The Special Theory of Relativity A Mathematical Approach

253 Pages·2016·4.22 MB·English

by  Farook Rahaman

Farook Rahaman
The Special
Theory of
Relativity
A Mathematical Approach
The Special Theory of Relativity
Farook Rahaman
The Special Theory
of Relativity
A Mathematical Approach
123
Farook Rahaman
Department of Mathematics
JadavpurUniversity
Kolkata, WestBengal
India
ISBN 978-81-322-2079-4 ISBN 978-81-322-2080-0 (eBook)
DOI 10.1007/978-81-322-2080-0
LibraryofCongressControlNumber:2014948764
SpringerNewDelhiHeidelbergNewYorkDordrechtLondon
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To my parents
Majeda Rahaman and Late Obaidur
Rahaman
and my son Rahil Miraj
Preface
In 1905, Albert Einstein wrote three papers which started three new branches in
physics; one of them was the special theory of relativity. The subject was soon
included as a compulsory subject in graduate and postgraduate courses of physics
and applied mathematics across the world. And then various eminent scientists
wrote several books on the topic.
This book is an outcome of a series of lectures delivered by the author to
graduatestudentsinmathematicsatJadavpurUniversity.Duringhislectures,many
studentsaskedseveralquestionswhichhelpedhimknowtheneedsofstudents.Itis,
therefore,awell-plannedtextbook,whosecontentsareorganizedinalogicalorder
where every topic has been dealt with in a simple and lucid manner. Suitable
problemswithhintsareincludedineachchaptermostlytakenfromquestionpapers
of several universities.
InChap.1,theauthorhasdiscussedthelimitationsofNewtonianmechanicsand
Galilean transformations. Some examples related to Galilean transformations have
been provided. It is shown that laws of mechanics are invariant but laws of elec-
trodynamics are not invariant under Galilean transformation. Now, assuming both
theGalileantransformationandelectrodynamicsgovernedbyMaxwell’sequations
aretrue,itisobvioustobelieve there exists auniqueprivilegedframeofreference
inwhichMaxwell’sequationsarevalidandinwhichlightpropagateswithconstant
velocityinalldirections.Scientiststriedtodeterminerelativevelocityoflightwith
respect to earth. For this purpose, the velocity of earth relative to some privileged
frame,knownasetherframe,wasessential.Therefore,scientistsperformedvarious
experiments involving electromagnetic waves. In this regard, the best and most
important experiment was performed by A.A. Michelson in 1881.
InChap.2,theauthorhasdescribedtheMichelson–Morleyexperiment.In1817,
J.A. Fresnel proposed that light is partially dragged along by a moving medium.
Using the ether hypothesis, Fresnel provided the formula for the effect of moving
medium on the velocity of light. In 1951, Fizeau experimentally confirmed this
effect.TheauthorhasdiscussedFizeau’sexperiment.Later,theauthorprovidedthe
relativistic concept of space and time. To develop the special theory of relativity,
Einsteinusedtwofundamentalpostulates:theprincipleofrelativityorequivalence
vii
viii Preface
andtheprincipleofconstancyofthespeedoflight.Theprincipleofrelativitystates
that the laws of nature are invariant under a particular group of space-time coor-
dinate transformations. Newton’s laws of motion are invariant under Galilean
transformation,butMaxwell’sequationsarenot,andEinsteinresolvedthisconflict
by replacing Galilean transformation with Lorentz transformation.
In Chap. 3, the author has provided three different methods for constructing
Lorentz transformation between two inertial frames of reference. The Lorentz
transformations have some interesting mathematical properties, particularly in
measurements of length and time. These new properties are not realized before.
Mathematical properties of Lorentz transformations—length contraction, time
dilation, relativity of simultaneity and their consequences—are discussed in
Chaps.4and5.Also,someparadoxesliketwinparadox,car–garageparadoxhave
been discussed in these chapters.
Various types of intervals as well as trajectories of particles in Minkowski’s
four-dimensional world can be described by diagrams known as space–time dia-
grams. To draw the diagram, one has to use a specific inertial frame in which one
axisindicatesthespacecoordinateandtheothereffectivelythetimeaxis.Chapter6
includes the geometrical representation of space–time.
Chapter 7 discusses the relativistic velocity transformations, relativistic accel-
erationtransformationsandrelativistictransformationsofthedirectioncosines.The
author has also discussed some applications of relativistic velocity and velocity
additionlaw:(i)explanationofindexofrefractionofmovingbodies(Fizeaueffect),
(ii) aberration of light and (iii) relativistic Doppler effect.
In Newtonian mechanics, physical parameters like position, velocity, momen-
tum,accelerationandforcehavethree-vectorforms.Itisnaturaltoextendthethree-
vectorformtoafour-vectorformintherelativisticmechanicsinordertosatisfythe
principle of relativity. Therefore, a fourth component of all physical parameters
should be introduced. In Euclidean geometry, a vector has three components. In
relativity, a vector has four components.
The world velocity or four velocity, Minkowski force or four force, four
momentumandrelativistickineticenergyhavebeendiscussedinChaps.8and10.
It is known that the fundamental quantities, length and time are dependent on the
observer. Therefore, it is expected that mass would be an observer-dependent
quantity. The author provides three methods for defining relativistic mass formula
in Chap. 9. The relativistic mass formula has been verified by many scientists;
however, he has discussed the experiment done by Guye and Lavanchy.
A short discussion on photon and Compton effect is provided in Chap. 11. A
force,ingeneral,isnotproportionaltoaccelerationincaseofthespecialtheoryof
relativity. The author has discussed force in the special theory of relativity and
covariant formula of the Newton’s law. Relativistic Lagrangian and Hamiltonian
are discussed in Chap. 12. He has also explained Lorentz transformation offorce
and relativistic harmonic oscillation. When charges are in motion, the electric and
magnetic fields are associated with this motion, which will have space and time
variation. This phenomenon is called electromagnetism. The study involves time-
dependentelectromagneticfieldsandthebehaviourofwhichisdescribedbyasetof
Preface ix
equations called Maxwell’s equations. In Chap. 13, the author has discussed rela-
tivistic electrodynamics of continuous medium. In Chap. 14, the author has pro-
vided a short note on relativistic mechanics of continuous medium (continua).
Finally,inappendixes,theauthorhasprovidedsomepreliminaryconceptsoftensor
algebra and action principle.
The author expresses his sincere gratitude to his wife Mrs. Pakizah Yasmin for
her patience and support during the gestation period of the manuscript. It is a
pleasure to thank Dr. M. Kalam, Indrani Karar, Iftikar Hossain Sardar and Mosiur
Rahaman for their technical assistance in preparation of the book. Finally, he is
thankful to painter Ibrahim Sardar for drawing the plots.
Farook Rahaman
Contents
1 Pre-relativity and Galilean Transformation . . . . . . . . . . . . . . . . . 1
1.1 Failure of Newtonian Mechanics. . . . . . . . . . . . . . . . . . . . . 1
1.2 Galilean Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Galilean Transformations in Vector Form. . . . . . . . . . . . . . . 4
1.4 Non-inertial Frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Galilean Transformation and Laws of Electrodynamics . . . . . 9
1.6 Attempts to Locate the Absolute Frame. . . . . . . . . . . . . . . . 9
2 Michelson–Morley Experiment and Velocity of Light. . . . . . . . . . 11
2.1 Attempts to Locate Special Privileged Frame . . . . . . . . . . . . 11
2.2 The Michelson–Morley Experiment (M–M) . . . . . . . . . . . . . 11
2.3 Phenomena of Aberration: Bradley’s Observation . . . . . . . . . 15
2.4 Fizeau’s Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 The Relativistic Concept of Space and Time . . . . . . . . . . . . 18
3 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Postulates of Special Theory of Relativity . . . . . . . . . . . . . . 21
3.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Lorentz Transformation Between Two Inertial
Frames of Reference (Non-axiomatic Approach) . . . 21
3.2.2 Axiomatic Derivation of Lorentz Transformation. . . 24
3.2.3 Lorentz Transformation Based on the Postulates
of Special Theory of Relativity. . . . . . . . . . . . . . . 27
3.3 The General Lorentz Transformations . . . . . . . . . . . . . . . . . 29
3.4 Thomas Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Mathematical Properties of Lorentz Transformations. . . . . . . . . . 35
4.1 Length Contraction (Lorentz-Fitzgerald Contraction) . . . . . . . 35
4.2 Time Dilation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Relativity of Simultaneity. . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Twin Paradox in Special Theory of Relativity. . . . . . . . . . . . 40
xi
xii Contents
4.5 Car–Garage Paradox in Special Theory of Relativity. . . . . . . 41
4.6 Real Example of Time Dilation . . . . . . . . . . . . . . . . . . . . . 42
4.7 Terrell Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 More Mathematical Properties of Lorentz Transformations . . . . . 53
5.1 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 The Interval Between Two Events Is Invariant Under
Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 Geometric Interpretation of Spacetime. . . . . . . . . . . . . . . . . . . . . 63
6.1 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Some Possible and Impossible World Lines . . . . . . . . . . . . . 65
6.3 Importance of Light Cone . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.4 Relationship Between Spacetime Diagrams
in S and S1 Frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.5 Geometrical Representation of Simultaneity,
Space Contraction and Time Dilation . . . . . . . . . . . . . . . . . 69
6.5.1 Simultaneity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.5.2 Space Contraction. . . . . . . . . . . . . . . . . . . . . . . . 69
6.5.3 Time Dilation. . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7 Relativistic Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . 73
7.1 Relativistic Velocity Addition. . . . . . . . . . . . . . . . . . . . . . . 73
7.2 Relativistic Velocity Transformations. . . . . . . . . . . . . . . . . . 75
7.3 Relativistic Acceleration Transformations. . . . . . . . . . . . . . . 76
7.4 Uniform Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.5 Relativistic Transformations of the Direction Cosines . . . . . . 78
7.6 Application of Relativistic Velocity and Velocity
Addition Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.6.1 The Fizeau Effect: The Fresnel’s Coefficient
of Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.6.2 Aberration of Light . . . . . . . . . . . . . . . . . . . . . . . 80
7.6.3 Relativistic Doppler Effect . . . . . . . . . . . . . . . . . . 81
8 Four Dimensional World. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.1 Four Dimensional Space–Time. . . . . . . . . . . . . . . . . . . . . . 89
8.2 Proper Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.3 World Velocity or Four Velocity. . . . . . . . . . . . . . . . . . . . . 92
8.4 Lorentz Transformation of Space and Time in Four
Vector Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97