Theory of Relativity

Text-only Preview

Theory of Relativity
Shashikant Phatak
National Institute of Science Education and Research
Bhubaneswar 751 005
November 15, 2010

2

Contents
1 Preliminaries
9
1.1 Space And Time . . . . . . . . . . . . . . . . . . . . . .
9
1.1.1 Space
. . . . . . . . . . . . . . . . . . . . . . . .
9
1.1.2 Time . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.1.3 Space-Time Continuum . . . . . . . . . . . . . .
11
1.2 Inertial Frames . . . . . . . . . . . . . . . . . . . . . . .
12
1.3 Galilean Relativity . . . . . . . . . . . . . . . . . . . . .
17
1.4 Light And Absolute space . . . . . . . . . . . . . . . . .
19
1.5 Electrodynamics . . . . . . . . . . . . . . . . . . . . . .
21
1.6 Electromagnetic waves and ether . . . . . . . . . . . .
24
1.7 Measurement of Velocity of Light . . . . . . . . . . . .
26
1.7.1 Galileo:
. . . . . . . . . . . . . . . . . . . . . . .
26
1.7.2 Romer-Huygen: . . . . . . . . . . . . . . . . . . .
27
1.7.3 Broadley: . . . . . . . . . . . . . . . . . . . . . .
28
1.7.4 Fizeau: . . . . . . . . . . . . . . . . . . . . . . . .
29
1.7.5 Electromagnetic Method . . . . . . . . . . . . . .
30
1.7.6 Latest Measurements . . . . . . . . . . . . . . .
30
1.8 Electrodynamics and Galilean relativity . . . . . . . .
31
3

4
CONTENTS
2 Search of Ether and Its frame
35
2.1 Michelson-Morley experiment . . . . . . . . . . . . . .
36
2.2 Electromagnetic experiments . . . . . . . . . . . . . . .
39
2.3 Extra-terrestrial Observations . . . . . . . . . . . . . .
41
2.4 Fizeau’s Experiment on Moving Media . . . . . . . . .
42
2.5 Ether Drag Theory . . . . . . . . . . . . . . . . . . . . .
42
2.6 Emission Theories . . . . . . . . . . . . . . . . . . . . .
43
3 Theoretical Developments Before Special theory
45
3.1 Lorentz-Fitzgerald Hypothesis . . . . . . . . . . . . . .
47
3.2 Voigt and Electromagnetic Wave Equation . . . . . . .
48
3.3 Lorentz’s theorem of Corresponding States
. . . . . .
49
3.4 Poincare’s Contribution . . . . . . . . . . . . . . . . . .
50
4 Special Theory of Relativity
53
4.1 The Principles of special relativity . . . . . . . . . . . .
55
4.2 The Lorentz Transformations . . . . . . . . . . . . . . .
57
4.3 Velocity Addition Formula
. . . . . . . . . . . . . . . .
61
4.4 Four Vectors . . . . . . . . . . . . . . . . . . . . . . . .
64
4.5 Group of Lorentz Transformations . . . . . . . . . . . .
67
4.6 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . .
68
4.7 Length Contraction . . . . . . . . . . . . . . . . . . . .
72
5 Space-Time Geometry
75
5.1 Scalars, Vectors and Tensors . . . . . . . . . . . . . . .
76
5.2 Tensors and Special Theory of Relativity . . . . . . . .
80
5.3 Constructing Tensors . . . . . . . . . . . . . . . . . . .
81

CONTENTS
5
5.4 Minkowski Space-Time Diagrams . . . . . . . . . . . .
84
5.5 Boost As A Rotation . . . . . . . . . . . . . . . . . . . .
86
5.6 Time Dilation and Length Contraction . . . . . . . . .
87
5.6.1 Time Dilation . . . . . . . . . . . . . . . . . . . .
87
5.6.2 Length Contraction . . . . . . . . . . . . . . . .
89
6 Newtonian Dynamics
91
6.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
6.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . . .
92
6.3 Reaction Kinematics . . . . . . . . . . . . . . . . . . . .
95
6.3.1 Particle decay . . . . . . . . . . . . . . . . . . . .
95
6.3.2 Two Particle Reactions . . . . . . . . . . . . . . .
96
6.4 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . .
97
6.5 Force Equation . . . . . . . . . . . . . . . . . . . . . . . 100
6.6 Currents and Densities . . . . . . . . . . . . . . . . . . 102
6.7 Energy-Momentum Tensor . . . . . . . . . . . . . . . . 105
7 Electrodynamics
109
7.1 Electromagnetic Current and Potential . . . . . . . . . 109
7.2 Electromagnetic Fields . . . . . . . . . . . . . . . . . . 111
7.3 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . 115
7.4 Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . 116
7.5 Energy Momentum Tensor . . . . . . . . . . . . . . . . 117
8 Relativistic Lagrangian Formulation
121
8.1 Particle Dynamics . . . . . . . . . . . . . . . . . . . . . 121
8.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6
CONTENTS
9 Principle of Equivalence
123
9.1 Inertial and Gravitational Masses . . . . . . . . . . . . 125
9.2 Principle of Equivalence . . . . . . . . . . . . . . . . . . 127
9.3 Equivalence Principle and Gravitational Field . . . . . 133
9.4 Weak Field (Newtonian) Limit
. . . . . . . . . . . . . . 139
9.5 Gravitational Time Dilation . . . . . . . . . . . . . . . . 143
9.6 Experimental Verification of Eq. Principle . . . . . . . 146
9.6.1 Measurement of Gravitational Time Dilation . . 147
9.6.2 Gravitational Frequency Shifts of Stellar Lines
149
9.6.3 Terrestrial Measurement of Gravitational Red
Shift . . . . . . . . . . . . . . . . . . . . . . . . . 150
10 Gravitation and Laws of Physics
153
10.1 Principle of General Covarience . . . . . . . . . . . . . 154
10.2 Transformation of Affine Connection . . . . . . . . . . 156
10.3 Covariant Derivatives . . . . . . . . . . . . . . . . . . . 158
10.4 Tensor Densities . . . . . . . . . . . . . . . . . . . . . . 162
10.5 Rules for Introducing Gravitation . . . . . . . . . . . . 164
10.6 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . 165
11 Rotation Group
167
11.1 Rotations in Two Dimensions: SO(2) . . . . . . . . . . 169
11.1.1Generator of SO(2) . . . . . . . . . . . . . . . . . 171
11.1.2Representations of SO(2) . . . . . . . . . . . . . 172
11.1.3Orthogonality and Completeness
. . . . . . . . 173
11.2 Parametrization of Rotation Matrices . . . . . . . . . . 174
11.2.1Axis-Angle Parametrization . . . . . . . . . . . . 174

CONTENTS
7
11.2.2Euler Angle Parametrization . . . . . . . . . . . 175
11.3 Generators of SO(3) . . . . . . . . . . . . . . . . . . . . 176
11.3.1Algebra of SO(3) . . . . . . . . . . . . . . . . . . 178
11.3.2Casimir Operators . . . . . . . . . . . . . . . . . 179
11.4 Rotation Group and SU(2) group
. . . . . . . . . . . . 180
11.5 Representations of Rotation Group . . . . . . . . . . . 182
12 The Poincare and Lorentz Groups
187
12.1 Homogeneous Proper Lorentz Group . . . . . . . . . . 188
12.2 Poincare Group . . . . . . . . . . . . . . . . . . . . . . . 191
12.3 Generators of Poincare Group . . . . . . . . . . . . . . 192
12.3.1Translations . . . . . . . . . . . . . . . . . . . . . 192
12.3.2Lorentz Transformations — Rotations . . . . . . 194
12.3.3Lorentz Transformations — Boosts . . . . . . . 196
12.4 Algebra of Poincare group . . . . . . . . . . . . . . . . . 197
12.5 Isomorphism of Lorentz group with SL(2) . . . . . . . 199
12.5.1Rotations . . . . . . . . . . . . . . . . . . . . . . 200
12.5.2Boosts . . . . . . . . . . . . . . . . . . . . . . . . 201
12.5.3Spinors . . . . . . . . . . . . . . . . . . . . . . . 202
12.6 Representations of Lorentz group . . . . . . . . . . . . 202
12.6.1Finite Dimensional Representations . . . . . . . 203
12.6.2Unitary Representations . . . . . . . . . . . . . 205
12.7 Unitary Representations of Poincare Group . . . . . . 206
12.7.1Null Vector Representations . . . . . . . . . . . 207
12.7.2Time-like Vector Representations . . . . . . . . 207
12.7.3The Spin . . . . . . . . . . . . . . . . . . . . . . . 208

8
CONTENTS
12.7.4Light-like Vector Representations . . . . . . . . 209
12.7.5Space-like Vector Representations: . . . . . . . 211

Chapter 1
Preliminaries
1.1
Space And Time
The ideas of space and time are generally taken for granted but one
needs to be more specific about what one means by space and time.
This is because when one talks of dynamics, one is considering
change of position of a particle or a body in time. So, one needs to
define the meaning of position as well as time precisely. There are
some philosophical questions associated with the concept of space
as well as time.
1.1.1
Space
When one thinks of a position of a particle, one thinks of space, en-
compassing everything. The question is, does the space exist even
in the absence of particles? That is, is space some tangible ob-
ject or is it just nothing filling everything? Consider two observers
moving with respect to each other. Each of these observers may be
studying the motion of a particle. So one can consider spaces for
each observer. Are these two spaces interpenetrating and indepen-
dent of each other? These are some of the philosophical questions
arising when one considers the idea of space.
Newton postulated that there is a space, which he called as ab-
solute space, encompassing whole universe. In this space, one can
9

10
CHAPTER 1. PRELIMINARIES
consider a coordinate system specifying each point in this space
and the position of a particle or a body is described by the val-
ues of its coordinates. Newton defined the absolute space to be
the space in which the distant stars appear to be stationary, that
is their angular positions appear to be fixed. The absolute space
exists even in the absence of material particles. That is, the abso-
lute space exists by itself. It is, however, not a material object and
cannot be detected by itself. When one studies dynamics of point
particles or extended bodies, one studies the change in the position
of the particle or the body as a function of time.
Physically, the absolute space may be visualized as a large box
of infinitely thin sides. The motion of a particle or an object is then
the time-dependence of the position of the particle or the object in
that box. The absolute space is then such a box with its sides be-
coming infinitely large. Obviously, the box, or the absolute space,
does not have any material content but it is required to describe the
motion or dynamics of objects. The laws of dynamics then essen-
tially describe the time dependence of the position of the particle
or the object.
Now, we may think of another box inside the first box and con-
sider this second box moving with respect to the first box. As be-
fore, we can consider these boxes to be infinitely large. We can
associate an observer with each of these boxes. Now these two ob-
servers may be observing a particle. The dynamics of the particles
may be described in terms of the time-dependence of the position
of the particle as observed by the two observers. So, we have now
two spaces defined, one associated with the first box, the absolute
space and the second associated with the second box. We can gen-
eralize and think of any number of such spaces. Does that mean
there are as many spaces as there are observers?
According to Newton, these is only one space, the absolute space
and a coordinate system may be associated with each of the moving
observers. The motion of the observer defines the relation between
the coordinates in the coordinate system of the observer and the
coordinates in the absolute space. This means that the coordinate
system of the observer plays a special role.