Preface Exposition and Paradoxes Organization of this Book... 5

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1 Contents I. A FIRST PASS 1 Preface Exposition and Paradoxes Organization of this Book Introduction to the Paradoxes Aristotle vs. Galileo Frames of Reference Straight-Line Trajectories in 3-Space Galilean Relativity Special Relativity: A First Pass A Symmetry Principle Lorentzian Relativity The Ubiquitous Shrinkage Constant Paradox: The Accommodating Universe Paradox: Time and Distance Asymmetry Paradox: The Traveling Twin Paradox: The Train in the Tunnel Paradox: The Pea-Shooter Paradox: The Bug and Rivet Exercises Clocks and Rods in Motion The Perfect Clock Synchronizing Clocks within a Single Frame Moving Clocks Run Slow, Moving Rods Shrink Exercises The Algebra of Frames Inertial Frames of Reference Vector Space Structure of Frames Several Parallel Moving Frames Six Rules for Frames Exercises The Graphing of Frames The Filmstrip Model of Spacetime Constant Velocities in Spacetime Worldlines are Parallel to the Home Frame Time Axis 71 ix

2 x CONTENTS 4.4 Simultaneous and Static Events Linearity of Line-of-Sight Functions Exercises II. GALILEAN TRANSFORMATIONS OF FRAMES 83 5 Galilean Transformations Key Ideas Galilean Spacetime Diagrams The Galilean Matrix Pattern of the Galilean Matrix Addition of Speeds via Matrices Addition of Speeds via Areas III. The SPEED OF LIGHT IS CONSTANT 96 6 Constant c in Spacetime Minkowski Spacetime Diagrams Constant c and Simultaneity How Constant c Destroys Simultaneity Exercise IV. LORENTZ TRANSFORMATIONS OF FRAMES Lorentz Transformations The Lorentz Matrix Pattern of the Lorentz Matrix The Lorentz Sum of Speeds Addition of Speeds via Matrices Addition of Speeds via Areas Exercises The Hyperbola of Time-Stamped Origins Invariance of Minkowski Length The Time-Stamped Origins Theorem Interpreting the Time-Stamped Origins Theorem Tangent Lines of Simultaneity Exercises V. GRAPHIC RESOLUTION OF THE PARADOXES The Accommodating Universe Paradox Preview

3 CONTENTS xi 9.2 Setup for the Minkowski Diagram Resolving the Accommodating Universe Exercises The Length-Time Comparison Paradoxes An Overview of the Paradoxes Resolving the Mutual Length-Time Paradoxes Summary Exercises The Twin Paradox An Overview of the Paradox A Simplifying Assumption Setup for the Minkowski Diagram Resolving the Twin Paradox General Relativity Confirmation Exercises The Train-Tunnel Paradox An Overview of the Paradox A Distance Lemma The Train-Tunnel Minkowski Diagram Explaining Mutual Contraction Resolving the Train-Tunnel Paradox Exercises The Pea-Shooter Paradox An Overview of the Paradox The Fizeau Experiment: Adding Speeds Exercises The Bug-Rivet Paradox The Minkowski Diagram Coordinates in the Minkowski Diagram The Slinky Connection Exercises VI. ENERGY AND MASS E = mc How We Came to This Place Speed-Dependent Mass: an Intuitive View Equivalence of Mass and Energy A Numerical Example

4 xii CONTENTS 15.5 Exercises VII. THE MATHEMATICS OF WAVES AND LIGHT The Nature of Waves Propagated Waves Speed of Rope Wave is Constant Shapes Traveling in One Dimension The Wave Equation in One Dimension Wave Propagation: The Skipping Stone Model The Doppler Effect in Spacetime Exercises Measuring the Speed of Light Early Thoughts on the Speed of Light Rømer: The Speed of Light is Finite Fizeau Measures the Speed of Light de Sitter: c Independent of Source Speed Michelson-Morley s Happy Failure Exercises VIII. MAXWELL S EQUATIONS Maxwell s Mathematical Toolkit Preface Language and Proportionality D Lengths & 2D Areas as 3D Vectors Orientations of Lines and Surfaces Vectors Modeling Reality Inner and Cross Products Riemann Sums and Integrals Integrals of the Inner Product Exercises Electric and Magnetic Fields Background Electric Forces: Coulomb s Law Electric Fields Magnetic Fields Magnetic Forces: Lorentz Forces How Thomson Discovers the Electron Electricity and Magnetism: Gauss Laws Flux of Vector Fields

5 CONTENTS xiii 20.2 Electric and Magnetic Flux Gauss Law for Electricity Gauss Law for Magnetism Exercises Towards Maxwell s Equations Biot-Savart Law: Magnetism from Electricity Quantitative Results for Biot-Savart Ampère s Law Maxwell Adds to Ampère s Law Faraday s Law: Electricity from Magnetism Lentz s Law: The Positive Side of Negativity Maxwell s Four Equations Exercises Electromagnetism: A Qualitative View Magnetic Waves from an Infinite Wire Wave Propagation The Geometry of Electromagnetism Electromagnetism: A Quantitative View Quantitative Preliminaries A Quantitative View of Propagation Theoretical Speed of Wave Propagation Maxwell s Calculation of c Mathematical Hits Exercises IX. FINAL THOUGHTS Epilogue: Final Thoughts A Coming of Age Einstein s Annus Mirabilis Comparing Relativities Against Conventional Wisdom Some Experimental Results Bad Assumption, Good Result A Limited Reality PIES Reality Exercises X. APPENDICES 358 A Linear Algebra Overview 359

6 xiv CONTENTS A.1 Mathematics as a Conduit to Reality A.2 Vector Spaces A.3 Functions A.4 Linear Functions and Matrices A.5 Eigenvectors and Eigenvalues B Hyperbolic Functions 376 B.1 Overview B.2 Even and Odd Functions B.3 Invariant Areas of Transformed Hyperbolas B.4 Exercises C Deconstructing a Moving Train 385 C.1 Motion Alters Age C.2 Minkowski Diagram for a Moving Train C.3 Exercises XI. SUPPLEMENTAL MATERIAL 388 D Dimensional Analysis 389 D.1 Unitless Quotients of Dimensions D.2 Dimensions in Fractions D.3 Exercises E Rings of Functions and Square Matrices 396 E.1 Associative, Binary Operations E.2 Rings over the Real Numbers E.3 The Ring of Matrices E.4 Exercises F The Scientific Method 406 F.1 Reality of the Unseen F.2 If-then Sentences F.3 Property Lists F.4 The Four-Step Scientific Method F.5 Is X a Duck? Applying the Scientific Method F.6 Whence the Scientific Method? F.7 The Logical Implication F.8 Induction vs. Deduction F.9 Necessary vs. Sufficient F.10 Uncertainty, Popper, and Derrida F.11 Popper Logic F.12 Exercises

7 CONTENTS xv G Logic of the Scientific Method 435 G.1 Implications Built from P, Q G.2 Equivalence of Implications G.3 Proof by Contradiction G.4 Exercises Bibliography 447 Index 451