Basic Relativity An Introductory Essay [Hrasko]

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