Baixe Física Moderna Tipler e outras Notas de estudo em PDF para Física, somente na Docsity! [et JU RV So Ralph A. Llewellyn Publisher: Clancy Marshall Senior Acquisitions Editor: Jessica Fiorillo Marketing Manager: Anthony Palmiotto Media Editors: Jeanette Picerno and Samantha Calamari Supplements Editor and Editorial Assistant: Janie Chan Senior Project Editor: Mary Louise Byrd Cover and Text Designer: Diana Blume Photo Editor: Ted Szczepanski Photo Researcher: Rae Grant Senior Illustration Coordinator: Bill Page Production Coordinator: Paul W. Rohloff Illustrations and Composition: Preparé Printing and Binding: Quebecor Printing Library of Congress Control Number: 2007931523 ISBN-13: 978-0-7167-7550-8 ISBN-10: 0-7167-7550-6 © 2008 by Paul A. Tipler and Ralph A. Llewellyn All rights reserved. Printed in the United States of America First printing W. H. 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PART 1 Relativity and Quantum Mechanics: The Foundations of Modern Physics 1 CHAPTER 1 Relativity I 3 1-1 The Experimental Basis of Relativity 4 Michelson-Morley Experiment 11 1-2 Einstein’s Postulates 11 1-3 The Lorentz Transformation 17 Calibrating the Spacetime Axes 28 1-4 Time Dilation and Length Contraction 29 1-5 The Doppler Effect 41 Transverse Doppler Effect 44 1-6 The Twin Paradox and OtherSurprises 45 The Case of the Identically Accelerated Twins 48 Superluminal Speeds 52 CHAPTER 2 Relativity II 65 2-1 Relativistic Momentum 66 2-2 Relativistic Energy 70 From Mechanics, AnotherSurprise 80 2-3 Mass/Energy Conversion and Binding Energy 81 2-4 Invariant Mass 84 Contents iv Contents 2-5 General Relativity 97 Deflection of Light in a Gravitational Field 100 Gravitational Redshift 103 Perihelion of Mercury’s Orbit 105 Delay of Light in a Gravitational Field 105 CHAPTER 3 Quantization of Charge, Light, and Energy 115 3-1 Quantization of Electric Charge 115 3-2 Blackbody Radiation 119 3-3 The Photoelectric Effect 127 3-4 X Rays and the Compton Effect 133 Derivation of Compton’s Equation 138 CHAPTER 4 The Nuclear Atom 147 4-1 Atomic Spectra 148 4-2 Rutherford’s Nuclear Model 150 Rutherford’s Prediction and Geiger and Marsden’s Results 156 4-3 The Bohr Model of the Hydrogen Atom 159 Giant Atoms 168 4-4 X-Ray Spectra 169 4-5 The Franck-Hertz Experiment 174 A Critique of BohrTheory and the “Old Quantum Mechanics” 176 CHAPTER 5 The Wavelike Properties of Particles 185 5-1 The de Broglie Hypothesis 185 5-2 Measurements of Particle Wavelengths 187 5-3 Wave Packets 196 5-4 The Probabilistic Interpretation of the Wave Function 202 5-5 The Uncertainty Principle 205 The Gamma-Ray Microscope 206 5-6 Some Consequences of the Uncertainty Principle 208 5-7 Wave-Particle Duality 212 Two-Slit Interference Pattern 213 CHAPTER 6 The Schrödinger Equation 221 6-1 The Schrödinger Equation in One Dimension 222 6-2 The Infinite Square Well 229 6-3 The Finite Square Well 238 Graphical Solution of the Finite Square Well 241 6-4 Expectation Values and Operators 242 Transitions Between Energy States 246 6-5 The Simple Harmonic Oscillator 246 Schrödinger’s Trick 249 Parity 250 6-6 Reflection and Transmission of Waves 250 Alpha Decay 258 NH 3 Atomic Clock 260 Tunnel Diode 260 CHAPTER 7 Atomic Physics 269 7-1 The Schrödinger Equation in Three Dimensions 269 7-2 Quantization of Angular Momentum and Energy in the Hydrogen Atom 272 7-3 The Hydrogen Atom Wave Functions 281 7-4 Electron Spin 285 Stern-Gerlach Experiment 288 7-5 Total Angular Momentum and the Spin-Orbit Effect 291 7-6 The Schrödinger Equation forTwo (or More) Particles 295 7-7 Ground States of Atoms: The PeriodicTable 297 7-8 Excited States and Spectra of Atoms 301 Multielectron Atoms 303 The Zeeman Effect 303 Frozen Light 304 Contents v viii Contents 12-3 Conservation Laws and Symmetries 580 When Is a Physical Quantity Conserved? 583 Resonances and Excited States 591 12-4 The Standard Model 591 Where Does the Proton Get Its Spin? 595 12-5 Beyond the Standard Model 605 Neutrino Oscillations and Mass 609 Theories of Everything 610 CHAPTER 13 Astrophysics and Cosmology 619 13-1 The Sun 619 Is There Life Elsewhere? 630 13-2 The Stars 630 The Celestial Sphere 636 13-3 The Evolution ofStars 639 13-4 Cataclysmic Events 644 13-5 Final States ofStars 647 13-6 Galaxies 653 13-7 Cosmology and Gravitation 662 13-8 Cosmology and the Evolution of the Universe 664 “Natural” Planck Units 673 Appendix A Table of Atomic Masses AP-1 Appendix B Mathematical Aids AP-16 B1 Probability Integrals AP-16 B2 Binomial and Exponential Series AP-18 B3 Diagrams of Crystal Unit Cells AP-19 Appendix C Electron Configurations AP-20 Appendix D Fundamental Physical Constants AP-26 Appendix E Conversion Factors AP-30 Appendix F Nobel Laureates in Physics AP-31 Answers AN-1 Index I-1 ix In preparing this new edition of Modern Physics, we have again relied heavily on themany helpful suggestions from a large team of reviewers and from a host of instruc- tor and student users of the earlier editions. Their advice reflected the discoveries that have further enlarged modern physics in the early years of this new century and took note of the evolution that is occurring in the teaching of physics in colleges and uni- versities. As the term modern physics has come to mean the physics of the modern era—relativity and quantum theory—we have heeded the advice of many users and reviewers and preserved the historical and cultural flavor of the book while being careful to maintain the mathematical level of the fourth edition. We continue to pro- vide the flexibility for instructors to match the book and its supporting ancillaries to a wide variety of teaching modes, including both one- and two-semester courses and media-enhanced courses. Features The successful features of the fourth edition have been retained, including the following: • The logical structure—beginning with an introduction to relativity and quantiza- tion and following with applications—has been continued. Opening the book with relativity has been endorsed by many reviewers and instructors. • As in the earlier editions, the end-of-chapter problems are separated into three sets based on difficulty, with the least difficult also grouped by chapter section. More than 10 percent of the problems in the fifth edition are new. The first edition’s Instructor’s Solutions Manual (ISM) with solutions, not just answers, to all end-of- chapter problems was the first such aid to accompany a physics (and not just a modern physics) textbook, and that leadership has been continued in this edition. The ISM is available in print or on CD for those adopting Modern Physics, fifth edition, for their classes. As with the previous edition, a paperback Student’s Solution Manual containing one-quarter of the solutions in the ISM is also available. • We have continued to include many examples in every chapter, a feature singled out by many instructors as a strength of the book. As before, we frequently use combined quantities such as , , and in to simplify many numerical calculations. • The summaries and reference lists at the end of every chapter have, of course, been retained and augmented, including the two-column format of the summaries, which improves their clarity. eV # nmke2Uchc Preface x Preface • We have continued the use of real data in figures, photos of real people and appa- ratus, and short quotations from many scientists who were key participants in the development of modern physics. These features, along with the Notes at the end of each chapter, bring to life many events in the history of science and help counter the too-prevalent view among students that physics is a dull, impersonal collection of facts and formulas. • More than two dozen Exploring sections, identified by an atom icon and dealing with text-related topics that captivate student interest such as superluminal speed and giant atoms, are distributed throughout the text. • The book’s Web site includes 30 MORE sections, which expand in depth on many text-related topics. These have been enthusiastically endorsed by both students and instructors and often serve as springboards for projects and alternate credit assign- ments. Identified by a laptop icon , each is introduced with a brief text box. • More than 125 questions intended to foster discussion and review of concepts are distributed throughout the book. These have received numerous positive comments from many instructors over the years, often citing how the questions encourage deeper thought about the topic. • Continued in the new edition are the Application Notes. These brief notes in the margins of many pages point to a few of the many benefits to society that have been made possible by a discovery or development in modern physics. New Features A number of new features are introduced in the fifth edition: • The “Astrophysics and Cosmology” chapter that was on the fourth edition’s Web site has been extensively rewritten and moved into the book as a new Chapter 13. Emphasis has been placed on presenting scientists’ current understanding of the evolution of the cosmos based on the research in this dynamic field. • The “Particle Physics” chapter has been substantially reorganized and rewritten focused on the remarkably successful Standard Model. As the new Chapter 12, it immediately precedes the new “Astrophysics and Cosmology” chapter to recog- nize the growing links between these active areas of current physics research. • The two chapters concerned with the theory and applications of nuclear physics have been integrated into a new Chapter 11, “Nuclear Physics.” Because of the renewed interest in nuclear power, that material in the fourth edition has been aug- mented and moved to a MORE section of the Web. • Recognizing the need for students on occasion to be able to quickly review key concepts from classical physics that relate to topics developed in modern physics, we have added a new Classical Concept Review (CCR) to the book’s Web site. Identified by a laptop icon in the margin near the pertinent modern physics topic of discussion, the CCR can be printed out to provide a convenient study sup- port booklet. • The Instructor’s Resource CD for the fifth edition contains all the illustrations from the book in both PowerPoint and JPEG format. Also included is a gallery of the astronomical images from Chapter 13 in the original image colors. • Several new MORE sections have been added to the book’s Web site, and a few for which interest has waned have been removed. Some Teaching Suggestions This book is designed to serve well in either one- or two-semester courses. The chap- ters in Part 2 are independent of one another and can be covered in any order. Some possible one-semester courses might consist of • Part 1, Chapters 1, 3, 4, 5, 6, 7; and Part 2, Chapters 11, 12 • Part 1, Chapters 3, 4, 5, 6, 7, 8; and Part 2, Chapters 9, 10 • Part 1, Chapters 1, 2, 3, 4, 5, 6, 7; and Part 2, Chapter 9 • Part 1, Chapters 1, 3, 4, 5, 6, 7; and Part 2, Chapters 11, 12, 13 Possible two-semester courses might consist of • Part 1, Chapters 1, 3, 4, 5, 6, 7; and Part 2, Chapters 9, 10, 11, 12, 13 • Part 1, Chapters 1, 2, 3, 4, 5, 6, 7, 8; and Part 2, Chapters 9, 10, 11, 12, 13 There is tremendous potential for individual student projects and alternate credit assignments based on the Exploring and, in particular, the MORE sections. The latter will encourage students to search for related sources on the Web. Acknowledgments Many people contributed to the success of the earlier editions of this book, and many more have helped with the development of the fifth edition. We owe our thanks to them all. Those who reviewed all or parts of this book, offering suggestions for the fifth edition, include Preface xiii Marco Battaglia University of California–Berkeley Mario Belloni Davidson College Eric D. Carlson Wake Forest University David Cinabro Wayne State University Carlo Dallapiccola University of Massachusetts–Amherst Anthony D. Dinsmore University of Massachusetts–Amherst Ian T. Durham Saint Anselm College Jason J. Engbrecht St. Olaf College Brian Fick Michigan Technological University Massimiliano Galeazzi University of Miami Hugh Gallagher Tufts University Richard Gelderman Western Kentucky University Tim Gfroerer Davidson College Torgny Gustafsson Rutgers University Scott Heinekamp Wells College Adrian Hightower Occidental College Mark Hollabaugh Normandale Community College Richard D. Holland II Southern Illinois University at Carbondale Bei-Lok Hu University of Maryland–College Park Dave Kieda University of Utah Steve Kraemer Catholic University of America Wolfgang Lorenzon University of Michigan xiv Preface Bryan A. Luther Concordia College at Moorhead Catherine Mader Hope College Kingshuk Majumdar Berea College Peter Moeck Portland State University Robert M. Morse University of Wisconsin–Madison Igor Ostrovskii University of Mississippi at Oxford Anne Reilly College of William and Mary David Reitze University of Florida Mark Riley Florida State University Nitin Samarth Pennsylvania State University Kate Scholberg Duke University Ben E. K. Sugerman Goucher College Rein Uritam Physics Department Boston College Ken Voss University of Miami Thad Walker University of Wisconsin–Madison Barry C. Walker University of Delaware Eric Wells Augustana College William R. Wharton Wheaton College Weldon J. Wilson University of Central Oklahoma R. W. M. Woodside University College of Fraser Valley We also thank the reviewers of the fourth and third editions. Their comments significantly influenced and shaped the fifth edition as well. For the fourth edition they were Darin Acosta, University of Florida; Jeeva Anandan, University of South Carolina; Gordon Aubrecht, Ohio State University; David A. Bahr, Bemidji State University; Patricia C. Boeshaar, Drew University; David P. Carico, California Polytechnic State University at San Luis Obispo; David Church, University of Washington; Wei Cui, Purdue University; Snezana Dalafave, College of New Jersey; Richard Gass, University of Cincinnati; David Gerdes, University of Michigan; Mark Hollabaugh, Normandale Community College; John L. Hubisz, North Carolina State University; Ronald E. Jodoin, Rochester Institute of Technology; Edward R. Kinney, University of Colorado at Boulder; Paul D. Lane, University of St. Thomas; Fernando J. Lopez-Lopez, Southwestern College; Dan MacIsaac, Northern Arizona University; Robert Pompi, SUNY at Binghamton; Warren Rogers, Westmont College; George Rutherford, Illinois State University; Nitin Samarth, Pennsylvania State University; Martin A. Sanzari, Fordham University; Earl E. Scime, West Virginia University; Gil Shapiro, University of California at Berkeley; Larry Solanch, Georgia College & State University; Francis M. Tam, Frostburg State University; Paul Tipton, University of Rochester; K. Thad Walker, University of Wisconsin at Madison; Edward A. Whittaker, Stevens Institute of Technology; Stephen Yerian, Xavier University; and Dean Zollman, Kansas State University. For the third edition, reviewers were Bill Bassichis, Texas A&M University; Brent Benson, Lehigh University; H. J. Biritz, Georgia Institute of Technology; Patrick Briggs, The Citadel; David A. Briodo, Boston College; Tony Buffa, California Polytechnic State University at San Luis Obispo; Duane Carmony, Purdue University; Ataur R. Chowdhury, University of Alaska at Fairbanks; Bill Fadner, University of Northern Colorado; Ron Gautreau, New Jersey Institute of Technology; Charles Glashauser, Rutgers–The State University of New Jersey; Roger Hanson, University of Northern Iowa; Gary G. Ihas, University of Florida; Yuichi Kubota, University of Minnesota; David Lamp, Texas Tech University; Philip Lippel, University of Texas at Arlington; A. E. Livingston, University of Notre Dame; Steve Meloma, Gustavus Adolphus College; Benedict Y. Oh, Pennsylvania State University; Paul Sokol, Pennsylvania State University; Thor F. Stromberg, New Mexico State University; Maurice Webb, University of Wisconsin at Madison; and Jesse Weil, University of Kentucky. All offered valuable suggestions for improvements, and we appreciate their help. In addition, we give a special thanks to all the physicists and students from around the world who took time to send us kind words about the third and fourth editions and offered suggestions for improvements. Finally, though certainly not least, we are grateful for the support, encouragement, and patience of our families throughout the project. We especially want to thank Mark Llewellyn for his preparation of the Instructor’s Solutions Manual and the Student’s Solutions Manual and for his numerous helpful suggestions from the very beginning of the project, Eric Llewellyn for his photographic and computer-generated images, David Jonsson at Uppsala University for his critical reading of every chapter of the fourth edition, and Jeanette Picerno for her imaginative work on the Web site. Finally, to the entire Modern Physics team at W. H. Freeman and Company goes our sincerest appreciation for their skill, hard work, understanding about deadlines, and support in bringing it all together. Paul A. Tipler, Ralph A. Llewellyn, Berkeley, California Oviedo, Florida Preface xv 2 Such optimism (or pessimism, depending on your point of view) turned out to be pre- mature, as there were already vexing cracks in the foundation of what we now refer to as classical physics. Two of these were described by Lord Kelvin, in his famous Baltimore Lectures in 1900, as the “two clouds” on the horizon of twentieth-century physics: the fail- ure of theory to account for the radiation spectrum emitted by a blackbody and the inex- plicable results of the Michelson-Morley experiment. Indeed, the breakdown of classical physics occurred in many different areas: the Michelson-Morley null result contradicted Newtonian relativity, the blackbody radiation spectrum contradicted predictions of thermo- dynamics, the photoelectric effect and the spectra of atoms could not be explained by elec- tromagnetic theory, and the exciting discoveries of x rays and radioactivity seemed to be outside the framework of classical physics entirely. The development of the theories of quan- tum mechanics and relativity in the early twentieth century not only dispelled Kelvin’s “dark clouds,” they provided answers to all of the puzzles listed here and many more. The ap- plications of these theories to such microscopic systems as atoms, molecules, nuclei, and fundamental particles and to macroscopic systems of solids, liquids, gases, and plasmas have given us a deep understanding of the intricate workings of nature and have revolu- tionized our way of life. In Part 1 we discuss the foundations of the physics of the modern era, relativity theory, and quantum mechanics. Chapter 1 examines the apparent conflict between Einstein’s prin- ciple of relativity and the observed constancy of the speed of light and shows how accepting the validity of both ideas led to the special theory of relativity. Chapter 2 discusses the relations connecting mass, energy, and momentum in special relativity and concludes with a brief dis- cussion of general relativity and some experimental tests of its predictions. In Chapters 3, 4, and 5 the development of quantum theory is traced from the earliest evidences of quantiza- tion to de Broglie’s hypothesis of electron waves. An elementary discussion of theSchrödinger equation is provided in Chapter 6, illustrated with applications to one-dimensional systems. Chapter 7 extends the application of quantum mechanics to many-particle systems and introduces the important new concepts of electron spin and the exclusion principle. Concluding the development, Chapter 8 discusses the wave mechanics of systems of large numbers of identical particles, underscoring the importance of the symmetry of wave func- tions. Beginning with Chapter 3, the chapters in Part 1 should be studied in sequence because each of Chapters 4 through 8 depends on the discussions, developments, and examples of the previous chapters. 3 The relativistic character of the laws of physics began to be apparent very earlyin the evolution of classical physics. Even before the time of Galileo and Newton, Nicolaus Copernicus1 had shown that the complicated and imprecise Aristotelian method of computing the motions of the planets, based on the assumption that Earth was located at the center of the universe, could be made much simpler, though no more accurate, if it were assumed that the planets move about the Sun instead of Earth. Although Copernicus did not publish his work until very late in life, it became widely known through correspondence with his contemporaries and helped pave the way for acceptance a century later of the heliocentric theory of planetary motion. While the Copernican theory led to a dramatic revolution in human thought, the aspect that concerns us here is that it did not consider the location of Earth to be special or favored in any way. Thus, the laws of physics discovered on Earth could apply equally well with any point taken as the center — i.e., the same equations would be obtained regardless of the origin of coordinates. This invariance of the equations that express the laws of physics is what we mean by the term relativity. We will begin this chapter by investigating briefly the relativity of Newton’s laws and then concentrate on the theory of relativity as developed by Albert Einstein (1879–1955). The theory of relativity consists of two rather different theories, the special theory and the general theory. The special theory, developed by Einstein and others in 1905, concerns the comparison of measurements made in different frames of reference moving with constant velocity relative to each other. Contrary to popu- lar opinion, the special theory is not difficult to understand. Its consequences, which can be derived with a minimum of mathematics, are applicable in a wide variety of situations in physics and engineering. On the other hand, the general theory, also developed by Einstein (around 1916), is concerned with accelerated reference frames and gravity. Although a thorough understanding of the general theory requires more sophisticated mathematics (e.g., tensor analysis), a number of its basic ideas and important predictions can be discussed at the level of this book. The general theory is of great importance in cosmology and in understanding events that occur in the 1-1 The Experimental Basis of Relativity 4 1-2 Einstein’s Postulates 11 1-3 The Lorenz Transformation 17 1-4 Time Dilation and Length Contraction 29 1-5 The Doppler Effect 41 1-6 The Twin Paradox and Other Surprises 45 Relativity I CHAPTER1 4 Chapter 1 Relativity I y S x z y S x z v vicinity of very large masses (e.g., stars) but is rarely encountered in other areas of physics and engineering. We will devote this chapter entirely to the special theory (often referred to as special relativity) and discuss the general theory in the final section of Chapter 2, following the sections concerned with special relativistic mechanics. 1-1 The Experimental Basis of Relativity Classical Relativity In 1687, with the publication of the Philosophiae Naturalis Principia Mathematica, Newton became the first person to generalize the observations of Galileo and others into the laws of motion that occupied much of your attention in introductory physics. The second of Newton’s three laws is 1-1 where is the acceleration of the mass m when acted upon by a net force F. Equation 1-1 also includes the first law, the law of inertia, by implication: if , then also, i.e., . (Recall that letters and symbols in boldface type are vectors.) As it turns out, Newton’s laws of motion only work correctly in inertial reference frames, that is, reference frames in which the law of inertia holds.2 They also have the remarkable property that they are invariant, or unchanged, in any reference frame that moves with constant velocity relative to an inertial frame. Thus, all inertial frames are equivalent—there is no special or favored inertial frame relative to which absolute measurements of space and time could be made. Two such inertial frames are illus- trated in Figure 1-1, arranged so that corresponding axes in S and are parallel and moves in the direction at velocity v for an observer in S (or S moves in the xxS S a 0dv>dt 0 F 0dv>dt a F m dv dt ma Figure 1-1 Inertial reference frame S is attached to Earth (the palm tree) and S to the cyclist. The corresponding axes of the frames are parallel, and S moves at speed v in the x direction of S. 1-1 The Experimental Basis of Relativity 7 Figure 1-5 Light source, mirror, and observer are moving with speed v relative to the ether. According to classical theory, the speed of light c, relative to the ether, would be c v relative to the observer for light moving from the source toward the mirror and c v for light reflecting from the mirror back toward the source. Observer Light source Mirror BA L v c + v c – v Albert A. Michelson, here playing pool in his later years, made the first accurate measurement of the speed of light while an instructor at the U.S. Naval Academy, where he had earlier been a cadet. [AIP Emilio Segrè Visual Archives.] techniques available at the time had an experimental accuracy of only about 1 part in 104, woefully insufficient to detect the predicted small effect. That single exception was the experiment of Michelson and Morley.5 Questions 1. What would the relative velocity of the inertial systems in Figure 1-4 need to be in order for the S observer to measure no net electromagnetic force on the charge q? 2. Discuss why the very large value for the speed of the electromagnetic waves would imply that the ether be rigid, i.e., have a large bulk modulus. The Michelson-Morley Experiment All waves that were known to nineteenth-century scientists required a medium in order to propagate. Surface waves moving across the ocean obviously require the water. Similarly, waves move along a plucked guitar string, across the surface of a struck drumhead, through Earth after an earthquake, and, indeed, in all materials acted upon by suitable forces. The speed of the waves depends on the properties of the medium and is derived relative to the medium. For example, the speed of sound waves in air, i.e., their absolute motion relative to still air, can be measured. The Doppler ef- fect for sound in air depends not only on the relative motion of the source and listener, but also on the motion of each relative to still air. Thus, it was natural for scientists of that time to expect the existence of some material like the ether to support the propa- gation of light and other electromagnetic waves and to expect that the absolute mo- tion of Earth through the ether should be detectable, despite the fact that the ether had not been observed previously. Michelson realized that although the effect of Earth’s motion on the results of any “out-and–back” speed of light measurement, such as shown generically in Figure 1-5, would be too small to measure directly, it should be possible to measure v2 c2 by a dif- ference measurement, using the interference property of the light waves as a sensitive “clock.” The apparatus that he designed to make the measurement is called the Michelson interferometer. The purpose of the Michelson-Morley experiment was to measure the speed of light relative to the interferometer (i.e., relative to Earth), thereby detecting Earth’s motion through the ether and thus verifying the latter’s existence. To illustrate how the interferometer works and the reasoning behind the experiment, let us first describe an analogous situation set in more familiar surroundings. > 8 Chapter 1 Relativity I EXAMPLE 1-1 A Boat Race Two equally matched rowers race each other over courses as shown in Figure 1-6a. Each oarsman rows at speed c in still water; the current in the river moves at speed v. Boat 1 goes from A to B, a distance L, and back. Boat 2 goes from A to C, also a distance L, and back. A, B, and C are marks on the riverbank. Which boat wins the race, or is it a tie? (Assume c v.) SOLUTION The winner is, of course, the boat that makes the round trip in the shortest time, so to discover which boat wins, we compute the time for each. Using the classical velocity transformation (Equations 1-3), the speed of 1 relative to the ground is , as shown in Figure 1-6b; thus the round-trip time t1 for boat 1 is 1-4 where we have used the binomial expansion. Boat 2 moves downstream at speed relative to the ground and returns at , also relative to the ground. The round-trip time t2 is thus 1-5 2L c 1 1 v2 c2 2L c a1 v2 c2 Á b t2 L c v L c v 2Lc c2 v2 c vc v 2L cA1 v 2 c2 2L c a1 v2 c2 b1/2 2L c a1 1 2 v2 c2 Á b t1 tASB tBSA L 2c2 v2 L 2c2 v2 2L 2c2 v2 (c2 v2)1>2 Ground Ground River C B A 1 2 L L v (a) (b) c 2 – v 2 v A→B c c 2 – v 2 v B →A c Figure 1-6 (a) The rowers both row at speed c in still water. (See Example 1-1.) The current in the river moves at speed v. Rower 1 goes from A to B and back to A, while rower 2 goes from A to C and back to A. (b) Rower 1 must point the bow upstream so that the sum of the velocity vectors c v results in the boat moving from A directly to B. His speed relative to the banks (i.e., points A and B) is then The same is true on the return trip.(c2 v2)1>2. 1-1 The Experimental Basis of Relativity 9 Figure 1-7 Drawing of Michelson-Morley apparatus used in their 1887 experiment. The optical parts were mounted on a 5 ft square sandstone slab, which was floated in mercury, thereby reducing the strains and vibrations during rotation that had affected the earlier experiments. Observations could be made in all directions by rotating the apparatus in the horizontal plane. [From R. S. Shankland, “The Michelson-Morley Experiment,” Copyright © November 1964 by Scientific American, Inc. All rights reserved.] Light source Telescope Mirrors Adjustable mirror Silvered glass plate Unsilvered glass plate Mirrors Mirrors 1 2 3 4 5 The Results Michelson and Morley carried out the experiment in 1887, repeating with a much-improved interferometer an inconclusive experiment that Michelson alone had performed in 1881 in Potsdam. The path length L on the new interferom- eter (Figure 1-7) was about 11 meters, obtained by a series of multiple reflections. Michelson’s interferometer is shown schematically in Figure 1-8a. The field of view seen by the observer consists of parallel alternately bright and dark interference bands, called fringes, as illustrated in Figure 1-8b. The two light beams in the inter- ferometer are exactly analogous to the two boats in Example 1-1, and Earth’s motion through the ether was expected to introduce a time (phase) difference as given by which, you may note, is the same result obtained in our discussion of the speed of light experiment in the Classical Concept Review. The difference ¢t between the round-trip times of the boats is then 1-6 The quantity is always positive; therefore, t2 t1 and rower 1 has the faster average speed and wins the race. Lv2>c3 ¢t t2 t1 2L c a1 v2 c2 b 2L c a1 1 2 v2 c2 b Lv2 c3 12 Chapter 1 Relativity I Figure 1-12 (a) Stationary light source S and a stationary observer R1, with a second observer R2 moving toward the source with speed v. (b) In the reference frame in which the observer R2 is at rest, the light source S and observer R1 move to the right with speed v. If absolute motion cannot be detected, the two views are equivalent. Since the speed of light does not depend on the motion of the source, observer R2 measures the same value for that speed as observer R1. S S v R2 R1 v R2 R1 v (a) (b) by any experiment. We can then consider the Michelson apparatus and Earth to be at rest. No fringe shift is expected when the interferometer is rotated 90°, since all di- rections are equivalent. The null result of the Michelson-Morley experiment is there- fore to be expected. It should be pointed out that Einstein did not set out to explain the Michelson-Morley experiment. His theory arose from his considerations of the theory of electricity and magnetism and the unusual property of electromagnetic waves that they propagate in a vacuum. In his first paper, which contains the complete theory of special relativity, he made only a passing reference to the experimental at- tempts to detect Earth’s motion through the ether, and in later years he could not re- call whether he was aware of the details of the Michelson-Morley experiment before he published his theory. The theory of special relativity was derived from two postulates proposed by Einstein in his 1905 paper: Postulate 1. The laws of physics are the same in all inertial reference frames. Postulate 2. The speed of light in a vacuum is equal to the value c, independent of the motion of the source. Postulate 1 is an extension of the Newtonian principle of relativity to include all types of physical measurements (not just measurements in mechanics). It implies that no inertial system is preferred over any other; hence, absolute motion cannot be de- tected. Postulate 2 describes a common property of all waves. For example, the speed of sound waves does not depend on the motion of the sound source. When an ap- proaching car sounds its horn, the frequency heard increases according to the Doppler effect, but the speed of the waves traveling through the air does not depend on the speed of the car. The speed of the waves depends only on the properties of the air, such as its temperature. The force of this postulate was to include light waves, for which experiments had found no propagation medium, together with all other waves, whose speed was known to be independent of the speed of the source. Recent analysis of the light curves of gamma-ray bursts that occur near the edge of the observable universe have shown the speed of light to be independent of the speed of the source to a preci- sion of one part in 1020. Although each postulate seems quite reasonable, many of the implications of the two together are surprising and seem to contradict common sense. One important im- plication of these postulates is that every observer measures the same value for the speed of light independent of the relative motion of the source and observer. Consider a light source S and two observers R1, at rest relative to S, and R2, moving toward S with speed v, as shown in Figure 1-12a. The speed of light measured by R1 is c 3 108 m s. What is the speed measured by R2? The answer is not c v, as one would expect based on Newtonian relativity. By postulate 1, Figure 1-12a is equiva- lent to Figure 1-12b, in which R2 is at rest and the source S and R1 are moving with speed v. That is, since absolute motion cannot be detected, it is not possible to say which is really moving and which is at rest. By postulate 2, the speed of light from a moving source is independent of the motion of the source. Thus, looking at Figure 1-12b, we see that R2 measures the speed of light to be c, just as R1 does. This result, that all observers measure the same value c for the speed of light, is often considered an alternative to Einstein’s second postulate. This result contradicts our intuition. Our intuitive ideas about relative velocities are approximations that hold only when the speeds are very small compared with the speed of light. Even in an airplane moving at the speed of sound, it is not possible to measure the speed of light accurately enough to distinguish the difference between the results c and c v, where v is the speed of the plane. In order to make such a > 1-2 Einstein’s Postulates 13 (Top) Albert Einstein in 1905 at the Bern, Switzerland, patent office. [Hebrew University of Jerusalem Albert Einstein Archives, courtesy AIP Emilio Segrè Visual Archives.] (Bottom) Clock tower and electric trolley in Bern on Kramstrasse, the street on which Einstein lived. If you are on the trolley moving away from the clock and look back at it, the light you see must catch up with you. If you move at nearly the speed of light, the clock you see will be slow. In this, Einstein saw a clue to the variability of time itself. [Underwood & Underwood/CORBIS.] distinction, we must either move with a very great velocity (much greater than that of sound) or make extremely accurate measurements, as in the Michelson-Morley ex- periment, and when we do, we will find, as Einstein pointed out in his original rela- tivity paper, that the contradictions are “only apparently irreconcilable.” Events and Observers In considering the consequences of Einstein’s postulates in greater depth, i.e., in de- veloping the theory of special relativity, we need to be certain that meanings of some important terms are crystal clear. First, there is the concept of an event. A physical event is something that happens, like the closing of a door, a lightning strike, the col- lision of two particles, your birth, or the explosion of a star. Every event occurs at some point in space and at some instant in time, but it is very important to recognize that events are independent of the particular inertial reference frame that we might use to describe them. Events do not “belong” to any reference frame. Events are described by observers who do belong to particular inertial frames of reference. Observers could be people (as in Section 1-1), electronic instruments, or other suitable recorders, but for our discussions in special relativity we are going to be very specific. Strictly speaking, the observer will be an array of recording clocks lo- cated throughout the inertial reference system. It may be helpful for you to think of the observer as a person who goes around reading out the memories of the recording clocks or receives records that have been transmitted from distant clocks, but always keep in mind that in reporting events, such a person is strictly limited to summarizing the data collected from the clock memories. The travel time of light precludes him from including in his report distant events that he may have seen by eye! It is in this sense that we will be using the word observer in our discussions. Each inertial reference frame may be thought of as being formed by a cubic three- dimensional lattice made of identical measuring rods (e.g., meter sticks) with a recording clock at each intersection as illustrated in Figure 1-13. The clocks are all identical, and we, of course, want them all to read the “same time” as one another at any instant; i.e., they must be synchronized. There are many ways to accomplish syn- chronization of the clocks, but a very straightforward way, made possible by the sec- ond postulate, is to use one of the clocks in the lattice as a standard, or reference clock. For convenience we will also use the location of the reference clock in the lattice as the coordinate origin for the reference frame. The reference clock is started with its indicator (hands, pointer, digital display) set at zero. At the instant it starts, it also sends out a flash of light that spreads out as a spherical wave in all directions. When the flash from the reference clock reaches the lattice clocks 1 meter away (notice that in Figure 1-13 there are six of them, two of which are off the edges of the figure), we want their indicators to read the time required for light to travel 1 m ( 1 299,792,458 s). This can be done simply by having an observer at each clock set that time on the in- dicator and then having the flash from the reference clock start them as it passes. The clocks 1 m from the origin now display the same time as the reference clock; i.e., they are all synchronized. In a similar fashion, all of the clocks throughout the inertial frame can be synchronized since the distance of any clock from the reference clock can be calculated from the space coordinates of its position in the lattice and the initial setting of its indicator will be the corresponding travel time for the reference light flash. This procedure can be used to synchronize the clocks in any inertial frame, but it does not synchronize the clocks in reference frames that move with respect to one another. Indeed, as we shall see shortly, clocks in relatively moving frames cannot in general be synchronized with one another. > 14 Chapter 1 Relativity I When an event occurs, its location and time are recorded instantly by the nearest clock. Suppose that an atom located at x 2 m, y 3 m, z 4 m in Figure 1-13 emits a tiny flash of light at t 21 s on the clock at that location. That event is recorded in space and in time or, as we will henceforth refer to it, in the spacetime coordinate sys- tem with the numbers (2,3,4,21). The observer may read out and analyze these data at his leisure, within the limits set by the information transmission time (i.e., the light travel time) from distant clocks. For example, the path of a particle moving through the lattice is revealed by analysis of the records showing the particle’s time of passage at each clock’s location. Distances between successive locations and the corresponding time dif- ferences make possible the determination of the particle’s velocity. Similar records of the spacetime coordinates of the particle’s path can, of course, also be made in any inertial frame moving relative to ours, but to compare the distances and time intervals measured in the two frames requires that we consider carefully the relativity of simultaneity. Relativity of Simultaneity Einstein’s postulates lead to a number of predictions about measurements made by ob- servers in inertial frames moving relative to one another that initially seem very strange, including some that appear paradoxical. Even so, these predictions have been experimentally verified; and nearly without exception, every paradox is resolved by an understanding of the relativity of simultaneity, which states that Two spatially separated events simultaneous in one reference frame are not, in general, simultaneous in another inertial frame moving relative to the first. Figure 1-13 Inertial reference frame formed from a lattice of measuring rods with a clock at each intersection. The clocks are all synchronized using a reference clock. In this diagram the measuring rods are shown to be 1 m long, but they could all be 1 cm, 1 or 1 km as required by the scale and precision of the measurements being considered. The three space dimensions are the clock positions. The fourth spacetime dimension, time, is shown by indicator readings on the clocks. m, x z y Reference clock 1-3 The Lorentz Transformation 17 Notice, too, in Figure 1-15 that C also concludes that the clock at A is ahead of the clock at B. This is important, and we will return to it in more detail in the next sec- tion. Figure 1-16 illustrates the relativity of simultaneity from a different perspective. Questions 3. In addition to that described above, what would be another possible method of synchronizing all of the clocks in an inertial reference system? 4. Using Figure 1-16d, explain how the spaceship observer concludes that Earth clocks are not synchronized. 1-3 The Lorentz Transformation We now consider a very important consequence of Einstein’s postulates, the general relation between the spacetime coordinates x, y, z, and t of an event as seen in refer- ence frame S and the coordinates x, y, z, and t of the same event as seen in reference frame S, which is moving with uniform velocity relative to S. For simplicity we will Earth view of Earth clocks(a) Spaceship view of spaceship clocks(b) AB AB Spaceship view of Earth clocks(d)Earth view of spaceship clocks(c) AB AB v v Figure 1-16 A light flash occurs on Earth midway between two Earth clocks. At the instant of the flash the midpoint of a passing spaceship coincides with the light source. (a) The Earth clocks record the lights’ arrival simultaneously and are thus synchronized. (b) Clocks at both ends of the spaceship also record the lights’ arrival simultaneously (Einstein’s second postulate) and they, too, are synchronized. (c) However, the Earth observer sees the light reach the clock at B before the light reaches the clock at A. Since the spaceship clocks read the same time when the light arrives, the Earth observer concludes that the clocks at A and B are not synchronized. (d) The spaceship observer similarly concludes that the Earth clocks are not synchronized. 18 Chapter 1 Relativity I consider only the special case in which the origins of the two coordinate systems are coincident at time t t 0 and S is moving, relative to S, with speed v along the x (or x) axis and with the y and z axes parallel, respectively, to the y and z axes, as shown in Figure 1-17. As we discussed earlier (Equation 1-2), the classical Galilean coordinate transformation is 1-2 which expresses coordinate measurements made by an observer in S in terms of those measured by an observer in S. The inverse transformation is and simply reflects the fact that the sign of the relative velocity of the reference frames is different for the two observers. The corresponding classical velocity trans- formation was given in Equation 1-3 and the acceleration, as we saw earlier, is invariant under a Galilean transformation. (For the rest of the discussion we will ig- nore the equations for y and z, which do not change in this special case of motion along the x and x axes.) These equations are consistent with experiment as long as v is much less than c. It should be clear that the classical velocity transformation is not consistent with the Einstein postulates of special relativity. If light moves along the x axis with speed c in S, Equation 1-3 implies that the speed in S is rather than . The Galilean transformation equations must therefore be modified to be consistent with Einstein’s postulates, but the result must reduce to the classical equations when v is much less than c. We will give a brief outline of one method of obtaining the rel- ativistic transformation that is called the Lorentz transformation, so named because of its original discovery by H. A. Lorentz.12 We assume the equation for x to be of the form 1-11 where is a constant that can depend upon v and c but not on the coordinates. If this equation is to reduce to the classical one, must approach 1 as v c approaches 0. The inverse transformation must look the same except for the sign of the velocity: 1-12 With the arrangement of the axes in Figure 1-17, there is no relative motion of the frames in the y and z directions; hence y y and z z. However, insertion of the as yet unknown multiplier modifies the classical transformation of time, t t. To see x (x vt) > x (x vt) ux œ cux œ c v x x vt y y z z t t x x vt y y z z t t Figure 1-17 Two inertial frames S and S with the latter moving at speed v in x direction of system S. Each set of axes shown is simply the coordinate axes of a lattice like that in Figure 1-13. Remember, there is a clock at each intersection. A short time before, the times represented by this diagram O and O were coincident and the lattices of S and S were intermeshed. y´ v S (xa, ta ) x z O y S´ (xb, tb ) x´ z´ O´ 1-3 The Lorentz Transformation 19 this, we substitute x from Equation 1-11 into Equation 1-12 and solve for t. The result is 1-13 Now let a flash of light start from the origin of S at t 0. Since we have assumed that the origins coincide at t t 0, the flash also starts at the origin of S at t 0. The flash expands from both origins as a spherical wave. The equation for the wave front according to an observer in S is 1-14 and according to an observer in S, it is 1-15 where both equations are consistent with the second postulate. Consistency with the first postulate means that the relativistic transformation that we seek must transform Equation 1-14 into Equation 1-15 and vice versa. For example, substituting Equations 1-11 and 1-13 into 1-15 results in Equation 1-14 if 1-16 where v c. Notice that 1 for v 0 and for v c. How this is done is illustrated in Example 1-2 below. EXAMPLE 1-2 Relativistic Transformation Multiplier Show that must be given by Equation 1-16 if Equation 1-15 is to be transformed into Equation 1-14 consis- tent with Einstein’s first postulate. SOLUTION Substituting Equations 1-11 and 1-13 into Equation 1-15 and noting that and in this case yields 1-17 To be consistent with the first postulate, Equation 1-15 must be identical to 1-12. This requires that the coefficient of the term in Equation 1-17 be equal to 1, that of the term be equal to , and that of the xt term be equal to 0. Any of those conditions can be used to determine , and all yield the same result. Using, for example, the coefficient of , we have from Equation 1-17 that 2 c2 2 (1 2)2 4v2 1 x2 c2t2 x2 2(x vt)2 y2 z2 c2 2 c t 1 2 2 x v d 2 z z y y S > 1 A1 v 2 c2 1 21 2 x2 y2 z2 c2t2 x2 y2 z2 c2t2 t c t (1 2) 2 x v d 22 Chapter 1 Relativity I from which we see that is given by or 1-22 and, if a particle has velocity components in the y and z directions, it is not difficult to find the components in S in a similar manner. Remember that this form of the velocity transformation is specific to the arrange- ment of the coordinate axes in Figure 1-17. Note, too, that when v c, i.e., when , the relativistic velocity transforms reduce to the classical velocity addition of Equation 1-3. Likewise, the inverse velocity transformation is 1-23 EXAMPLE 1-4 Relative Speeds of Cosmic Rays Suppose that two cosmic ray pro- tons approach Earth from opposite directions, as shown in Figure 1-18a. The speeds relative to Earth are measured to be and . What is Earth’s velocity relative to each proton, and what is the velocity of each proton relative to the other? v2 0.8cv1 0.6c ux uœx va1 vuœx c2 b uy u œ y a1 vuœx c2 b uz u œ z a1 vuœx c2 b v>c 0 V uœy uy a1 vux c2 b uœz uz a1 vuz c2 b uœx ux v 1 vux c2 uœx dx dt (dx vdt) adt vdx c2 b (dx>dt v)1 v c2 dx dt uœx Figure 1-18 (a) Two cosmic ray protons approach Earth from opposite directions at speeds v1 and v2 with respect to Earth. (b) Attaching an inertial frame to each particle and Earth enables one to visualize the several relative speeds involved and apply the velocity transformation correctly. v2v1 S´ 1 2 x´ S´́ x´́ S Earth x (a) (b) SOLUTION Consider each particle and Earth to be inertial reference frames S, S, and S with their respective x axes parallel as in Figure 1-18b. With this arrangement and . Thus, the speed of Earth measured in S is and the speed of Earth measured in S is .vflEx 0.8c vœEx 0.6cv2 u2x 0.8c v1 u1x 0.6c 1-3 The Lorentz Transformation 23 To find the speed of proton 2 with respect to proton 1, we apply Equation 1-22 to compute i.e., the speed of particle 2 in S. Its speed in S has been measured to be , where the S system has relative speed with respect to S. Thus, substituting into Equation 1-22, we obtain and the first proton measures the second to be approaching (moving in the direction) at 0.95c. The observer in S must of course make a consistent measurement, i.e., find the speed of proton 1 to be 0.95c in the x direction. This can be readily shown by a second application of Equation 1-22 to compute Questions 5. The Lorentz transformation for y and z is the same as the classical result: y y and z z. Yet the relativistic velocity transformation does not give the classical result and Explain. 6. Since the velocity components of a moving particle are different in relatively moving frames, the directions of the velocity vectors are also different in general. Explain why the fact that observers in S and S measure different directions for a particle’s motion is not an inconsistency in their observations. Spacetime Diagrams The relativistic discovery that time intervals between events are not the same for all observers in different inertial reference frames underscores the four-dimensional character of spacetime. With the diagrams that we have used thus far, it is difficult to depict and visualize on the two-dimensional page events that occur at different times, since each diagram is equivalent to a snapshot of spacetime at a particular instant. Showing events as a function of time typically requires a series of diagrams, such as Figures 1-14, 1-15, and 1-16, but even then our attention tends to be drawn to the space coordinate systems rather than the events, whereas it is the events that are fun- damental. This difficulty is removed in special relativity with a simple yet powerful graphing method called the spacetime diagram. (This is just a new name given to the t vs. x graphs that you first began to use when you discussed motion in introductory physics.) On the spacetime diagram we can graph both the space and time coordi- nates of many events in one or more inertial frames, albeit with one limitation. Since the page offers only two dimensions for graphing, we suppress, or ignore for now, two of the space dimensions, in particular y and z. With our choice of the relative motion of inertial frames along the x axis, y y and z z anyhow. (This is one of the reasons we made that convenient choice a few pages back, the other reason being mathematical simplicity.) This means that for the time being, we are limiting our attention to one space dimension and to time, i.e., to events that occur, regardless of uz u œ z .uy u œ y ufl1x 0.6c (0.8c) 1 (0.6c)(0.8c)>c2 1.4c1.48 0.95c ufl1x : x uœ2x 0.8c (0.6c) 1 (0.6c)(0.8c)>c2 1.4c1.48 0.95c v1 0.6cu2x 0.8c uœ2x , 24 Chapter 1 Relativity I when, along one line in space. Should we need the other two dimensions, e.g., in a consideration of velocity vector transformations, we can always use the Lorentz transformation equations. In a spacetime diagram the space location of each event is plotted along the x axis horizontally and the time is plotted vertically. From the three-dimensional array of measuring rods and clocks in Figure 1-13, we will use only those located on the x axis, as in Figure 1-19. (See, things are simpler already!) Since events that exhibit relativis- tic effects generally occur at high speeds, it will be convenient to multiply the time scale by the speed of light (a constant), which enables us to use the same units and scale on both the space and time axes, e.g., meters of distance and meters of light travel time.13 The time axis is, therefore, c times the time t in seconds, i.e., ct. As we will see shortly, this choice prevents events from clustering about the axes and makes possible the straightforward addition of other inertial frames into the diagram. As time advances, notice that in Figure 1-19 each clock in the array moves verti- cally upward along the dotted lines. Thus, as events A, B, C, and D occur in spacetime, one of the clocks of the array is at (or very near) each event when it happens. Remember that the clocks in the reference frame are synchronized, and so the differ- ence in the readings of clocks located at each event records the proper time interval be- tween the events. (See Example 1-3.) In the figure, events A and D occur at the same place (x 2 m) but at different times. The time interval between them measured on clock 2 is the proper time interval since clock 2 is located at both events. Events A and B occur at different locations but at the same time (i.e., simultaneously in this frame). Event C occurred before the present since ct 1 m. For this discussion we will con- sider the time that the coordinate origins coincide, , to be the present. Worldlines in Spacetime Particles moving in space trace out a line in the spacetime diagram called the worldline of the particle. The worldline is the “trajectory” of the par- ticle on a ct versus x graph. To illustrate, consider four particles moving in space (not spacetime) as shown in Figure 1-20a, which shows the array of synchronized clocks on the x axis and the space trajectories of four particles, each starting at x 0 and mov- ing at some constant speed during 3 m of time. Figure 1-20b shows the worldline for each of the particles in spacetime. Notice that constant speed means that the worldline has constant slope; i.e., it is a straight line (slope ¢t ¢x 1 (¢x ¢t) 1 speed).>>>> ct ct 0 Figure 1-19 Spacetime diagram for an inertial reference frame S. Two of the space dimensions (y and z) are suppressed. The units on both the space and time axes are the same, meters. A meter of time means the time required for light to travel one meter, i.e., 3.3 109 s. ct (m) x (m)–3 –2 –1 0 1 2 3 2 B C A D 1 –1 3 1-3 The Lorentz Transformation 27 space and time. For example, the clock at x 1 m in Figure 1-22b passed the point x 0 at about ct 1.5 m as the x axis of S moved both upward and to the right in S. Remember, as time advances, the array of synchronized clocks and measuring rods that are the x axis also move upward, so that, for example, when ct 1, the origin of S (x 0, ct 0) has moved vt (v c)ct ct to the right along the x axis. Question 7. Explain how the spacetime diagram in Figure 1-22b would appear drawn by an observer in S. > 1 1 2 3 4 0.866 0.866 2 3 4–1 –1 ct (m) ct´ (m) (b) 4 4 3 2 1 3 2 1 –1 –1 x (m) x´ (m) 1 2 3 4–1 ct (m) ct´ (m)(a) 4 3 2 1 –1 x (m) x´ (m) Figure 1-22 Spacetime diagram of S showing S moving at speed v 0.5c in the x direction. The diagram is drawn with t t 0 when the origins of S and S coincided. The dashed line shows the worldline of a light flash that passed through the point x 0 at t 0 heading in the x direction. Its slope equals 1 in both S and S. The ct and x axes of S have slopes of and respectively. (a) Calibrating the axes of S as described in the text allows the grid of coordinates to be drawn on S. Interpretation is facilitated by remembering that (b) shows the system S as it is observed in the spacetime diagram of S. 0.5, 1> 2 28 Chapter 1 Relativity I 2 1 –2 –1 2 1 –2 –1 B A C C´ A´ B´ ct ct´ x x´ Platform (S frame) Train (S´ frame) ct´ (B´) ct´ (A´) Figure 1-23 Spacetime equivalent of Figure 1-15, showing the spacetime diagram for the system S in which the platform is at rest. Measurements made by observers in S are read from the primed axes. EXAMPLE 1-6 Simultaneity in Spacetime Use the train-platform example of Figure 1-15 and a suitable spacetime diagram to show that events simultaneous in one frame are not simultaneous in a frame moving relative to the first. (This is the corollary to the relativity of simultaneity that we first demonstrated in the previous section using Figure 1-15.) SOLUTION Suppose a train is passing a station platform at speed v and an observer C at the mid- point of the platform, system S, announces that light flashes will be emitted at clocks A and B located at opposite ends of the platform at t 0. Let the train, system S, be a rocket train with v 0.5c. As in the earlier discussion, clocks at C and C both read 0 as C passes C. Figure 1-23 shows this situation. It is the spacetime equivalent of Figure 1-15. Two events occur, the light flashes. The flashes are simultaneous in S since both occur at ct 0. In S, however, the event at A occurred at ct(A) (see Figure 1-23), about 1.2 ct units before ct 0, and the event at B occurred at ct(B), about 1.2 ct units after ct 0. Thus, the flashes are not simultaneous in S and A occurs before B, as we also saw in Figure 1-15. EXPLORING Calibrating the Spacetime Axes By calibrating the coordinate axes of S consistent with the Lorentz transformation, we will be able to read the coordinates of events and calculate space and time intervals between events as measured in both S and S directly from the diagram, in addition to calculating them from Equations 1-18 and 1-19. The calibration of the S axes is 1-4 Time Dilation and Length Contraction 29 straightforward and is accomplished as follows. The locus of points, e.g., with x 1 m, is a line parallel to the ct axis through the point x 1 m, ct 0, just as we saw ear- lier that the ct axis was the locus of those points with x 0 through the point x 0, ct 0. Substituting these values into the Lorentz transformation for x, we see that the line through x 1 m intercepts the x axis, i.e., the line where ct 0 at 1-24 or, in general, In Figure 1-22b, where 0.5, the line x 1 m intercepts the x axis at x 0.866 m. Similarly, if x 2 m, x 1.73 m; if x 3 m, x 2.60 m; and so on. The ct axis is calibrated in a precisely equivalent manner. The locus of points with ct 1 m is a line parallel to the x axis through the point ct 1 m, x 0. Using the Lorentz transformation, the intercept of that line with the ct axis (where x 0) is found as follows: which can also be written as 1-25 or ct ct for x 0. Thus, for ct 1 m, we have 1 ct or ct (1 2) and, again in general, ct ct(1 2) . The x ct coordinate grid is shown in Figure 1-22b. Notice in Figure 1-22b that the clocks located in S are not found to be synchro- nized by observers in S, even though they are synchronized in S. This is exactly the conclusion that we arrived at in the discussion of the lightning striking the train and platform. In addition, those with positive x coordinates are behind the S reference clock and those with negative x coordinates are ahead, the differences being greatest for those clocks farthest away. This is a direct consequence of the Lorentz transforma- tion of the time coordinate—i.e., when ct 0 in Equation 1-25, ct x. Note, too, that the slope of the worldline of the light beam equals 1 in S as well as in S, as required by the second postulate. 1-4 Time Dilation and Length Contraction The results of correct measurements of the time and space intervals between events do not depend upon the kind of apparatus used for the measurements or on the events themselves. We are free therefore to choose any events and measuring apparatus that will help us understand the application of the Einstein postulates to the results of mea- surements. As you have already seen from previous examples, convenient events in relativity are those that produce light flashes. A convenient, simple such clock is a light clock, pictured schematically in Figure 1-24. A photocell detects the light pulse and sends a voltage pulse to an oscilloscope, which produces a vertical deflection of the oscilloscope’s trace. The phosphorescent material on the face of the oscilloscope tube gives a persistent light that can be observed visually, photographed, or recorded electronically. The time between two light flashes is determined by measuring the #1>2 1>2 ct (ct x) t (t vx>c2) x x21 2 1 x or x 1> 21 2x (x vt) (x ct) Figure 1-26 Spacetime diagram illustrating time dilation. The dashed line is the worldline of a light flash emitted at x 0 and reflected back to that point by a mirror at x 1 m. 0.5. 1 2 ct´ (m) 3 2 1 x (m) ct (m) ct = 2γ 2 1 1 2 x´ (m) 32 Chapter 1 Relativity I EXAMPLE 1-7 Spatial Separation of Events Two events occur at the same point at times and in S, which moves with speed v relative to S. What is the spatial separation of these events measured in S? SOLUTION tœ2t œ 1 xœ0 1. The location of the events in S is given by the Lorentz inverse transformation Equation 1-19: x (x vt) 2. The spatial separation of the two events is then¢x x2 x1 ¢x (xœ0 vt œ 2) (x œ 0 vt œ 1) 3. The terms cancel: xœ0 ¢x v(t œ 2 t œ 1) v¢t œ 4. Since ¢t is the proper time interval , Equation 1-26 yields ¢x v v¢t 5. Using the situation in Figure 1-26 as a numerical example, where 0.5 and 1.15, we have 1.15 m ¢x v c ¢(ct) (1.15)(0.5)(2) EXAMPLE 1-8 The Pregnant Elephant14 Elephants have a gestation period of 21 months. Suppose that a freshly impregnated elephant is placed on a spaceship and sent toward a distant space jungle at v 0.75c. If we monitor radio transmissions from the spaceship, how long after launch might we expect to hear the first squeal- ing trumpet from the newborn calf? SOLUTION 1. In S, the rest frame of the elephant, the time interval from launch to birth, is 21 months. In the Earth frame S the time interval is given by Equation 1-26: ¢t1 , 31.7 months 1 21 (0.75)2 (21 months) ¢t1 1 21 2 2. At that time the radio signal announcing the happy event starts toward Earth at speed c, but from where? Using the result of Example 1-7, since launch the spaceship has moved ¢x in S, given by where c month is the distance light travels in one month. # 23.8 c # months (1.51)(0.75)(21 c # months) ¢x vt ct 3. Notice that there is no need to con- vert ¢x into meters since our interest is in how long it will take the radio signal to travel this distance in S. That time is given by¢t2 , 23.8 months 23.8 c # months>c¢t2 ¢x>c 4. Thus, the good news will arrive at Earth at time ¢t after launch where 55.5 months 31.7 23.8 ¢t ¢t1 ¢t2 Figure 1-27 Sketch of the spacetime diagram for Example 1-8. The colored line is the worldline of the pregnant elephant. The worldline of the radio signal is the dashed line at 45° toward the upper left. 0.75. ct (c • mo) 55.5 Radio signal Elephant 31.7 0 23.80 (c • mo) x ct´ x´ 1-4 Time Dilation and Length Contraction 33 Question 8. You are standing on a corner and a friend is driving past in an automobile. Both of you note the times when the car passes two different intersections and determine from your watch readings the time that elapses between the two events. Which of you has determined the proper time interval? The time dilation of Equation 1-26 is easy to see in a spacetime diagram such as Figure 1-26, using the same round trip for a light pulse used above. Let the light flash leave x 0 at ct 0 when the S and S origins coincided. The flash travels to x 1 m, reflects from a mirror located there, and returns to x 0. Let 0.5. The dotted line shows the worldline of the light beam, reflecting at (x 1, ct 1) and returning to x 0 at ct 2 m. Note that the S observer records the latter event at ; i.e., the observer in S sees the S clock running slow. Experimental tests of the time dilation prediction have been performed using macroscopic clocks, in particular, accurate atomic clocks. In 1975, C. O. Alley con- ducted a test of both general and special relativity in which a set of atomic clocks were carried by a U.S. Navy antisubmarine patrol aircraft while it flew back and forth over the same path for 15 hours at altitudes between 8000 m and 10,000 m over Chesapeake Bay. The clocks in the plane were compared by laser pulses with an iden- tical group of clocks on the ground. (See Figure 1-13 for one way such a comparison might be done.) Since the experiment was primarily intended to test the gravitational effect on clocks predicted by general relativity (see Section 2-5), the aircraft was deliberately flown at the rather sedate average speed of 270 knots (140 m s) 4.7 107c to minimize the time dilation due to the relative speeds of the clocks. Even so, after Alley deducted the effect of gravitation as predicted by general relativity, the airborne clocks lost an average of 5.6 109 s due to the relative speed during the 15-hour flight. This result agrees with the prediction of spe- cial relativity, 5.7 109 s, to within 2 percent, even at this low relative speed. The experimental results leave little basis for fur- ther debate as to whether traveling clocks of all kinds lose time on a round trip. They do. Length Contraction A phenomenon closely related to time dilation is length contrac- tion. The length of an object measured in the reference frame in which the object is at rest is called its proper length Lp. In a ref- erence frame in which the object is moving, the measured length parallel to the direction of motion is shorter than its proper length. Consider a rod at rest in the frame S with one end at and the other end at as illustrated in Figure 1-28. The length of the rod in this frame is its proper length Some care mustLp x œ 2 x œ 1 . xœ1 , xœ2 > ct 2 m Remarks: This result, too, is readily obtained from a spacetime diagram. Figure 1-27 illustrates the general appearance of the spacetime diagram for this example, showing the elephant’s worldline and the worldline of the radio signal. x (m)(x1) (x2) ct (m) ct ′ (m) x ′ (m) 1 2 1 2 L Lp 1 (x ′)1 2 (x ′)2 Figure 1-28 A measuring rod, a meter stick in this case, lies at rest in S between and System S moves with relative to S. Since the rod is in motion, S must measure the locations of the ends of the rod x2 and x1 simultaneously in order to have made a valid length measurement. L is obviously shorter than Lp . By direct measurement from the diagram (use a millimeter scale) L>Lp 0.61 1> . 0.79xœ1 1 m. xœ2 2 m 34 Chapter 1 Relativity I be taken to find the length of the rod in frame S. In this frame, the rod is moving to the right with speed v, the speed of frame S. The length of the rod in frame S is defined as where x2 is the position of one end at some time t2 and x1 is the position of the other end at the same time as measured in frame S. Since the rod is at rest in S, need not equal Equation 1-18 is convenient to use to cal- culate at some time t because it relates x, x, and t, whereas Equation 1-19 is not convenient because it relates x, x, and t: Since we obtain or 1-28 Thus, the length of a rod is smaller when it is measured in a frame with respect to which it is moving. Before Einstein’s paper was published, Lorentz and G. FitzGerald had independently shown that the null result of the Michelson-Morley experiment could be explained by assuming that the lengths in the direction of the interferometer’s motion contracted by the amount given in Equation 1-28. For that reason, the length contraction is often called the Lorentz-FitzGerald contraction. EXAMPLE 1-9 Speed of S A stick that has a proper length of 1 m moves in a direction parallel to its length with speed v relative to you. The length of the stick as measured by you is 0.914 m. What is the speed v? SOLUTION L 1 Lp A1 v 2 c2 Lp x2 x1 1 (xœ2 x œ 1) A1 v 2 c2 (xœ2 x œ 1) xœ2 x œ 1 (x2 x1) t2 t1 , xœ2 (x2 vt2) and x œ 1 (x1 vt1) x2 x1 tœ1 .t œ 2 t1 t2 L x2 x1 , L Lp 1. The length of the stick measured in a frame relative to which it is mov- ing with speed v is related to its proper length by Equation 1-28: Lp L 2. Rearranging to solve for : 1 m 0.914 m 1 21 v2>c23. Substituting the values of Lp and L: v 0.406c v2 0.165c2 v2>c2 1 0.835 0.165 1 v2>c2 (0.914)2 0.835 21 v2>c2 0.9144. Solving for v: 1-4 Time Dilation and Length Contraction 37 we expect to observe at sea level in the same time interval? According to the nonrel- ativistic prediction, the time it takes for these muons to travel 9000 m is (9000 m) 0.998c 30 s, which is 15 lifetimes. Substituting N0 10 8 and t 15 into Equation 1-29, we obtain We would thus expect all but about 31 of the original 100 million muons to decay before reaching sea level. According to the relativistic prediction, Earth must travel only the contracted dis- tance of 600 m in the rest frame of the muon. This takes only 2 s 1. Therefore, the number of muons expected at sea level is Thus relativity predicts that we would observe 36.8 million muons in the same time interval. Experiments of this type have confirmed the relativistic predictions. The Spacetime Interval We have seen earlier in this section that time intervals and lengths ( space intervals), quantities that were absolutes, or invariants, for relatively moving observers using the classical Galilean coordinate transformation, are not invariants in special relativity. The Lorentz transformation and the relativity of simultaneity lead observers in iner- tial frames to conclude that lengths moving relative to them are contracted and time intervals are stretched, both by the factor . The question naturally arises: Is there any quantity involving the space and time coordinates that is invariant under a Lorentz transformation? The answer to that question is yes, and as it happens, we have already dealt with a special case of that invariant quantity when we first obtained the correct form of the Lorentz transformation. It is called the spacetime interval, or usually just the interval, ¢s, and is given by 1-30 or, specializing it to the one-space-dimensional systems that we have been discussing, 1-31 It may help to think of Equations 1-30 and 1-31 like this: The interval ¢s is the only measurable quantity describing pairs of events in spacetime for which observers in all inertial frames will obtain the same numerical value. The negative sign in Equations 1-30 and 1-31 implies that (¢s)2 may be posi- tive, negative, or zero depending on the relative sizes of the time and space separa- tions. With the sign of (¢s)2, nature is telling us about the causal relation between the two events. Notice that whichever of the three possibilities characterizes a pair for one observer, it does so for all observers since ¢s is invariant. The interval is called time- like if the time separation is the larger and spacelike if the space separation predomi- nates. If the two terms are equal, so that ¢s 0, then it is called lightlike. [interval]2 [separation in time]2 [separation in space]2 (¢s)2 (c¢t)2 (¢x)2 (¢s)2 (c¢t)2 [¢x2 ¢y2 ¢z2] N 108e1 3.68 107 N 108e15 30.6 > Experiments with muons moving near the speed of light are performed at many accelerator laboratories throughout the world despite their short mean life. Time dilation results in much longer mean lives relative to the laboratory, providing plenty of time to do experiments. 38 Chapter 1 Relativity I Timelike Interval Consider a material particle15 or object, e.g., the elephant in Figure 1-27, that moves relative to S. Since no material particle has ever been measured traveling faster than light, particles always travel less than 1 m of distance in 1 m of light travel time. We saw that to be the case in Example 1-8, where the time interval between launch and birth of the baby was 31.7 months on the S clock, during which time the elephant had moved a distance of 23.8c months. Equation 1-31 then yields (c¢t)2 (¢x)2 (31.7c)2 (23.8c)2 (21.0c)2 (¢s)2, and the interval in S is ¢s 21.0 c months. The time interval term being the larger, ¢s is a timelike in- terval and we say that material particles have timelike worldlines. Such worldlines lie within the shaded area of the spacetime diagram in Figure 1-21. Note that in the ele- phant’s frame S the separation in space between the launch and birth is zero and ¢t is 21.0 months. Thus ¢s 21.0 c months in S, too. That is what we mean by the interval being invariant: observers in both S and S measure the same number for the separation of the two events in spacetime. The proper time interval between two events can be determined from Equations 1-31 using space and time measurements made in any inertial frame since we can write that equation as Since ¢t when ¢x 0—i.e., for the time interval recorded on a clock in a system moving such that the clock is located at each event as it occurs—in that case 1-32 Notice that this yields the correct proper time 21.0 months in the elephant example. Spacelike Interval When two events are separated in space by an interval whose square is greater than the value of (c¢t )2, then ¢s is called spacelike. In that case it is convenient for us to write Equation 1-31 in the form 1-33 so that, as with timelike intervals, (¢s)2 is not negative.16 Events that are spacelike occur sufficiently far apart in space and close together in time that no inertial frame could move fast enough to carry a clock from one event to the other. For example, sup- pose two observers in Earth frame S, one in San Francisco and one in London, agree to each generate a light flash at the same instant, so that c¢t 0 m in S and ¢x 1.08 107 m. For any other inertial frame (c¢t )2 0, and we see from Equation 1-33 that (¢x)2 must be greater than (1.08 107)2 in order that ¢s be invariant. In other words, 1.08 107 m is as close in space as the two events can be in any system; con- sequently, it will not be possible to find a system moving fast enough to move a clock from one event to the other. A speed greater than c, in this case infinitely greater, would be needed. Notice that the value of ¢s Lp, the proper length. Just as with the proper time interval , measurements of space and time intervals in any inertial sys- tem can be used to determine Lp. (¢s)2 (¢x)2 (c¢t)2 2(¢t)2 (¢x>c)2 22 0 ¢s c ¢s c 2(¢t)2 (¢x>c)2 # # # 1-4 Time Dilation and Length Contraction 39 Lightlike (or Null) Interval The relation between two events is lightlike if s in Equation 1-31 equals zero. In that case 1-34 and a light pulse that leaves the first event as it occurs will just reach the second as it occurs. The existence of the lightlike interval in relativity has no counterpart in the world of our everyday experience, where the geometry of space is Euclidean. In order for the distance between two points in space to be zero, the separation of the points in each of the three space dimensions must be zero. However, in spacetime the interval be- tween two events may be zero, even though the intervals in space and time may indi- vidually be quite large. Notice, too, that pairs of events separated by lightlike intervals have both the proper time interval and proper length equal to zero since s 0. Things that move at the speed of light17 have lightlike worldlines. As we saw ear- lier (see Figure 1-22), the worldline of light bisects the angles between the ct and x axes in a spacetime diagram. Timelike intervals lie in the shaded areas of Figure 1-32 and share the common characteristic that their relative order in time is the same for observers in all inertial systems. Events A and B in Figure 1-32 are such a pair. Observers in both S and S agree that A occurs before B, although they of course mea- sure different values for the space and time separations. Causal events, i.e., events that depend upon or affect one another in some fashion, such as your birth and that of your mother, have timelike intervals. On the other hand, the temporal order of events with spacelike intervals, such as A and C in Figure 1-32, depends upon the relative motion of the systems. As you can see in the diagram, A occurs before C in S, but C occurs first in S. Thus, the relative order of pairs of events is absolute in the shaded areas but elsewhere may be in either order. c¢t ¢x Figure 1-32 The relative temporal order of events for pairs characterized by timelike intervals, such as A and B, is the same for all inertial observers. Events in the upper shaded area will all occur in the future of A; those in the lower shaded area occurred in As past. Events whose intervals are spacelike, such as A and C, can be measured as occurring in either order, depending on the relative motion of the frames. Thus, C occurs after A in S but before A in S. ct B Absolute future Absolute past Worldline of light moving in +x direction Worldline of light moving in –x direction C A ct´ ctC ctB x x´ ct B́ ct Ć The use of Doppler radar to track weather systems is a direct application of special relativity. 42 Chapter 1 Relativity I diagram of S, the system in which A and B are at rest. The source is located at x 0 (x axis is not shown), and, of course, its worldline is the ct axis. Let the source emit a train of N electromagnetic waves in each direction beginning when the S and S ori- gins were coincident. First, let’s consider the train of waves headed toward A. During the time t over which the source emits the N waves, the first wave emitted will have traveled a distance ct and the source itself a distance vt in S. Thus, the N waves are seen by the observer at A to occupy a distance ct vt and, correspondingly, their wavelength is given by and the frequency is The frequency of the source in S, called the proper frequency, is given by where t is measured in S, the rest system of the source. The time interval t is the proper time interval since the light waves, in particular the first and the Nth, are all emitted at x 0; hence x 0 between the first and the Nth in S. Thus, t and t are related by Equation 1-26 for time dilation, so t t, and when the source and receiver are moving toward each other, the observer A in S measures the frequency 1-35 or 1-36 This differs from the classical equation only in the addition of the time dilation factor. Note that f fo for the source and observer approaching each other. Since for visible light this corresponds to a shift toward the blue part of the spectrum, it is called a blueshift. Suppose the source and receiver are moving away from each other, as for ob- server B in Figure 1-34b. Observer B, in S, sees the N waves occupying a distance , and the same analysis shows that observer B in S measures the frequency 1-37 Notice that f f0 for the observer and source receding from each other. Since for vis- ible light this corresponds to a shift toward the red part of the spectrum, it is called a redshift. It is left as a problem for you to show that the same results are obtained when the analysis is done in the frame in which the source is at rest. In the event that (i.e., ), as is often the case for light sources mov- ing on Earth, useful (and easily remembered) approximations of Equations 1-36 and 1-37 can be obtained. Using Equation 1-36 as an example and rewriting it in the form f f0(1 ) 1>2(1 )1>2 V 1v V c f 21 2 1 f0 A 1 1 f0 (receding) c¢t v¢t f 21 2 1 f0 A 1 1 f0 (approaching) f 1 1 f0¢t ¢t f0 1 1 f0 c> N>¢t, f c cN (c v)¢t 1 1 N ¢t f c> c¢t v¢t N 1-5 The Doppler Effect 43 the two quantities in parentheses can be expanded by the binomial theorem to yield Multiplying out and discarding terms of higher order than yields (approaching) and, similarly, (receding) and in both situations, where f f0 f. EXAMPLE 1-12 Rotation of the Sun The Sun rotates at the equator once in about 25.4 days. The Sun’s radius is 7.0 108 m. Compute the Doppler effect that you would expect to observe at the left and right limbs (edges) of the Sun near the equa- tor for light of wavelength 550 nm 550 109 m (yellow light). Is this a redshift or a blueshift? SOLUTION The speed of limbs v (circumference)/(time for one revolution) or so we may use the approximation equations. Using we have that or Hz. Since fo c o (3 10 8 m s) (550 109) 5.45 1014 Hz, f represents a frac- tional change in frequency of or about one part in 105. It is a redshift for the receding limb, a blueshift for the approaching one. Doppler Effect of Starlight In 1929 E. P. Hubble became the first astronomer to suggest that the universe is expanding.18 He made that suggestion and offered a simple equation to describe the expansion on the basis of measurements of the Doppler shift of the frequencies of light emitted toward us by distant galaxies. Light from distant galaxies is always shifted toward frequencies lower than those emitted by similar sources nearby. Since the general expression connecting the frequency f and wavelength of light is c f the shift corresponds to longer wavelengths. As noted above, the color red is on the longer-wavelength side of the visible spectrum (see Chapter 4), so the redshift is used to describe the Doppler effect for a receding source. Similarly, blueshift describes light emitted by stars, typically stars in our galaxy, that are approaching us. Astronomers define the redshift of light from astronomical sources by the expression z (fo f) f, where fo frequency measured in the frame of the star or galaxy and f frequency measured at the receiver on Earth. This allows us to write v c in terms of z as 1-38 (z 1)2 1 (z 1)2 1 > > , , >>> ¢f 2000>550 109 3.64 109¢f fo c> o , ¢f>f0 ,v V c, v 2 R T 2 (7.0 108) m 25.4 d # 3600 s>h # 24 h>d 2000 m>s ƒ¢f>f0 ƒ f>f0 1 f>f0 1 f f0a1 12 18 2 Á b a1 12 38 2 Á b 44 Chapter 1 Relativity I Equation 1-37 is the appropriate one to use for such calculations, rather than the approximations, since galactic recession velocities can be quite large. For example, the quasar 2000-330 has a measured z 3.78, which implies from Equation 1-38 that it is receding from Earth due to the expansion of space at 0.91c. (See Chapter 13.) EXAMPLE 1-13 Redshift of Starlight The longest wavelength of light emitted by hydrogen in the Balmer series (see Chapter 4) has a wavelength of 656 nm. In light from a distant galaxy, this wavelength is measured as 1458 nm. Find the speed at which the galaxy is receding from Earth. SOLUTION o f A 1 1 f01. The recession speed is the v in Since this is a redshift and Equation 1-37 applies: o , v>c. f A 1 1 f f0 o 2. Rewriting Equation 1-37 in terms of the wavelengths: a 656 nm 1458 nm b 2 0.202 1 1 a 0 b 23. Squaring both sides and substituting val- ues for and : o 0.798 1.202 0.664 1.202 1 0.202 0.798 1 (0.202)(1 )4. Solving for : v c 0.664c5. The galaxy is thus receding at speed v, where EXPLORING Transverse Doppler Effect Our discussion of the Doppler effect in Section 1-5 involved only one space dimension, wherein the source, observer, and the direction of the relative motion all lie on the x axis. In three space dimensions, where they may not be colinear, a more complete analysis, though beyond the scope of our discussion, makes only a small change in Equation 1-35. If the source moves along the positive x axis but the observer views the light emitted at some angle with the x axis, as shown in Figure 1-34c, Equation 1-35 becomes 1-35a When 0, this becomes the equation for the source and receiver approaching, and when the equation becomes that for the source and receiver receding. Equation 1-35a also makes the quite surprising prediction that even when viewed perpendicular , f f0 1 1 cos 1-6 The Twin Paradox and Other Surprises 47 A correct analysis can be made using the invariant interval s from Equation 1-31 rewritten as where the left side is constant and equal to the proper time interval squared, and the right side refers to measurements made in any inertial frame. Thus, Ulysses along each of his worldlines in Figure 1-35a has x 0 and, of course, measures t 3 y, or 6 y for the round trip. Homer, on the other hand, measures and since (x c)2 is always positive, he always measures t . In this situation x 0.8ct, so or or 10 y for the round trip, as we saw earlier. The reason that the twins’ situations can- not be treated symmetrically is because the special theory of relativity can predict the behavior of accelerated systems, such as Ulysses at the turnaround, provided that in the formulation of the physical laws we take the view of an inertial, i.e., unacceler- ated, observer such as Homer. That’s what we have done. Thus, we cannot do the same analysis in the rest frame of Ulysses’ spaceship because it does not remain in an inertial frame during the round trip; hence, it falls outside of the special theory, and no paradox arises. The laws of physics can be reformulated so as to be invariant for accelerated observers, which is the role of general relativity (see Chapter 2), but the result is the same: Ulysses returns younger than Homer by just the amount calculated above. EXAMPLE 1-14 Twin Paradox and the Doppler Effect This example, first suggested by C. G. Darwin,20 may help you understand what each twin sees during Ulysses’ journey. Homer and Ulysses agree that once each year, on the anniversary of the launch date of Ulysses’ spaceship (when their clocks were together), each twin will send a light signal to the other. Figure 1-35b shows the light signals each sends. Based on our discussion above, Homer sends 10 light flashes (the ct axis, Homer’s worldline, is divided into 10 equal interval corresponding to the 10 years of the jour- ney on Homer’s clock) and Ulysses sends 6 light flashes (each of Ulysses’ world- lines is divided into 3 equal intervals corresponding to 3 years on Ulysses’ clock). Note that each transmits his final light flash as they are reunited at B. Although each transmits light signals with a frequency of 1 per year, they obviously do not receive them at that frequency. For example, Ulysses sees no signals from Homer during the first three years! How can we explain the observed frequencies? ¢t 3 0.6 5 y (¢t)2(0.36) (3)2 (¢t)2 (3 y)2 (0.8c¢t>c)2 > (¢t)2 ()2 a¢x c b 2 ()2, a¢s c b 2 (¢t)2 a¢x c b 2 48 Chapter 1 Relativity I Question 10. The different ages of the twins upon being reunited are an example of the relativity of simultaneity that was discussed earlier. Explain how that accounts for the fact that their biological clocks are no longer synchronized. More It is the relativity of simultaneity, not their different accelerations, that is responsible for the age difference between the twins. This is readily illustrated in The Case of the Identically Accelerated Twins, which can be found on the home page: www.whfreeman.com/tiplermodern- physics5e. See also Figure 1-36 here. The Pole and Barn Paradox An interesting problem involving length contraction, reported initially by W. Rindler, involves putting a long pole into a short barn. One version, from E. F. Taylor and J. A. Wheeler,22 goes as follows. A runner carries a pole 10 m long toward the open front door of a small barn 5 m long. A farmer stands near the barn so that he can see both the front and the back doors of the barn, the latter being a closed swinging door, as shown in Figure 1-37a. The runner carrying the pole at speed v enters the barn, and at some instant the farmer sees the pole completely contained in the barn and closes the SOLUTION The Doppler effect provides the explanation. As the twins (and clocks) recede from each other the frequency of their signals is reduced from the proper frequency f0 according to Equation 1-37 and we have which is exactly what both twins see (refer to Figure 1-35b): Homer receives 3 flashes in the first 9 years and Ulysses 1 flash in his first 3 years; i.e., f (1 3)f0 for both. After the turnaround they are approaching each other and Equation 1-38 yields and again this agrees with what the twins see: Homer receives 3 flashes during the final (10th) year and Ulysses receives 9 flashes during his final 3 years; i.e., f 3fo for both. f f0 A 1 1 A 1 0.8 1 0.8 3 > f f0 A 1 1 A 1 0.8 1 0.8 1 3 1-6 The Twin Paradox and Other Surprises 49 Figure 1-37 (a) A runner carrying a 10-m pole moves quickly enough so that the farmer will see the pole entirely contained in the barn. The spacetime diagrams from the point of view of the farmer’s inertial frame (b) and that of the runner (c). The resolution of the paradox is in the fact that the events of interest, shown by the large dots in each diagram, are simultaneous in S but not in S. Pole entirely within barn (ct = 5.8 m) Front end of pole enters barn door at ct = 0 Front door of barn Rear door of barn Back end of pole Pole Front end of pole Rear doorFront door Front of pole leaves barn Back of pole enters barn Back of pole Front of pole (c)(b) (a) 10 1050–5 –5 ct x (m) S 10 5 50–10 ct´ x´ (m) S´ Front of pole enters barn door at ct´ = 0 Pole front door, thus putting a 10-m pole into a 5-m barn. The minimum speed of the run- ner v that is necessary for the farmer to accomplish this feat can be computed from Equation 1-28, giving the relativistic length contraction L Lp , where Lp proper length of the pole (10 m) and L length of the pole measured by the farmer, to be equal to the length of the barn (5 m). Therefore, we have v 0.866c or 0.866 v2>c2 1 (5>10)2 0.75 1 v2>c2 (5>10)2 1 21 v2>c2 LpL 105 > In determining the brightness of stars and galaxies, a critical parameter in understanding them, astronomers must correct for the headlight effect, particularly at high velocities relative to Earth. 52 Chapter 1 Relativity I Considering the half of the light emitted by the source in S into the forward hemisphere, i.e., rays with between note that Equation 1-41 restricts the an- gles measured in S for those rays (50 percent of all the light) to lie between For example, for the observer in S would see half of the total light emitted by the source in S to lie between i.e., in a cone of half angle 60° whose axis is along the direction of the velocity of the source. For values of near unity is very small, e.g., yields 8.1°. This means that the observer in S sees half of all the light emitted by the source to be concentrated into a forward cone with that half angle. (See Figure 1-38b.) Note, too, that the remaining 50 percent of the emitted light is distributed throughout the remaining 344° of the two-dimensional diagram.23 As a result of the headlight effect, light from a directly approaching source appears far more intense than that from the same source at rest. For the same reason, light from a directly receding source will appear much dimmer than that from the same source at rest. This result has substantial applications in experimental particle physics and astrophysics. Question 12. Notice from Equation 1-41 that some light emitted by the moving source into the rear hemisphere is seen by the observer in S as having been emitted into the forward hemisphere. Explain how that can be, using physical arguments. EXPLORING Superluminal Speeds We conclude this chapter with a few comments about things that move faster than light. The Lorentz transformations (Equations 1-18 and 1-19) have no meaning in the event that the relative speeds of two inertial frames exceed the speed of light. This is gener- ally taken to be a prohibition on the moving of mass, energy, and information faster than c. However, it is possible for certain processes to proceed at speeds greater than c and for the speeds of moving objects to appear to be greater than c without contradict- ing relativity theory. A common example of the first of these is the motion of the point where the blades of a giant pair of scissors intersect as the scissors are quickly closed, sometimes called the scissors paradox. Figure 1-39 shows the situation. A long straight rod (one blade) makes an angle with the x axis (the second blade) and moves in the y direction at constant speed vy y t. During time t, the intersection of the blades, point P, moves to the right a distance x. Note from the figure that > .99 60°, 0.5, cos1 . >2, Figure 1-39 As the long straight rod moves vertically downward, the intersection of the “blades,” point P, moves toward the right at speed vp x t. In terms of vy and vp vy>tan ., > x θ y P vy vy Δy Δx 1-6 The Twin Paradox and Other Surprises 53 y x The speed with which P moves to the right is 1-42 or Since as it will always be possible to find a value of close enough to zero so that vp c for any (nonzero) value of vy . As real scissors are closed, the angle gets progressively smaller, so in principle all that one needs for vp c are long blades so that Question 13. Use a diagram like Figure 1-32 to explain why the motion of point P cannot be used to convey information to observers along the blades. The point P in the scissors paradox is, of course, a geometrical point, not a mate- rial object, so it is perhaps not surprising that it could appear to move at speeds greater than c. As an example of an object with mass appearing to do so, consider a tiny mete- orite moving through space directly toward you at high speed v. As it enters Earth’s at- mosphere, about 9 km above the surface, frictional heating causes it to glow and the first light from the glow starts toward your eye. After some time t the frictional heat- ing has evaporated all of the meteorite’s matter, the glow is extinguished, and its final light starts toward your eye, as illustrated in Figure 1-40. During the time between the first and the final glow, the meteorite traveled a distance vt. During that same time in- terval light from the first glow has traveled toward your eye a distance ct. Thus, the space interval between the first and final glows is given by and the visual time interval at your eye teye between the arrival of the first and final light is and, finally, the apparent visual speed va that you record is 1-43 Clearly, yields va c and any larger yields va c. For example, a mete- orite approaching you at v 0.8c is perceived to be moving at va 4c. Certain galactic 0.5 va v¢t ¢teye v¢t ¢t(1 ) c 1 ¢teye ¢y>c ¢t(c v)c ¢t(1 ) ¢y c¢t v¢t ¢t(c v) S 0. S 0,tan S 0 vp vy tan vp ¢x>¢t ¢x¢y>vy vy¢x¢x tan tan .> Meteorite first glow Last glow wave front First glow wave front Eye v v Δt c Δt (c – v) Δt Figure 1-40 A meteorite moves directly toward the observer’s eye at speed v. The spatial distance between the wave fronts is (c v)t as they move at c, so the time interval between their arrival at the observer is not t, but teye , which is and the apparent speed of approach is va v¢t>¢teye c>(1 ). (1 )¢t, (c v)¢t>c 54 Chapter 1 Relativity I Superluminal Motion in M87 Jet 1994 1995 1996 1997 1998 6.0c 5.5c 6.1c 6.0c structures may also be observed to move at superluminal speeds, as the sequence of im- ages of the jet from galaxy M87 in Figure 1-41 illustrates. As a final example of things that move faster than c, it has been proposed that par- ticles with mass might exist whose speeds would always be faster than light speed. One basis for this suggestion is an appealing symmetry: ordinary particles always have v c, and photons and other massless particles have v c, so the existence of particles with v c would give a sort of satisfying completeness to the classification of particles. Called tachyons, their existence would present relativity with serious but not necessar- ily insurmountable problems of infinite creation energies and causality paradoxes, e.g., alteration of history. (See the next example.) No compelling theoretical arguments pre- clude their existence and eventual discovery; however, to date, all experimental searches for tachyons24 have failed to detect them, and the limits set by those experi- ments indicate that it is highly unlikely they exist. EXAMPLE 1-15 Tachyons and Reversing History Use tachyons and an appropriate spacetime diagram to show how the existence of such particles might be used to change history and, hence, alter the future, leading to a paradox. SOLUTION In a spacetime diagram of the laboratory frame S the worldline of a particle with v c created at the origin traveling in the x direction makes an angle less than 45° with the x axis; i.e., it is below the light worldline, as shown in Figure 1-42. After some time the tachyon reaches a tachyon detector mounted on a spaceship moving rapidly away at v c in the x direction. The spaceship frame S is shown in the figure at P. The detector immediately creates a new tachyon, sending it off in the x direction and, of course, into the future of S, i.e., with ct 0. The second tachyon returns to the laboratory at x 0 but at a time ct before the first tachyon was emitted, having traveled into the past of S to point M, where ct 0. Having sent an object into our own past, we would then have the ability to alter events that occur after M and produce causal contradictions. For example, the laboratory tachyon detector could be coupled to equipment that created the first tachyon via a computer programmed to Figure 1-41 Superluminal motion has been detected in a number of cosmic objects. This sequence of images taken by the Hubble Space Telescope shows apparent motion at six times the speed of light in galaxy M87. The jet streaming from the galaxy’s nucleus (the bright round region at the far left in the bar image at the top) is about 5000 long. The boxed region is enlarged. The slanting lines track the moving features and indicate the apparent speeds in each region. [John Biretta, Space Telescope Science Institute.] c # y Notes 57 Notes 1. Polish astronomer, 1473–1543. His book describing heliocentric (i.e., Sun-centered) orbits for the planets was published only a few weeks before his death. He had hesitated to release it for many years, fearing that it might be con- sidered heretical. It is not known whether or not he saw the published book. 2. Events are described by measurements made in a coordi- nate system that defines a frame of reference. The question was, Where is the reference frame in which the law of inertia is valid? Newton knew that no rotating system, e.g., Earth or the Sun, would work and suggested the distant “fixed stars” as the fundamental inertial reference frame. 3. The speed of light is exactly 299,792,458 m s. The value is set by the definition of the standard meter as being the dis- tance light travels in 1 299,792,458 s. 4. Over time, an entire continuous spectrum of electromag- netic waves has been discovered, ranging from extremely low- frequency (radio) waves to extremely high-frequency waves (gamma rays), all moving at speed c. 5. Albert A. Michelson (1852–1931), an American experi- mental physicist whose development of precision optical in- struments and their use in precise measurements of the speed of light and the length of the standard meter earned him the Nobel Prize in 1907. Edward W. Morley (1838–1923), an American chemist and physicist and professor at Western Reserve College when Michelson was a professor at the nearby Case School of Applied Science. 6. Albert A. Michelson and Edward W. Morley, The American Journal of Science, XXXIV, no. 203, November 1887. 7. Note that the width depends on the small angle between and M1 . A very small angle results in relatively few wide fringes, a larger angle in many narrow fringes. 8. Since the source producing the waves, the sodium lamp, was at rest relative to the interferometer, the frequency would be constant. 9. T. S. Jaseja, A. Javan, J. Murray, and C. H. Townes, Physical Review, 133, A1221 (1964). 10. A. Brillet and J. Hall, Physical Review Letters, 42, 549 (1979). 11. Annalen der Physik, 17, 841(1905). For a translation from the original German, see the collection of original papers by Lorentz, Einstein, Minkowski, and Weyl (Dover, New York, 1923). 12. Hendrik Antoon Lorentz (1853–1928), Dutch theoretical physicist, discovered the Lorentz transformation empirically while investigating the fact that Maxwell’s equations are not in- variant under a Galilean transformation, although he did not rec- ognize its importance at the time. An expert on electromagnetic M œ2 > > theory, he was one of the first to suggest that atoms of matter might consist of charged particles whose oscillations could ac- count for the emission of light. Lorentz used this hypothesis to explain the splitting of spectral lines in a magnetic field discov- ered by his student Pieter Zeeman, with whom he shared the 1902 Nobel Prize. 13. One meter of light travel time is the time for light to travel 1 m, i.e., ct 1 m, or t 1 m/3.00 108 m s 3.3 109 s. Similarly, 1 cm of light travel time is ct 1 cm, or t 3.3 1011 s, and so on. 14. This example is adapted from a problem in H. Ohanian, Modern Physics (Englewood Cliffs, NJ: Prentice Hall, 1987). 15. Any particle that has mass. 16. Equation 1-31 would lead to imaginary values of s for spacelike intervals, an apparent problem. However, the geom- etry of spacetime is not Euclidean, but Lorentzian. While a consideration of Lorentz geometry is beyond the scope of this chapter, suffice it to say that it enables us to write (s)2 for spacelike intervals as in Equation 1-33. 17. There are only two such things: photons (including those of visible light), which will be introduced in Chapter 3, and gravitons, which are the particles that transmit the gravita- tional force. 18. Edwin P. Hubble, Proceedings of the National Academy of Sciences, 15, 168 (1929). 19. Walter Kündig, Physical Review, 129, 2371 (1963). 20. C. G. Darwin, Nature, 180, 976 (1957). 21. S. P. Boughn, American Journal of Physics, 57, 791 (1989). 22. E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2d ed. (New York: W. H. Freeman and Co., 1992). 23. Seen in three space dimensions by the observer in S, 50 percent of the light is concentrated in 0.06 steradian of 4 - steradian solid angle around the moving source. 24. T. Alväger and M. N. Kreisler, “Quest for Faster-Than- Light Particles,” Physical Review, 171, 1357 (1968). 25. Paul Ehrenfest (1880–1933), Austrian physicist and pro- fessor at the University of Leiden (the Netherlands), longtime friend and correspondent of Einstein, about whom, upon his death, Einstein wrote, “[He was] the best teacher in our pro- fession I have ever known.” 26. This experiment is described in J. C. Hafele and R. E. Keating, Science, 177, 166 (1972). Although not as accurate as the experiment described in Section 1-4, its results sup- ported the relativistic prediction. 27. R. Shaw, American Journal of Physics, 30, 72 (1962). > 58 Chapter 1 Relativity I Problems Level I Section 1-1 The Experimental Basis of Relativity 1-1. In episode 5 of Star Wars, the Empire’s spaceships launch probe droids throughout the galaxy to seek the base of the Rebel Alliance. Suppose a spaceship moving at 2.3 108 m s toward Hoth (site of the rebel base) launches a probe droid toward Hoth at 2.1 108 m s relative to the spaceship. According to Galilean relativity, (a) What is the speed of the droid relative to Hoth? (b) If rebel astronomers are watching the approaching spaceship through a telescope, will they see the probe before it lands on Hoth? 1-2. In one series of measurements of the speed of light, Michelson used a path length L of 27.4 km (17 mi). (a) What is the time needed for light to make the round trip of distance 2L? (b) What is the classical correction term in seconds in Equation 1-5, assuming Earth’s speed is v 104c? (c) From about 1600 measurements, Michelson arrived at a result for the speed of light of 299,796 4 km s. Is this experimental value accurate enough to be sensitive to the cor- rection term in Equation 1-5? 1-3. A shift of one fringe in the Michelson-Morley experiment would result from a difference of one wavelength or a change of one period of vibration in the round-trip travel of the light when the interferometer is rotated by 90°. What speed would Michelson have computed for Earth’s motion through the ether had the experiment seen a shift of one fringe? 1-4. In the “old days” (circa 1935) pilots used to race small, relatively high-powered airplanes around courses marked by a pylon on the ground at each end of the course. Suppose two such evenly matched racers fly at airspeeds of 130 mph. (Remember, this was a long time ago!) Each flies one complete round trip of 25 miles, but their courses are perpendicular to each other and there is a 20-mph wind blowing steadily parallel to one course. (a) Which pilot wins the race and by how much? (b) Relative to the axes of their respective courses, what headings must the two pilots use? 1-5. Paul Ehrenfest25 suggested the following thought experiment to illustrate the dramati- cally different observations that might be expected, dependent on whether light moved relative to a stationary ether or according to Einstein’s second postulate: Suppose that you are seated at the center of a huge dark sphere with a radius of 3 108 m and with its inner surface highly reflective. A source at the center emits a very brief flash of light that moves outward through the darkness with uniform intensity as an expanding spherical wave. What would you see during the first 3 seconds after the emission of the flash if (a) the sphere moved through the ether at a constant 30 km s and (b) if Einstein’s second postulate is correct? 1-6. Einstein reported that as a boy he wondered about the following puzzle. If you hold a mirror at arm’s length and look at your reflection, what will happen as you begin to run? In par- ticular, suppose you run with speed v 0.99c. Will you still be able to see yourself? If so, what would your image look like, and why? 1-7. Verify by calculation that the result of the Michelson-Morley experiment places an upper limit on Earth’s speed relative to the ether of about 5 km s. 1-8. Consider two inertial reference frames. When an observer in each frame measures the following quantities, which measurements made by the two observers must yield the same re- sults? Explain your reason for each answer. (a) The distance between two events (b) The value of the mass of a proton (c) The speed of light (d) The time interval between two events (e) Newton’s first law ( f ) The order of the elements in the periodic table (g) The value of the electron charge > > > > > Problems 59 Section 1-2 Einstein’s Postulates 1-9. Assume that the train shown in Figure 1-14 is 1.0 km long as measured by the observer at C and is moving at 150 km h. What time interval between the arrival of the wave fronts at C is measured by the observer at C in S? 1-10. Suppose that A, B, and C are at rest in frame S, which moves with respect to S at speed v in the x direction. Let B be located exactly midway between A and C. At t 0 a light flash occurs at B and expands outward as a spherical wave. (a) According to an observer in S, do the wave fronts arrive at A and C simultaneously? (b) According to an observer in S, do the wave fronts arrive at A and C simultaneously? (c) If you answered no to either (a) or (b), what is the difference in their arrival times and at which point did the front arrive first? Section 1-3 The Lorentz Transformation 1-11. Make a graph of the relativistic factor as a function of Use at least 10 values of ranging from 0 up to 0.995. 1-12. Two events happen at the same point in frame S at times and (a) Use Equation 1-19 to show that in frame S, the time interval between the events is greater than by a factor (b) Why is Equation 1-18 less convenient than Equation 1-19 for this problem? 1-13. Suppose that an event occurs in inertial frame S with coordinates x 75 m, y 18 m, z 4.0 m at t 2.0 105 s. The inertial frame S moves in the x direction with v 0.85c. The origins of S and S coincided at t t 0. (a) What are the coordinates of the event in S? (b) Use the inverse transformation on the results of (a) to obtain the original coordinates. 1-14. Show that the null effect of the Michelson-Morley experiment can be accounted for if the interferometer arm parallel to the motion is shortened by a factor of 1-15. Two spaceships are approaching each other. (a) If the speed of each is 0.9c relative to Earth, what is the speed of one relative to the other? (b) If the speed of each relative to Earth is 30,000 m s (about 100 times the speed of sound), what is the speed of one relative to the other? 1-16. Starting with the Lorentz transformation for the components of the velocity (Equation 1-23), derive the transformation for the components of the acceleration. 1-17. Consider a clock at rest at the origin of the laboratory frame. (a) Draw a spacetime dia- gram that illustrates that this clock ticks slow when observed from the reference frame of a rocket moving with respect to the laboratory at v 0.8c. (b) When 10 s have elapsed on the rocket clock, how many have ticked by on the lab clock? 1-18. A light beam moves along the y axis with speed c in frame S, which is moving to the right with speed v relative to frame S. (a) Find ux and uy , the x and y components of the veloc- ity of the light beam in frame S. (b) Show that the magnitude of the velocity of the light beam in S is c. 1-19. A particle moves with speed 0.9c along the x axis of frame S, which moves with speed 0.9c in the positive x direction relative to frame S. Frame S moves with speed 0.9c in the pos- itive x direction relative to frame S. (a) Find the speed of the particle relative to frame S. (b) Find the speed of the particle relative to frame S. Section 1-4 Time Dilation and Length Contraction 1-20. Use the binomial expansion to derive the following results for values of and use when applicable in the problems that follow. (a) (b) (c) 1-21. How great must the relative speed of two observers be for their time-interval measure- ments to differ by 1 percent (see Problem 1-20)? 1 1 1 1 2 v2 c2 1 1 1 2 v2 c2 1 1 2 v2 c2 v V c > (1 v2>c2)1>2. . tœ2 t œ 1 tœ2 .t œ 1x œ 0 v>c. 1>(1 v2>c2)1>2 > 62 Chapter 1 Relativity I to account for the absence of the expected shift of 0.4 of a fringe width? (Assume the diameter of atoms to be about m.) 1-45. Observers in reference frame S see an explosion located at x1 480 m. A second explo- sion occurs 5 later at x2 1200 m. In reference frame S, which is moving along the x axis at speed v, the explosions occur at the same point in space. (a) Draw a spacetime diagram de- scribing this situation. (b) Determine v from the diagram. (c) Calibrate the ct axis and deter- mine the separation in time in between the two explosions as measured in S. (d) Verify your results by calculation. 1-46. Two spaceships, each 100 m long when measured at rest, travel toward each other with speeds of 0.85c relative to Earth. (a) How long is each ship as measured by someone on Earth? (b) How fast is each ship traveling as measured by an observer on the other? (c) How long is one ship when measured by an observer on the other? (d) At time t 0 on Earth, the fronts of the ships are together as they just begin to pass each other. At what time on Earth are their ends together? (e) Sketch accurately scaled diagrams in the frame of one of the ships showing the passing of the other ship. 1-47. If v is much less than c, the Doppler frequency shift is approximately given by both classically and relativistically. A radar transmitter-receiver bounces a signal off an aircraft and observes a fractional increase in the frequency of What is the speed of the aircraft? (Assume the aircraft to be moving directly toward the transmitter.) 1-48. The null result of the Michelson-Morley experiment could be explained if the speed of light depended on the motion of the source relative to the observer. Consider a binary eclipsing star system, that is, a pair of stars orbiting their common center of mass with Earth lying in the orbital plane of the system, as is very nearly the case for the binary system Algol (see More sec- tion about the Michelson-Morley experiment). Assume that the stars in the system have circu- lar orbits with a period of 115 days and that one of the stars’ orbital speed is 32 km s (about the same as Earth’s orbital speed around the Sun). If the suggestion above were true, astronomers would simultaneously see two images of the star in opposition, i.e., on opposite sides of its orbit. What is the minimum distance L from Earth to the binary for this phenomenon to occur? 1-49. Frames S and S are moving relative to each other along the x and x axes. They set their clocks to t t 0 when their origins coincide. In frame S, event 1 occurs at x1 1 c y and t1 1 y and event 2 occurs at x2 2.0 c y and t2 0.5 y. These events occur simultaneously in frame S. (a) Find the magnitude and direction of the velocity of S relative to S. (b) At what time do both of these events occur as measured in S? (c) Compute the spacetime interval s between the events. (d) Is the interval spacelike, timelike, or lightlike? (e) What is the proper distance Lp between the events? 1-50. Do Problem 1-49 parts (a) and (b) using a spacetime diagram. 1-51. An observer in frame S standing at the origin observes two flashes of colored light sep- arated spatially by x 2400 m. A blue flash occurs first, followed by a red flash 5 later. An observer in S moving along the x axis at speed v relative to S also observes the flashes 5 apart and with a separation of 2400 m, but the red flash is observed first. Find the magnitude and direction of v. 1-52. A cosmic-ray proton streaks through the lab with velocity 0.85c at an angle of 50° with the x direction (in the xy plane of the lab). Compute the magnitude and direction of the pro- ton’s velocity when viewed from frame S moving with Level III 1-53. A meter stick is parallel to the x axis in S and is moving in the y direction at constant speed vy . From the viewpoint of S show that the meter stick will appear tilted at an angle with respect to the x axis of S moving in the x direction at Compute the angle measured in S. 1-54. The equation for the spherical wave front of a light pulse that begins at the origin at time t 0 is Using the Lorentz transformation, show that such a light pulse also has a spherical wave front in S by showing that in S. 1-55. An interesting paradox has been suggested by R. Shaw27 that goes like this. A very thin steel plate with a circular hole 1 m in diameter centered on the y axis lies parallel to the xz plane xœ2 yœ2 zœ2 (ct)2 0 x2 y2 z2 (ct)2 0. 0.65. 0.72. s s # # > ¢f>f0 8 107.¢f>f0 , s s 1010 Problems 63 y x z vy vy β = v/c Figure 1-45 in frame S and moves in the y direction at constant speed vy as illustrated in Figure 1-45. A meter stick lying on the x axis moves in the x direction with The steel plate ar- rives at the y 0 plane at the same instant that the center of the meter stick reaches the origin of S. Since the meter stick is observed by observers in S to be contracted, it passes through the 1-m hole in the plate with no problem. A paradox appears to arise when one considers that an observer in S, the rest system of the meter stick, measures the diameter of the hole in the plate to be contracted in the x dimension and, hence, becomes too small to pass the meter stick, re- sulting in a collision. Resolve the paradox. Will there be a collision? 1-56. Two events in S are separated by a distance and a time (a) Use the Lorentz transformation to show that in frame S, which is moving with speed v relative to S, the time separation is (b) Show that the events can be simultaneous in frame S only if D is greater than cT. (c) If one of the events is the cause of the other, the sep- aration D must be less than cT since is the smallest time that a signal can take to travel from x1 to x2 in frame S. Show that if D is less than cT, is greater than in all reference frames. (d) Suppose that a signal could be sent with speed c c so that in frame S the cause precedes the effect by the time Show that there is then a reference frame moving with speed v less than c in which the effect precedes the cause. 1-57. Two observers agree to test time dilation. They use identical clocks and one observer in frame S moves with speed v 0.6c relative to the other observer in frame S. When their ori- gins coincide, they start their clocks. They agree to send a signal when their clocks read 60 min and to send a confirmation signal when each receives the other’s signal. (a) When does the ob- server in S receive the first signal from the observer in S. (b) When does he receive the confir- mation signal? (c) Make a table showing the times in S when the observer sent the first signal, received the first signal, and received the confirmation signal. How does this table compare with one constructed by the observer in S? 1-58. The compact disk in a CD-ROM drive rotates with angular speed There is a clock at the center of the disk and one at a distance r from the center. In an inertial reference frame, the clock at distance r is moving with speed Show that from time dilation in special rela- tivity, time intervals to for the clock at rest and tr for the moving clock are related by if 1-59. Two rockets, A and B, leave a space station with velocity vectors vA and vB relative to the station frame S, perpendicular to each other. (a) Determine the velocity of A relative to B, vBA . (b) Determine the velocity of B relative to A, vAB . (c) Explain why vAB and vBA do not point in opposite directions. 1-60. Suppose a system S consisting of a cubic lattice of meter sticks and synchronized clocks, e.g., the eight clocks closest to you in Figure 1-13, moves from left to right (the x direction) at high speed. The meter sticks parallel to the x direction are, of course, contracted and the cube r V c ¢tr ¢to ¢to r22 2c2 u r. . T D>c. t œ 1t œ 2 D>ct2 t1 (T vD>c 2). T t2 t1 .D x2 x1 v>c. 64 Chapter 1 Relativity I would be measured by an observer in a system S to be foreshortened in that direction. However, recalling that your eye constructs images from light waves that reach it simultaneously, not those leaving the source simultaneously, sketch what your eye would see in this case. Scale con- tractions and show any angles accurately. (Assume the moving cube to be farther than 10 m from your eye.) 1-61. Figure 1-11b (in the More section about the Michelson-Morley experiment) shows an eclipsing binary. Suppose the period of the motion is T and the binary is a distance L from Earth, where L is sufficiently large so that points A and B in Figure 1-11b are a half orbit apart. Consider the motion of one of the stars and (a) show that the star would appear to move from A to B in time and from B to A in time assuming classical velocity addition applies to light, i.e., that emission theories of light were correct. (b) What rotational period would cause the star to appear to be at both A and B simultaneously? 1-62. Show that if a particle moves at an angle with respect to the x axis with speed u in sys- tem S, it moves at an angle with the x axis in S given by 1-63. Like jets emitted from some galaxies (see Figure 1-41), some distant astronomical ob- jects can appear to travel at speeds greater than c across our line of sight. Suppose distant galaxy AB15 moving with velocity v at an angle with respect to the direction toward Earth emits two bright flashes of light separated by time t on the galaxy AB15 local clock. Show that (a) the time interval and (b) the apparent speed of AB15 measured by ob- servers on Earth is (c) For compute the value of for which vapp c. 0.75,vapp ¢xEarth ¢tEarth sin 1 cos . ¢tEarth ¢t(1 cos) tan sin (cos v>u) T>2 2Lv>(c2 v2),T>2 2Lv>(c2 v2) 2-1 Relativistic Momentum 67 Thus, in S the total y component of classical momentum is not zero. Since the y com- ponents of the velocities are reversed in an elastic collision, momentum as defined by p mu is not conserved in S. Analysis of this problem in S leads to the same conclusion (Figure 2-1b), since the roles of A and B are simply interchanged.2 In the classical limit momentum is conserved, of course, because in that limit and The reason for defining momentum as mu in classical mechanics is that this quantity is conserved when there is no external force, as in our collision example. We now see that this quantity is conserved only in the approximation We will define the relativistic momentum p of a particle to have the following properties: 1. p is conserved in collisions. 2. p approaches mu as approaches zero. Let’s apply the first of these conditions to the collision of the two balls that we just discussed, noting two important points. First, for each observer in Figure 2-1, the speed of each ball is unchanged by the elastic collision. It is either (for the ob- server’s own ball) or (for the other ball). Second, the failure of the conservation of momentum in the collision we described can’t be due to the velocities because we used the Lorentz transformation to find the y components. It must have something to do with the mass! Let us write down the conservation of the y compo- nent of the momentum as observed in S, keeping the masses of the two balls straight by writing for the S observer’s own ball and for the observer’s ball. 2-3 Equation 2-3 can be readily rewritten as 2-4 If is small compared to the relative speed v of the reference frames, then it follows from Equation 2-2 that and, therefore, If we can now imagine the limiting case where i.e., where each ball is at rest in its “home” frame so that the collision becomes a “grazing” one as B moves past A at speed v u, then we conclude from Equations 2-2 and 2-4 that in order for Equation 2-3 to hold, i.e., for the momentum to be conserved, or 2-5 Equation 2-5 says that the observer in S measures the mass of ball B, moving relative to him at speed u, as equal to times the rest mass of the ball, or its mass measured in the frame in which it is at rest. Notice that observers always mea- sure the mass of an object that is in motion with respect to them to be larger than the value measured when the object is at rest. 1>(1 v2>c2)1>2 m(u) m 21 v2>c2 m(u v) m(u0 0) u0 u021 v2>c2 u0 S 0, u v.uyB V v u0 m(u) m(u0) u0 uyB (after collision)(before collision) m(u0)u0 m(u)uyB m(u0)u0 m(u)uyB Sm(u)m(u0) (u2y v 2)1>2 u u0 u>c v V c. uyB u0 . 1v V c, S´ S B A A v x´ x y´ y (b) S´ S B v x´ x y´ y u0 u0 (a) The design and construction of large particle accelerators throughout the world, such as CERN’s LHC, are based directly on the relativistic expressions for momentum and energy. Figure 2-1 (a) Elastic collision of two identical balls as seen in frame S. The vertical component of the velocity of ball B is in S if it is u0 in S. (b) The same collision as seen in S. In this frame, ball A has vertical component of velocity u0> . u0> 68 Chapter 2 Relativity II 4mc 3mc 2mc mc Relativistic momentum 0 0 0.2 0.4 0.6 0.8 1.0 u/c p Figure 2-2 Relativistic momentum as given by Equation 2-6 versus where u speed of the object relative to an observer. The magnitude of the momentum p is plotted in units of mc. The fainter dashed line shows the classical momentum mu for comparison. u>c, Thus, we see that the law of conservation of momentum will be valid in relativ- ity, provided that we write the momentum p of an object with rest mass m moving with velocity u relative to an inertial system S to be 2-6 where u is the speed of the particle. We therefore take this equation as the definition of relativistic momentum. It is clear that this definition meets our second criterion be- cause the denominator approaches 1 when u is much less than c. From this definition, the momenta of the two balls A and B in Figure 2-1 as seen in S are where and It is similarly straightforward to show that Because of the similarity of the factor and in the Lorentz transformation, Equation 2-6 is often written 2-7 This use of the symbol for two different quantities causes some confusion; the notation is standard, however, and simplifies many of the equations. We will use this notation except when we are also considering transformations between reference frames. Then, to avoid confusion, we will write out the factor and reserve for where v is the relative speed of the frames. Figure 2-2 shows a graph of the magnitude of p as a function of The quantity m(u) in Equation 2-5 is sometimes called the relativistic mass; however, we will avoid using the term or a symbol for relativistic mass: in this book m always refers to the mass u>c.1>(1 v2>c2)1>2, 1>(1 u 2>c2)1>2 p mu with 1 21 u2>c2 1>11 u2>c2pyB pyA . uxB v.uyB u0(1 v2>c2)1>2 pyA mu0 21 u20>c2 pyB muyB21 (u2xB u2yB)>c2 p mu 21 u2>c2 2-1 Relativistic Momentum 69 measured in the rest frame. In this we are following Einstein’s view. In a letter to a colleague in 1948 he wrote:3 It is not good to introduce the concept of mass of a body for which no clear definition can be given. It is better to introduce no other mass than “the rest mass” m. Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion. M m>(1 v 2>c 2)1>2 EXAMPLE 2-2 Momentum of a Rocket A high-speed interplanetary probe with a mass m 50,000 kg has been sent toward Pluto at a speed u 0.8c. What is its momentum as measured by Mission Control on Earth? If, preparatory to landing on Pluto, the probe’s speed is reduced to 0.4c, by how much does its momentum change? EXAMPLE 2-1 Measured Values of Moving Mass For what value of will the measured mass of an object exceed the rest mass by a given fraction f? SOLUTION From Equation 2-5 we see that Solving for or from which we can compute the table of values below or the value of for any other f. Note that the value of that results in a given fractional increase f in the measured value of the mass is independent of m. A diesel locomotive moving at a particular will be observed to have the same f as a proton moving with that u>c.u>c u>c u>cu>c 2f(f 2) f 1 1 u2>c2 1 (f 1)2 ¡ u2>c2 1 1 (f 1)2 u>c, f m m m 1 1 21 u2>c2 1 m u>c f Example 1012 1.4 106 jet fighter aircraft 5 109 0.0001 Earth’s orbital speed 0.0001 0.014 50-eV electron 0.01 (1%) 0.14 quasar 3C273 1.0 (100%) 0.87 quasar 0Q172 10 0.996 muons from cosmic rays 100 0.99995 some cosmic ray protons u>c 72 Chapter 2 Relativity II 1.0 0.5 Nonrelativistic Relativistic 0.0 0 4 5321 Kinetic energy (MeV) u –– c 2 1– 2 Ek = mu 2 Ek = mc 2 – 1 1––––––––– 1 – (u/c)2 Equation 2-9 defines the relativistic kinetic energy. Notice that, as we warned earlier, is not or even This is strikingly evident in Figure 2-3. However, consistent with our second condition on the relativistic total energy E, Equation 2-9 does approach when We can check this assertion by noting that for expanding by the binomial theorem yields and thus The expression for kinetic energy in Equation 2-9 consists of two terms. One term, depends on the speed of the particle (through the factor ), and the other term, is independent of the speed. The quantity is called the rest energy of the particle, i.e., the energy associated with the rest mass m. The relativistic total en- ergy E is then defined as the sum of the kinetic energy and the rest energy. 2-10 Thus, the work done by a net force increases the energy of the system from the rest energy to (or increases the measured mass from m to ). For a particle at rest relative to an observer, and Equation 2-10 becomes perhaps the most widely recognized equation in all of physics, Einstein’s famous E mc2. When Equation 2-10 can be written as E 1 2 mu2 mc2 u V c, Ek 0, m mc2mc2 E Ek mc 2 mc2 mc2 21 u2>c2 mc2mc2, mc2, Ek mc 2a1 1 2 u2 c2 Á 1b 1 2 mu2 a1 u2 c2 b1>2 1 1 2 u2 c2 Á u>c V 1, u V c.mu2>2 mu2>2.mu2>2Ek Figure 2-3 Experimental confirmation of the relativistic relation for kinetic energy. Electrons were accelerated to energies up to several MeV in large electric fields, and their velocities were determined by measuring their time of flight over 8.4 m. Note that when the velocity the relativistic and nonrelativistic (i.e., classical) relations are indistinguishable. [W. Bertozzi, American Journal of Physics, 32, 551 (1964).] u V c, Before the development of relativity theory, it was thought that mass was a con- served quantity;4 consequently, m would always be the same before and after an interaction or event and would therefore be constant. Since the zero of energy is arbitrary, we are always free to include an additive constant; therefore, our definition of the relativistic total energy reduces to the classical kinetic energy for and our second condition on E is satisfied.5 Be very careful to understand Equation 2-10 correctly. It defines the total energy E, and E is what we are seeking to conserve for isolated systems in all inertial frames, not and not Remember, too, the distinction between conserved quantities and invariant quantities. The former have the same value before and after an interaction in a particular reference frame. The latter have the same value when measured by ob- servers in different reference frames. Thus, we are not requiring observers in relatively moving inertial frames to measure the same values for E, but rather that E be un- changed in interactions as measured in each frame. To assist us in showing that E as defined by Equation 2-10 is conserved in relativity, we will first see how E and p transform between inertial reference frames. Lorentz Transformation of E and p Consider a particle of rest mass m that has an arbitrary velocity u with respect to frame S, as shown in Figure 2-4. System is a second inertial frame moving in the x direction. The particle’s momentum and energy are given in the S and systems, respectively, by, In S: 2-11 where In S: 2-12 where Developing the Lorentz transformation for E and p requires that we first express in terms of quantities measured in S. (We could just as well express in terms of primed quantities. Since this is relativity, it makes no difference which we choose.) The result is 2-13 1 21 u2>c2 (1 vux>c2)21 u2>c2 where now 121 v2>c2 1>21 u2>c2 pœz mu œ z pœy mu œ y pœx mu œ x E mc2 1>21 u2>c2 pz muz py muy px mux E mc2 S S mc2.Ek u V c mc2 2-2 Relativistic Energy 73 y´S S´ z y u x, x´ z´ v Figure 2-4 Particle of mass m moves with velocity u measured in S. System S moves in the x direction at speed v. The Lorentz velocity transformation makes possible determination of the relations connecting measurements of the total energy and the components of the momentum in the two frames of reference. 74 Chapter 2 Relativity II Substituting Equation 2-13 into the expression for in Equation 2-12 yields The first term in the brackets you will recognize as E, and the second term, canceling the factors, as from Equation 2-11. Thus, we have 2-14 Similarly, substituting Equation 2-13 and the velocity transformation for into the expression for in Equations 2-12 yields The first term in the brackets is from Equation 2-11, and, because the second term is Thus, we have 2-15 Using the same approach, we can show (Problem 2-46) that Together these relations are the Lorentz transformation for momentum and energy: 2-16 The inverse transformation is 2-17 with Note the striking similarity between Equations 2-16 and 2-17 and the Lorentz transformation of the space and time coordinates, Equations 1-18 and 1-19. The mo- mentum transforms in relativity exactly like and the total en- ergy E transforms like the time t. We will return to this remarkable result and related matters shortly, but first let’s do some examples and then, as promised, show that the energy as defined by Equation 2-10 is conserved in relativity. EXAMPLE 2-3 Transforming Energy and Momentum Suppose a micrometeorite of mass 109 kg moves past Earth at a speed of 0.01c. What values will be measured for the energy and momentum of the particle by an observer in a system moving relative to Earth at 0.5c in the same direction as the micrometeorite? S r(x, y, z),p(px , py , pz) 1 21 v2>c2 121 2 E (E vpœx) pz p œ z px (p œ x vE>c2) py pœy E (E vpx) p œ z pz pœx (px vE>c2) pœy py pœy py and p œ z pz pœx (px vE>c2)vE>c 2.E>c2, m(1 u2>c2)1>2 px pœx muœx 21 u2>c2 c mux21 u2>c2 mv21 u2>c2 d pœx uœx E (E vpx) vpxc 2 E mc2 21 u2>c2 c mc221 u2>c2 mc2vux>c221 u2>c2 d E 2-2 Relativistic Energy 77 Using the total energy as defined by Equation 2-10, we have in : Before collision: 2-18 After collision: 2-19 Energy will be conserved in if i.e., if 2-20 This is ensured by the validity of conservation of momentum, in particular by Equation 2-5, and so energy is conserved in (The validity of Equation 2-20 is im- portant and not trivial. We will consider it in more detail in Example 2-7.) To see if energy as we have defined it is also conserved in S, we transform to S using the inverse transform, Equation 2-17. We then have in S: Before collision: 2-21 After collision: 2-22 The energy will be conserved in S and, therefore, the law of conservation of energy will hold in all inertial frames if i.e., if 2-23 which, like Equation 2-20, is ensured by Equation 2-5. Thus, we conclude that the en- ergy as defined by Equation 2-10 is consistent with a relativistically invariant law of con- servation of energy, satisfying the first of the conditions set forth at the beginning of this section. While this demonstration was not a general one, since that is beyond the scope of our discussions, you may be assured that our conclusion is quite generally valid. Question 3. Explain why the result of Example 2-4 does not mean that energy conservation is violated. a 2mc221 u2>c2 b Mc2 Ebefore Eafter , Eafter (Mc 2 upœx) Mc 2 since again pœx 0 Ebefore a 2mc221 u2>c2 b since pœx 0 Ebefore a 2mc221 u2>c2 upœxb Ebefore (E œ before vp œ x) S. 2mc2 21 u2>c2 Mc2 Eœbefore E œ after ,S Eœafter Mc 2 2mc2 21 u2>c2 Eœbefore mc2 21 u2>c2 mc221 u2>c2 S 78 Chapter 2 Relativity II EXAMPLE 2-5 Mass of Cosmic Ray Muons In Chapter 1, muons produced as sec- ondary particles by cosmic rays were used to illustrate both the relativistic length contraction and time dilation resulting from their high speed relative to observers on Earth. That speed is about 0.998c. If the rest energy of a muon is 105.7 MeV, what will observers on Earth measure for the total energy of a cosmic ray–produced muon? What will they measure for its mass? SOLUTION The electron volt (eV), the amount of energy acquired by a particle with electric charge equal in magnitude to that on an electron (e) accelerated through a potential difference of 1 volt, is a convenient unit in physics, as you may have learned. It is defined as 2-24 Commonly used multiples of the eV are the keV (103 eV), the MeV (106 eV), the GeV (109 eV), and the TeV (1012 eV). Many experiments in physics involve the measurement and analysis of the energy and/or momentum of particles and systems of particles, and Equation 2-10 allows us to express the masses of particles in en- ergy units rather than the SI unit of mass, the kilogram. That and the convenient size of the eV facilitate6 numerous calculations. For example, the mass of an electron is 9.11 1031 kg. Its rest energy is given by or or The mass of the particle is often expressed with the same number thus: Now, applying the above to the muons produced by cosmic rays, each has a total energy E given by and a measured mass (see Equation 2-5) of The dependence of the measured mass on the speed of the particle has been verified by numerous experiments. Figure 2-7 illustrates a few of those results. m E>c2 1670 MeV>c2 E 1670 MeV E mc2 1 21 (0.998c)2>c2 105.7 MeVc2 c2 m E c2 0.511 MeV>c2 mass of the electron E 0.511 MeV rest energy of the electron E 8.19 1014 J 1 1.602 1019 J>eV 5.11 105 eV E mc2 9.11 1031 kg # c2 8.19 1014 J 1.0 eV 1.602 1019 C 1.0 V 1.602 1019 J 2-2 Relativistic Energy 79 2.0 3.0 4.0 5.0 6.0 1.0 0 0.2 0.4 0.6 0.8 1.0 γm /m u/c Figure 2-7 A few of the many experimental measurements of the mass of electrons as a function of their speed The data points are plotted onto Equation 2-5, the solid line. The data points represent the work of W. Kaufmann ( , 1901), A. H. Bucherer (, 1908), and W. Bertozzi (●, 1964). Note that Kaufmann’s work preceded the appearance of Einstein’s 1905 paper on special relativity. Kaufmann used an incorrect mass for the electron and interpreted his results as support for classical theory. [Adapted from Figure 3-4 in R. Resnick and D. Halliday, Basic Concepts in Relativity and Early Quantum Theory, 2d ed. (New York: Macmillan, 1992).] u>c. EXAMPLE 2-6 Change in the Solar Mass Compute the rate at which the Sun is losing mass, given that the mean radius R of Earth’s orbit is 1.50 108 km and the intensity of solar radiation at Earth (called the solar constant) is 1.36 103 W m2. SOLUTION 1. The conversion of mass into energy, a consequence of conservation of energy in relativity, is implied by Equation 2-10. With u 0 that equation becomes 2. Assuming that the Sun radiates uniformly over a sphere of radius R, the total power radiated by the Sun is given by 3. Thus, every second the Sun emits 3.85 1026 J, which, from Equation 2-10, is the result of converting an amount of mass given by Remarks: Thus, the Sun is losing 4.3 109 kg of mass (about 4 million metric tons) every second! If this rate of mass loss remains constant (which it will for the next few billion years) and with a fusion mass-to-energy conversion efficiency of about 1 percent, the Sun’s present mass of about 2.0 1030 kg will “only” last for about 1011 more years! 4.3 109 kg 3.85 1026 J (3.00 108 m>s)2 m E>c2 3.85 1026 J>s 4 (1.50 1011 m)2(1.36 103 W>m2) (4 R2)(1.36 103 W>m2)P (area of the sphere)(solar constant) E mc2 >