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The use of “implied warranty” and certain “exclusions” vary from state to state, and might not apply to the purchaser of this product. special.indb ii 28-04-2016 20:40:16 THE SPECIAL THEORY OF RELATIVITY An Introduction Dennis Morris MERCURY LEARNING AND INFORMATION Dulles, Virginia Boston, Massachusetts New Delhi special.indb iii 28-04-2016 20:40:16 Reprint and Revision Copyright ©2016 by Mercury Learning And Information LLC All rights reserved. Original title and copyright: Empty Space Is Amazing Stuff: The Special Theory of Relativity and the Nature of Space. Copyright ©2013 by The Pantaneto Press. All rights reserved. Published by The Pantaneto Press. 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The sole obligation of Mercury Learning and Information to the purchaser is to replace the book, based on defective materials or faulty workmanship, but not based on the operation or functionality of the product. special.indb iv 28-04-2016 20:40:16 CONTENTS Introduction ix Chapter 1 An Overview of the Theory of Special Relativity 1 1.1 Physics is Invariant Under Rotation 1 1.2 Space and Time are Not Separate Things 2 1.3 Mass Dimensions 3 1.4 Directions in Space-time 3 1.5 The Constancy of the Speed of Light 5 1.6 Symmetry and Noether’s Theorem 11 1.7 Noether’s Theorem 12 Worked Examples 14 Exercises 14 Chapter 2 The Results of Special Relativity Without Detailed Explanation 15 Exercises 29 Chapter 3 Special Relativity in Physics 33 3.1 The History of Special Relativity 34 Chapter 4 The Nature of Space 41 4.1 Possible Types of Distance Functions 42 4.2 What is Empty Space? 44 4.3 Views of the Nature of Space 47 Exercises 53 Chapter 5 Physical Constants 55 Chapter 6 Numbers 61 Exercises 63 Chapter 7 Comments on Matrices 65 7.1 Symmetric Matrices and Anti-symmetric Matrices 67 7.2 Rotation Matrices 69 Exercises 71 Chapter 8 Introduction to Finite Groups 73 8.1 How to Find Finite Groups 74 8.2 Sub-groups and Types of Rotation 77 8.3 Geometric Spaces from Finite Groups 78 8.4 More on Groups 78 8.5 Non-commutativity and the Space in Which We Sit 78 special.indb v 28-04-2016 20:40:16 vi • The Special Theory of Relativity 8.6 The Ginite Group Matrix 79 8.7 Special Relativity is in C2 80 8.8 The Classification of the Finite Groups 82 Exercises 83 Chapter 9 Trigonometric Functions 85 9.1 Circles Defined 86 9.2 Rotation Matrices 89 9.3 The Two Types of 2-dimensional Space 92 9.4 Trigonometric Identities 93 9.5 Gamma 96 9.6 The Graphs of the Trigonometric Functions 99 Exercises 102 Chapter 10 Introduction to Vectors 105 10.1 Orthogonality and Perpendicularity 114 10.2 Differentiation (the Standard View) 115 10.3 Differentiation with Matrices 117 10.4 Differentiation of Vector Fields and Scalar Fields 118 10.5 Potentials 121 10.6 Five Fundamental Vectors 122 Exercises 124 Chapter 11 The Nature of Velocity 125 11.1 Acceleration 125 11.2 A Clockwise Angle Equals an Counterclockwise Angle 126 Exercises 128 Chapter 12 Simultaneity 129 12.1 Another View 130 12.2 Cause and Effect – Ordered Events 131 12.3 Back to Simultaneity 134 12.4 Comparison of Observations 135 Worked Examples 136 Exercises 138 Chapter 13 The Lorentz Transformation 139 13.1 Circles 139 13.2 The Lorentz Transformation 141 13.3 Time Dilation 142 13.4 We All Move Through Space-time at The Speed of Light 146 13.5 The Feynman Clock 148 13.6 Time Dilation by Distance Function 148 13.7 Experimental Evidence 149 special.indb vi 28-04-2016 20:40:16 Contents • vii 13.8 Length Contraction 151 13.9 Experimental Evidence 154 13.10 Getting Technical 154 13.11 A Few Useful Identities 155 Worked Examples 156 Exercises 156 Chapter 14 Velocity and Acceleration Transformations 159 14.1 Acceleration Transformation 161 14.2 First Derivation of the Acceleration Transformation 161 14.3 Second Derivation of the Acceleration Transformation 162 Exercises 163 Chapter 15 The Nature of Straight Lines and the Twin Paradox 165 15.1 The Calculus of Variations 165 15.2 Another View 170 15.3 Yet Another View 171 15.4 The Twin Paradox 171 15.5 The Pole and Barn Paradox 174 Exercises 175 Chapter 16 4–Vectors 177 16.1 The Standard Presentation of 4-vectors 178 16.2 Differentiation of 4-vectors 181 16.3 4-velocity 182 16.4 4-acceleration 184 Exercises 189 Chapter 17 4-Momentum and Relativistic Mass 191 17.1 About Mass 192 17.2 Momentum and Energy Conservation in Special Relativity 193 17.3 Colliding Sticky Buds and the Center of Mass Reference Frame 201 17.4 Center of Mass Reference Frame 203 17.5 4-Force 203 17.6 Electromagnetism 206 Worked Examples 207 Exercises 208 Chapter 18 Doing it With Matrices 211 18.1 Differentiating the Velocity Vector 212 18.2 Momentum 218 18.3 Force 220 special.indb vii 28-04-2016 20:40:16 viii • The Special Theory of Relativity 18.4 A Little Food for Thought 221 18.5 Five Fundamental Vectors Again 222 Chapter 19 Electromagnetism 225 19.1 The Maxwell Equations I 227 19.2 The Invariance of Electric Charge 232 19.3 Magnetic Effects 234 19.4 Electromagnetic Waves 237 19.5 Moving Electric Charges 239 19.6 4-tensors 240 19.7 4-potential 241 19.8 The Maxwell Equations II 249 19.9 The Components of Electromagnetism 252 Exercise 253 Chapter 20 Quaternions 255 20.1 Introduction to Quaternions 257 20.2 Quaternion Matrix Forms 259 20.3 The C2 × C2 Finite Group and the Space in Which We Sit 262 20.4 Non-commutative Spaces 264 20.5 The Space in Which We Sit 267 Exercise 269 Chapter 21 Electromagnetism With Quaternions 271 21.1 Why No Anti-matter in our Classical universe 274 21.2 The Inhomogeneous Maxwell Equations 274 Chapter 22 4-Dimensional Space-time and the Lorentz Group 277 22.1 The Lorentz Group 278 22.2 Physics is Invariant Under All Rotations in the Lorentz Group 278 22.3 The Standard Presentation of the Lorentz Group, SO(3, 1), in 4 space 280 22.4 Splitting the Lorentz Group 284 22.5 A General View 285 Chapter 23 The Expanding Universe and the Cosmic Background Radiation 287 23.1 The Cosmic Background Radiation 293 Chapter 24 Concluding Remarks 297 Bibliography 301 Index 303 special.indb viii 28-04-2016 20:40:16 INTRODUCTION This book is written to give interested readers or fi rst year under- graduates a comprehensive understanding of the theory of special relativity.