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

481 Pages·2014·6.62 MB·English

by  John de Pillis

Illustrated
Special Relativity
Through Its Paradoxes
A Fusion of Linear Algebra,
Graphics, and Reality
c 2013 by the Mathematical Association of America, Inc.
(cid:13)
Library of Congress Catalog Card Number 2013956313
Electronic edition ISBN: 978-1-61444-517-3
Illustrated
Special Relativity
Through Its Paradoxes
A Fusion of Linear Algebra,
Graphics, and Reality
John de Pillis
Professor of Mathematics
University of California, Riverside
and
Jos´e Wudka
Professor of Physics
University of California, Riverside
Illustrations and animations by John dePillis
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Contents
I. A FIRST PASS 1
Preface 2
0.1 Exposition and Paradoxes . . . . . . . . . . . . . . . . 2
0.2 Organization of this Book . . . . . . . . . . . . . . . . 5
1 Introduction to the Paradoxes 11
1.1 Aristotle vs. Galileo . . . . . . . . . . . . . . . . . . . 11
1.2 Frames of Reference . . . . . . . . . . . . . . . . . . . 12
1.3 Straight-Line Trajectories in 3-Space . . . . . . . . . . 13
1.4 GalileanRelativity . . . . . . . . . . . . . . . . . . . . 14
1.5 Special Relativity: A First Pass . . . . . . . . . . . . . 16
1.6 A Symmetry Principle . . . . . . . . . . . . . . . . . . 18
1.7 Lorentzian Relativity . . . . . . . . . . . . . . . . . . . 19
1.8 The Ubiquitous Shrinkage Constant . . . . . . . . . . 19
1.9 Paradox: The Accommodating Universe . . . . . . . . 22
1.10 Paradox: Time and Distance Asymmetry . . . . . . . 26
1.11 Paradox: The Traveling Twin . . . . . . . . . . . . . . 29
1.12 Paradox: The Train in the Tunnel . . . . . . . . . . . 33
1.13 Paradox: The Pea-Shooter . . . . . . . . . . . . . . . . 36
1.14 Paradox: The Bug and Rivet . . . . . . . . . . . . . . 40
1.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 42
2 Clocks and Rods in Motion 43
2.1 The Perfect Clock . . . . . . . . . . . . . . . . . . . . 43
2.2 Synchronizing Clocks within a Single Frame . . . . . . 45
2.3 MovingClocks Run Slow,Moving Rods Shrink . . . . 47
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 The Algebra of Frames 54
3.1 Inertial Frames of Reference . . . . . . . . . . . . . . . 54
3.2 Vector Space Structure of Frames . . . . . . . . . . . . 55
3.3 Several Parallel MovingFrames . . . . . . . . . . . . . 56
3.4 Six Rules for Frames . . . . . . . . . . . . . . . . . . . 58
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 The Graphing of Frames 66
4.1 The Filmstrip Model of Spacetime . . . . . . . . . . . 66
4.2 Constant Velocities in Spacetime . . . . . . . . . . . . 69
4.3 Worldlinesare Parallelto the Home Frame Time Axis 71
ix