Modern Aspects Of Relativity

by Eckehard W Mielke

Overview

Today, Relativity is becoming an integrated aspect of engineering fields. Its application to the Global Positioning System (GPS), extends in usage from smart watches to the navigation of cars, airplanes (drones) and even autonomous tractors. In rather expensive particle accelerators, physicists are everyday 'playing' with Relativistic Billiards, common to the betatrons of cancer therapy using electrons. Computer programs, such as 'ray tracing' methods, are enhanced to simulate objects in relativistic motion, which now offer us relativistic visualizations of accretion disks around compact, astrophysical objects like Black Holes.Against the backdrop of the applications explained throughout the chapters, this book takes on a practical and intuitive approach in introducing the Lorentz invariance of light propagation and space-time concepts. The book begins with simple mathematics, like the classical Pythagoras formula for energy-momentum 'triangles'. Later, readers will find the intuitive vector calculus reemerging in the expansion of full relativistic expressions. Prepared with instructive diagrams of recent experiments, even the layperson can grasp the essential study of Relativity and marvel at its applications within this book.

Product Details

ISBN-13:	9789811244049
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/21/2022
Pages:	208
Product dimensions:	6.00(w) x 9.00(h) x 0.50(d)

Table of Contents

Preface v

1 Introduction: "Dawn of Time" 1

1.1 The Concept of Time in Ancient Cultures 1

1.2 The Clock Story 3

1.3 The Definition of the Second 6

1.3.1 Precursors of the second 7

1.3.2 First standards 8

1.4 What is the Most Accurate Clock in the Universe? 9

1.5 A Brief Biography of Albert Einstein 11

1.5.1 1905: A special year for physics 14

1.6 Einstein and Mileva Marie: A Failed Collaboration? 17

1.6.1 Ludek Zakel: A son "in theory" 19

References 21

2 Invariant Light Velocity 23

2.1 Concepts of Light Propagation 23

2.2 Romer and the Moons of Jupiter 26

2.3 Terrestrial Measurements of Fizeau 28

2.4 The Michelson-Morley Interferometer 29

2.5 Decisive Lorentz Contraction 33

2.6 Isotropy of Light 34

2.6.1 Comparison with the velocity of neutrinos and gravitational waves 34

2.7 Superluminal Velocities? 36

2.7.1 Tunnel effect 37

2.8 Homework 38

References 38

3 Time Dilation 39

3.1 Idealized Einstein Clock 39

3.1.1 Comparison of light, clocks in different inertial frames 40

3.2 Time Dilation of Muons 42

3.3 Gravitational Time Dilation 44

3.3.1 Hafele and Keating experiment 45

3.4 The Apparent Twin Paradox 46

3.5 Time Dilation Measured by Optical Clocks 48

3.6 The GPS Navigation System 51

3.7 Homework 53

References 53

4 Relativistic Shape of Mechanics 55

4.1 Relativistic Momentum and Force 55

4.2 Relativistic Energy 56

4.2.1 Unitary transformation in QED 58

4.2.2 Examples 60

4.3 Homework 60

Reference 61

5 Spacetime Diagrams 63

5.1 Spacetime: Minkowski Diagrams 65

5.1.1 Observer at rest 67

5.1.2 Observer in relative motion: Lorentz transformations 68

5.2 Spacetime Interval 69

5.2.1 The calibration of the axes 69

5.2.2 Example: Space rocket 71

5.2.3 Hyperbolic intersections 72

5.2.4 Lorentz contraction 74

5.3 Simultaneity 74

5.4 Temporal Order: Past, Present, Future, and Causality 75

5.4.1 Times machines? 77

5.5 Homework 78

References 80

6 Rapidity 81

6.1 Galilean Relativity 81

6.2 Lorentz Transformations 82

6.3 Rapidity in 2D 84

6.4 Generalized Lorentz Transformations 86

6.5 Velocity: The General Lorentz Transformations 87

6.5.1 Aberration 89

6.6 Fizeau Interferometer 91

6.6.1 "Aether drag" experiment in the undergraduate lab 93

6.6.2 Drag coefficient from Einstein's addition of velocities 93

6.7 Homework 95

References 96

7 Relativistic Kinematics 97

7.1 Four-Dimensional Cartesian Coordinates 97

7.1.1 Invariant distance 97

7.1.2 Proper time 98

7.1.3 Time dilation 98

7.2 Four-D Velocities 98

7.2.1 Quadratic invariant 99

7.3 Acceleration in 4D 99

7.4 Relativistic Momentum 100

7.5 Quadratic Invariant for the Momentum 101

7.6 Relativistic Energy Corrections 102

7.7 Relativistic Shape of Mechanics 103

7.8 Spinning Top 106

7.9 Group Velocity of a Particle 107

7.10 Homework 108

References 109

8 Relativistic Billiards 111

8.1 Collision of Identical Particles 111

8.1.1 Elastic collisions 114

8.2 Mandelstam Variables 117

8.2.1 Center of mass 120

8.3 Inelastic Collision of Particles in the CM Frame 121

8.3.1 Scattering angle 125

8.4 Compton Effect 126

8.5 Homework 129

Reference 130

9 Electrodynamics in 4D 131

9.1 The Poincare Transformations 131

9.2 Electric Currents in 4D 133

9.3 Unifying Vector Potential 134

9.3.1 Aharonov-Bohm effect 135

9.3.2 Electromagnetic fields derived from potentials 136

9.3.3 Magnetic moments 138

9.4 Maxwell's Equations in Matter 138

9.5 Electric and Magnetic Fields of a Charge in Uniform Motion 139

9.5.1 Nanoscopically thin nickel needle as "fake" magnetic monopole 141

9.6 Lorentz Force 143

9.6.1 Synchrotron 145

9.7 Homework 147

References 148

10 Relativistic Ray Tracing 149

10.1 Relativistic Doppler Effect 149

10.2 Aberration 152

10.2.1 Bradley discovered the aberration of light 153

10.2.2 Superluminal velocities in twin jets? 154

10.3 Invisibility of the Lorentz Contraction 155

10.3.1 Lampa-Terrell-Penrose rotation 158

10.4 Visualization: Angular Compression and Increased Intensity 158

10.5 Ray Tracing in General Relativity 161

10.5.1 Relativistic jet emanating from the galaxy M87 162

10.6 Homework 164

References 164

Appendix A Spacetime Groups 167

A.1 Group Axioms 167

A.2 Rotations and Euler Angles 167

A.3 Homework 169

A.4 Inhomogeneous Galilean Group 169

A.5 Homework 170

A.6 Poincare Transformations 170

A.7 Lorentz Group 170

A.7.1 Lorentz group parameters 171

A.7.2 Algebraic conditions 172

A.7.3 Lorentz factor 173

A.8 Homogeneous and Isotropic Spacetime Transformations 173

Reference 173

Appendix B Units of the International System 175

B.1 Homework 180

References 180

Bibliography 181

Index 189