantaneous rest frame of the particle and the common x direction of I and I 0 is parallel to v at the moment chosen. Then V ¼ vx v; and we have ð1 À v2 =c2 Þ3=2 ay : a0y ¼ 1 À v2 =c2 For the z-component we obtain similarly a0z ¼ az : 1 À v2 =c2 a0x ¼ ax : and vy ¼ vz ¼ 0 ð2:18:2Þ

In Sect. 1.7 we pointed out that in the rest frame which is now identical to the primed one equations of motion must have their original Newtonian form ma0 = F0 . If in this equation the primed components are expressed through the unprimed ones we obtain the equation of motion valid in the unprimed frame where the point mass at the given moment of time is moving in x-direction5: m 0 ax ¼ F x ; ð1 À v2 =c2 Þ3=2 m 0 ay ¼ F y ; 1 À v2 =c2 m 0 az ¼ F z : 1 À v2 =c2

ð2:18:3Þ

The primed components of the force must also be expressed through the corresponding unprimed components but this can only be done after having speciﬁed its mathematical form (see Sect. 2.21).

2.19 The Energy–Momentum Four Vector
Four-component quantities whose primed components are expressed through the unprimed ones via Lorentz-transformations are called four-vectors. The coordinates ðct; x; y; zÞ and the coordinate differentials in Minkowski coordinates constitute a four-vector. In the expressions (1.7.9) and (1.10.1) of the momentum and energy the coordinate time can be replaced, using (2.10.4), with the proper time:
5

In Sect. 2.21 we will dispose of this limitation.

2.19

The Energy–Momentum Four Vector

67

m dx dy dz dx dy dz ; ; ; ; p ¼ðpx ; py ; pz Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼m ds ds ds 1 À v2 =c2 dt dt dt mc2 dðctÞ : E ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ mc 2 =c2 ds 1Àv

ð2:19:1Þ

Under Lorentz-transformations ðdðctÞ; dx; dy; dzÞ behave as components of a four-vector while m and ds remain unchanged. Therefore, ðE=c; px ; py ; pz Þ is also a four-vector, called the energy–momentum four vector or four-momentum, whose primed and unprimed components are related to each other by Lorentztransformation: 9 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E0 ¼ pEÀVpx 2 > 2 > 1ÀV =c > = V px À 2 E 0 c ð2:19:2Þ px ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > 1ÀV 2 =c2 > > ; p0y ¼ py ; p0z ¼ pz : It has to be stressed that the kinetic energy K of a point mass is not a component of any four-vector. The knowledge of the mass–energy relation is, therefore, crucial for being able to form a four-vector which contains the momentum of the particle as three of its components. To deﬁne this four-vector without the preliminary clariﬁcation of the relation between the rest energy and the mass of the particle would be a mere hindsight without physical motivation. For the momentum four-vector the invariant, corresponding to (2.5.2), is the quadratic expression E2 =c2 À p2 À p2 À p2 : Its value, which is the same in any x y z inertial frame, is easily found in the rest frame where p = 0 and E = E0 = mc2: E2 E2 À p2 À p2 À p2 2 À p2 ¼ m2 c2 : x y z 2 c c ð2:19:3Þ

The validity of this relation can be veriﬁed also by substituting (2.19.1) into its left hand side and making use of (2.5.2). The mass is, therefore, invariant under Lorentz-transformation. Solving this equation with respect to the energy we obtain the formula qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2:19:4Þ E ¼ c p2 þ m 2 c 2 which corresponds to the Newtonian expression K = p2/2m but includes the rest energy too. The relativistic equivalent of the Newtonian relation v = p/m is arrived at by dividing (1.7.9) by (1.10.1): v¼ c2 p: E ð2:19:5Þ

The most signiﬁcant property of the energy and the momentum is their conservation in isolated systems and their mathematical expression should have been derived from this requirement. This could be done within the Lagrangian approach

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2 The Lorentz-Transformation

which would conﬁrm the correctness of (1.7.9) and (1.10.1). The use of them will ! now be illustrated on the example of the decay Po210 À Pb206 þ a (cf. Sect. 1.11). The problem is to calculate the velocity of the a-particle from the known masses of the three particles. Let the masses of Po210 ; Pb206 and the a-particle be equal to M, m and l, respectively. When the decaying polonium nucleus is at rest, the momenta of the lead nucleus and the a-particle will be of the same magnitude p and opposite direction. Then the conservation of energy is expressed by the equation Mc2 ¼ c qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p 2 þ m 2 c 2 þ c p 2 þ l2 c 2 ;

from which p can be determined. The velocity of the a-particle can then be qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ calculated by means of the formula v ¼ c2 p=c p2 þ l2 c2 : Since in beta-decay the number of the ﬁnal particles is three, energy and momentum conservation are not sufﬁcient to ﬁx their velocity unambiguously. Hence beta-electrons have, unlike a-particles in the above decay, continuous velocity distribution within a range determined by the conservation laws.

2.20 Massless Particles
If in (2.19.3) we put the mass equal to zero and use (2.19.5) too we arrive at the equations E ¼ cp; v ¼ cn; ð2:20:1Þ

where n = p/p. Particles whose energy and momentum obey these relations are permanently in state of motion with the speed of light with respect to any inertial frame. In this respect they differ signiﬁcantly from particles with mass however small but different from zero. This contrast is closely related, through (2.19.3), to the sharp distinction between timelike and lightlike spacetime intervals. Since particles of this kind are never at rest it is meaningless to speak of their rest frames. But, according to Sect. 1.7, mass is the measure of inertia of a particle at rest. Therefore, m = 0 particles are genuinely massless in the sense that the notion of mass is inapplicable to them (while the term ‘zero mass particles’ would suggest that they possess mass equal to zero). As an obvious corrolary: mass–energy relation is inapplicable to massless particles. How the energy of a massless particle is altered when it is observed from an inertial frame, moving toward it? Consider the standard setting of inertial frames I and I 0 , and assume that a massless particle of energy E moves in the negative direction of the x-axis of I . Its momentum is then equal to p = (- E/c, 0, 0). According to Lorentz-transformations, in I 0 its energy E0 is larger than E:

2.20

Massless Particles

69

EþVE E À Vpx c E ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1 À V 2 =c2 1 À V 2 =c2
0

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ V =c E: 1 À V =c

ð2:20:2Þ

This increase in energy cannot be attributed to the change in the velocity since the latter is equal to the same c in both I and I 0 . A massless particle must, therefore, posses a property, other than velocity, which determines its energy and does change when going from one inertial frame to the other. For an explanation return to the discussion of the Doppler-effect in Sect. 1.2. When (1.2.7) is applied to the receiver which is approaching the emitter [in this case V has to be replaced by (- V)], the ratio m0 /m is found identical to E0 /E obtained from (2.20.2)! The conclusion from this coincidence is that the ratio of the energy of a massless particle to the frequency of a light wave is independent of the inertial frame chosen: E/m = konstans. If, therefore, frequency was among the attributes of massless particles then it might be the required property we are looking for. This demand is indeed fulﬁlled in quantum theory but in the reverse direction, since light waves are endowed with particle properties: photons are massless particles associated with light waves. The value of the energy-to-frequency ratio is the Planck-constant h. It follows from the ﬁrst equation of (2.20.1) that, beside energy, photons posses momentum too: p = E/c = hm /c = h/k.

2.21 The Transformation of the Electromagnetic Field
According to the widespread belief Einstein discovered relativity theory in the course of his analysis of the negative result of the Michelson–Morley experiment. But this is a mere urban legend which is refuted by both the text of Einstein’s original paper On The Electrodynamics of Moving Bodies and his later recollections. In the ﬁrst paragraph of this paper Einstein stated the motivation of his study with his usual clarity:
It is known that Maxwell’s electrodynamics—as usually understood at the present time— when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric ﬁeld with a certain deﬁnite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric ﬁeld arises in the neighbourhood of the magnet. In the conductor, however, we ﬁnd an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

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The embarassing fact here was that, in spite of the sharp difference between the underlying physical pictures, the calculation based on the Maxwell equations lead precisely to the same force in both cases, proving that the asymmetry is indeed not ‘inherent in the phenomena’. But this seemed only an accidental coincidence rather than a necessary consequence of the equivalence of the rest frames of the magnet and the conductor since Maxwell-equations are not invariant with respect to the Galilei-transformations. Einstein’s primary aim in his paper was, therefore, to replace Galilei-transformations with a generalization of them which ensure the same mathematical form of the Maxwell-equations in all inertial frames and reduce to Galilei-transformations in those domains of experience where these latter transformations have been proved to work well. Since the value of the light speed may be deduced from the Maxwell-equations, Einstein perhaps started from the reasonable expectation that, if such transformations do indeed exist, they should lead automatically to constancy of the light speed and, moreover, they could possibly be found as the consequence of this constancy. This is at least as good an argument in favour of the constancy of light velocity as the negative result of the Michelson–Morley experiment. Maxwell-equations and their invariance will not be discussed here. But it has to be noted that in his attempts to ﬁnd the correct transformations of these equations one of Einstein’s difﬁcult task was to ﬁnd the transformation of the electric and magnetic ﬁeld from one inertial frame to the other. For a standard setting of the coordinate systems these transformations written in the International System of Units (SI) are as follows: E0x ¼ Ex ; B0x ¼ Bx ; Ey À VBz E0y ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 1 À V 2 =c2
V By þ c2 Ez B0y ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 1 À V 2 =c2

Ez þ VBy E0z ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V 2 =c2
V Bz À c2 Ey B0z ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 1 À V 2 =c2

ð2:21:1Þ

ð2:21:2Þ

In Sect. 2.18 we have left open the problem of how to express the primed components of the force in terms of the unprimed ones, valid in the inertial frame in which the particle is moving. For a point charge the answer can now be given. Since a charge at rest is acted upon solely by the Coulomb-force, in (2.18.3) we can specify the force as F0 = QE0 . Then, using (2.21.1), we obtain the following equation of motion for a point charge of magnitude Q and mass m: m ax ¼ QEx ; ð1 À v2 =c2 Þ3=2 m Ey À vBz ay ¼ Q pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À v2 =c2 1 À v2 =c2 m Ez þ vBy az ¼ Q pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 1 À v2 =c2 1 À v2 =c2

ð2:21:3Þ

2.21

The Transformation of the Electromagnetic Field

71

These equations are often cast into the form mk ax ¼ QEx ; m? ay ¼ QðEy À vBz Þ m? az ¼ QðEz þ vBy Þ; where mk ¼ m ð1 À v2 =c2 Þ3=2 and m m? ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À v2 =c2

are the longitudinal and transverse masses, respectively. Equations (2.21.3) can be written as a single vector equation which is applicable when the force is of arbitrary direction: ! m 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a ¼ Q E À 2 ðE Á vÞv þ ðv Â BÞ ð2:21:4Þ c 1 À v2 =c2 (the Á and the 9 denote scalar and vector product). The validity of this equation follows from the fact that for v ¼ ðv; 0; 0Þ it reduces to (2.21.3). When v2 =c2 ( 1 the force on the right hand side becomes equal to the sum of the Coulomb and Lorentz forces.

2.22 The Thomas-Precession
The amazing property of the gyroscope is that if no external torque is acted upon it the orientation of its spin axis remains ﬁxed, regardless of the motion of the platform on which it is mounted. If, for example, the gyroscope frame is pinned on a disc at the distance r from its axis and its spinning axis is directed toward the point P at the wall of the laboratory then it will continue to point at P even after the disc is made to rotate. Thanks to this property gyroscopes became indispensable part of navigation systems. When e.g. the axis of the gyroscope of an airplane navigation system is seen to decline 5° to the right around the vertical axis than the cockpit crew knows that the aircraft has made a 5° turn to the left. But this behaviour of the gyroscope is true only in Newtonian physics. According to relativity theory gyroscopes, moving on curvilinear trajectories, do not preserve the orientation of their spin axis. For example if mounted on a rotating disc their spin direction will precess with respect to the laboratory around the rotation axis of the disc. The angular velocity of this motion known as Thomasprecession is given by the formula " # 1 xT ¼ x½1 À cðvÞ ¼ x 1 À pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ð2:22:1Þ 1 À v2 =c2

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2 The Lorentz-Transformation

Since xT is obviously negative the sense of Thomas-precession is opposite to that of the rotation of the support. For the gyroscope mounted on the rotating disc we have v = rx but (2.22.1) remains valid in the general case too. If the acceleration of the carrier of the gyroscope is not collinear to its velocity, i.e. when it rotates with some instantaneous angular velocity x in the plane S which contains this pair of vectors then Thomas-precession will take place in S with the angular velocity given by (2.22.1).6 If aircrafts could ﬂy with speed comparable to light velocity then Thomas-precession would render navigation by means of gyroscopes less straightforward since the true deviation in the course of the aircraft would be less then indicated by the deﬂection of the gyroscope with respect to the cockpit.

2.23 The Sagnac Effect
In this section another phenomenon on a rotating disc will be discussed. Imagine a source of light placed at the point A1 on the disc’s circumference wich is capable to emit simultaneously a pair of light pulses along the tangent both in the direction of the rotation velocity and opposite to it. A series of mirrors reﬂects the signals to keep their track along the perimeter. The point on the disc opposite to A1 will be denoted by A2 and a pair of diametrically opposite points B1 and B2 will also be marked on the ground. Suppose that the light ﬂash at A1 takes place at the moment when A1 passes by B1. The two signals emitted in diametrically opposite directions will meet again at B2 after having travelled a distance p R along the circumference (R is the radius of the disc). In the laboratory frame, wich is an inertial one by assumption, the time required to cover this distance is equal to Dt ¼ pR=c: The point of encounter on the disc will obviously have an angular displacement with respect to A2 equal to xDt where x is the angular velocity of rotation. Therefore, the pulse wich propagated in the counter rotation direction travelled only a distance ðp À xDtÞR with respect to the disc. Accordingly, the distance covered by the pulse of opposite direction is equal to ðp þ xDtÞR: The path difference is, therefore, equal to 2Dt Â xR ¼ 2Sx=c where S = p R2 is the area enclosed by the paths of the signals. As we see, the path difference is proportional to the instantaneous angular velocity of the device with respect to the inertial frames. In optical gyroscopes this path difference is monitored continuously in order to keep track of the orientation of the equipment in space. Instruments of this kind are as useful for purposes of navigation as their mechanical predecessors.

À Á In the general case the angular velocity vector is equal to x ¼ v12 v Â dv : When this is dt substituted into (2.22.1) we obtain for the Thomas-precession angular velocity vector the À Á À Á c expression xT ¼ ð1 À cÞ v12 v Â dv ¼ À 1þc c12 v Â dv : dt ds
6

2.23

The Sagnac Effect

73

The use of moving mirrors in the above description may raise doubt on the validity of the formula Dt ¼ pR=c since the velocity of the light pulses might perhaps be inﬂuenced by the motion of the mirrors which reﬂect it. Such an inﬂuence is, however, excluded by the basic principle of relativity theory, according to which the velocity of light in inertial frames is equal to c irrespective of its source. As a matter of fact, in gyroscopes available in commerce light pulses are replaced by monochromatic laser beams of wavelength k and mirrors by ﬁber optics. The path difference can then be converted into a phase shift Du ¼ ð2p=kÞ Â ð2Sx=cÞ ð2:23:1Þ

which can be measured by interferometric methods. The phase shift itself is known as the Sagnac effect. Both mechanical and optical gyroscopes are suitable for the laboratory demonstration of the Earth’s rotation. This fact alone is sufﬁcient to assert that on the surface of the Earth light velocity is not isotropic.

Chapter 3

General Relativity

Abstract The idea of geometrization of the gravity is elucidated. The place of the inertial frames in the theory is explained. GP-B experiment, light deﬂection, perihelion precession and gravitational red shift are discussed. Keywords Weight

Á Mass Á Gravity Á Geodetic Á Local frames

3.1 Gravitational and Inertial Mass
The mass on the left hand side of the Newtonian equation of motion ma ¼ F and the masses in the expression FG ¼ GmM=r2 of the inverse-square law of universal gravitation share a physical dimension but their physical meanings are in a sense opposite to each other. The mass which multiplies acceleration is the measure of the body’s resistance to any type of accelerating inﬂuence (including gravity). It is called the inertial mass of the body and will be denoted simply by m. The masses participating in the gravitational force formula express, on the contrary, the willingness of either body to subject itself to the accelerating power of the other in this special (gravitational) type of interaction. Masses of this kind are known as gravitational masses and will be distinguished by an asterisk from their inertial counterparts: FG ¼ GmÃ M Ã =r2 . The inertial mass of a body can be measured by e.g. comparing the centrifugal force acting on it on a rotating disc with the centrifugal force operating on the international kilogram prototype under similar conditions. The essential point is that since in such a procedure the weight of the bodies plays no role it is the inertial mass which is measured. Elastic collision can also be employed to compare the inertial mass of a body with that of the prototype. On the contrary, any measurement of the gravitational mass must utilize the weight of the object. Therefore, the gravitational mass of a body is equal to the gravitational mass of the international kilogram prototype if under identical conditions they
P. Hraskó, Basic Relativity, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-17810-8_3, Ó Péter Hraskó 2011 75

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3 General Relativity

produce identical stretches of a spring-balance. From these deﬁnitions it is clear that both the inertial and gravitational masses of the prototype are by deﬁnition equal to 1 kg. Moreover, any body made of the same material as the prototype has equal inertial and gravitational mass (but not necessarily equal to 1 kg). But in Newtonian physics no compelling reason is found in favour of the equality of the two types of mass of a body whose material composition is different from that of the prototype. Experience, however, is strongly in favour of the equality of the inertial and gravitational masses for all existing bodies. Experiments designed to determine the ratio mÃ =m (experiments of Loránd Eötvös using a torsion balance invented by him and their modern versions) prove this equality with extremely high precision but the equality mÃ ¼ m ﬁnds its natural place only in general relativity theory. Violation of this equality would lead to important consequences some of which will now be brieﬂy surveyed. (1) The equation of motion for a free-falling body under normal earth-bound conditions would take the form ma ¼ ÀmÃ GM Ã ¼ ÀmÃ g R2

(g is the gravitational acceleration of the prototype and R is the radius of the Earth). Hence, gravitational acceleration a¼ mÃ g m

would no longer be the same for all bodies. (2) The formula for the angular frequency of the mathematical pendulum would be of the form rﬃﬃﬃﬃﬃﬃﬃﬃﬃ mÃ g x¼ m l and, therefore, its period would depend, beside its length l, on the material composition of the swinging body also (before Eötvös this method was employed to measure the ratio mÃ =m). (3) If mÃ =m was different from 1 no weightlessness would be experienced in freely orbiting spaceships. A satellite, performing free rotationless motion on a circular orbit, is not an inertial frame because of the centripetal acceleration ac , acting within. Objects of inertial mass m experience, therefore, an outward directed radial inertial force mac in it. But these same objects are subjected to the gravitational attraction GmÃ M Ã =r2 of the Earth too. When the inertial and gravitational masses of all bodies (including the spacecraft capsule itself) are exactly equal, the two forces acting on them compensate each other since the radius of revolution is deﬁned by the equation ac ¼ GM=r2 common to all bodies belonging to the spacecraft. If, however, the ratio of mÃ to m had diverse magnitude depending on the material composition then bodies would be acted upon by forces of various strengths directed either toward or opposite the Earth.

3.1 Gravitational and Inertial Mass

77

The circular shape of the orbit is by no means a necessary condition of apparent weihtlessness. This phenomenon should be observed in spacecrafts orbiting freely (i.e. with engines switched off) on trajectories of any shape. This is the direct consequence of the fact that, when m ¼ mÃ , under these circumstances mass drops out of the Newtonian equations of motion.

3.2 The Equivalence Principle
If the equality m ¼ mÃ is indeed generally true then, as we have just noticed, the mass of the body, moving freely or under the action of gravity alone, is absent from the Newtonian equation of motion. For free motion this equation is simply m€ ¼ 0 and the mass can obviously be omitted. When gravitation is also present, r the zero on the right hand side must be replaced by the expression of the gravitational force proportional to the gravitational mass mÃ of the moving body and the masses are not cancelled out. But in the case of their equality cancellation becomes possible and again no parameter remains in the equation which would be characteristic of the moving body. From this point of view, therefore, gravitational motion is similar to free motion. But world lines in the latter case (the solutions of the equation € ¼ 0) are r straight lines which are pure geometrical objects. This almost banal consequence of the equation ma ¼ F prompted Einstein to pose the following surprizing question1: is it not possible that the world lines of freely gravitating bodies are, from a suitable point of view, also purely geometric objects i.e. ‘straight lines’? Is it conceivable that ﬂying tennis-balls, the Moon, the planets and artiﬁcial satellites are all moving on ‘straight lines’? Einstein gave afﬁrmative answer to this perplexing question which he based on the assumption that domains of spacetime where gravitation is present are curved four dimensional manifolds and freely moving bodies follow the ‘straightest path’ compatible with their curvature. Since for human perception even three dimensional curved space presents an insurmountable obstacle, the elucidation of the nature of curved manifolds must be based on the example of curved two-dimensional surfaces. ‘Straight lines’ do indeed exist on them since between pairs of not very distant points there is always found a unique shortest path called geodesic.2 On a sphere, for example, the geodesics are the main circles. Though sphere is a surface of very regular shape, even its geodesics posses properties uncommon with ordinary straight lines on a plane: they cross twice each other. The mathematically elaborated form of the theory permits in real spacetime geodesics as diverse as belonging to a tennis-ball or a planet.
1

That was of course not as simple as it seems. It took Einstein several years of painstaking efforts to cast the problem of gravitation into this geometrical form. 2 Straight lines on curved manifolds are called geodesics. The term ‘straight line’ is reserved to the geodesics on a plane (or on multidimensional analogues of the plane).

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According to general theory of relativity, therefore, no such physical entity as ‘gravitational force ﬁeld’ exists. Gravitational phenomena are rather manifestations of a hitherto unsuspected geometrical property of spacetime: its deformation around massive heavenly bodies.3 The world lines of bodies (e.g. planets) moving freely around them are the geodesics in this deformed spacetime (geodesic hypothesis). This is a free inertial motion in the same sense as the motion along the straight world lines in ﬂat (free from gravity) spacetime of special relativity. There is, however, a problem here. At the end of the previous section we have explained that the weightlessness in a freely orbiting spacecraft is only apparent, caused by the compensation of the weight of bodies in it by the outward directed inertial force mac . But if gravitational attraction was indeed absent then, contrary to experience, this inertial force would remain unbalanced. The solution of general relativity to this puzzle is that objects belonging to spacecrafts in free motion are not in fact exposed to the action of either gravitation (weight) or inertial forces because such spacecrafts are true inertial frames. This is a perfectly logical corollary to the geodesic hypothesis since in Newtonian physics too objects in free motion are at the same time inertial frames. Therefore, the conclusion drawn by Einstein from the equality m ¼ mÃ is in fact the generalization of the two basic Newtonian properties of free motion to gravitation: the trajectories are rectilinear (geodesics) in both cases and reference frames, following them (without rotation), are inertial frames. This last principle is known as the equivalence principle. These ideas were transformed from a program into a theory when, after eight years of painful reﬂexion, Einstein discovered that system of nonlinear partial differential equations which describe the deformation of spacetime around celestial bodies such as the Sun (Einstein-equations). By about this time mathematicians had already clariﬁed how to calculate geodesics in manifolds of given deformation. It then became possible to calculate the orbits of planets within the framework of the Einstein’s concepts, without resorting to the Newtonian law of universal gravity. This totally new approach conﬁrmed the correctness of Keplerian orbits but only as accurate ﬁrst approximations. The corrections to them are small but fully accessible to observation. Their veriﬁcation proved unequivocally the superiority of general relativity over the Newtonian law of universal gravity.

3.3 The Meaning of the Relation m* 5 m
In the Newtonian framework this equality means that every body possesses both an inertial and a gravitational mass and, according to the experience, they are always

3

The source of the spacetime curvature around these bodies is in fact not their mass but the more complex ten-component quantity known as energy-momentum tensor. In the case of a star at rest its main component is star’s rest energy which is proportional to its mass.

3.3 The Meaning of the Relation m* = m

79

equal to each other to a high degree of accuracy. From the point of view of general relativity, however, the correct interpretation of this empirical equality is that objects posses only inertial mass m which automatically appears wherever the Newtonian concept of gravity requires the presence of the gravitational mass mÃ . As it has already been stressed in the previous section, world lines of the free gravitational motion are geodesics in the spacetime deformed due to the inﬂuence of large bodies. Geodesics (the analogues of straight lines in Cartesian space) are purely geometric objects and so their equations (the geodesic equation) contain no physical parameter which would characterize the bodies moving on them. Around the Sun, for example, in the domain of the planetary orbits the geodesic equation turns out to have in a very good approximation the form d2 r GM ¼ À 3 r: 2 dt r ð3:3:1Þ

This equation allows us, after proper speciﬁcation of the initial conditions, to determine the motion of point masses (planets), i.e. to calculate the function r ¼ rðtÞ. The appearance of the gravitational constant G and the mass M of the Sun in this equation is the consequence of the fact that the Einstein-equation for the spacetime domain around the Sun contains these parameters and, therefore, the deformation of spacetime also depends on them. Since the geodesic equation (3.3.1) is speciﬁc to this domain these parameters appear necessarily in it. This equation would be identical to the Newtonian equation of motion of a test body in the gravitational ﬁeld of the Sun if its left and right hand sides contained the body’s inertial and gravitational mass m and mÃ respectively. But if the latters were of different magnitude this equation of motion would not be the consequence of (3.3.1) since the validity of an equation is destroyed when its sides are multiplied by different numbers. Since the acceleration on the left hand side is multiplied always by the inertial mass m independently of the nature of the accelerating force, the right hand side should also be multiplied by m. This is how in general relativity the place of the gravitational mass mÃ is automatically taken up by the inertial mass m.

3.4 Locality of the Inertial Frames
In Newtonian physics and special relativity the spatial extension of the inertial frames may in principle be arbitrarily large but actually they are real physical objects of ﬁnite dimensions. The coordinate system, however, attached to them can in principle be extended up to inﬁnity. The spatial part of Minkowski coordinates is a Cartesian system whose axes are inﬁnitely long in both directions. In this sense in Newtonian physics and special relativity the inertial frames are global. But from what had so far been said about general relativity it must be clear that the inertial frames in a deformed spacetime should be local. A freely gravitating

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spacecraft in rotationless motion can be taken to a good approximation for an inertial frame only if its dimensions are of limited magnitude. Mental extension of local inertial frames to global ones is an illegitimate intellectual operation. The satellites of the Earth or the planets of the Sun, the stone which is thrown up perform free inertial motion. If to think of an inertial frame as large as the Solar System was a permissible abstraction then all these objects could be related to it and should perform uniform rectilinear motion. If general relativity theory is correct, nothing in Nature corresponds to the notion of such global inertial frames. As a matter of fact, the inertial frames, occurring in Newtonian physics, are of very limited extension with only one exception which is, however, of paramount importance: it is the inertial frame implicit in the Newtonian theory of planetary motion. The theory’s basic equation is (3.3.1) on whose right hand side no inertial force is present. Hence, the equation is only valid in an inertial frame whether or not this is stated explicitly. Therefore, Newtonian theory of gravitation assumes that an inertial frame however large is a meaningful concept. On the contrary, in general relativity planetary orbits are calculated without such an assumption. We have outlined above the steps of this calculation which requires only the choice of a coordinate system adapted to the spherical symmetry and stationarity of the Sun without assuming the existence of a reference frame of any kind. The local inertial frames of general relativity posses all the genuine properties of the inertial frames as discussed in Sect. 1.1. In particular, it is true with respect to them (and only them) that light speed is of equal magnitude c in any direction. Owing to the lack of global inertial frames, no general statement can be formulated concerning the propagation of light in the interstellar space. It must be calculated in each particular case in essentially the same manner as calculations of planetary orbits are performed. Therefore, in the deformed spacetime of general relativity it is impossible to synchronize distant clocks by means of light signals for the same reason as in the accelerating reference frames in the ﬂat spacetime of special relativity. As a consequence, no global Minkowski coordinates are compatible with such spacetimes (just as no Cartesian coordinates exist on a sphere). As in special relativity, distant simultaneity remains contingent on the choice of the coordinate system. Judged by the equality of their coordinate times, with an event here now a whole segment of events can be simultaneous there in a distant point. This is the same ‘interval of simultaneity’ which we have already met with in the example of the Mars rover (Sect. 1.3), but when gravity is taken into consideration, the deformation of spacetime makes to apprehend its structure more difﬁcult. In Sect. 1.1 we have already hinted at the curious property of inertial frames in general relativity that their relative motion is different from being uniform rectilinear. We can now see with more clarity the content of this statement and that one of the main tasks of general relativity is to clarify how local inertial frames are moving. General relativity is, in a sense, the general theory of the inertial frames.

3.5 The Weight

81

3.5 The Weight
According to general relativity gravitation is not a force but the curvature of spacetime. Where then the weight of bodies comes from? When standing on the ground, we are not at rest in any inertial frame. The latters are freely falling objects and, therefore, we would only be at rest with respect to one of them if we were standing in a freely falling lift (Einstein lift)— but then we wouldn’t experience the weight of a body held in our hands. When we are standing ﬁrmly on the ground our rest frame is accelerating upward with respect to our instantaneous rest frame of inertia with an acceleration of g ¼ 9:81 m=s2 and, consequently, bodies of mass m in our hands experience a downward directed inertial force of magnitude mg. This force is called weight. The name of the equivalence principle is rooted in this interpretation of the weight. It is meant to express that ‘weight is equivalent to the inertial force’.

3.6 The GP-B Experiment
Thanks to the GP-B (Gravity Probe B) experiment of NASA we are now in possession of an important experimental evidence to the effect that an inertial frame of the dimensions of the Earth is indeed an empty abstraction. In this experiment the orientation of the spin axes of four rotating balls, serving as ﬂywheels of gyroscopes ﬁxed in a satellite, were followed up during a year. Owing to cardan suspension applied in gyroscopes, the behaviour of the balls from the point of view of the orientation of their spin axis is the same as if they were ﬂoating in the satellite and revolving freely round the Earth. Consider, therefore, a rotating ball which is orbiting the Earth. It will be a good approximation to assume that this system is isolated from the rest of the world. Consecutive positions of the ball are seen on Fig. 3.1. It moves on a circle which passes above the geographical poles as indeed was the case in the GP-B experiment. If at the initial moment of time the spin axis of the ball is parallel to the axis of rotation of the Earth then, according to Newtonian mechanics, both axes must remain parallel to each other in later times too. This becomes evident if the motion is referred to an inertial frame in which both the magnitude and direction of the angular momentum of a body remain constant if no torque is acting on it. The only possible origin of a torque applied to the revolving ball is Earth’s gravitational attraction. But if the ball is of precisely spherical shape then this torque is equal to zero and the ball’s spin axis remains indeed pointing continuously in the same direction. But what if experiment contradicts this expectation and a slow precession of the spin axis is observed? If experimental errors can be safely excluded the only possibility is to admit that it is not permissible to view the motion in its relation to an inertial frame because nothing in Nature presumably corresponds to this concept.

82 Fig. 3.1 Motion of the gyroscope in Newtonian gravity

3 General Relativity

The conclusion drawn from GP-B experiment is that the spin axis of the rotating ball does not remain parallel to itself but performs indeed very slow rotation (precession) in the plane of the orbit in positive direction (i.e. in the same sense as it revolves around the Earth). But even this minute deviation from the Newtonian prediction presents strong direct evidence against global inertial frames. Notice that this conclusion is based solely on the notion of the inertial frames without resort to the special content of either Newton’s law of universal gravitation or Einstein’s general relativity. In the latter’s framework the rotation of the spin axis of freely orbiting rotating bodies has long been predicted under the name of geodesic precession whose angular velocity is given by the formula rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 3GM xG ¼ þx 1 À 1 À 2 ; c r

ð3:6:1Þ

where x and r are the angular velocity and radius of the orbit and M is the mass of the Earth. Applied to the GP-B experiment this formula gives for the angular velocity of the geodesic precession the value 6 seconds in a year which has been veriﬁed to an accuracy of about 1%. The sense of the observed precession was positive as required by (3.6.1). The spin axis should, however, be conﬁned to the orbital plane only if the angular momentum of the Earth is neglected. Formula (3.6.1) refers to this approximation. Under the inﬂuence of the angular momentum of the central body the spin axis departs the orbital plane but the angular velocity of this motion called

3.6 The GP-B Experiment

83

drag4 is 170 times slower then that of the geodesic precession. For the separation of the drag from geodesic precession the most convenient orbit is that whose plane contains the angular momentum vector of the Earth. The polar orbit in the GP-B experiment was selected to meet this requirement. Since, however, the accuracy of the measurement is estimated about 1% no conclusion can be drawn from it about the magnitude of the drag.

3.7 Light Deﬂection
An unexpected direct consequence of the principle of equivalence is that a light beam should be deﬂected under the inﬂuence of gravitation. If equivalence principle is true then a freely falling Einstein lift is a local inertial frame with respect to which light propagates along straight line. Consider a light ray which is horizontal with respect to the lift. Since the latter is accelerating toward the center of the Earth, the ray under consideration will be deﬂected downward with respect to an observer standing on the ground. This deﬂection is the consequence of the gravitational acceleration g of the lift, therefore, light deﬂection must be attributed to the same origin: the gravitation exerted by the Earth. This conclusion from the equivalence principle was made by Einstein as early as in 1907 in a paper where outlines of general relativity were ﬁrst envisaged. But only eight years later was he able to calculate the angle of deﬂection in the vicinity of the Sun. It turned out equal to 1.6 s for the ray grazing the Sun and even smaller for rays farther apart from it. This angle can be measured by observing the radio waves of radio stars when they disappear behind the Sun and reappear again. Observations of this type verify the theory’s prediction with an accuracy of about 1%. According to theory the effect of spacetime deformation on light rays around Sun is mathematically equivalent to the effect of a condenser with varying in space refractive index n ¼ 1 þ 2GM=c2 r: the closer light travels to the Sun the more it deﬂects toward it. Astronomical objects of larger dimension (e.g. clusters of galaxies) ‘bend’ light rays similarly (gravitational lensing). It is tempting to interpret light deﬂection on the basis of the corpuscular theory of light as the effect on ‘corpuscles’ of Sun’s gravitational attraction. As a matter of fact, at the beginning of nineteenth century the german naturalist J. G. von Soldner predicted light deﬂection in the framework of Newton’s universal law of gravity based on this physical picture. For the angle of deﬂection he obtained precisely the half of the relativistic value calculated by Einstein in 1915 (Soldner shouldn’t have known the mass of the corpuscules since it drops out of the equations of motion if their inertial and gravitational masses are equal, as it was tacitly assumed by him). Believers of the popular view on mass–energy relation
4

The meaning of this term is that rotation of the Earth drags the spin axis with it out of the plane of revolution.

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(Sect. 1.12) often bring forward essentially this same interpretation asserting that it is only ‘the rest mass’ of the photon which is equal to zero, its ‘relativistic mass’ being given by the relation hm=c2 where hm is its energy. This ‘explanation’ is, however, grossly misleading since, as it has been clariﬁed in Sect. 2.20, the mass– energy relation is not applicable to massless particles and, even if it were, this picture can account only for half of the value of the deﬂection. The interpretation based on the analogy with light refraction by a spherically symmetric medium is much closer to the truth.

3.8 Perihelion Precession
Keplerian planetary orbits are ellipses of deﬁnite orientation with respect to ﬁxed stars. The most convincing veriﬁcation of Newton’s universal law of gravity is that the laws of Kepler follow from it provided mutual gravitation attraction of planets is neglected as compared to the much stronger attraction from the side of the Sun. It is in this approximation that planetary orbits are stationary ellipses whose perihelion, which is their point nearest to Sun, is seen from the center of the Sun at a ﬁxed point on the sky. However, owing to mutual attraction of the planets, Keplerian ellipses undergo slow continuous change of their shape and orientation. In particular, their perihelion is shifting continuously on the sky (perihelion precession). This phenomenon is best seen on Mercury’s orbit which is the most elongated one among planetary orbits. In the middle of nineteenth century French astronomer Urbain Jean Joseph Le Verrier investigated the action of the other planets on Mercury. According to the principle of additivity of small perturbations, the perihelion precession of Mercury’s orbit is determined by the sum of planets’ individual contributions. He found that on the basis of Newtonian gravitation the rate of precession should be equal to 527 s in a century, but the observed shift actually exceeded this value by an amount of 38 s in a century. Later S. Newcomb on the basis of an improved analysis concluded that Newtonian theory accounts for only 534 s in a century of the total observed shift which is equal to 575 s in a century.5 The discrepancy between the predicted and observed values is far beyond experimental uncertainties. To appreciate the precision of these observations remember that the apparent diameter of the Sun and the Moon is about 30’ which is 45 times larger then the anomaly of 41 seconds accumulated in a century. In the following decades various ad hoc explanations for the Mercury anomaly had been suggested but neither of them was successful. In general relativity it is automatically settled without any additional assumption. The solutions of the

5

Expressed in ecliptic longitude the rate of perihelion precession contains the additional constant 5037 s in a century which is the rate of displacement of the origin of the ecliptic longitude (of the vernal equinox) in opposite direction.

3.8 Perihelion Precession

85

precise geodetic equations in the Sun’s deformed spacetime differ slightly from the ﬁxed Keplerian ellipses and can be viewed as slowly rotating ones. For the shift of the perihelion the formula Du ¼ 3p Á 2GM 1 Á rad/revolution c2 a ð3:8:1Þ

is obtained in which a is the ellipse’s major axis. Speciﬁed to Mercury (3.8.1) leads to the rate of precession equal to 42.95 s in a century in excellent agreement with the observations.

3.9 Gravitational Red Shift
Gravitational red shift is a phenomenon akin to transverse Doppler-effect. In Sect. 1.2 this latter phenomenon was discussed for the special case when the emitter was at rest and the receiver revolved around it on a circular orbit. Suppose now that both the emitter end the receiver are revolving on concentric circles with identical angular velocities. Let the radius and rotation velocity of the emitter and receiver be re ; Ve and rr ; Vr respectively. The equality of their angular velocity can be expressed as Vr =Ve ¼ rr =re and for the sake of deﬁniteness assume that rr [ re . In the course of this motion the relative position of the emitter and the receiver remains obviously unaltered, therefore, according to prerelativistic conceptions, no Doppler-effect should to occur. In relativity theory, however, because of time dilation, a shift in frequency must arise. The emitter’s signals follow each other at proper time intervals Te (with frequency me ¼ 1=Te ). Owing to time dilation, Te is smaller than the corresponding period T measured in coordinate time: Te ¼ T =ce pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where ce ¼ 1= 1 À Ve 2 =c2 . As in Sect. 1.2, the coordinate time t is the time which appears in the equations, describing the motion (revolution) of the emitter and the receiver. The signals arrive at the receiver with the same coordinate time period T (with the frequency m ¼ 1=T ) as they left the emitter. In the proper time of the receiver this corresponds to the period Tr ¼ T =cr , where cr ¼ 1= pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À Vr 2 =c2 . Clocks, comoving with the emitter and receiver, show of course Te and Tr respectively. If T is eliminated from the equations Te ¼ T =ce and Tr ¼ T =cr we arrive at the equation ce Te ¼ cr Tr from which the formula for the transverse Doppler-effect in the geometry under consideration is obtained: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À Ve 2 =c2 mr cr ¼ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ð3:9:1Þ me ce 1 À Vr 2 =c2 When rr [ re this ratio is greater than unity. Imagine now that the emitter is arranged on the ground at some point of the Equator and the receiver is at a height z above it. Because of the rotation of the Earth this arrangement realizes the situation described above with Ve ¼ XR; Vr ¼

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XðR þ zÞ where R is the radius of the Earth and X ¼ 2p/day is the angular velocity of its rotation. But the huge mass of the Earth deforms spacetime around it (as testiﬁed by the weight of bodies). As we have already noted in Sect. 2.11, proper time is affected, beside velocity, by spacetime deformation too. According to general relativity theory, the modiﬁed formula for the proper time around the qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Earth is given by (2.11.2) as ds ¼ dt 1 À v2 =c2 þ 2U=c2 . Instead of (3.9.1), therefore, the correct value of the ratio mr =me is given by the formula qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 À Ve =c2 þ 2Ue =c2 mr ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ me 1 À Vr2 =c2 þ 2Ur =c2 in which Ue ¼ GM=R and Ur ¼ GM=ðR þ zÞ. Substitution of the numerical values for G; M; R and X shows that the velocity terms V 2 =c2 are negligibly small with respect to the gravitational terms 2U=c2 . Dropping them, we obtain for the gravitational red shift the formula sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À 2GM=c2 R mv gz ¼ 1 À 2 þ oð1=c4 Þ ¼ ð3:9:2Þ c ma 1 À 2GM=c2 ðR þ zÞ in which g ¼ GM=R2 is the gravitational acceleration on the Earth’s surface. The simple ﬁnal form is the result of a Taylor-expansion in 1=c2 . From (3.9.2) we obtain for the relative frequency shift the expression Dm mr À me gz ¼ ¼ 2: mr c mr ð3:9:3Þ

Beside series’ of discrete signals, these formulae are applicable to the frequency of monochromatic light beams as well. When the receiver is situated above the emitter we have mr =me \1 and one speaks of gravitational red shift. If their positions are interchanged the beam suffers a ‘blue shift’ because of the increase of its frequency. The term ‘gravitational red shift’ for the effect originates from the early hopes to observe the shift in the radiation of the Sun which is travelling ‘upward’ with respect to Sun and is shifted toward red. For the ﬁrst time the existence of the gravitational red shift was demonstrated by R. Pound and G. Rebka in 1960, in a laboratory experiment with the help of the Mössbauer-effect. The distance between the emitter and the receiver was equal to 22.5 m in which case, according to (3.9.3), the expected relative frequency shift is only about 2 Á 10À15 . Twenty-ﬁrst century technology permits one to observe the effect at a height difference less than 1 m. As explained above, formula (3.9.2) for the gravitational red shift comes from (2.11.2) which for U ¼ 0 describes the transverse Doppler-effect in special relativity and for v ¼ 0 represents gravitational red shift. However, applying (3.9.2) to gravitational red shift, we make the tacit assumption that, while travelling from the emitter to the receiver in curved spacetime, the frequency m of a light beam

3.9 Gravitational Red Shift

87

remains unchanged, as was the case in the pure Doppler-effect (see above). In the framework of general relativity rigorous proof can be given to the effect that in spacetimes which are constant in time, this is always the case.6 The frequency m refers to the coordinate time which is the time, occurring in the equations of trajectories as e.g. in h ¼ gt2 =2 for free falling. Having this property of light propagation in mind, gravitational red shift can be viewed as the consequence of ‘slowing down of the speed of time by gravitation’. Indeed, the inequality mr \me means that while at a height z the time elapsed between two consecutive signals is equal to, say, 2 s, on the ground level the time passed by the same pair is only 1 s. Since on the way between the two points the time T (the frequency m ¼ 1=T ) remains unchanged, this can only mean that at the ground level, where gravitation is stronger, clocks go slower than at the height z. This is the basis for the variant of the twin paradox sometimes called tower effect: If Alice and Bob are twins and Bob goes upstairs a high tower and remains there for a time, while Alice is staying at rest downstairs, he turns out, when comes down, older than Alice. Doppler-effect and gravitational red shift are very general phenomena not restricted to light signals. When, for example, the emitter is replaced by a machine gun, ﬁring at a rate me , bullets will be hitting the target at the place of the receiver at the rate mr determined by (3.9.2). This generality is the direct consequence of the fact that it is the speed of the ﬂow of time which is responsible for the observed effect. Advocates of the popular view on the mass–energy relation claim that gravitational red shift is the consequence of this relation. Their argumentation starts with the assertion that, while the rest mass of the photon is zero, its relativistic mass is equal to the kinetic energy of the photon divided by c2 . At the ground level, the reasoning continues, where the photon is born, its potential energy is zero, therefore its energy hme is a purely kinetical one. Hence its mass is equal to hme =c2 . At a height z, where it is detected, its potential energy is then equal to hme =c2 Â gz, and so its kinetic energy should have decreased to hme À hme =c2 Â gz ¼ hme ð1 À gz=c2 Þ. Since this energy is by deﬁnition equal to hmr , equating them we obtain the relation mr ¼ me ð1 À gz=c2 Þ which is identical to (3.9.2). This argumentation is, however, unacceptable. Already its starting point is untenable since the mass–energy relation is inapplicable to massless particles (see Sect. 2.20). Details of the argumentation (separation of the photon energy into kinetic and potential part, identiﬁcation of hm with the former) need justiﬁcation in the framework of quantum electrodynamics. The scope of the ‘explanation’ is, moreover, extremely narrow. It has nothing to say about the example with the machine gun which is the mechanical analogue of the gravitational red shift. This indicates clearly that it hopelessly misses the point.

6

The analogue of this theorem in geometric optics is that when light in static conditions passes from one medium into another with a different index of refraction its frequency remains constant.

Chapter 4

Concluding Remarks

The reader perhaps misses the term ‘ether’ which did not appear up to this moment. The omission was a deliberate one. Ether has been abandoned for the same reason as books on modern thermodynamics usually refrain from mentioning caloricum. Both ether and caloricum were important stages in the history of physics but the recognition that neither of them exists was an important step forward. In textbooks whose purpose is a concise presentation of physics in its contemporary state there remains no place for either of them. At the same time it is certainly true that familiarity with their role in the formation of science is instrumental in the fuller appreciation of the concepts of modern physics. The consequence of this omission is that no detailed description of the Michelson–Morley experiment has been given. The aim of this experiment, rightly honored by Nobel-prize, was to determine the velocity of the Earth in the ether— which is now known not to exist. Today it counts a commonplace to assert that we owe this knowledge just to the Michelson–Morley experiment but this is not true: nobody before Einstein drew this conclusion from it.1 It is relativity theory from which we know that there is no ether. Indeed, this theory is an ‘operating system under which’ the whole of modern physics is succesfully running. It is, moreover, based on the assumption that light has speed equal to the same c in any direction in every inertial frame which contradicts plainly to the idea of a medium-like carrier of it. As we have already seen in Sect. 2.21, the assertion that relativity theory has emerged from the analysis of the Michelson–Morley experiment, is false also historically. In spite of all this, Michelson–Morley experiment has remained an important evidence in favour of relativity theory. But it is a highly dangerous practice to take it for the sole buttress of the theory as it is usually done. As a consequence of this unfortunate practice it is widely believed that relativity theory survives only as far

The conclusion drawn by Lorentz was that motion with respect to the ether makes clocks to go slower and length of bodies along the motion to become shorter.

1

P. Hraskó, Basic Relativity, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-17810-8_4, Ó Péter Hraskó 2011

89

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as the Michelson–Morley experiment stands fast. If in a repetition of it a ‘positive effect’ due to the motion of the Earth was observed the theory would be dead. As a matter of fact, in the ﬁrst quarter of 20. century positive effect was observed in a series of experiments performed by C. Miller (at one time president of the American Physical Society) but this fact did not undermine the positions of relativity theory. This theory is, in fact, veriﬁed by the totality of modern physics and Miller’s result should be confronted with that huge body of mutually intertwined evidences. One is, therefore, enforced to admit that Miller’s data should suffer from a deﬁciency of unknown origin. Some have pointed out that, owing to the rotation and revolution of the Earth, a laboratory on its surface is not an inertial frame. Others attributed the positive effect to statistical ﬂuctuations or to the inﬂuence of temperature variations. It was also a deliberate choice to introduce the perplexing effects of relativity theory, such as time dilation and Lorentz contraction, through ad hoc thought experiments rather than to derive them in a systematic way from Lorentz-transformations. Following this path, one hopes to abandon the false impression that these effects are mere ‘appearances’, resulting from the manipulation with clocks (which are even not there), rather than the ‘true’ phenomena themselves. Time dilation, for example, has been deduced from the Doppler-effect. The analysis was based on the equivalence of the inertial observers (which is an easily acceptable assumption) and the constancy of light speed (which contradicts intuition). Time dilation, whose role can be grasped relatively easily, emerges as a kind of bridge between these two ingredients. Frequency shift due to time dilation is felt in a sense ‘more real’ than that in the original prerelativistic effect which is the consequence of the mere variation of the distance between the emitter and the observer. On the other hand, it should have been made clear that ‘manipulations with nonexisting clocks’ are an indispensable prerequisite for the derivation of Lorentztransformations which are more than effective computational tools: they provide the basis for the proof of the inner consistency of the theory.2 They are formulated in terms of the coordinate time whose introduction has been split into two steps. In Chap. 1 the emphasis was made on the similarity (within a given inertial frame) between the coordinate time of relativity theory and the Newtonian t learned in secondary school physics. Relativity of simultaneity was explained, of course, at the very beginning of the text, but only at a phenomenological level. Its substantiation had been postponed to Chap. 2; it is certainly beneﬁcial for the reader if he/she is aware of what is to be substantiated. One must, ﬁnally, face the question of whether or not relativity theory is refutable? Attempts to refute the theory obtain, as a rule, much wider publicity than attempts to elucidate its content. For an answer we have to recourse to historical analogues. The best example at hand is Newtonian mechanics. Is (was) it

2

Problems of consistency have remained untouched in the present text since they would require elaborate mathematics.

4 Concluding Remarks

91

refutable? Yes and no. In possession of relativity theory it can be hardly denied that Newtonian physics has been proved erroneous and the theory of Einstein ‘overthrew’ it. But why then engineers are still taught to Newtonian physics? Orbits of most satellites are calculated on the basis of Newton’s law of gravity, relativity theory is needed only when extreme accuracy is required (as e.g. in GPS satellites). Refutation is, therefore, not quite the proper expression for what happened to Newtonian physics. Relativity theory has expounded its limitations but within these limits it has retained its truth and applicability. The two milestones on its boundary are (1) velocities must be much smaller than the velocity of light, and (2) the relative change of the internal energy must be negligibly small (Dm=m ( 1). Something like this will certainly happen with relativity theory too but it will come from a direction as unexpected as E0 ¼ mc2 burst in on physics. Nobody can foresee it. It will probably emerge from attempts to solve important physical puzzles rather than from the desire to generalize the existing theory.

Chapter 5

Selected Problems to Chapter 1

(1) A light source is continuously emitting sharp signals with a period of T0 . The signals are reﬂected by a mirror, receding from the source with the constant velocity V . Calculate the period T of the reﬂected signals according to both Newtonian physics and relativity theory. Solution 1 Let us solve the problem ﬁrst in the Newtonian framework. Assume that the signals hit the mirror with the period T 0 . According to (1.2.3) T 0 ¼ T0 =ð1 À V =cÞ. From the point of view of the refracted signals the mirror is the emitter. Therefore, using (1.2.1), we have T ¼ T 0 ð1 þ V =cÞ. Excluding T 0 from these equations we obtain the solution in the Newtonian case: T ¼ T0 1 þ V =c : 1 À V =c ð1Þ

In relativity theory the reasoning remains the same with the only difference that in both steps formula (1.2.6) should be used: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ V =c 1 þ V =c T 0 ¼ T0 and T ¼ T 0 : 1 À V =c 1 À V =c From these we arrive at the same formula (1) as before. We have obtained, therefore, identical results in both approaches. This could have been foreseen since both T0 and T can be measured with the same clock and so time dilation plays no role in their connection. It is only T 0 which is sensitive to time dilation and for T 0 we have indeed obtained different formulae. Solution 2 Let us begin again with the Newtonian case. Figure 1 shows the path of the mirror and those of the light signals. From the shaded triangle we obtain T À T0 T À T0 : h ¼ V T0 þ ¼c 2 2

P. Hraskó, Basic Relativity, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-17810-8_5, Ó Péter Hraskó 2011

93

94

5 Selected Problems to Chapter 1

Solving this with respect to T the result (1) is recovered. This formula remains valid in the relativistic case too since time dilation does not play role in the problem. (2) According to Einstein’s train thought experiment (Sect. 1.3) in the rest frame I of the platform the explosion at the rear of the car takes place earlier than at its front. Calculate the time difference Dt in I for a car of proper length l0 and velocity V . pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Solution As seen from I the length of the car is equal to l0 1 À V 2 =c2 . Therefore, at the moment the light ﬂash is given off both ends of the car are at a pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ distance 1 l0 1 À V 2 =c2 from the signals. The distance between the forward signal 2 and the front end on the one hand and the backward signal and the rear end on the other decreases with the rate ðc À V Þ and ðc þ V Þ respectively (see Sect. 1.6). These distances are, therefore, consumed in time intervals pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ l0 1 À V 2 =c2 l0 1 À V 2 =c2 ; and Dtr ¼ : Dtf ¼ 2ðc À V Þ 2ðc þ V Þ

Fig. 1 Graphical solution of Problem 1

5 Selected Problems to Chapter 1

95

Dt is equal to the difference of these intervals: 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 V Á l0 2 =c2 Á À Dt ¼ Dtf À Dtr ¼ l0 1 À V ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ð2Þ 2 1 À V 2 =c2 2 cÀV cþV c

(3) In Problem 2 light signals can be replaced by pistol bullets: from two pistols at the center of the car a simultaneous pair of shots is ﬁred off toward the ends of the car. The speed of the bullets is smaller than c. Calculate the time difference Dt of their hits. Solution 1 In the rest frame of the car both bullets will hit the ends of the car at the same moment as was the case with light signals. Therefore, formula (2) of Problem 2 remains valid for revolver bullets too. Solution 2 Let the muzzle velocity is equal to v. For the sake of deﬁniteness assume that v [ V . The magnitudes of bullets’ velocities ﬂying toward the front end the rear end of the car are, with respect to the rest frame of the platform, equal to Àv þ V vþV ¼ vÀV : vf ¼ ; and vr ¼ 2 2 1 þ vV =c 1 À vV =c 1 À vV =c2 The duration of their ﬂying is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ l0 1 À V 2 =c2 Dtf ¼ 2ðvf À V Þ The denominators are vf À V ¼ vð1 À V 2 =c2 Þ 1 þ vV =c2 and vr þ V ¼ vð1 À V 2 =c2 Þ : 1 À vV =c2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V 2 =c2 and Dtr ¼ : 2ðvr þ V Þ l0

The time difference Dt is equal to the difference between Dtf and Dtr : Á À ÁÃ l0 1 ÂÀ Dt ¼ Dtf À Dtr ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Á Á 1 þ vV =c2 À 1 À vV =c2 2 1 À V 2 =c2 v l0 V ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c2 1 À V 2 =c2 which is indeed identical to the result of the previous problem. In the Newtonian approach we would have vf ¼ v þ V ; vr ¼ v À V ; l0 l0 Dtf ¼ 2ðvfl0ÀV Þ ¼ 2v ; Dtr ¼ 2ðvrl0 Þ ¼ 2v ; þV and Dt ¼ 0 as should be.

96

5 Selected Problems to Chapter 1

(4) A cyclist is riding along the railway embankment with the velocity v. A train is coming from the opposite direction. Its velocity and proper length are V and l0 . How long takes it for the cyclist to pass by the train as shown by his/her clock? Give the answer in both Newtonian and relativistic physics. Solution In the Newtonian physics the answer is l0 =ðV þ vÞ. The relativistic calculation is most simple in the rest frame of the embankment. At the moment when the front end of the train passes by the cyclist the rear end is at a distance pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ l0 1 À V 2 =c2 from him/her. The rate of its decrease is equal to V þ v and so it is consumed in the coordinate time interval pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V 2 =c2 : Dt ¼ l0 V þv The corresponding proper time interval elapsed on the cyclist’s clock is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À V 2 =c2 Á 1 À v2 =c2 2 =c2 ¼ l Ds ¼ Dt Á 1 À v : 0 V þv

ð3Þ

The calculation performed in the rest frame of the train or the cyclist is a bit more complicated but leads, of course, to the same result (the proper time Ds is invariant). In the rest frame I 0 of the train, for example, the velocity of the cyclist is equal to v0 ¼ vþV : 1 þ vV =c2

The length of the train here is l0 , hence the coordinate time interval, during which the cyclist is riding alongside the train, is equal to Dt0 ¼ l0 : v0

This corresponds to a proper time interval equal to pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À v02 =c2 Ds ¼ l0 : v0 If the expression for v0 is substituted here, we obtain after a little algebra the solution (3) again. (5) An inﬁnitely massive wall moves with the constant velocity V (V \c=2) in the inertial frame I . A ball, ﬂying in the same direction with the velocity 2V , rebounds elastically from it. Determine the velocity u of the ball after rebound in Newtonian and relativistic physics.

5 Selected Problems to Chapter 1

97

Solution In both cases the calculation is based on the fact that in the rest frame I 0 of the wall the magnitude of the ball’s velocity remains unchanged. The steps are summarized is the following table:
Ball’s velocity 0. 1. 2. 3. In I before rebound In I 0 before rebound In I 0 after rebound In I after rebound Newtonian 2V 2V À V ¼ V ÀV ÀV À ðÀV Þ ¼ 0 Relativistic 2V
2V ÀV 1À2V ÁV =c2 V ¼ 1À2V 2 =c2

V À 1À2V 2 =c2 À 1À
V

1À2V 2 =c2
À V2 2 1À2V =c

ÀðÀV Þ ðÀV Þ

V =c ¼ À2V 1À3V 2 =c2

2

2

In the Newtonian case, therefore, the ball remains after the rebound at rest while, according to relativistic calculation, its velocity becomes equal to u ¼ À2V V 2 =c2 : 1 À 3V 2 =c2

The Newtonian answer is recovered from this in the limit c ! 1. Another limiting case is when the ball and the wall are replaced by a light signal and a mirror. In that case V ¼ c=2 and we obtain u ¼ Àc as should be. The last problem is a bit more involved than the previous ones: (6) On a straight highway a column of cars is travelling with the uniform velocity v in positive direction. Cars follow closely each other, in every short time interval Ds a car is passing through every cross section of the highway. (a) How many cars are found in the average on the segment AB of length l ) v Á Ds? (b) How many cars are found on this same segment AB if viewed from an inertial frame with respect to which the highway is moving with the velocity V in positive direction? Solution Let us denote the average number of cars in the ﬁrst case (i.e. with respect to the rest frame I of the highway) by N. Any given car, after it has passed the initial point A of AB, remains within this segment during a time interval of T ¼ l=v. In this time T =Ds cars passes by A. Therefore, N¼ T l ¼ : Ds v Á Ds ð4Þ

The corresponding number of cars with respect to the moving inertial frame I 0 0 will be denoted by Nþ (the index þ serves to indicate that both the velocity v of the cars in I and the velocity of the highway in I 0 are of the same direction). At ﬁrst

98

5 Selected Problems to Chapter 1

0 sight one is inclined to think that Nþ does not differ from N but a short reﬂexion reveals that, because of the relativity of simultaneity, they cannot be the same. To clarify this point, let us rephrase our previous reasoning. Give the name Car1 to the car chosen as the ﬁrst one. We have to count the number of cars passing A until Car1 reaches B. At this moment of time we shut an imagined gate at A. Formula (4) gives the number of those cars which are found at this moment of time within AB. In I 0 the same procedure should be applied. But those events which are simultaneous with respect to I 0 , if viewed from I , take place at different moments pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ of time: the event at A takes place by the time Dt ¼ V Á l0 = c2 1 À V 2 =c2 later

than the event at B. (See Problem 2. I 0 corresponds to the train which now travels in negative direction.) This formula contains the length l0 of AB as measured in I 0 , since we are now considering the time difference in I of a pair of events which are simultaneous in I 0 . As a consequence of the Lorentz contraction we have l0 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ l 1 À V 2 =c2 and so Dt ¼ V Á l=c2 . Therefore, Car1 is found within AB during the time interval T 0 ¼ T þ Dt rather than T ¼ l=v. During that time T 0 =Ds cars pass by A. Hence, T0 1 l vV vV 0 ðT þ DtÞ ¼ Á 1þ 2 ¼NÁ 1þ 2 : ¼ Nþ ¼ ð5Þ v Á Ds c c Ds Ds There is another argument, leading to this same result, but it does not show up with such a clarity the role of the relativity of simultaneity. It is based on the 0 transformation of (4) so that it give Nþ instead of N. To this end we must take into pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ account Lorentz contraction l ! l 1 À V 2 =c2 , time dilation ðDs ! Dt ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ds= 1 À V 2 =c2 Þ and the relativistic law of velocity addition, according to which the velocity of the cars in I 0 is equal to u¼ vþV : 1 þ vV =c2 ð6Þ

The rate of decrease of the distance between Car1 and the point B is ðu À V Þ, this difference must be put in place of v in the denominator of (4). Having performed all these replacements in (4) we arrive after a little algebra at (5) again. When the direction of v and V is opposite to each other we obtain vV 0 NÀ ¼ N Á 1 À 2 : ð7Þ c If a pair of events simultaneous in I 0 is viewed from I , then in this case the event in A precedes the event in B. This is why the number of cars on AB is diminished. This seemingly artiﬁcial problem has an interesting application. But before considering it we must dispense with the following paradox. Assume that the highway is inﬁnitely long and is bordered by trees: as viewed from I , every

5 Selected Problems to Chapter 1

99

segment of length l (as e.g. the segment AB itself) contains n trees. Then the number of cars per a tree is equal to N=n. Let us now change the point of view from I to I 0 . The number of trees on the original segments of length l (on AB for example) remains unchanged but the 0 number of cars becomes equal to NÆ and the number of cars per a tree is modiﬁed 2 by a factor of ð1 Æ vV =c Þ, depending on the relative direction of v and V . But this is impossible because the number of neither the cars nor the trees is inﬂuenced by the choice of the inertial frame they are viewed from. The solution of the paradox is that an inﬁnitely long highway with uniform distribution of moving cars on it is not an idealization of anything because cars moving on a real highway of ﬁnite length cannot have uniform distribution. On a closed path, however, the density of both the trees and cars may be uniform. Let us, therefore, complete the segment AB up to a square ABCD through every point of which in every time interval Ds a car is passing. According to (4), in I the number of cars on the sides of the square is in the average equal to N ¼ l=ðDs Á vÞ. In that I 0 however, with respect to which the square is moving with velocity V in the direction of the cars’ motion on the side AB, the average number of cars on the 0 side AB and CD is equal to NÆ ¼ N ð1 Æ vV =c2 Þ respectively. The number of cars on the sides BC and DA remains equal to N because if a pair of events on the endpoints of a segment perpendicular to V is simultaneous in I , it remains simultaneous in I 0 too. The total number of cars on a closed highway is, therefore, independent of the reference frame they are observed from. The motion of the highway as a whole leads only to the redistribution of the cars between the sides (parts) of it. This is the consequence of the motion of the cars because the number of trees remains the same on each side. Let us now replace the highway with a metallic frame in which an electric current is ﬂowing (see Fig. 2; the source of the applied voltage is suppressed, one may think of a superconducting frame). The cars and the trees correspond to the moving electrons and ﬁxed ions respectively. Applying our formulae, the current can be written as

Fig. 2 The current loop moving with the velocity V

100

5 Selected Problems to Chapter 1

I¼

e eNv ¼ : Ds l

ð8Þ

Here e is the absolute value of the electron charge, the velocity v of the cars has become the velocity of the electrons. Since the electron charge is negative, the direction of the current is opposite to that of the cars. In its rest frame the magnetic dipole moment m of this loop is parallel to the z-axis (see the ﬁgure): mz ¼ I Á l2 ¼ eNvl: ð9Þ

In I 0 , with respect to which it moves with the velocity V in x-direction, it acquires an electric dipole moment too which is parallel to the y-axis:
0 py ¼ ðÀeÞ Á ðNþ À NÞ Á l ¼ Àe Á

vV V N Á l ¼ À 2 mz : c2 c

ð10Þ

This is the result of the redistribution of the electrons: the electron surplus on AB is compensated by the corresponding deﬁcit on CD. If we take into account that V is the x component of the velocity vector V then (10) can be written, using cross product, as a vector equation: p¼ 1 ðV Â mÞ: c2 ð11Þ

The seemingly artiﬁcial problem lead us, therefore, to the conclusion that moving magnetic dipoles acquire electric dipole moment. As we have seen this is also the consequence of the relativity of simultaneity.

Index

p0meson, 23

A Acceleration gravitational, 75 transformation of Acceleration suppression, 19, 21 Alpha decay, 27 energy and momentum conservation in, 68

B Bell’s thought experiment, 62 Beta-decay, 68

C Causality paradox, 45, 47 Charge conservation Coordinate system, 1, 29, 38 attached, 29, 33, 56 Coordinate time, 29, 31, 32, 34, 51–54, 90 in accelerated frame of reference on the Earth, 60 on the rotating disc Coulomb-force, 70

E Eötvös, 76 Einstein lift, 81, 83 Einstein, 2, 8, 22, 25, 69, 78, 82, 89, 91 Einstein-equations, 78 Electric moment acquired by moving magnetic moment Electromagnetic ﬁeld transformation of Energy, 26 Energy conservation, 26 Energy–momentum four vector, 68 Energy-momentum tensor, 78 Equation of motion Newtonian, 3, 17, 77 in accelerating frame of a point charge, 70 relativistic, 19 Equivalence principle, 77, 78, 81, 83 Ether, 89 Event(s), 29 lightlike pair of, 39 spacelike pair of, 40, 48 timelike pair of, 40, 47

D Doppler-effect, 4, 50, 69, 90 longitudinal, 7 transverse, 7, 85 Drag, 83

F Force Coriolis, 2 inertial, 1, 78 true, 2 Four-vector, 66 Frequency its constancy in static spacetime

101

102 G Galilei transformation, 40 Galilei, G. experiment with ralling balls Galilei-transformation, 34, 70 homogeneity linearity of Gamow, G., 63 General relativity, 3, 78 observable consequencies of Geodesic, 77, 78 Geodesic equation, 79 Geodesic hypothesis, 78 Geodesic precession, 82 GPS, 53 Gravitation source of Gravitational ﬁeld nonexistence of Gravitational lensing, 83 Gravitational red shift, 85, 86 Pound-Rebka experiment Gravity Probe A (GP-A) experiment, 52 Gravity Probe B (GP-B) experiment Gyroscope, 32, 71, 81 optical, 72

Index Light corpuscular theory of, 83 Light cone, 43, 44 Light deﬂection, 27, 83 as refraction, 84 Light speedsee speed of light, 3, 31, 33, 34, 70, 80, 90 Light velocity isotropy, 2, 33, 34 on Earth Lightlike future and past, 43 Logical contradiction, 8, 46 Lorentz contraction, 13–15, 19, 39, 60–64, 90, 98 Lorentz, H. A, 90 Lorentz-force, 71 Lorentz-transformation, 27, 34, 35, 43, 44, 64–67 linearity and homogeneity, 37

I Ideal clock, 30, 52 Inertial frames, 1, 2, 31, 34, 78 equivalence of, 3, 14 global, 79, 80 for planetary motion, 80 isolated, 2 local, 79, 83 relative motion of Instantaneous rest frame, 14, 17, 51 Internal combustion engine, 25 Internal energy, see rest energy

K Kaufmann experiments, 19 Kinetic energy, 20, 23, 25, 67 Newtonian, 21 relativistic, 21

M Manifold, 38, 39, 77, 78 euclidean and pseudoeuclidean, 40 Mars rover, 9, 44, 80 Mass, 17, 20, 22, 23, 25–27, 38, 40 inertial and gravitational, 75, 79 equality of, 76 invariance of, 48, 54, 70 longitudinal and transversal, 71 Mass conservation, 25, 26 popular view on Mass–energy relation, 20, 23, 24, 26, 27, 38, 67, 68 popular view on, 26, 84, 87 Massless particles, 68, 69 transformation of the energy of Maxwell equations, 2, 70 in Minkowski coordinates, 34 Metric, 38 Michelson–Morley experiment, 69, 70, 89, 90 Miller, C, 90 Minkowski coordinates, 33, 34, 38, 52, 54, 80 Minkowski diagram, see spacetime diagram Momentum, 19 Momentum conservation, 23, 68

L Le Verrier, 84 Length, 13, 14, 30, 37, 43, 52, 54, 59, 61–64, 76, 89, 94, 96–99 vs distance, 17

N Newcomb, S, 84 Newton’s law, 82, 91 Newtonian time, 12

Index P Penrose–Terrel effect, 64 Perihelion precession, 84 Photon, 69 relativistic mass of, 84 Planck-constant, 69 Positronium, 23 Proper time, 11, 12, 17, 18, 32, 37, 51–53, 55, 96 inﬂuenced by gravitation invariance of Pseudorotation

103 Speed of time slowing down by gravitation, 87 Squared distance, 38, 51 invariance of, 38 Standard setting, 32–35 freedom in, 35 Superluminal signal, 45, 47 Synchronization, 31 Einsteinian, V, 27, 31, 32, 34, 38, 54 circularity of, 32 in accelerating frame of reference, 54, 80 in general relativity, 80 method of, 30 Newtonian, 31, 32, 34 subtlities of, 52

R Rate of change of the distance, 15, 64 Red shift, see gravitational red shift Reference frames accelerating, 34, 59 Relative velocity, 15, 64 Relativistic mass, 20, 26 fallacy, 20 Relativity of simultaneity, see simultaneity Rest energy, 22, 25, 67, 78n Rest mass, 20 Rotating disc, 59, 71

S Sagnac effect, 3n, 72, 73 Simultaneity, 8, 9, 13, 14, 39, 48, 49, 61, 80, 90, 98 interval of, 10, 11, 44, 80 Soldner von, J. G., 83 Spacetime, 29 as manifold, see manifold curved, 77, 86 domains in, 43 Spacetime diagram, 40, 47, 48, 61 nonrelativistic parallel projection on, 41, 44 relativistic, 42 scales on, 41, 43 Spacetime intervals, 39, 68 on spacetime diagrams, 42 Special relativity ﬁrst postulate of, 3, 72 refutability of, 89, 90 second postulate of, 10 Speed of light, 3, 54 constancy of, 3, 31–33, 70, 90 in general relativity, 79 measurement of, 3 reﬂected from moving mirror, 72

T Tachyon, 48 Thomas-precession, 71, 72n Time dilation, V, 7, 19, 20, 31, 59, 85, 90, 91, 94, 98 symmetry of, 8, 48 Timelike future and past, 44 Tompkins, Mr, 63 Total energy, 23, 26 Tower effect, 87 Train thought experiment, 9, 14, 36, 37, 40, 94 Twin paradox, 12, 31, 31n, 52, 53 as seen by the accelerating twin, 12, 53–59 the magnitude of, 52, 54

U Universal gravity law, 78, 83

V Velocity addition formula, V, 15, 16, 18n, 46, 64, 98 Velocity reversal, 55 lasting effect, 58 deformation, 59 desynchronization, 58 vernal equinox displacement of, 84n Virtual clocks, 33 set S of, 32, 33

W Wave packet, 27 energy and momentum, 22 Weight, 75, 78, 81

104

Index Z Zero mass particles, see massless particles

W (cont.)
Weightlessness, 76, 78 World line, 40, 77

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