Table of Contents
Preface vii
Special Theory of Relativity 1
1 Galilean Transformations 3
1.1 Introduction 3
1.2 Galilean Transformations 4
1.3 Newtonian Mechanics and the Principle of Galilean Relativity 5
1.3.1 Electrodynamics and Galilean Relativity 6
2 Lorentz Transformations 9
2.1 The Two Postulates of Relativity 9
2.2 Lorentz-Fitz Gerald Contraction 12
2.3 Time Dilation 13
2.4 Proper Time 15
2.5 Transformation of Particle Velocities 16
3 Relativistic Mechanics 17
3.1 Momentum and Energy 17
3.2 Application of Relativistic Mechanics 22
3.2.1 Doppler Shift 22
3.3 Scattering Kinematics 23
3.3.1 Two-Particle Scattering 23
3.4 Motion of a Charged Particle in a Uniform Magnetic Field 27
3.5 Problems Related to Chapters 2 and 3 30
4 Loitentz Transformations (General Case) 39
4.1 Lorentz Transformations (General Case) (in Four Dimensional Space Time) 39
4.2 Light Cone 43
4.3 Lorentz Boost Transformation 44
4.4 Vectors and Tensors 45
5 Four-Velocity: Minkowski Force 49
5.1 Four-Velocity 49
5.2 Minkowski Force 50
5.3 Problems 53
6 Covariant Form Of Electrodynamics 55
6.1 Electromagnetic Field Tensor 55
6.1.1 Lorentz Force 57
6.2 Transformation of E and B Under Lorentz Transformation 59
6.3 Electromagnetic Field of a Moving Charge 60
6.4 Scalar and Vector Potential of a Moving Charge 61
6.5 Covariant Form of Maxwell's Equations 62
6.6 Energy-Momentum Tensor of an Electromagnetic Field 63
6.7 Problems 65
7 Spin 71
7.1 Relativistic Equations of Motion for Spin in a Uniform External Electromagnetic Field 71
7.2 Problem 75
8 Space Time Groups and their Representations 77
8.1 Matrix Representation of Lorent-z Transformation 77
8.2 Invariance: Representations of a Group 80
8.3 Poincaré Group and its Representations 82
8.4 Poincaré Group and Physical States 93
8.5 Scale Invariance 97
8.5.1 Scale Transformation 98
8.5.2 Conformal Group 99
8.6 Energy Momentum Tensor 101
8.7 Super-symmetry (SUSY) 103
8.8 SUSY Quantum Mechanics 105
8.8.1 SUSY Vacuum 107
8.9 Super Lie Algebra 107
8.9.1 Two-component Spinors 109
8.9.2 Spinor Charges 112
8.10 Super symmetric Multiplets 116
8.11 Problems 120
General Theory of Relativity: Riemannian Geometry; Curved Space Time 131
9 Tensor Analysis and Affine Connection 133
9.1 Introduction 133
9.2 Metric Tensor, Tensors, Tensor Densities 134
9.2.1 Tensors. Tensor Densities 136
9.3 Covariant Derivative, Affine Connection, Christoffel Symbol 139
9.4 Gradient, Curl and Divergence 145
9.5 Problems 146
10 Geodesic and Equivalence Principle 151
10.1 Geodesic Equation 151
10.2 Equivalence Principle 153
10.3 Weak Field and Low Velocity Limit: Gravity as a Metric Phenomenon 155
10.4 Problems 158
11 Curvature Tensor and Einstein's Field Equations 161
11.1 Curvature Tensor 161
11.2 Einstein's Field Equations 165
11.3 Newtonian Limit of Field Equations 167
11.4 Problems 168
12 The Schwarzsohild, Friedmann Robertson Walker Metric 173
12.1 Introduction 173
12.2 The Schwarzschild Metric 174
12.3 Friedmannn-Robertson-Walker (FRW) Metric 178
12.4 Cosmology ISO
12.4.1 Cosmological Principle ISO
12.4.2 Standard Model of Cosmology 182
12.4.3 Friedmann-Lemaitre Equations 184
12.4.4 Observational Cosmology 186
12.5 Problems 191
Appendix 199
A.1 Covariant Derivative; Parallel Displacement of a Vector 199
A.2 Hot Big Bang: Thermal History of the Universe 201
A.2.1 Thermal Equilibrium 201
A.2.2 The Radiation Era 203
A.3 Fundamental Units 208
Bibliography 213
