Illustrated Special Relativity Through Its Paradoxes: Standard Edition: A Fusion of Linear Algebra, Graphics, and Reality

Description

This accessible work, with its plethora of full-color illustrations by the author, shows that linear algebra --- actually, 2x2 matrices --- provide a natural language for special relativity. The book includes an overview of linear algebra with all basic definitions and necessary theorems. There are exercises with hints for each chapter along with supplemental animations at special-relativity-illustrated.com. Since Einstein acknowledged his debt to Clerk Maxwell in his seminal 1905 paper introducing the theory of special relativity, we fully develop Maxwell's four equations that unify the theories of electricity, optics, and magnetism. Using just two laboratory measurements, these equations lead to a simple calculation for the frame-independent speed of electromagnetic waves in a vacuum. (Maxwell himself was unaware that light was a special electromagnetic wave.) Before analyzing the paradoxes, we establish their linear algebraic context. Inertial frames become ( 2-dimensional vector spaces ) whose ordered spacetime pairs ( x , t ) are linked by “line-of-sight” linear transformations. These are the Galilean transformations in classical physics, and the Lorentz transformations in the more general relativistic physics. The Lorentz transformation is easily derived once we show how a novel swiveled line theorem, ( a geometric concept ) is equivalent to the speed of light being invariant for all observers a ( a physical concept ). Six paradoxes are all analyzed using Minkowski spacetime diagrams. These are (1) The Accommodating Universe paradox, (2) Time and distance asymmetry between frames, (3) The Twin paradox, (4) The Train-Tunnel paradox, (5) The Pea-Shooter paradox, and the lesser known (6) Bug-Rivet paradox. The Bug-Rivet paradox, animated by the author at Special-Relativity-Illustrated.com, presents another proof that rigidity is incompatible with special relativity. E = mc2 finds a simple derivation using only the relativistic addition of speeds ( the Pea-Shooter paradox ), conservation of momentum, and a power series. Finally, three appendices contain the self-contained overview of linear algebra, key properties of hyperbolic functions used to add relativistic speeds graphically, and a deconstruction of a moving train that proves the non-intuitive fact that when a moving train pulls into a station, its front car is always younger than its rear car, even though the front car has been in the station for a longer time. Both this standard edition (red cover) and the Deluxe edition (blue cover) contain all the previous topics. The Deluxe edition (blue cover) will add 74 pages containing chapters on Dimensional Analysis. Mathematical Rings, which also shows why a minus x minus is positive. The Scientific Method, a self-correcting intellectual invention. Mathematical Logic outlines the “algebraic” structure of thought. From this we learn that Sherlock Holmes almost never deduced anything! Early Attempts to Measure the Speed of Light, and how these primitive efforts were uncannily accurate. A bonus in this chapter is a 20-second experiment that allows the reader to measure the speed of light using any kitchen microwave. Cover ... 1 S Title ... 2 Copyright ... 3 c 2013 by the Mathematical Association of America, Inc. ... 3 Library of Congress Catalog Card Number 2013956313 ... 3 Electronic edition ISBN: 978-1-61444-517-3 ... 3 Illustrated Special Relativity Through Its Paradoxes: A Fusion of Linear Algebra,Graphics, and Reality ... 4 Council on Publications and Communications ... 5 Contents ... 10 Dedication ... 17 I. A First Pass ... 18 Preface ... 19 0.1 Exposition and Paradoxes ... 19 0.2 Organization of this Book ... 22 1 Introduction to the Paradoxes ... 28 1.1 Aristotle vs. Galileo ... 28 1.2 Frames of Reference ... 29 1.3 Straight-Line Trajectories in 3-Space ... 30 1.4 Galilean Relativity ... 31 1.5 Special Relativity: A First Pass ... 33 1.6 A Symmetry Principle ... 35 1.7 Lorentzian Relativity ... 36 1.8 The Ubiquitous Shrinkage Constant ... 36 1.9 Paradox: The Accommodating Universe ... 39 1.10 Paradox: Time and Distance Asymmetry ... 43 1.11 Paradox: The Traveling Twin ... 46 1.12 Paradox: The Train in the Tunnel ... 50 1.13 Paradox: The Pea-Shooter ... 53 1.14 Paradox: The Bug and Rivet ... 57 1.15 Exercises ... 59 2 Clocks and Rods in Motion ... 60 2.1 The Perfect Clock ... 60 2.2 Synchronizing Clocks within a Single Frame ... 62 2.3 Moving Clocks Run Slow, Moving Rods Shrink ... 64 2.4 Exercises ... 67 3 The Algebra of Frames ... 71 3.1 Inertial Frames of Reference ... 71 3.2 Vector Space Structure of Frames ... 72 3.3 Several Parallel Moving Frames ... 73 3.4 Six Rules for Frames ... 75 3.5 Exercises ... 81 4 The Graphing of Frames ... 83 4.1 The Filmstrip Model of Spacetime ... 83 4.2 Constant Velocities in Spacetime ... 86 4.3 Worldlines are Parallel to the Home Frame Time Axis ... 88 4.4 Simultaneous and Static Events ... 89 4.5 Linearity of Line-of-Sight Functions ... 92 4.6 Exercises ... 97 II. Galilean Transformations of Frames ... 100 5 Galilean Transformations ... 101 5.1 Key Ideas ... 101 5.2 Galilean Spacetime Diagrams ... 102 5.3 The Galilean Matrix ... 103 5.4 Pattern of the Galilean Matrix ... 106 5.5 Addition of Speeds via Matrices ... 107 5.6 Addition of Speeds via Areas ... 110 III. The Speed of Light is ConstantCh ... 113 6 Constant c in Spacetime ... 114 6.1 Minkowski Spacetime Diagrams ... 114 6.2 Constant c and Simultaneity ... 115 6.3 How Constant c Destroys Simultaneity ... 117 Summary ... 119 6.4 Exercise ... 119 IV. Lorentz Transformations of Frames ... 120 7 Lorentz Transformations ... 121 7.1 The Lorentz Matrix ... 121 7.2 Pattern of the Lorentz Matrix ... 126 7.3 The Lorentz Sum of Speeds ... 126 7.4 Addition of Speeds via Matrices ... 128 7.5 Addition of Speeds via Areas ... 129 7.6 Exercises ... 133 8 The Hyperbola of Time-Stamped Origins ... 135 8.1 Invariance of Minkowski Length ... 135 8.2 The Time-Stamped Origins Theorem ... 137 8.3 Interpreting the Time-Stamped Origins Theorem ... 138 8.4 Tangent Lines of Simultaneity ... 139 8.5 Exercises ... 142 V. Graphic Resolutionof the Paradoxes ... 143 9 The Accommodating Universe Paradox ... 144 9.1 Preview ... 144 9.2 Setup for the Minkowski Diagram ... 144 9.3 Resolving the Accommodating Universe ... 146 9.4 Exercises ... 148 10 The Length-Time Comparison Paradoxes ... 149 10.1 An Overview of the Paradoxes ... 149 10.2 Resolving the Mutual Length-Time Paradoxes ... 154 10.3 Summary ... 155 10.4 Exercises ... 156 11 The Twin Paradox ... 158 11.1 An Overview of the Paradox ... 158 11.2 A Simplifying Assumption ... 158 11.3 Setup for the Minkowski Diagram ... 160 11.4 Resolving the Twin Paradox ... 160 11.5 General Relativity Con?rmation ... 164 11.6 Exercises ... 168 12 The Train-Tunnel Paradox ... 170 12.1 An Overview of the Paradox ... 170 12.2 A Distance Lemma ... 172 12.3 The Train-Tunnel Minkowski Diagram ... 175 12.4 Explaining Mutual Contraction ... 176 12.5 Resolving the Train-Tunnel Paradox ... 177 12.6 Exercises ... 178 13 The Pea-Shooter Paradox ... 180 13.1 An Overview of the Paradox ... 180 13.2 The Fizeau Experiment: Adding Speeds ... 181 13.3 Exercises ... 185 14 The Bug-Rivet Paradox ... 188 14.1 The Minkowski Diagram ... 188 14.2 Coordinates in the Minkowski Diagram ... 191 14.3 The Slinky Connection ... 196 14.4 Exercises ... 198 VI. Energy and MassCh ... 201 15 E = mc 2 ... 202 15.1 How We Came to This Place ... 202 15.2 Speed-Dependent Mass: an Intuitive View ... 203 15.3 Equivalence of Mass and Energy ... 207 15.4 A Numerical Example ... 209 15.5 Exercises ... 211 VII. The Mathematics of Waves and Light ... 215 16 The Nature of Waves ... 216 16.1 Propagated Waves ... 216 16.2 Speed of Rope Wave is Constant ... 221 16.3 Shapes Traveling in One Dimension ... 221 16.4 The Wave Equation in One Dimension ... 226 16.5 Wave Propagation: The Skipping Stone Model ... 227 16.6 The Doppler E?ect in Spacetime ... 230 16.7 Exercises ... 233 17 Measuring the Speed of Light ... 234 17.1 Early Thoughts on the Speed of Light ... 234 17.2 Rømer: The Speed of Light is Finite ... 235 17.3 Fizeau Measures the Speed of Light ... 238 17.4 de Sitter: c Independent of Source Speed ... 242 17.5 Michelson-Morley’s Happy Failure ... 245 17.6 Exercises ... 251 VIII. Maxwell’s Equations ... 255 18 Maxwell’s Mathematical Toolkit ... 256 18.1 Preface ... 256 18.2 Language and Proportionality ... 257 18.3 1D Lengths & 2D Areas as 3D Vectors ... 258 18.4 Orientations of Lines and Surfaces ... 261 18.5 Vectors Modeling Reality ... 264 18.6 Inner and Cross Products ... 266 18.7 Riemann Sums and Integrals ... 268 18.8 Integrals of the Inner Product ... 272 18.9 Exercises ... 276 19 Electric and Magnetic Fields ... 279 19.1 Background ... 279 19.2 Electric Forces: Coulomb’s Law ... 280 19.3 Electric Fields ... 283 19.4 Magnetic Fields ... 285 19.5 Magnetic Forces: Lorentz Forces ... 287 19.6 How Thomson Discovers the Electron ... 289 20 Electricity and Magnetism: Gauss’ Laws ... 293 20.1 Flux of Vector Fields ... 293 20.2 Electric and Magnetic Flux ... 296 20.3 Gauss’ Law for Electricity ... 297 20.4 Gauss’ Law for Magnetism ... 299 20.5 Exercises ... 300 21 Towards Maxwell’s Equations ... 306 21.1 Biot-Savart Law: Magnetism from Electricity ... 306 21.2 Quantitative Results for Biot-Savart ... 307 21.3 Amp`ere’s Law ... 309 21.4 Maxwell Adds to Amp`ere’s Law ... 311 21.5 Faraday’s Law: Electricity from Magnetism ... 313 21.6 Lentz’s Law: The Positive Side of Negativity ... 314 21.7 Maxwell’s Four Equations ... 316 21.8 Exercises ... 318 22 Electromagnetism: A Qualitative View ... 321 22.1 Magnetic Waves from an In?nite Wire ... 321 22.2 Wave Propagation ... 323 22.3 The Geometry of Electromagnetism ... 327 23 Electromagnetism: A Quantitative View ... 330 23.1 Quantitative Preliminaries ... 330 23.2 A Quantitative View of Propagation ... 333 23.3 Theoretical Speed of Wave Propagation ... 340 23.4 Maxwell’s Calculation of c ... 343 23.5 Mathematical Hits ... 345 23.6 Exercises ... 348 IX. Final Thoughts ... 351 24 Epilogue: Final Thoughts ... 352 24.1 A Coming of Age ... 352 24.2 Einstein’s Annus Mirabilis ... 354 24.3 Comparing Relativities ... 356 24.4 Against Conventional Wisdom ... 360 24.5 Some Experimental Results ... 362 24.6 Bad Assumption, Good Result ... 365 24.7 A Limited Reality ... 366 24.8 PIES Reality ... 370 24.9 Exercises ... 372 X. Appendices ... 375 A Linear Algebra Overview ... 376 A.1 Mathematics as a Conduit to Reality ... 376 A.2 Vector Spaces ... 377 A.3 Functions ... 383 A.4 Linear Functions and Matrices ... 385 A.5 Eigenvectors and Eigenvalues ... 391 B Hyperbolic Functions ... 393 B.1 Overview ... 393 B.2 Even and Odd Functions ... 394 B.3 Invariant Areas of Transformed Hyperbolas ... 395 B.4 Exercises ... 400 C Deconstructing a Moving Train ... 402 C.1 Motion Alters Age ... 402 C.2 Minkowski Diagram for a Moving Train ... 402 C.3 Exercises ... 404 XI. Supplemental Material Online ... 405 D Dimensional Analysis ... 406 D.1 Unitless Quotients of Dimensions ... 406 D.2 Dimensions in Fractions ... 407 D.3 Exercises ... 410 E Rings of Functions and Square Matrices ... 413 E.1 Associative, Binary Operations ... 413 E.2 Rings over the Real Numbers ... 416 E.3 The Ring of Matrices ... 417 E.4 Exercises ... 418 F The Scienti?c Method ... 423 F.1 Reality of the Unseen ... 423 F.2 If-then Sentences ... 427 F.3 Property Lists ... 429 F.4 The Four-Step Scienti?c Method ... 430 F.5 Is X a Duck? Applying the Scienti?c Method. ... 431 F.6 Whence the Scienti?c Method? ... 434 F.7 The Logical Implication ... 436 F.8 Induction vs. Deduction ... 436 F.9 Necessary vs. Su?cient ... 437 F.10 Uncertainty, Popper, and Derrida. ... 439 F.11 Implications and Falsi?ability of Karl Popper ... 442 F.12 Exercises ... 444 G Logic of the Scienti?c Method ... 452 G.1 Implications Built from P, Q ... 452 G.2 Equivalence of Implications ... 455 G.3 Proof by Contradiction ... 457 G.4 Exercises ... 460 Bibliography ... 464 Index ... 468