Table Of ContentUndergraduate Lecture Notes in Physics
Costas Christodoulides
The Special
Theory of
Relativity
Foundations, Theory, Verification,
Applications
Undergraduate Lecture Notes in Physics
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Costas Christodoulides
The Special Theory
of Relativity
fi
Foundations, Theory, Veri cation,
Applications
123
Costas Christodoulides
Department ofPhysics, Schoolof Applied
MathematicalandPhysical Sciences
National Technical University of Athens
Athens
Greece
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©Tziolas Publications, 2014.
ISSN 2192-4791 ISSN 2192-4805 (electronic)
Undergraduate Lecture Notesin Physics
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Preface
The book at hand has developed from the lecture notes used in the teaching of the
SpecialTheoryofRelativityformanyyearsbytheauthor,originallytoengineering
students and finally to students of Physics and Mathematics at the School of
Mathematical and Physical Sciences of the National Technical University of
Athens. About half the material contained in the book was covered in 25 hourly
lectures during the second semester of the first year of studies.
As a textbook, the book has some special characteristics: The proofs of the
theorems are given in adequate detail, many figures are used, many examples are
used for the comprehension of the theory and many problems are suggested for
solution by the reader. The detailed solutions of all the problems are given at the
end of the book. However, it must be understood that, for maximum benefit, the
reader should really try to solve a problem before resorting to the solution given.
Asassistanceinthisdirection,theanswerstoalltheproblemsaregiven,aswellas
appropriate hints or suggestions on how to solve the problems.
Special attention has been paid to presenting the historical approach of Physics
to the Special Theory of Relativity, to its experimental foundation and to the
experiments performed in order totest thevalidity ofthetheory.Forthis reason, a
large number of experiments are described in detail.
The Special Theory of Relativity found applications in a large number of
problems,mainlyinPhysics,andanattemptismadetopresentthemostimportant
of them, which every physicist should be familiar with. Many of the applications
are developed in the Examples and Problems and the reader should pay particular
attention to them.
In order that the reader should take advantage of the ease of access to the
scientificliteraturemadepossiblebytoday’stechnology,referencesaregiventothe
original articles and review articles concerning each topic.
ThebookgivesafairlycompletepresentationoftheSpecialTheoryofRelativity
for a first approach to the subject at the undergraduate university level. A basic
course on the Special Theory of Relativity could consist of Chaps. 2, 3, 6 and 9,
supplemented by appropriate applications of the theory from Chaps. 4, 5 and 7.
v
vi Preface
Additional,moredemandingtopicsmaybefoundintherestofthechaptersandin
the appendices. The mathematics needed is that of the first year of a degree in
Physics. Whenever necessary, a brief presentation of additional mathematics is
given in the text.
Itishopedthatthebookwillbeausefuladditiontotheexistingliteratureonthis
fascinating subject.
Athens Costas Christodoulides
November 2015
Contents
1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Main Landmarks in the Development of the Special
Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ThePrincipleofRelativityofGalileo.Galileo’sInvariance
Hypothesis.TheLawofInertia.InertialFramesofReference. . . 4
1.2.1 The Galilean Transformation . . . . . . . . . . . . . . . . . . 5
1.3 Rømer and the Speed of Light . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Newton’s Laws of Motion. Inertia and Inertial Frames
of Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 The Invariance of Newton’s Second Law of Motion
Under the Galilean Transformation . . . . . . . . . . . . . . 11
1.5 The Aberration of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5.1 The Discovery by Bradley of the Aberration
of Stellar Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Arago’s Measurements Concerning the Constancy
of the Speed of Light from Stars. . . . . . . . . . . . . . . . . . . . . . 18
1.7 Measurements of the Speed of Light in the Laboratory . . . . . . 21
1.7.1 The Possibility of Dependence of the Speed
of Light in Vacuum on Its Frequency . . . . . . . . . . . . 23
1.8 Attempts to Measure the Dragging of Aether
by Moving Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.8.1 The Experiment of Fizeau . . . . . . . . . . . . . . . . . . . . 24
1.8.2 The Experiment of Hoek . . . . . . . . . . . . . . . . . . . . . 27
1.9 Maxwell’s Equations and the Wave Equation. . . . . . . . . . . . . 28
1.10 The Experiment of Michelson and Morley. . . . . . . . . . . . . . . 30
1.11 The Lorentz-FitzGerald Contraction Hypothesis . . . . . . . . . . . 37
1.12 The Increase of the Mass of the Electron with Speed. . . . . . . . 38
1.13 The Invariance of Maxwell’s Equations and the Lorentz
Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.14 The Formulation of the Special Theory of Relativity. . . . . . . . 43
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
vii
viii Contents
2 Prolegomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1 Inertial Frames of Reference. . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1.1 The Earth, the Sun and the Galaxy as Inertial Frames
of Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2 The Calibration of a Frame of Reference and the
Synchronization of Its Clocks. . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 The Relativity of Simultaneity . . . . . . . . . . . . . . . . . . . . . . . 51
2.4 The Relativity of Time and Length. . . . . . . . . . . . . . . . . . . . 53
2.4.1 The Invariance of the Dimensions of a Body
Which Are Perpendicular to Its Velocity . . . . . . . . . . 53
2.4.2 The Relativity of Time . . . . . . . . . . . . . . . . . . . . . . 54
2.4.3 The Relativity of Length . . . . . . . . . . . . . . . . . . . . . 56
2.5 The Inevitability of the Special Theory of Relativity . . . . . . . . 58
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 The Lorentz Transformation for the Coordinates
of an Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.1 The Contraction of Length. . . . . . . . . . . . . . . . . . . . 69
3.1.2 The Dilation of Time. . . . . . . . . . . . . . . . . . . . . . . . 76
3.2 The Transformation of Velocity . . . . . . . . . . . . . . . . . . . . . . 83
3.2.1 The Transformation of the Components
of Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2.2 The Transformation of the Magnitude of Velocity. . . . 84
3.2.3 The Transformation of the Lorentz Factor, γ. . . . . . . . 86
3.2.4 Speed c as an Upper Limit for Speeds. . . . . . . . . . . . 87
3.3 The Transformation of Acceleration . . . . . . . . . . . . . . . . . . . 91
3.3.1 Proper Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . 92
4 Applications of Relativistic Kinematics. . . . . . . . . . . . . . . . . . . . . 97
4.1 The ‘Meson’ Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.1.1 Experimental Verification of the Dilation
of Time with Muon Experiments at CERN. . . . . . . . . 102
4.2 The Apparent Focusing of Fast Charged Particle Beams
Due to the Dilation of Time. . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 The Sagnac Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4 Clocks Moving Around the Earth . . . . . . . . . . . . . . . . . . . . . 109
4.5 The Experiment of Hafele and Keating . . . . . . . . . . . . . . . . . 116
4.6 Einstein’s Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.7 The Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.8 Motion with a Constant Proper Acceleration.
Hyperbolic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.8.1 The Dilation of Time and Journeys in Space . . . . . . . 126
Contents ix
4.9 Two Successive Lorentz Transformations.
The Wigner Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.9.1 The Thomas Precession. . . . . . . . . . . . . . . . . . . . . . 133
4.9.2 The Thomas Precession in the Cases
of a Planetary or a Stellar System. . . . . . . . . . . . . . . 135
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Optical Phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.1 The Aberration of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2 Fizeau’s Experiment. Fresnel’s Aether Dragging Theory . . . . . 141
5.3 The Doppler Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.3.1 The Longitudinal Doppler Effect. . . . . . . . . . . . . . . . 143
5.3.2 The General Doppler Effect . . . . . . . . . . . . . . . . . . . 146
5.3.3 The Experimental Verification of the Relativistic
Terms of the Doppler Effect. . . . . . . . . . . . . . . . . . . 152
5.4 Relativistic Beaming or the Headlight Effect . . . . . . . . . . . . . 155
5.4.1 Synchrotron Radiation or Magnetic
Bremsstrahlung. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.5 The Forces Exerted by Light . . . . . . . . . . . . . . . . . . . . . . . . 161
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6 Relativistic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.1 The Definition of Relativistic Momentum.
Relativistic Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.2 Relativistic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.3 The Relationship Between Momentum and Energy . . . . . . . . . 170
6.3.1 Units of Energy, Mass and Momentum . . . . . . . . . . . 170
6.4 Classical Approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.5 Particles with Zero Rest Mass. . . . . . . . . . . . . . . . . . . . . . . . 175
6.6 The Conservation of Momentum and of Energy . . . . . . . . . . . 176
6.7 The Equivalence of Mass and Energy . . . . . . . . . . . . . . . . . . 178
6.7.1 The Validity of the Conservation of Momentum
and Energy During the Transmutation of Nuclei
and the Annihilation and Creation of Particles . . . . . . 179
6.8 The Transformation of Momentum and Energy. . . . . . . . . . . . 186
6.9 The Zero-Momentum Frame of Reference . . . . . . . . . . . . . . . 189
6.10 The Transformation of the Total Momentum
and the Total Energy of a System of Particles . . . . . . . . . . . . 193
6.10.1 The Invariance of the Quantity E2(cid:1)c2P2
for a System of Particles . . . . . . . . . . . . . . . . . . . . . 194
6.11 The Collision of Two Identical Particles . . . . . . . . . . . . . . . . 197
6.11.1 Experimental Test of the Special Theory
of Relativity with Colliding Electrons . . . . . . . . . . . . 200
6.12 The Transformation of Force . . . . . . . . . . . . . . . . . . . . . . . . 200
