zotero/storage/ID9KU2IB/.zotero-ft-cache

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Fundamental Constants

Quantity

Symbol Value(s)

Elementary charge

e

1.6022 ϫ 10Ϫ19 C

Speed of light in vacuum

c

2.9979 ϫ 108 m/s

Permeability of vacuum (magnetic constant)

m0

4p ϫ 10Ϫ7 N ؒ AϪ2

Permittivity of vacuum (electric constant)

P0

8.8542 ϫ 10Ϫ12 F ؒ mϪ1

Gravitation constant

G

6.6738 ϫ 10Ϫ11 N ؒ m2 ؒ kgϪ2

Planck constant

h

6.6261 ϫ 10Ϫ34 J ؒ s

4.1357 ϫ 10Ϫ15 eV ؒ s

Avogadro constant Boltzmann constant

NA

6.0221 ϫ 1023 molϪ1

k

1.3807 ϫ 10Ϫ23 J ؒ KϪ1

Stefan-Boltzmann constant

s

5.6704 ϫ 10Ϫ8 W ؒ mϪ2 ؒ KϪ4

Atomic mass unit

u

1.66053886 ϫ 10Ϫ27 kg

931.494061 MeV/c 2

Particle Masses

Particle
Electron Muon Proton Neutron Deuteron a particle

kg
9.1094 ϫ 10Ϫ31 1.8835 ϫ 10Ϫ28 1.6726 ϫ 10Ϫ27 1.6749 ϫ 10Ϫ27 3.3436 ϫ 10Ϫ27 6.6447 ϫ 10Ϫ27

Mass in units of
MeV/c 2
0.51100 105.66 938.27 939.57 1875.61 3727.38

u
5.4858 ϫ 10Ϫ4 0.11343 1.00728 1.00866 2.01355 4.00151

Conversion Factors
1 y ϭ 3.156 ϫ 107 s 1 lightyear ϭ 9.461 ϫ 1015 m 1 cal ϭ 4.186 J 1 MeV/c ϭ 5.344 ϫ 10Ϫ22 kg ؒ m/s 1 eV ϭ 1.6022 ϫ 10Ϫ19 J

1 T ϭ 104 G 1 Ci ϭ 3.7 ϫ 1010 Bq 1 barn ϭ 10Ϫ28 m2 1 u ϭ 1.66054 ϫ 10Ϫ27 kg

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Useful Combinations of Constants

U ϭ h/2p ϭ 1.0546 ϫ 10Ϫ34 J ؒ s ϭ 6.5821 ϫ 10Ϫ16 eV ؒ s

hc ϭ 1.9864 ϫ 10Ϫ25 J ؒ m ϭ 1239.8 eV ؒ nm

Uc ϭ 3.1615 ϫ 10Ϫ26 J ؒ m ϭ 197.33 eV ؒ nm

# # 1
4pP0

ϭ 8.9876

ϫ 109

N

m2

CϪ2

Compton

wavelength

lc

h ϭ mec

ϭ 2.4263

ϫ 10Ϫ12

m

# # e2
4pP0

ϭ

2.3071

ϫ

10Ϫ28

J

m ϭ 1.4400 ϫ 10Ϫ9 eV

m

Fine

structure

constant

a

ϭ

e 2 4pP0 Uc

ϭ

0.0072974

Ϸ

1 137

Bohr

magneton

mB

ϭ

e U 2me

ϭ

9.2740

ϫ

10Ϫ24

J/T

ϭ

5.7884

ϫ

10Ϫ5

eV/T

Nuclear

magneton

mN

ϭ

e U 2mp

ϭ

5.0508

ϫ

10Ϫ27

J/T

ϭ 3.1525 ϫ 10Ϫ 8 eV/T

Bohr

radius

a0

ϭ

4pP0 U2 mee 2

ϭ

5.2918

ϫ

10Ϫ11

m

Hydrogen

ground

state

E0

ϭ

e 2 8pP0a0

ϭ

13.606

eV

ϭ

2.1799

ϫ

10Ϫ18

J

Rydberg

constant

Rq

ϭ

a2mec 2h

ϭ 1.09737

ϫ 107

mϪ1

Hydrogen

Rydberg

RH

ϭ

m me R q

ϭ

1.09678

ϫ

107

mϪ1

Gas constant R ϭ NAk ϭ 8.3145 J ؒ molϪ1 ؒ KϪ1

# Magnetic

flux

quantum

£0

h ϭ 2e

ϭ 2.0678

ϫ 10Ϫ15

T

m2

Classical electron radius re ϭ a2a0 ϭ 2.8179 ϫ 10Ϫ15 m

kT

ϭ

2.5249

ϫ

10Ϫ2

eV

Ϸ

1 40

eV

at

T

ϭ

293

K

Note: The latest values of the fundamental constants can be found at the National Institute of Standards and Technology website at http://physics.nist.gov/cuu/Constants

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Modern Physics
For Scientists and Engineers
Fourth Edition
Stephen T. Thornton
University of Virginia
Andrew Rex
University of Puget Sound

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Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

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This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest.

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Modern Physics for Scientists and Engineers, Fourth Edition Stephen T. Thornton, Andrew Rex
Publisher, Physics and Astronomy: Charles Hartford
Development Editor: Ed Dodd Associate Development Editor:
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Printed in the United States of America 1 2 3 4 5 6 7 15 14 13 12 11

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Contents Overview

1 The Birth of Modern Physics 1 2 Special Theory of Relativity 19 3 The Experimental Basis of Quantum Physics 84 4 Structure of the Atom 127 5 Wave Properties of Matter and Quantum Mechanics I 162 6 Quantum Mechanics II 201 7 The Hydrogen Atom 241 8 Atomic Physics 272 9 Statistical Physics 298 10 Molecules, Lasers, and Solids 339 11 Semiconductor Theory and Devices 392 12 The Atomic Nucleus 431 13 Nuclear Interactions and Applications 475 14 Particle Physics 519 15 General Relativity 555 16 Cosmology and Modern Astrophysics—
The Beginning and the End 577
Appendices A-1 Answers to Selected Odd-Numbered Problems A-45 Index I-1
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iii
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Contents

Preface xii

Chapter

1

The Birth of Modern Physics 1
1.1 Classical Physics of the 1890s 2 Mechanics 3 Electromagnetism 4 Thermodynamics 5
1.2 The Kinetic Theory of Gases 5 1.3 Waves and Particles 8 1.4 Conservation Laws and Fundamental
Forces 10 Fundamental Forces 10
1.5 The Atomic Theory of Matter 13 1.6 Unresolved Questions of 1895 and New
Horizons 15 On the Horizon 17 Summary 18

Chapter

2

Special Theory of Relativity 19
2.1 The Apparent Need for Ether 20 2.2 The Michelson-Morley Experiment 21

2.3 Einstein’s Postulates 26
2.4 The Lorentz Transformation 29
2.5 Time Dilation and Length Contraction 31 Time Dilation 31 Length Contraction 35
2.6 Addition of Velocities 38
2.7 Experimental Verification 42 Muon Decay 42 Atomic Clock Measurement 43 Velocity Addition 45 Testing Lorentz Symmetry 46
2.8 Twin Paradox 46
2.9 Spacetime 48
2.10 Doppler Effect 52 Special Topic: Applications of the Doppler Effect 54
2.11 Relativistic Momentum 58
2.12 Relativistic Energy 62 Total Energy and Rest Energy 64 Equivalence of Mass and Energy 65 Relationship of Energy and Momentum 66 Massless Particles 67
2.13 Computations in Modern Physics 68 Binding Energy 70
2.14 Electromagnetism and Relativity 73
Summary 75

iv
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Contents

v

Chapter

3

Chapter

5

The Experimental Basis of Quantum Physics 84
3.1 Discovery of the X Ray and the Electron 84
3.2 Determination of Electron Charge 88 3.3 Line Spectra 91
Special Topic: The Discovery of Helium 93
3.4 Quantization 95 3.5 Blackbody Radiation 96 3.6 Photoelectric Effect 102
Experimental Results of Photoelectric Effect 103 Classical Interpretation 105 Einstein’s Theory 107 Quantum Interpretation 107
3.7 X-Ray Production 110 3.8 Compton Effect 113 3.9 Pair Production and Annihilation 117
Summary 121

Chapter

4

Structure of the Atom 127
4.1 The Atomic Models of Thomson and Rutherford 128
4.2 Rutherford Scattering 131 Special Topic: Lord Rutherford of Nelson 134
4.3 The Classical Atomic Model 139 4.4 The Bohr Model of the Hydrogen
Atom 141 The Correspondence Principle 146
4.5 Successes and Failures of the Bohr Model 147 Reduced Mass Correction 148 Other Limitations 150
4.6 Characteristic X-Ray Spectra and Atomic Number 151
4.7 Atomic Excitation by Electrons 154 Summary 157

Wave Properties of Matter and Quantum Mechanics I 162
5.1 X-Ray Scattering 163 5.2 De Broglie Waves 168
Bohr’s Quantization Condition 169 Special Topic: Cavendish Laboratory 170 5.3 Electron Scattering 172 5.4 Wave Motion 175 5.5 Waves or Particles? 182 5.6 Uncertainty Principle 186 5.7 Probability, Wave Functions, and the Copenhagen Interpretation 191 The Copenhagen Interpretation 192 5.8 Particle in a Box 194 Summary 196

Chapter

6

Quantum Mechanics II 201
6.1 The Schrödinger Wave Equation 202 Normalization and Probability 204 Properties of Valid Wave Functions 206 Time-Independent Schrödinger Wave Equation 206
6.2 Expectation Values 209
6.3 Infinite Square-Well Potential 212
6.4 Finite Square-Well Potential 216
6.5 Three-Dimensional Infinite-Potential Well 218
6.6 Simple Harmonic Oscillator 220
6.7 Barriers and Tunneling 226 Potential Barrier with E Ͼ V0 226 Potential Barrier with E Ͻ V0 227 Potential Well 231 Alpha-Particle Decay 231 Special Topic: Scanning Probe Microscopes 232
Summary 235

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vi

Contents

Chapter

7

The Hydrogen Atom 241
7.1 Application of the Schrödinger Equation to the Hydrogen Atom 241
7.2 Solution of the Schrödinger Equation for Hydrogen 242 Separation of Variables 243 Solution of the Radial Equation 244 Solution of the Angular and Azimuthal Equations 246
7.3 Quantum Numbers 248 Principal Quantum Number n 249 Orbital Angular Momentum Quantum Number / 250 Magnetic Quantum Number m/ 251
7.4 Magnetic Effects on Atomic Spectra— The Normal Zeeman Effect 253
7.5 Intrinsic Spin 258 Special Topic: Hydrogen and the 21-cm Line Transition 260
7.6 Energy Levels and Electron Probabilities 260 Selection Rules 262 Probability Distribution Functions 263
Summary 268

Chapter

8

Atomic Physics 272
8.1 Atomic Structure and the Periodic Table 273 Inert Gases 278 Alkalis 278 Alkaline Earths 278 Halogens 279 Transition Metals 279 Lanthanides 279 Special Topic: Rydberg Atoms 280 Actinides 281
8.2 Total Angular Momentum 281 Single-Electron Atoms 281 Many-Electron Atoms 285 LS Coupling 286 jj Coupling 289

8.3 Anomalous Zeeman Effect 292 Summary 295

Chapter

9

Statistical Physics 298
9.1 Historical Overview 299
9.2 Maxwell Velocity Distribution 301
9.3 Equipartition Theorem 303
9.4 Maxwell Speed Distribution 307
9.5 Classical and Quantum Statistics 311 Classical Distributions 311 Quantum Distributions 312
9.6 Fermi-Dirac Statistics 315 Introduction to Fermi-Dirac Theory 315 Classical Theory of Electrical Conduction 316 Quantum Theory of Electrical Conduction 317
9.7 Bose-Einstein Statistics 323 Blackbody Radiation 323 Liquid Helium 325 Special Topic: Superfluid 3He 328 Symmetry of Boson Wave Functions 331 Bose-Einstein Condensation in Gases 332 Summary 334

10 Chapter
Molecules, Lasers, and Solids 339
10.1 Molecular Bonding and Spectra 340 Molecular Bonds 340 Rotational States 341 Vibrational States 342 Vibration and Rotation Combined 344
10.2 Stimulated Emission and Lasers 347 Scientific Applications of Lasers 352 Holography 353 Quantum Entanglement, Teleportation, and Information 354 Other Laser Applications 355
10.3 Structural Properties of Solids 356 10.4 Thermal and Magnetic Properties
of Solids 359 Thermal Expansion 359 Thermal Conductivity 361

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Contents vii

Magnetic Properties

363

Diamagnetism 364

Paramagnetism 365 Ferromagnetism 366 Antiferromagnetism and Ferrimagnetism 367

10.5 Superconductivity 367 The Search for a Higher Tc 374 Special Topic: Low-Temperature Methods 378
Other Classes of Superconductors 380

10.6

Applications of Superconductivity 380 Josephson Junctions 381 Maglev 382 Generation and Transmission of Electricity 383 Other Scientific and Medical Applications 383

Summary 385

11 Chapter

Semiconductor Theory and Devices 392

11.1 11.2 11.3
11.4

Band Theory of Solids 392 Kronig-Penney Model 395 Band Theory and Conductivity 397
Semiconductor Theory 397 Special Topic: The Quantum Hall Effect 402 Thermoelectric Effect 404
Semiconductor Devices 406 Diodes 406 Rectifiers 408 Zener Diodes 408 Light-Emitting Diodes 409 Photovoltaic Cells 409 Transistors 413 Field Effect Transistors 415 Schottky Barriers 416 Semiconductor Lasers 417 Integrated Circuits 418
Nanotechnology 421 Carbon Nanotubes 421 Nanoscale Electronics 422 Quantum Dots 424 Nanotechnology and the Life Sciences 425 Information Science 426
Summary 426

12 Chapter
The Atomic Nucleus 431
12.1 Discovery of the Neutron 431 12.2 Nuclear Properties 434
Sizes and Shapes of Nuclei 435 Nuclear Density 437 Intrinsic Spin 437 Intrinsic Magnetic Moment 437 Nuclear Magnetic Resonance 438 12.3 The Deuteron 439 12.4 Nuclear Forces 441 12.5 Nuclear Stability 442 Nuclear Models 448 12.6 Radioactive Decay 449 12.7 Alpha, Beta, and Gamma Decay 452 Alpha Decay 453 Beta Decay 456 Special Topic: Neutrino Detection 458 Gamma Decay 462 12.8 Radioactive Nuclides 464 Time Dating Using Lead Isotopes 466 Radioactive Carbon Dating 467 Special Topic: The Formation and Age of the Earth 468 Summary 470
13 Chapter
Nuclear Interactions and Applications 475
13.1 Nuclear Reactions 475 Cross Sections 478
13.2 Reaction Kinematics 480 13.3 Reaction Mechanisms 482
The Compound Nucleus 483 Direct Reactions 486 13.4 Fission 486 Induced Fission 487 Thermal Neutron Fission 488 Chain Reactions 489

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viii Contents

13.5 Fission Reactors 490 Nuclear Reactor Problems 493 Breeder Reactors 494 Future Nuclear Power Systems 495 Special Topic: Early Fission Reactors 496
13.6 Fusion 499 Formation of Elements 499 Nuclear Fusion on Earth 501 Controlled Thermonuclear Reactions 502
13.7 Special Applications 505 Medicine 505 Archaeology 507 Art 507 Crime Detection 507 Mining and Oil 508 Materials 508 Small Power Systems 510 New Elements 510 Special Topic: The Search for New Elements 512
Summary 514
14 Chapter
Particle Physics 519
14.1 Early Discoveries 520 The Positron 520 Yukawa’s Meson 521
14.2 The Fundamental Interactions 523
14.3 Classification of Particles 526 Leptons 527 Hadrons 528 Particles and Lifetimes 530
14.4 Conservation Laws and Symmetries 532 Baryon Conservation 532 Lepton Conservation 533 Strangeness 534 Symmetries 535
14.5 Quarks 536 Quark Description of Particles 537 Color 539 Confinement 539
14.6 The Families of Matter 541
14.7 Beyond the Standard Model 541 Neutrino Oscillations 542 Matter-Antimatter 542 Grand Unifying Theories 543 Special Topic: Experimental Ingenuity 544

14.8 Accelerators 546 Synchrotrons 547 Linear Accelerators 547 Fixed-Target Accelerators 548 Colliders 549
Summary 551
15 Chapter
General Relativity 555
15.1 Tenets of General Relativity 555 Principle of Equivalence 556 Spacetime Curvature 558
15.2 Tests of General Relativity 560 Bending of Light 560 Gravitational Redshift 561 Perihelion Shift of Mercury 562 Light Retardation 563
15.3 Gravitational Waves 564 15.4 Black Holes 565
Special Topic: Gravitational Wave Detection 566 15.5 Frame Dragging 572 Summary 573
16 Chapter
Cosmology and Modern Astrophysics—The Beginning and the End 577
16.1 Evidence of the Big Bang 578 Hubble’s Measurements 578 Cosmic Microwave Background Radiation 581 Nucleosynthesis 581 Olbers’ Paradox 583
16.2 The Big Bang 583 16.3 Stellar Evolution 588
The Ultimate Fate of Stars 589 Special Topic: Planck’s Time, Length, and Mass 591 16.4 Astronomical Objects 592 Active Galactic Nuclei and Quasars 593 Gamma Ray Astrophysics 594 Novae and Supernovae 595

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Contents

ix

16.5 Problems with the Big Bang 599 The Inflationary Universe 599 The Lingering Problems 600
16.6 The Age of the Universe 603 Age of Chemical Elements 603 Age of Astronomical Objects 603 Cosmological Determinations 604 Universe Age Conclusion 607
16.7 The Standard Model of Cosmology 607 16.8 The Future 609
The Demise of the Sun 609 Special Topic: Future of Space Telescopes 610 The Future of the Universe 610 Are Other Earths Out There? 611 Summary 612
Appendix 1 Fundamental Constants A-1
Appendix 2 Conversion Factors A-2
Appendix 3 Mathematical Relations A-4

Appendix 5 Mean Values and Distributions A-7

Appendix 6 Probability Integrals
Ύq
In ϭ x n exp1Ϫax 2 2 dx A-9 0

Appendix 7

Ύ Integrals of the Type

q xnϪ1 dx 0 ex Ϫ 1

A-12

Appendix 8 Atomic Mass Table A-14

Appendix 9 Nobel Laureates in Physics A-37
Answers to Selected Odd-Numbered
Problems A-45
Index I-1

Appendix 4 Periodic Table of the Elements A-6

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x
03721_fm_i-xiv.indd x

Preface
Our objective in writing this book was to produce a textbook for a modern physics
course of either one or two semesters for physics and engineering students. Such a course normally follows a full-year, introductory, calculus-based physics course for freshmen or sophomores. Before each edition we have the publisher send a questionnaire to users of modern physics books to see what needed to be changed or added. Most users like our textbook as is, especially the complete coverage of topics including the early quantum theory, subfields of physics, general relativity, and cosmology/astrophysics. Our book continues to be useful for either a one- or two-semester modern physics course. We have made no major changes in the order of subjects in the fourth edition.
Coverage
The first edition of our text established a trend for a contemporary approach to the exciting, thriving, and changing field of modern science. After briefly visiting the status of physics at the turn of the last century, we cover relativity and quantum theory, the basis of any study of modern physics. Almost all areas of science depend on quantum theory and the methods of experimental physics. We have included the name Quantum Mechanics in two of our chapter titles (Chapters 5 and 6) to emphasize the quantum connection. The latter part of the book is devoted to the subfields of physics (atomic, condensed matter, nuclear, and particle) and the exciting fields of cosmology and astrophysics. Our experience is that science and engineering majors particularly enjoy the study of modern physics after the sometimes-laborious study of classical mechanics, thermodynamics, electricity, magnetism, and optics. The level of mathematics is not difficult for the most part, and students feel they are finally getting to the frontiers of physics. We have brought the study of modern physics alive by presenting many current applications and challenges in physics, for example, nanoscience, high-temperature superconductors, quantum teleportation, neutrino mass and oscillations, missing dark mass and energy in the universe, gamma-ray bursts, holography, quantum dots, and nuclear fusion. Modern physics texts need to be updated periodically to include recent advances. Although we have emphasized modern applications, we also provide the sound theoretical basis for quantum theory that will be needed by physics majors in their upper division and graduate courses.
Changes for the Fourth Edition
Our book continues to be the most complete and up-to-date textbook in the modern physics market for sophomores/juniors. We have made several changes for the fourth edition to aid the student in learning modern physics. We have added additional end-ofchapter questions and problems and have modified many problems from earlier editions,
10/3/11 4:25 PM

with an emphasis on including more real-world problems with current research applications whenever possible. We continue to have a larger number of questions and problems than competing textbooks, and for users of the robust Thornton/Rex Modern Physics for Scientists and Engineers, third edition course in WebAssign, we have a correlation guide of the fourth edition problems to that third edition course.
We have added additional examples to the already large number in the text. The pedagogical changes made for the third edition were highly successful. To encourage and support conceptual thinking by students, we continue to use conceptual examples and strategy discussion in the numerical examples. Examples with numerical solutions include a discussion of what needs to be accomplished in the example, the procedure to go through to find the answer, and relevant equations that will be needed. We present the example solutions in some detail, showing enough steps so that students can follow the solution to the end.
We continue to provide a significant number of photos and biographies of scientists who have made contributions to modern physics. We have done this to give students a perspective of the background, education, trials, and efforts of these scientists. We have also updated many of the Special Topic boxes, which we believe provide accurate and useful descriptions of the excitement of scientific discoveries, both past and current.
Chapter-by-Chapter Changes We have rewritten some sections in order to make the explanations clearer to the student. Some material has been deleted, and new material has been added. In particular we added new results that have been reported since the third edition. This is especially true for the chapters on the subfields of physics, Chapters 8–16. We have covered the most important subjects of modern physics, but we realize that in order to cover everything, the book would have to be much longer, which is not what our users want. Our intention is to keep the level of the textbook at the sophomore/ junior undergraduate level. We think it is important for instructors to be able to supplement the book whenever they choose—especially to cover those topics in which they themselves are expert. Particular changes by chapter include the following:
• Chapter 2: we have updated the search for violations of Lorentz symmetry and added some discussion about four vectors.
• Chapter 3: we have rewritten the discussion of the Rayleigh-Jeans formula and Planck’s discovery.
• Chapter 9: we improved the discussion about the symmetry of boson wave functions and its application to the Fermi exclusion principle and Bose-Einstein condensates.
• Chapter 10: we added a discussion of classes of superconductors and have updated our discussion concerning applications of superconductivity. The latter includes how superconductors are now being used to determine several fundamental constants.
• Chapter 11: we added more discussion about solar energy, Blu-ray DVD devices, increasing the number of transistors on a microchip using new semiconductor materials, graphene, and quantum dots. Our section on nanotechnology is especially complete.
• Chapter 12: we updated our discussion on neutrino detection and neutrino mass, added a description of nuclear magnetic resonance, and upgraded our discussion on using radioactive decay to study the oldest terrestrial materials.
• Chapter 13: we updated our discussion about nuclear power plants operating in the United States and the world and presented a discussion of possible new, improved reactors. We discussed the tsunami-induced tragedy at the Fukushima Daiichi nuclear power plant in Japan and added to our discussion of searches for new elements and their discoveries.
• Chapter 14: we upgraded our description of particle physics, improved and expanded the discussion on Feynman diagrams, updated the search for the Higgs boson, discussed new experiments on neutrino oscillations, and added discussion on matter-antimatter, supersymmetry, string theory, and M-theory. We mention that the LHC has begun operation as the Fermilab Tevatron accelerator is shutting down.
• Chapter 15: we improved our discussion on gravitational wave detection, added to our discussion on black holes, and included the final results of the Gravity Probe B satellite.
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Preface

xi

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xii Preface
03721_fm_i-xiv.indd xii

• Chapter 16: we changed the chapter name from Cosmology to Cosmology and Modern Astrophysics, because of the continued importance of the subject in modern physics. Our third edition of the textbook already had an excellent discussion and correct information about the age of the universe, dark matter, and dark energy, but Chapter 16 still has the most changes of any chapter, due to the current pace of research in the field. We have upgraded information and added discussion about Olbers’ paradox, discovery of the cosmic microwave background, gamma ray astrophysics, standard model of cosmology, the future of space telescopes, and the future of the universe (Big Freeze, Big Crunch, Big Rip, Big Bounce, etc).
Teaching Suggestions
The text has been used extensively in its first three editions in courses at our home institutions. These include a one-semester course for physics and engineering students at the University of Virginia and a two-semester course for physics and pre-engineering students at the University of Puget Sound. These are representative of the one- and two-semester modern physics courses taught elsewhere. Both one- and two-semester courses should cover the material through the establishment of the periodic table in Chapter 8 with few exceptions. We have eliminated the denoting of optional sections, because we believe that depends on the wishes of the instructor, but we feel Sections 2.4, 4.2, 6.4, 6.6, 7.2, 7.6, 8.2, and 8.3 from the first nine chapters might be optional. Our suggestions for the one- and two-semester courses (3 or 4 credit hours per semester) are then
One-semester: Chapters 1–9 and selected other material as chosen by the instructor Two-semester: Chapters 1–16 with supplementary material as desired, with possible student projects
An Internet-based, distance-learning version of the course is offered by one of the authors every summer (Physics 2620, 4 credit hours) at the University of Virginia that covers all chapters of the textbook, with emphasis on Chapters 1–8. Homework problems and exams are given on WebAssign. The course can be taken by a student located anywhere there is an Internet connection. See http://modern.physics.virginia.edu/course/ for details.
Features
End-of-Chapter Problems
The 1166 questions and problems (258 questions and 908 problems) are more than in competing textbooks. Such a large number of questions and problems allows the instructor to make different homework assignments year after year without having to repeat problems. A correlation guide to the Thornton/Rex Modern Physics for Scientists and Engineers, third edition course in WebAssign is available via the Instructor’s companion website (www.cengage.com/physics/thornton4e). We have tried to provide thought-provoking questions that have actual answers. In this edition we have focused on adding problems that have real-world or current research applications. The end-of-chapter problems have been separated by section, and general problems are included at the end to allow assimilation of the material. The easier problems are generally listed first within a section, and the more difficult ones are noted by a shaded blue square behind the problem number. A few computer-based problems are given in the text, but no computer disk supplement is provided, because many computer software programs are commercially available.
Solutions Manuals
PDF files of the Instructor’s Solutions Manual are available to the instructor on the Instructor’s Resource CD-ROM (by contacting your local Brooks/Cole—Cengage sales representative). This manual contains the solution to every end-of-chapter problem and has been checked by at
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least two physics professors. The answers to selected odd-numbered problems are given at the end of the textbook itself. A Student Solutions Manual that contains the solutions to about 25% of the end-of-chapter problems is also available for sale to the students.
Instructor’s Resource CD-ROM for Thornton/Rex’s Modern Physics for Scientists and Engineers, Fourth Edition
Available to adopters is the Modern Physics for Scientists and Engineers Instructor’s Resource CD-ROM. This CD-ROM includes PowerPoint® lecture outlines and also contains 200 pieces of line art from the text. It also features PDF files of the Instructor’s Solutions Manual. Please guard this CD and do not let anyone have access to it. When end-of-chapter problem solutions find their way to the internet for sale, learning by students deteriorates because of the temptation to look up the solution.
Text Format
The two-color format helps to present clear illustrations and to highlight material in the text; for example, important and useful equations are highlighted in blue, and the most important part of each illustration is rendered in thick blue lines. Blue margin notes help guide the student to the important points, and the margins allow students to make their own notes. The first time key words or topics are introduced they are set in boldface, and italics are used for emphasis.
Examples
Although we had a large number of worked examples in the third edition, we have added new ones in this edition. The examples are written and presented in the manner in which students are expected to work the end-of-chapter problems: that is, to develop a conceptual understanding and strategy before attempting a numerical solution. Problem solving does not come easily for most students, especially the problems requiring several steps (that is, not simply plugging numbers into one equation). We expect that the many text examples with varying degrees of difficulty will help students.
Special Topic Boxes
Users have encouraged us to keep the Special Topic boxes. We believe both students and professors find them interesting, because they add some insight and detail into the excitement of physics. We have updated the material to keep them current.
History
We include historical aspects of modern physics that some students will find interesting and that others can simply ignore. We continue to include photos and biographies of scientists who have made significant contributions to modern physics. We believe this helps to enliven and humanize the material.
Website
Students can access the book’s companion website at www.cengagebrain.com/shop/ ISBN/9781133103721. This site features student study aids such as outlines, summaries, and conceptual questions for each chapter. Instructors will also find downloadable PowerPoint lectures and images for use in classroom lecture presentation. Students may also access the authors’ websites at http://www.modern.physics.virginia.edu/ and http://www. pugetsound.edu/faculty-pages/rex where the authors will post errata, present new exciting results, and give links to sites that have particularly interesting features like simulations and photos, among other things.
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xiv Preface

Acknowledgments

We acknowledge the assistance of many persons who have helped with this text. There are too many that helped us with the first three editions to list here, but the book would not have been possible without them. We acknowledge the professional staff at Brooks/Cole, Cengage Learning who helped make this fourth edition a useful, popular, and attractive text. They include Developmental Editor Ed Dodd and Senior Content Project Manager Cathy Brooks, who kept the production process on track, and Physics Publisher Charlie Hartford for his support, guidance, and encouragement. Elizabeth Budd did a superb job with the copyediting. We also want to thank Jeff Somers and the staff of Graphic World Inc. for their skilled efforts. We also want to thank the many individuals who gave us critical reviews and suggestions since the first edition. We especially would like to thank Michael Hood (Mt. San Antonio College) and Carol Hood (Augusta State University) for their help, especially with the Cosmology and Modern Astrophysics chapter. In preparing this fourth edition, we owe a special debt of gratitude to the following reviewers:

Jose D’Arruda, University of North Carolina, Pembroke David Church, Texas A & M University Hardin R. Dunham, Odessa College Paul A. Heiney, University of Pennsylvania Paul Keyes, Wayne State University Cody Martin, College of Menominee Nation

Prior to our work on this revision, we conducted a survey of professors to gauge how they taught their classes. In all, 78 professors responded with many insightful comments, and we would like to thank them for their feedback and suggestions.
We especially want to acknowledge the valuable help of Richard R. Bukrey of Loyola University of Chicago who helped us in many ways through his enlightening reviews, careful manuscript proofing, and checking of the end-of-chapter problem solutions in the first two editions, and to Thushara Perera of Illinois Wesleyan University, and Paul Weber of University of Puget Sound, for their accuracy review of the fourth edition. We also thank Allen Flora of Hood College for assuming the task of preparing and checking problem solutions for the third and fourth editions.

Stephen T. Thornton University of Virginia Charlottesville, Virginia stt@virginia.edu

Andrew Rex University of Puget Sound Tacoma, Washington rex@pugetsound.edu

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The Birth of Modern Physics
The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote. . . . Our future discoveries must be looked for in the sixth place of decimals.
Albert A. Michelson, 1894
There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.
William Thomson (Lord Kelvin), 1900
A lthough the Greek scholars Aristotle and Eratosthenes performed measure-
ments and calculations that today we would call physics, the discipline of physics has its roots in the work of Galileo and Newton and others in the scientific revolution of the sixteenth and seventeenth centuries. The knowledge and practice of physics grew steadily for 200 to 300 years until another revolution in physics took place, which is the subject of this book. Physicists distinguish classical physics, which was mostly developed before 1895, from modern physics, which is based on discoveries made after 1895. The precise year is unimportant, but monumental changes occurred in physics around 1900.
The long reign of Queen Victoria of England, from 1837 to 1901, saw considerable changes in social, political, and intellectual realms, but perhaps none so important as the remarkable achievements that occurred in physics. For example, the description and predictions of electromagnetism by Maxwell are partly responsible for the rapid telecommunications of today. It was also during this period that thermodynamics rose to become an exact science. None of these achievements, however, have had the ramifications of the discoveries and applications of modern physics that would occur in the twentieth century. The world would never be the same.
In this chapter we briefly review the status of physics around 1895, including Newton’s laws, Maxwell’s equations, and the laws of thermodynamics. These results are just as important today as they were over a hundred years ago. Arguments by scientists concerning the interpretation of experimental data using

11
CHAPTER 1

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2

Chapter 1 The Birth of Modern Physics

wave and particle descriptions that seemed to have been resolved 200 years ago were reopened in the twentieth century. Today we look back on the evidence of the late nineteenth century and wonder how anyone could have doubted the validity of the atomic view of matter. The fundamental interactions of gravity, electricity, and magnetism were thought to be well understood in 1895. Physicists continued to be driven by the goal of understanding fundamental laws throughout the twentieth century. This is demonstrated by the fact that other fundamental forces (specifically the nuclear and weak interactions) have been added, and in some cases—curious as it may seem—various forces have even been combined. The search for the holy grail of fundamental interactions continues unabated today.
We finish this chapter with a status report on physics just before 1900. The few problems not then understood would be the basis for decades of fruitful investigations and discoveries continuing into the twenty-first century. We hope you find this chapter interesting both for the physics presented and for the historical account of some of the most exciting scientific discoveries of the modern era.

Early successes of science Classical conservation laws

1.1 Classical Physics of the 1890s
Scientists and engineers of the late nineteenth century were indeed rather smug. They thought they had just about everything under control (see the quotes from Michelson and Kelvin on page 1). The best scientists of the day were highly recognized and rewarded. Public lectures were frequent. Some scientists had easy access to their political leaders, partly because science and engineering had benefited their war machines, but also because of the many useful technical advances. Basic research was recognized as important because of the commercial and military applications of scientific discoveries. Although there were only primitive automobiles and no airplanes in 1895, advances in these modes of transportation were soon to follow. A few people already had telephones, and plans for widespread distribution of electricity were under way.
Based on their success with what we now call macroscopic classical results, scientists felt that given enough time and resources, they could explain just about anything. They did recognize some difficult questions they still couldn’t answer; for example, they didn’t clearly understand the structure of matter— that was under intensive investigation. Nevertheless, on a macroscopic scale, they knew how to build efficient engines. Ships plied the lakes, seas, and oceans of the world. Travel between the countries of Europe was frequent and easy by train. Many scientists were born in one country, educated in one or two others, and eventually worked in still other countries. The most recent ideas traveled relatively quickly among the centers of research. Except for some isolated scientists, of whom Einstein is the most notable example, discoveries were quickly and easily shared. Scientific journals were becoming accessible.
The ideas of classical physics are just as important and useful today as they were at the end of the nineteenth century. For example, they allow us to build automobiles and produce electricity. The conservation laws of energy, linear momentum, angular momentum, and charge can be stated as follows:
Conservation of energy: The total sum of energy (in all its forms) is conserved in all interactions.
Conservation of linear momentum: In the absence of external forces, linear momentum is conserved in all interactions (vector relation).

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1.1 Classical Physics of the 1890s

3

Conservation of angular momentum: In the absence of external torque, angular momentum is conserved in all interactions (vector relation).
Conservation of charge: Electric charge is conserved in all interactions.
A nineteenth-century scientist might have added the conservation of mass to this list, but we know it not to be valid today (you will find out why in Chapter 2). These conservation laws are reflected in the laws of mechanics, electromagnetism, and thermodynamics. Electricity and magnetism, separate subjects for hundreds of years, were combined by James Clerk Maxwell (1831–1879) in his four equations. Maxwell showed optics to be a special case of electromagnetism. Waves, which permeated mechanics and optics, were known to be an important component of nature. Many natural phenomena could be explained by wave motion using the laws of physics.

Mechanics
The laws of mechanics were developed over hundreds of years by many researchers. Important contributions were made by astronomers because of the great interest in the heavenly bodies. Galileo (1564–1642) may rightfully be called the first great experimenter. His experiments and observations laid the groundwork for the important discoveries to follow during the next 200 years.
Isaac Newton (1642–1727) was certainly the greatest scientist of his time and one of the best the world has ever seen. His discoveries were in the fields of mathematics, astronomy, and physics and include gravitation, optics, motion, and forces.
We owe to Newton our present understanding of motion. He understood clearly the relationships among position, displacement, velocity, and acceleration. He understood how motion was possible and that a body at rest was just a special case of a body having constant velocity. It may not be so apparent to us today, but we should not forget the tremendous unification that Newton made when he pointed out that the motions of the planets about our sun can be understood by the same laws that explain motion on Earth, like apples falling from trees or a soccer ball being shot toward a goal. Newton was able to elucidate

Galileo, the first great experimenter
Newton, the greatest scientist of his time

Galileo Galilei (1564–1642) was born, educated, and worked in Italy. Often said to be the “father of physics” because of his careful experimentation, he is shown here performing experiments by rolling balls on an inclined plane. He is perhaps best known for his experiments on motion, the development of the telescope, and his many astronomical discoveries. He came into disfavor with the Catholic Church for his belief in the Copernican theory. He was finally cleared of heresy by Pope John Paul II in 1992, 350 years after his death.

Scala/Art Resource, NY

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Courtesy of Bausch & Lomb Optical Co. and the AIP Niels Bohr Library.

4

Chapter 1 The Birth of Modern Physics

Newton’s laws
Isaac Newton (1642–1727), the great English physicist and mathematician, did most of his work at Cambridge where he was educated and became the Lucasian Professor of Mathematics. He was known not only for his work on the laws of motion but also as a founder of optics. His useful works are too numerous to list here, but it should be mentioned that he spent a considerable amount of his time on alchemy, theology, and the spiritual universe. His writings on these subjects, which were dear to him, were quite unorthodox. This painting shows him performing experiments with light.
Maxwell’s equations

carefully the relationship between net force and acceleration, and his concepts were stated in three laws that bear his name today:

Newton’s first law: An object in motion with a constant velocity will continue in motion unless acted upon by some net external force. A body at rest is just a special case of Newton’s first law with zero velocity. Newton’s first law is often called the law of inertia and is also used to describe inertial reference frames.

Newton’s second law: The acceleration a of a body is proportional to the net external force F and inversely proportional to the mass m of the body. It is stated mathematically as

F ϭ ma

(1.1a)

A more general statement* relates force to the time rate of change of the linear momentum p .

F

ϭ

dp dt

(1.1b)

Newton’s third law: The force exerted by body 1 on body 2 is equal in magnitude and

opposite in direction to the force that body 2 exerts on body 1. If the force on body 2 by body 1 is denoted by F21, then Newton’s third law is written as

F21 ϭ ϪF12

(1.2)

It is often called the law of action and reaction.

These three laws develop the concept of force. Using that concept together with the concepts of velocity v , acceleration a , linear momentum p , rotation (angular velocity v and angular acceleration a), and angular momentum L , we
can describe the complex motion of bodies.

Electromagnetism

Electromagnetism developed over a long period of time. Important contributions were made by Charles Coulomb (1736–1806), Hans Christian Oersted (1777– 1851), Thomas Young (1773–1829), André Ampère (1775–1836), Michael Faraday (1791–1867), Joseph Henry (1797–1878), James Clerk Maxwell (1831–1879), and Heinrich Hertz (1857–1894). Maxwell showed that electricity and magnetism were intimately connected and were related by a change in the inertial frame of reference. His work also led to the understanding of electromagnetic radiation, of which light and optics are special cases. Maxwell’s four equations, together with the Lorentz force law, explain much of electromagnetism.

Gauss’s law for electricity Gauss’s law for magnetism Faraday’s law

ΏE

#

dA

ϭ

q P0

ΏB # dA ϭ 0

ΏE

#

ds

ϭ

Ϫ

d£B dt

(1.3) (1.4) (1.5)

* It is a remarkable fact that Newton wrote his second law not as F ϭ ma, but as F ϭ d(mv )/dt, thus taking into account mass flow and change in velocity. This has applications in both fluid mechanics and rocket propulsion.

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1.2 The Kinetic Theory of Gases

5

Ώ Generalized Ampere’s law

#B

ds

ϭ

m0P0

d£ dt

E

ϩ

m0I

(1.6)

Lorentz force law

F ϭ qE ϩ qv ϫ B

(1.7)

Maxwell’s equations indicate that charges and currents create fields, and in turn, these fields can create other fields, both electric and magnetic.

Thermodynamics

Thermodynamics deals with temperature T, heat Q , work W, and the internal energy of systems U. The understanding of the concepts used in thermodynamics— such as pressure P, volume V, temperature, thermal equilibrium, heat, entropy, and especially energy—was slow in coming. We can understand the concepts of pressure and volume as mechanical properties, but the concept of temperature must be carefully considered. We have learned that the internal energy of a system of noninteracting point masses depends only on the temperature.
Important contributions to thermodynamics were made by Benjamin Thompson (Count Rumford, 1753–1814), Sadi Carnot (1796–1832), James Joule (1818–1889), Rudolf Clausius (1822–1888), and William Thomson (Lord Kelvin, 1824–1907). The primary results of thermodynamics can be described in two laws:

First law of thermodynamics: The change in the internal energy ⌬U of a system is equal to the heat Q added to the system plus the work W done on the system.

¢U ϭ Q ϩ W

(1.8)

The first law of thermodynamics generalizes the conservation of energy by including heat.

Second law of thermodynamics: It is not possible to convert heat completely into work without some other change taking place. Various forms of the second law state similar, but slightly different, results. For example, it is not possible to build a perfect engine or a perfect refrigerator. It is not possible to build a perpetual motion machine. Heat does not spontaneously flow from a colder body to a hotter body without some other change taking place. The second law forbids all these from happening. The first law states the conservation of energy, but the second law says what kinds of energy processes cannot take place. For example, it is possible to completely convert work into heat, but not vice versa, without some other change taking place.

Two other “laws” of thermodynamics are sometimes expressed. One is called the “zeroth” law, and it is useful in understanding temperature. It states that if two thermal systems are in thermodynamic equilibrium with a third system, they are in equilibrium with each other. We can state it more simply by saying that two systems at the same temperature as a third system have the same temperature as each other. This concept was not explicitly stated until the twentieth century. The “third” law of thermodynamics expresses that it is not possible to achieve an absolute zero temperature.

Laws of thermodynamics

1.2 The Kinetic Theory of Gases
We understand now that gases are composed of atoms and molecules in rapid motion, bouncing off each other and the walls, but in the 1890s this had just gained acceptance. The kinetic theory of gases is related to thermodynamics and

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6

Chapter 1 The Birth of Modern Physics

Ideal gas equation
Figure 1.1 Molecules inside a closed container are shown colliding with the walls and with each other. The motions of a few molecules are indicated by the arrows. The number of molecules inside the container is huge.
Statistical thermodynamics
Equipartition theorem
Internal energy

to the atomic theory of matter, which we discuss in Section 1.5. Experiments were relatively easy to perform on gases, and the Irish chemist Robert Boyle (1627– 1691) showed around 1662 that the pressure times the volume of a gas was constant for a constant temperature. The relation PV ϭ constant (for constant T ) is now referred to as Boyle’s law. The French physicist Jacques Charles (1746–1823) found that V/T ϭ constant (at constant pressure), referred to as Charles’s law. Joseph Louis Gay-Lussac (1778–1850) later produced the same result, and the law is sometimes associated with his name. If we combine these two laws, we obtain the ideal gas equation

PV ϭ nRT

(1.9)

where n is the number of moles and R is the ideal gas constant, 8.31 J/mol # K.

In 1811 the Italian physicist Amedeo Avogadro (1776–1856) proposed that

equal volumes of gases at the same temperature and pressure contained equal

numbers of molecules. This hypothesis was so far ahead of its time that it was not

accepted for many years. The famous English chemist John Dalton opposed the

idea because he apparently misunderstood the difference between atoms and

molecules. Considering the rudimentary nature of the atomic theory of matter

at the time, this was not surprising.

Daniel Bernoulli (1700–1782) apparently originated the kinetic theory of

gases in 1738, but his results were generally ignored. Many scientists, including

Newton, Laplace, Davy, Herapath, and Waterston, had contributed to the devel-

opment of kinetic theory by 1850. Theoretical calculations were being compared

with experiments, and by 1895 the kinetic theory of gases was widely accepted.

The statistical interpretation of thermodynamics was made in the latter half of the

nineteenth century by Maxwell, the Austrian physicist Ludwig Boltzmann (1844–

1906), and the American physicist J. Willard Gibbs (1839–1903).

In introductory physics classes, the kinetic theory of gases is usually

taught by applying Newton’s laws to the collisions that a molecule makes with

other molecules and with the walls. A representation of a few molecules col-

liding is shown in Figure 1.1. In the simple model of an ideal gas, only elastic

collisions are considered. By taking averages over the collisions of many mol-

ecules, the ideal gas law, Equation (1.9), is revealed. The average kinetic

energy of the molecules is shown to be linearly proportional to the tempera-

ture, and the internal energy U is

U

ϭ

nNA8K

9

ϭ

3 2

nRT

(1.10)

where n is the number of moles of gas, NA is Avogadro’s number, 8K 9 is the average kinetic energy of a molecule, and R is the ideal gas constant. This relation ignores any nontranslational contributions to the molecular energy, such as rotations and vibrations.
However, energy is not represented only by translational motion. It became clear that all degrees of freedom, including rotational and vibrational, were also capable of carrying energy. The equipartition theorem states that each degree of freedom of a molecule has an average energy of kT/2, where k is the Boltzmann constant (k ϭ R/NA). Translational motion has three degrees of freedom, and rotational and vibrational modes can also be excited at higher temperatures. If there are f degrees of freedom, then Equation (1.10) becomes

f U ϭ 2 nRT

(1.11)

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1.2 The Kinetic Theory of Gases

7

The molar (n ϭ 1) heat capacity c V at constant volume for an ideal gas is the rate of change in internal energy with respect to change in temperature and is given by

c V

ϭ

3 2

R

(1.12) Heat capacity

The experimental quantity c V /R is plotted versus temperature for hydrogen in Figure 1.2. The ratio c V /R is equal to 3/2 for low temperatures, where only translational kinetic energy is important, but it rises to 5/2 at 300 K, where rotations occur for H2, and finally reaches 7/2, because of vibrations at still higher temperatures, before the molecule dissociates. Although the kinetic theory of gases fails to predict specific heats for real gases, it leads to models that can be used on a gas-bygas basis. Kinetic theory is also able to provide useful information on other properties such as diffusion, speed of sound, mean free path, and collision frequency.
In the 1850s Maxwell derived a relation for the distribution of speeds of the molecules in gases. The distribution of speeds f (v) is given as a function of the speed and the temperature by the equation

f

1v2

ϭ

4pN

a

m 2pkT

b

3

/2
v

2e

Ϫmv

2

/2kT

(1.13)

Maxwell’s speed distribution

where m is the mass of a molecule and T is the temperature. This result is plotted for nitrogen in Figure 1.3 for temperatures of 300 K, 1000 K, and 4000 K. The peak of each distribution is the most probable speed of a gas molecule for the given temperature. In 1895 measurement was not precise enough to confirm Maxwell’s distribution, and it was not confirmed experimentally until 1921.
By 1895 Boltzmann had made Maxwell’s calculation more rigorous, and the general relation is called the Maxwell-Boltzmann distribution. The distribution can be used to find the root-mean-square speed vrms,

vrms

ϭ

28v 2 9

ϭ

3kT Bm

(1.14)

which shows the relationship of the energy to the temperature for an ideal gas:

U

ϭ

nNA8K

9

ϭ

m 8v2 nNA 2

9

ϭ

nNA

m3kT 2m

ϭ

3 2

nRT

(1.15)

This was the result of Equation (1.10).

c V/R

4

72–

3

Vibration 52–

2

Rotation

32–

1 Translation

020

50 100 200 500 1000 2000 5000 10,000

Temperature (K)

Figure 1.2 The molar heat capacity at constant volume (c V) divided by R (c V/R is dimensionless) is displayed as a function of temperature for hydrogen gas. Note that as the temperature increases, the rotational and vibrational modes become important. This experimental result is consistent with the equipartition theorem, which adds kT/2 of energy per molecule (RT/2 per mole) for each degree of freedom.

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8

Chapter 1 The Birth of Modern Physics

Figure 1.3 The Maxwell distribution of molecular speeds (for nitrogen), f (v), is shown as a function of speed for three temperatures.

f(v)

Most probable speed (300 K) 300 K
Nitrogen 1000 K

4000 K

00

1000

2000

3000

4000

v (m/s)

Energy transport
Nature of light: waves or particles?

1.3 Waves and Particles
We first learned the concepts of velocity, acceleration, force, momentum, and energy in introductory physics by using a single particle with its mass concentrated in one small point. In order to adequately describe nature, we add twoand three-dimensional bodies and rotations and vibrations. However, many aspects of physics can still be treated as if the bodies are simple particles. In particular, the kinetic energy of a moving particle is one way that energy can be transported from one place to another.
But we have found that many natural phenomena can be explained only in terms of waves, which are traveling disturbances that carry energy. This description includes standing waves, which are superpositions of traveling waves. Most waves, like water waves and sound waves, need an elastic medium in which to move. Curiously enough, matter is not transported in waves—but energy is. Mass may oscillate, but it doesn’t actually propagate along with the wave. Two examples are a cork and a boat on water. As a water wave passes, the cork gains energy as it moves up and down, and after the wave passes, the cork remains. The boat also reacts to the wave, but it primarily rocks back and forth, throwing around things that are not fixed on the boat. The boat obtains considerable kinetic energy from the wave. After the wave passes, the boat eventually returns to rest.
Waves and particles were the subject of disagreement as early as the seventeenth century, when there were two competing theories of the nature of light. Newton supported the idea that light consisted of corpuscles (or particles). He performed extensive experiments on light for many years and finally published his book Opticks in 1704. Geometrical optics uses straight-line, particle-like trajectories called rays to explain familiar phenomena such as reflection and refraction. Geometrical optics was also able to explain the apparent observation of sharp shadows. The competing theory considered light as a wave phenomenon. Its strongest proponent was the Dutch physicist Christian Huygens (1629–1695), who presented his theory in 1678. The wave theory could also explain reflection and refraction, but it could not explain the sharp shadows observed. Experimental physics of the 1600s and 1700s was not able to discern between the two competing theories. Huygens’s poor health and other duties kept him from working on optics much after 1678. Although Newton did not feel strongly about his corpuscular

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1.3 Waves and Particles

9

theory, the magnitude of his reputation caused it to be almost universally accepted for more than a hundred years and throughout most of the eighteenth century.
Finally, in 1802, the English physician Thomas Young (1773–1829) announced the results of his two-slit interference experiment, indicating that light behaved as a wave. Even after this singular event, the corpuscular theory had its supporters. During the next few years Young and, independently, Augustin Fresnel (1788–1827) performed several experiments that clearly showed that light behaved as a wave. By 1830 most physicists believed in the wave theory—some 150 years after Newton performed his first experiments on light.
One final experiment indicated that the corpuscular theory was difficult to accept. Let c be the speed of light in vacuum and v be the speed of light in another medium. If light behaves as a particle, then to explain refraction, light must speed up when going through denser material (v Ͼ c). The wave theory of Huygens predicts just the opposite (v Ͻ c). The measurements of the speed of light in various media were slowly improving, and finally, in 1850, Foucault showed that light traveled more slowly in water than in air. The corpuscular theory seemed incorrect. Newton would probably have been surprised that his weakly held beliefs lasted as long as they did. Now we realize that geometrical optics is correct only if the wavelength of light is much smaller than the size of the obstacles and apertures that the light encounters.
Figure 1.4 shows the “shadows” or diffraction patterns from light falling on sharp edges. In Figure 1.4a the alternating black and white lines can be seen all around the razor blade’s edges. Figure 1.4b is a highly magnified photo of the diffraction from a sharp edge. The bright and dark regions can be understood only if light is a wave and not a particle. The physicists of 200 to 300 years ago apparently did not observe such phenomena. They believed that shadows were sharp, and only the particle nature of light could explain their observations.
In the 1860s Maxwell showed that electromagnetic waves consist of oscillating electric and magnetic fields. Visible light covers just a narrow range of the total electromagnetic spectrum, and all electromagnetic radiation travels at the speed of light c in free space, given by

c ϭ 1 ϭ lf 2m0P0

(1.16)

where l is the wavelength and f is the frequency. The fundamental constants m0 and P0 are defined in electricity and magnetism and reveal the connection to the speed of light. In 1887 the German physicist Heinrich Hertz (1857–1894) suc-

ceeded in generating and detecting electromagnetic waves having wavelengths

Courtesy Ken Kay/Fundamental Photographs New York The Atlas of Optical Phenomena by Michael Cagnet, Mauric Francon, and Jean Claude Thrierr p. 32 © 1962 Springer-Verlag New York.

(a)
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Figure 1.4 In contradiction to

what scientists thought in the sev-

enteenth century, shadows are

not sharp, but show dramatic dif-

fraction patterns—as seen here

(a) for a razor blade and (b) for

(b)

a highly magnified sharp edge.

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10 Chapter 1 The Birth of Modern Physics

far outside the visible range (l Ϸ 5 m). The properties of these waves were just as Maxwell had predicted. His results continue to have far-reaching effects in modern telecommunications: cable TV, cell phones, lasers, fiber optics, wireless Internet, and so on.
Some unresolved issues about electromagnetic waves in the 1890s eventually led to one of the two great modern theories, the theory of relativity (see Section 1.6 and Chapter 2). Waves play a central and essential role in the other great modern physics theory, quantum mechanics, which is sometimes called wave mechanics. Because waves play such a central role in modern physics, we review their properties in Chapter 5.

p؊ p

K؉

K؊ ⌳

⍀؊

K0

K؊ ϩ p K0 ϩ K؉ ϩ ⍀؊ K؊

1.4 Conservation Laws and Fundamental Forces
Conservation laws are the guiding principles of physics. The application of a few laws explains a vast quantity of physical phenomena. We listed the conservation laws of classical physics in Section 1.1. They include energy, linear momentum, angular momentum, and charge. Each of these is extremely useful in introductory physics. We use linear momentum when studying collisions, and the conservation laws when examining dynamics. We have seen the concept of the conservation of energy change. At first we had only the conservation of kinetic energy in a force-free region. Then we added potential energy and formed the conservation of mechanical energy. In our study of thermodynamics, we added internal energy, and so on. The study of electrical circuits was made easier by the conservation of charge flow at each junction and the conservation of energy throughout all the circuit elements.
Much of what we know about conservation laws and fundamental forces has been learned within the last hundred years. In our study of modern physics we will find that mass is added to the conservation of energy, and the result is sometimes called the conservation of mass-energy, although the term conservation of energy is still sufficient and generally used. When we study elementary particles we will add the conservation of baryons and the conservation of leptons. Closely related to conservation laws are invariance principles. Some parameters are invariant in some interactions or in specific systems but not in others. Examples include time reversal, parity, and distance. We will study the Newtonian or Galilean invariance and find it lacking in our study of relativity; a new invariance principle will be needed. In our study of nuclear and elementary particles, conservation laws and invariance principles will often be used (see Figure 1.5).

Figure 1.5 The conservation laws of momentum and energy are invaluable in untangling complex particle reactions like the one shown here, where a 5-GeV KϪ meson interacts with a proton at rest to produce an ⍀Ϫ in a bubble chamber. The uncharged K0 is not observed. Notice the curved paths of the charged particles in the magnetic field. Such reactions are explained in Chapter 14.

Fundamental Forces
In introductory physics, we often begin our study of forces by examining the reaction of a mass at the end of a spring, because the spring force can be easily calibrated. We subsequently learn about tension, friction, gravity, surface, electrical, and magnetic forces. Despite the seemingly complex array of forces, we presently believe there are only three fundamental forces. All the other forces can be derived from them. These three forces are the gravitational, electroweak, and strong forces. Some physicists refer to the electroweak interaction as separate electromagnetic and weak forces because the unification occurs only at very high energies. The approximate strengths and ranges of the three fundamental forces are listed in Table 1.1. Physicists sometimes use the term interaction when

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1.4 Conservation Laws and Fundamental Forces 11

T a b l e 1.1 Fundamental Forces

Interaction

Relative Strength *

Range

Strong Electroweak f Electromagnetic
Weak Gravitational

1
10Ϫ2 10Ϫ9 10Ϫ39

Short, ϳ10Ϫ15 m
Long, 1/r 2 Short, ϳ10Ϫ15 m Long, 1/r 2

*These strengths are quoted for neutrons and/or protons in close proximity.

referring to the fundamental forces because it is the overall interaction among the constituents of a system that is of interest.
The gravitational force is the weakest. It is the force of mutual attraction between masses and, according to Newton, is given by

Fg

ϭ

ϪG

m1m2 r2

rˆ

(1.17)

where m1 and m2 are two point masses, G is the gravitational constant, r is the distance between the masses, and rˆ is a unit vector directed along the line between the two point masses (attractive force). The gravitational force is noticeably effective only on a macroscopic scale, but it has tremendous importance: it is the force that keeps Earth rotating about our source of life energy—the sun— and that keeps us and our atmosphere anchored to the ground. Gravity is a long-range force that diminishes as 1/r 2.
The primary component of the electroweak force is electromagnetic. The other component is the weak interaction, which is responsible for beta decay in nuclei, among other processes. In the 1970s Sheldon Glashow, Steven Weinberg, and Abdus Salam predicted that the electromagnetic and weak forces were in fact facets of the same force. Their theory predicted the existence of new particles, called W and Z bosons, which were discovered in 1983. We discuss bosons and the experiment in Chapter 14. For all practical purposes, the weak interaction is effective in the nucleus only over distances the size of 10Ϫ15 m. Except when dealing with very high energies, physicists mostly treat nature as if the electromagnetic and weak forces were separate. Therefore, you will sometimes see references to the four fundamental forces (gravity, strong, electromagnetic, and weak).
The electromagnetic force is responsible for holding atoms together, for friction, for contact forces, for tension, and for electrical and optical signals. It is responsible for all chemical and biological processes, including cellular structure and nerve processes. The list is long because the electromagnetic force is responsible for practically all nongravitational forces that we experience. The electrostatic, or Coulomb, force between two point charges q1 and q2, separated by a distance r, is given by

FC

ϭ

1 4pP0

q1q2 r2

rˆ

(1.18)

The easiest way to remember the vector direction is that like charges repel and unlike charges attract. Moving charges also create and react to magnetic fields [see Equation (1.7)].

Gravitational interaction
Weak interaction
Electromagnetic interaction Coulomb force

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12 Chapter 1 The Birth of Modern Physics

Strong interaction Unification of forces

The third fundamental force, the strong force, is the one holding the nucleus together. It is the strongest of all the forces, but it is effective only over short distances—on the order of 10Ϫ15 m. The strong force is so strong that it easily binds two protons inside a nucleus even though the electrical force of repulsion over the tiny confined space is huge. The strong force is able to contain dozens of protons inside the nucleus before the electrical force of repulsion becomes strong enough to cause nuclear decay. We study the strong force extensively in this book, learning that neutrons and protons are composed of quarks, and that the part of the strong force acting between quarks has the unusual name of color force.
Physicists strive to combine forces into more fundamental ones. Centuries ago the forces responsible for friction, contact, and tension were all believed to be different. Today we know they are all part of the electroweak force. Two hundred years ago scientists thought the electrical and magnetic forces were independent, but after a series of experiments, physicists slowly began to see their connection. This culminated in the 1860s in Maxwell’s work, which clearly showed they were but part of one force and at the same time explained light and other radiation. Figure 1.6 is a diagram of the unification of forces over time. Newton certainly had an inspiration when he was able to unify the planetary motions with the apple falling from the tree. We will see in Chapter 15 that Einstein was even able to link gravity with space and time.
The further unification of forces currently remains one of the most active research fields. Considerable efforts have been made to unify the electroweak and strong forces through the grand unified theories, or GUTs. A leading GUT is the mathematically complex string theory. Several predictions of these theories have not yet been verified experimentally (for example, the instability of the proton and the existence of magnetic monopoles). We present some of the exciting research areas in present-day physics throughout this book, because these

Figure 1.6 The three fundamental forces (shown in the heavy boxes) are themselves unifications of forces that were once believed to be fundamental. Present research is under way (see blue lines) to further unify the fundamental forces into a single force.

Electrical

Faraday

Magnetic

Weak

Light

Maxwell

Electromagnetic

Astronomical motion

Terrestrial motion

Glashow, Salam, and Weinberg
Electroweak

Strong

Newton Gravitation

Space Time
Einstein

Grand Unification
Single Force

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1.5 The Atomic Theory of Matter 13
topics are the ones you will someday read about on the front pages of newspapers and in the weekly news magazines and perhaps will contribute to in your own careers.

1.5 The Atomic Theory of Matter

Today the idea that matter is composed of tiny particles called atoms is taught in elementary school and expounded throughout later schooling. We are told that the Greek philosophers Democritus and Leucippus proposed the concept of atoms as early as 450 b.c. The smallest piece of matter, which could not be subdivided further, was called an atom, after the Greek word atomos, meaning “indivisible.” Physicists do not discredit the early Greek philosophers for thinking that the basic entity of life consisted of atoms. For centuries, scientists were called “natural philosophers,” and in this tradition the highest university degree American scientists receive is a Ph.D., which stands for doctor of philosophy.
Not many new ideas were proposed about atoms until the seventeenth century, when scientists started trying to understand the properties and laws of gases. The work of Boyle, Charles, and Gay-Lussac presupposed the interactions of tiny particles in gases. Chemists and physical chemists made many important advances. In 1799 the French chemist Proust (1754–1826) proposed the law of definite proportions, which states that when two or more elements combine to form a compound, the proportions by weight (or mass) of the elements are always the same. Water (H2O) is always formed of one part hydrogen and eight parts oxygen by mass.
The English chemist John Dalton (1766–1844) is given most of the credit for originating the modern atomic theory of matter. In 1803 he proposed that the atomic theory of matter could explain the law of definite proportions if the elements are composed of atoms. Each element has atoms that are physically and chemically characteristic. The concept of atomic weights (or masses) was the key to the atomic theory.
In 1811 the Italian physicist Avogadro proposed the existence of molecules, consisting of individual or combined atoms. He stated without proof that all gases
contain the same number of molecules in equal volumes at the same temperature and pressure. Avogadro’s ideas were ridiculed by Dalton and others who could not imagine that atoms of the same element could combine. If this could happen, they argued, then all the atoms of a gas would combine to form a liquid. The concept of molecules and atoms was indeed difficult to imagine, but finally, in 1858, the Italian chemist Cannizzaro (1826–1910) solved the problem and showed how Avogadro’s ideas could be used to find atomic masses. Today we think of an atom as the smallest unit of matter that can be identified with a particular element. A molecule can be a single atom or a combination of two or more atoms of either like or dissimilar elements. Molecules can consist of thousands of atoms.
The number of molecules in one gram-molecular weight of a particular element (6.023 ϫ 1023 molecules/mol) is called Avogadro’s number (NA). For example, one mole of hydrogen (H2) has a mass of about 2 g and one mole of carbon has a mass of about 12 g; one mole of each substance consists of 6.023 ϫ 1023 atoms. Avogadro’s number was not even estimated until 1865, and it was finally accurately measured by Perrin, as we discuss at the end of this section.
During the mid-1800s the kinetic theory of gases was being developed, and because it was based on the concept of atoms, its successes gave validity to the

Dalton, the father of the atomic theory
Avogadro’s number

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14 Chapter 1 The Birth of Modern Physics

Opposition to atomic theory

atomic theory. The experimental results of specific heats, Maxwell speed distribution, and transport phenomena (see the discussion in Section 1.2) all supported the concept of the atomic theory.
In 1827 the English botanist Robert Brown (1773–1858) observed with a microscope the motion of tiny pollen grains suspended in water. The pollen appeared to dance around in random motion, while the water was still. At first the motion (now called Brownian motion) was ascribed to convection or organic matter, but eventually it was observed to occur for any tiny particle suspended in liquid. The explanation according to the atomic theory is that the molecules in the liquid are constantly bombarding the tiny grains. A satisfactory explanation was not given until the twentieth century (by Einstein).
Although it may appear, according to the preceding discussion, that the atomic theory of matter was universally accepted by the end of the nineteenth century, that was not the case. Certainly most physicists believed in it, but there was still opposition. A principal leader in the antiatomic movement was the renowned Austrian physicist Ernst Mach. Mach was an absolute positivist, believing in the reality of nothing but our own sensations. A simplified version of his line of reasoning would be that because we have never seen an atom, we cannot say anything about its reality. The Nobel Prize–winning German physical chemist Wilhelm Ostwald supported Mach philosophically but also had more practical arguments on his side. In 1900 there were difficulties in understanding radioactivity, x rays, discrete spectral lines, and how atoms formed molecules and solids. Ostwald contended that we should therefore think of atoms as hypothetical constructs, useful for bookkeeping in chemical reactions.
On the other hand, there were many believers in the atomic theory. Max Planck, the originator of quantum theory, grudgingly accepted the atomic theory of matter because his radiation law supported the existence of submicroscopic quanta. Boltzmann was convinced that atoms must exist, mainly because they were necessary in his statistical mechanics. It is said that Boltzmann committed suicide in 1905 partly because he was despondent that so many people rejected his theory. Today we have pictures of the atom (see Figure 1.7) that would

Figure 1.7 This scanning tunneling microscope photo, called the “stadium corral,” shows 76 individually placed iron atoms on a copper surface. The IBM researchers were trying to contain and modify electron density, observed by the wave patterns, by surrounding the electrons inside the quantum “corral.” Researchers are thus able to study the quantum behavior of electrons. See also the Special Topic on Scanning Probe Microscopes in Chapter 6.
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Courtesy of International Business Machines.

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1.6 Unresolved Questions of 1895 and New Horizons 15

undoubtedly have convinced even Mach, who died in 1916 still unconvinced of the validity of the atomic theory.
Overwhelming evidence for the existence of atoms was finally presented in the first decade of the twentieth century. First, Einstein, in one of his three famous papers published in 1905 (the others were about special relativity and the photoelectric effect), provided an explanation of the Brownian motion observed almost 80 years earlier by Robert Brown. Einstein explained the motion in terms of molecular motion and presented theoretical calculations for the random walk problem. A random walk (often called the drunkard’s walk) is a statistical process that determines how far from its initial position a tiny grain may be after many random molecular collisions. Einstein was able to determine the approximate masses and sizes of atoms and molecules from experimental data.
Finally, in 1908, the French physicist Jean Perrin (1870–1942) presented data from an experiment designed using kinetic theory that agreed with Einstein’s predictions. Perrin’s experimental method of observing many particles of different sizes is a classic work, for which he received the Nobel Prize for Physics in 1926. His experiment utilized four types of measurements. Each was consistent with the atomic theory, and each gave a quantitative determination of Avogadro’s number—the first accurate measurements that had been made. Since 1908 the atomic theory of matter has been accepted by practically everyone.

Overwhelming evidence of atomic theory

1.6 Unresolved Questions of 1895 and New Horizons
We choose 1895 as a convenient time to separate the periods of classical and modern physics, although this is an arbitrary choice based on discoveries made in 1895–1897. The thousand or so physicists living in 1895 were rightfully proud of the status of their profession. The precise experimental method was firmly established. Theories were available that could explain many observed phenomena. In large part, scientists were busy measuring and understanding such physical parameters as specific heats, densities, compressibility, resistivity, indices of refraction, and permeabilities. The pervasive feeling was that, given enough time, everything in nature could be understood by applying the careful thinking and experimental techniques of physics. The field of mechanics was in particularly good shape, and its application had led to the stunning successes of the kinetic theory of gases and statistical thermodynamics.
In hindsight we can see now that this euphoria of success applied only to the macroscopic world. Objects of human dimensions such as automobiles, steam engines, airplanes, telephones, and electric lights either existed or were soon to appear and were triumphs of science and technology. However, the atomic theory of matter was not universally accepted, and what made up an atom was purely conjecture. The structure of matter was unknown.
There were certainly problems that physicists could not resolve. Only a few of the deepest thinkers seemed to be concerned with them. Lord Kelvin, in a speech in 1900 to the Royal Institution, referred to “two clouds on the horizon.” These were the electromagnetic medium and the failure of classical physics to explain blackbody radiation. We mention these and other problems here. Their solutions were soon to lead to two of the greatest breakthroughs in human thought ever recorded—the theories of quantum physics and of relativity.

Experiment and reasoning Clouds on the horizon

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16 Chapter 1 The Birth of Modern Physics

Electromagnetic Medium. The waves that were well known and understood by physicists all had media in which the waves propagated. Water waves traveled in water, and sound waves traveled in any material. It was natural for nineteenthcentury physicists to assume that electromagnetic waves also traveled in a medium, and this medium was called the ether. Several experiments, the most notable of which were done by Michelson, had sought to detect the ether without success. An extremely careful experiment by Michelson and Morley in 1887 was so sensitive, it should have revealed the effects of the ether. Subsequent experiments to check other possibilities were also negative. In 1895 some physicists were concerned that the elusive ether could not be detected. Was there an alternative explanation?

William Thomson (Lord Kelvin, 1824–1907) was born in Belfast, Ireland, and at age 10 entered the University of Glasgow in Scotland where his father was a professor of mathematics. He graduated from the University of Cambridge and, at age 22, accepted the chair of natural philosophy (later called physics) at the University of Glasgow, where he finished his illustrious 53-year career, finally resigning in 1899 at age 75. Lord Kelvin’s contributions to nineteenth-century science were far reaching, and he made contributions in electricity, magnetism, thermodynamics, hydrodynamics, and geophysics. He was involved in the successful laying of the transatlantic cable. He was arguably the preeminent scientist of the latter part of the nineteenth century. He was particularly well known for his prediction of the Earth’s age, which would later turn out to be inaccurate (see Chapter 12).
Ultraviolet catastrophe: infinite emissivity

Electrodynamics. The other difficulty with Maxwell’s electromagnetic theory had to do with the electric and magnetic fields as seen and felt by moving bodies. What appears as an electric field in one reference system may appear as a magnetic field in another system moving with respect to the first. Although the relationship between electric and magnetic fields seemed to be understood by using Maxwell’s equations, the equations do not keep the same form under a Galilean transformation [see Equations (2.1) and (2.2)], a situation that concerned both Hertz and Lorentz. Hertz unfortunately died in 1894 at the young age of 36 and never experienced the modern physics revolution. The Dutch physicist Hendrik Lorentz (1853–1928), on the other hand, proposed a radical idea that solved the electrodynamics problem: space was contracted along the direction of motion of the body. George FitzGerald in Ireland independently proposed the same concept. The Lorentz-FitzGerald hypothesis, proposed in 1892, was a precursor to Einstein’s theory advanced in 1905 (see Chapter 2).
Blackbody Radiation. In 1895 thermodynamics was on a strong footing; it had achieved much success. One of the interesting experiments in thermodynamics concerns an object, called a blackbody, that absorbs the entire spectrum of electromagnetic radiation incident on it. An enclosure with a small hole serves as a blackbody, because all the radiation entering the hole is absorbed. A blackbody also emits radiation, and the emission spectrum shows the electromagnetic power emitted per unit area. The radiation emitted covers all frequencies, each with its own intensity. Precise measurements were carried out to determine the spectrum of blackbody radiation, such as that shown in Figure 1.8. Blackbody radiation was a fundamental issue, because the emission spectrum is independent of the body itself—it is characteristic of all blackbodies.
Many physicists of the period—including Kirchhoff, Stefan, Boltzmann, Rubens, Pringsheim, Lummer, Wien, Lord Rayleigh, Jeans, and Planck—had worked on the problem. It was possible to understand the spectrum both at the low-frequency end and at the high-frequency end, but no single theory could account for the entire spectrum. When the most modern theory of the day (the equipartition of energy applied to standing waves in a cavity) was applied to the problem, the result led to an infinite emissivity (or energy density) for high frequencies. The failure of the theory was known as the “ultraviolet catastrophe.” The solution of the problem by Max Planck in 1900 would shake the very foundations of physics.

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Intensity per unit wavelength

lmax 3000 K 2500 K

1.6 Unresolved Questions of 1895and New Horizons 17
Figure 1.8 The blackbody spectrum, showing the emission spectrum of radiation emitted from a blackbody as a function of the radiation wavelength. Different curves are produced for different temperatures, but they are independent of the type of blackbody cavity. The intensity peaks at lmax.

2000 K

0

1

2

3

4

l (mm)

On the Horizon
During the years 1895–1897 there were four discoveries that were all going to require deeper understanding of the atom. The first was the discovery of x rays by the German physicist Wilhelm Röntgen (1845–1923) in November 1895. Next came the accidental discovery of radioactivity by the French physicist Henri Becquerel (1852–1908), who in February 1896 placed uranium salt next to a carefully wrapped photographic plate. When the plate was developed, a silhouette of the uranium salt was evident—indicating the presence of a very penetrating ray.
The third discovery, that of the electron, was actually the work of several physicists over a period of years. Michael Faraday, as early as 1833, observed a gas discharge glow—evidence of electrons. Over the next few years, several scientists detected evidence of particles, called cathode rays, being emitted from charged cathodes. In 1896 Perrin proved that cathode rays were negatively charged. The discovery of the electron, however, is generally credited to the British physicist J. J. Thomson (1856–1940), who in 1897 isolated the electron (cathode ray) and measured its velocity and its ratio of charge to mass.
The final important discovery of the period was made by the Dutch physicist Pieter Zeeman (1865–1943), who in 1896 found that a single spectral line was sometimes separated into two or three lines when the sample was placed in a magnetic field. The (normal) Zeeman effect was quickly explained by Lorentz as the result of light being emitted by the motion of electrons inside the atom. Zeeman and Lorentz showed that the frequency of the light was affected by the magnetic field according to the classical laws of electromagnetism.
The unresolved issues of 1895 and the important discoveries of 1895–1897 bring us to the subject of this book, Modern Physics. In 1900 Max Planck completed his radiation law, which solved the blackbody problem but required that energy be quantized. In 1905 Einstein presented his three important papers on Brownian motion, the photoelectric effect, and special relativity. While the work of Planck and Einstein may have solved the problems of the nineteenth-century physicists, they broadened the horizons of physics and have kept physicists active ever since.

Discovery of x rays Discovery of radioactivity
Discovery of the electron
Discovery of the Zeeman effect

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18 Chapter 1 The Birth of Modern Physics
Summary

Physicists of the 1890s felt that almost anything in nature could be explained by the application of careful experimental methods and intellectual thought. The application of mechanics to the kinetic theory of gases and statistical thermodynamics, for example, was a great success.
The particle viewpoint of light had prevailed for over a hundred years, mostly because of the weakly held belief of the great Newton, but in the early 1800s the nature of light was resolved in favor of waves. In the 1860s Maxwell showed that his electromagnetic theory predicted a much wider frequency range of electromagnetic radiation than the visible optical phenomena. In the twentieth century, the question of waves versus particles was to reappear.
The conservation laws of energy, momentum, angular momentum, and charge are well established. The three fundamental forces are gravitational, electroweak, and strong. Over the years many forces have been unified into these three. Physicists are actively pursuing attempts to unify these three forces into only two or even just one single fundamental force.

The atomic theory of matter assumes atoms are the smallest unit of matter that is identified with a characteristic element. Molecules are composed of atoms, which can be from different elements. The kinetic theory of gases assumes the atomic theory is correct, and the development of the two theories proceeded together. The atomic theory of matter was not fully accepted until around 1910, by which time Einstein had explained Brownian motion and Perrin had published overwhelming experimental evidence.
The year 1895 saw several outstanding problems that seemed to worry only a few physicists. These problems included the inability to detect an electromagnetic medium, the difficulty in understanding the electrodynamics of moving bodies, and blackbody radiation. Four important discoveries during the period 1895–1897 were to signal the atomic age: x rays, radioactivity, the electron, and the splitting of spectral lines (Zeeman effect). The understanding of these problems and discoveries (among others) is the object of this book on modern physics.

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Special Theory of Relativity

219

It was found that there was no displacement of the interference fringes, so that the result of the experiment was negative and would, therefore, show that there is still a difficulty in the theory itself. . . .
Albert Michelson, Light Waves and Their Uses, 1907

CHAPTER

One of the great theories of physics appeared early in the twentieth century
when Albert Einstein presented his special theory of relativity in 1905. We learned in introductory physics that Newton’s laws of motion must be measured relative to some reference frame. A reference frame is called an inertial frame if Newton’s laws are valid in that frame. If a body subject to no net external force moves in a straight line with constant velocity, then the coordinate system attached to that body defines an inertial frame. If Newton’s laws are valid in one reference frame, then they are also valid in a reference frame moving at a uniform velocity relative to the first system. This is known as the Newtonian principle of relativity or Galilean invariance.
Newton showed that it was not possible to determine absolute motion in space by any experiment, so he decided to use relative motion. In addition, the Newtonian concepts of time and space are completely separable. Consider two inertial reference frames, K and KЈ, that move along their x and x œ axes, respectively, with uniform relative velocity v as shown in Figure 2.1. We show system KЈ moving to the right with velocity v with respect to system K, which is fixed or

Inertial frame Galilean invariance

y

y′

K′ K
O′ O

v
x′ x

Figure 2.1 Two inertial systems are moving with relative speed v along their x axes. We show the system K at rest and the system KЈ moving with speed v relative to the system K .

z′ z
19

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20 Chapter 2 Special Theory of Relativity

Galilean transformation

stationary somewhere. One result of the relativity theory is that there are no fixed, absolute frames of reference. We use the term fixed to refer to a system that is fixed on a particular object, such as a planet, star, or spaceship that itself is moving in space. The transformation of the coordinates of a point in one system to the other system is given by

x œ ϭ x Ϫ vt

yœ ϭ y

(2.1)

zœ ϭ z

Similarly, the inverse transformation is given by

x ϭ x œ ϩ vt

y ϭ yœ

(2.2)

z ϭ zœ

where we have set t ϭ t œ because Newton considered time to be absolute. Equations (2.1) and (2.2) are known as the Galilean transformation. Newton’s laws of motion are invariant under a Galilean transformation; that is, they have the same form in both systems K and KЈ.
In the late nineteenth century Albert Einstein was concerned that although Newton’s laws of motion had the same form under a Galilean transformation, Maxwell’s equations did not. Einstein believed so strongly in Maxwell’s equations that he showed there was a significant problem in our understanding of the Newtonian principle of relativity. In 1905 he published ideas that rocked the very foundations of physics and science. He proposed that space and time are not separate and that Newton’s laws are only an approximation. This special theory of relativity and its ramifications are the subject of this chapter. We begin by presenting the experimental situation historically—showing why a problem existed and what was done to try to rectify the situation. Then we discuss Einstein’s two postulates on which the special theory is based. The interrelation of space and time is discussed, and several amazing and remarkable predictions based on the new theory are shown.
As the concepts of relativity became used more often in everyday research and development, it became essential to understand the transformation of momentum, force, and energy. Here we study relativistic dynamics and the relationship between mass and energy, which leads to one of the most famous equations in physics and a new conservation law of mass-energy. Finally, we return to electromagnetism to investigate the effects of relativity. We learn that Maxwell’s equations don’t require change, and electric and magnetic effects are relative, depending on the observer. We leave until Chapter 15 our discussion of Einstein’s general theory of relativity.

2.1 The Apparent Need for Ether
Thomas Young, an English physicist and physician, performed his famous experiments on the interference of light in 1802. A decade later, the French physicist and engineer Augustin Fresnel published his calculations showing the detailed understanding of interference, diffraction, and polarization. Because all known waves (other than light) require a medium in which to propagate (water waves have water, sound waves have, for example, air, and so on), it was naturally

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2.2 The Michelson-Morley Experiment 21

assumed that light also required a medium, even though light was apparently able to travel in vacuum through outer space. This medium was called the luminiferous ether or just ether for short, and it must have some amazing properties. The ether had to have such a low density that planets could pass through it, seemingly for eternity, with no apparent loss of orbit position. Its elasticity must be strong enough to pass waves of incredibly high speeds!
The electromagnetic theory of light (1860s) of the Scottish mathematical physicist James Clerk Maxwell shows that the speed of light in different media depends only on the electric and magnetic properties of matter. In vacuum, the speed of light is given by v ϭ c ϭ 1 / 1m0P0, where m0 and P0 are the permeability and permittivity of free space, respectively. The properties of the ether, as proposed by Maxwell in 1873, must be consistent with electromagnetic theory, and the feeling was that to be able to discern the ether’s various properties required only a sensitive enough experiment. The concept of ether was well accepted by 1880.
When Maxwell presented his electromagnetic theory, scientists were so confident in the laws of classical physics that they immediately pursued the aspects of Maxwell’s theory that were in contradiction with those laws. As it turned out, this investigation led to a new, deeper understanding of nature. Maxwell’s equations predict the velocity of light in a vacuum to be c. If we have a flashbulb go off in the moving system KЈ, an observer in system KЈ measures the speed of the light pulse to be c. However, if we make use of Equation (2.1) to find the relation between speeds, we find the speed measured in system K to be c ϩ v, where v is the relative speed of the two systems. However, Maxwell’s equations don’t differentiate between these two systems. Physicists of the late nineteenth century proposed that there must be one preferred inertial reference frame in which the ether was stationary and that in this system the speed of light was c. In the other systems, the speed of light would indeed be affected by the relative speed of the reference system. Because the speed of light was known to be so enormous, 3 ϫ 108 m/s, no experiment had as yet been able to discern an effect due to the relative speed v. The ether frame would in fact be an absolute standard, from which other measurements could be made. Scientists set out to find the effects of the ether.
2.2 The Michelson-Morley Experiment
The Earth orbits around the sun at a high orbital speed, about 10Ϫ4c, so an obvious experiment is to try to find the effects of the Earth’s motion through the ether. Even though we don’t know how fast the sun might be moving through the ether, the Earth’s orbital velocity changes significantly throughout the year because of its change in direction, even if its orbital speed is nearly constant.
Albert Michelson (1852–1931) performed perhaps the most significant American physics experiment of the 1800s. Michelson, who was the first U.S. citizen to receive the Nobel Prize in Physics (1907), was an ingenious scientist who built an extremely precise device called an interferometer, which measures the phase difference between two light waves. Michelson used his interferometer to detect the difference in the speed of light passing through the ether in different directions. The basic technique is shown in Figure 2.2. Initially, it is assumed that one of the interferometer arms (AC) is parallel to the motion of the Earth through the ether. Light leaves the source S and passes through the glass plate at A. Because the back of A is partially silvered, part of the light is reflected,

The concept of ether
Albert A. Michelson (1852– 1931) shown at his desk at the University of Chicago in 1927. He was born in Prussia but came to the United States when he was two years old. He was educated at the U.S. Naval Academy and later returned on the faculty. Michelson had appointments at several American universities including the Case School of Applied Science, Cleveland, in 1883; Clark University, Worcester, Massachusetts, in 1890; and the University of Chicago in 1892 until his retirement in 1929. During World War I he returned to the U.S. Navy, where he developed a rangefinder for ships. He spent his retirement years in Pasadena, California, where he continued to measure the speed of light at Mount Wilson.

03721_ch02_019-083.indd 21

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22 Chapter 2 Special Theory of Relativity

Mirror M2

D
Optical path length ᐉ2

Ether drift
v Partially silvered mirror

Figure 2.2 A schematic diagram of Michelson’s interferometer experiment. Light of a single wavelength is partially reflected and partially transmitted by the glass at A. The light is subsequently reflected by mirrors at C and D, and, after reflection or transmission again at A, enters the telescope at E. Interference fringes are visible to the observer at E.

S
Monochromatic light source

Mirror

A

B

C

M1

Compensator

Optical path length ᐉ1

E

eventually going to the mirror at D, and part of the light travels through A on to the mirror at C. The light is reflected at the mirrors C and D and comes back to the partially silvered mirror A, where part of the light from each path passes on to the telescope and eye at E. The compensator is added at B to make sure both light paths pass through equal thicknesses of glass. Interference fringes can be found by using a bright light source such as sodium, with the light filtered to make it monochromatic, and the apparatus is adjusted for maximum intensity of the light at E. We will show that the fringe pattern should shift if the apparatus is rotated through 90° such that arm AD becomes parallel to the motion of the Earth through the ether and arm AC is perpendicular to the motion.
We let the optical path lengths of AC and AD be denoted by /1 and /2, respectively. The observed interference pattern consists of alternating bright and dark bands, corresponding to constructive and destructive interference, respectively (Figure 2.3). For constructive interference, the difference between the two

From L. S. Swenson, Jr., Invention and Discovery 43 (Fall 1987).

Figure 2.3 Interference fringes as they would appear in the eyepiece of the Michelson-Morley experiment.
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2.2 The Michelson-Morley Experiment 23

path lengths (to and from the mirrors) is given by some number of wavelengths, 21/1 Ϫ /2 2 ϭ nl, where l is the wavelength of the light and n is an integer.
The expected shift in the interference pattern can be calculated by determining the time difference between the two paths. When the light travels from A to C, the velocity of light according to the Galilean transformation is c ϩ v, because the ether carries the light along with it. On the return journey from C to A the velocity is c Ϫ v, because the light travels opposite to the path of the ether. The total time for the round-trip journey to mirror M1 is t1:

t1

ϭ

c

/1 ϩ

v

ϩ

c

/1 Ϫ

v

ϭ

2c /1 c2 Ϫ v2

ϭ

2/1 c

a

1

Ϫ

1 v 2 /c 2

b

Now imagine what happens to the light that is reflected from mirror M2. If the

light is pointed directly at point D, the ether will carry the light with it, and the

light misses the mirror, much as the wind can affect the flight of an arrow. If a

swimmer (who can swim with speed v2 in still water) wants to swim across a swiftly

moving river (speed v1), the swimmer must start heading upriver, so that when

the current carries her downstream, she will move directly across the river. Care-

ful

reasoning

shows

that

the

swimmer’s

velocity

is

2v

2 2

Ϫ

v

2 1

throughout

her

journey (Problem 4). Thus the time t2 for the light to pass to mirror M2 at D and

back is

t2

ϭ

2/2 2c 2 Ϫ

v2

ϭ

2/2 c

21

1 Ϫ v2/c2

The time difference between the two journeys ⌬t is

¢t

ϭ

t2

Ϫ

t1

ϭ

2 c

a

21

/2 Ϫ v2/c2

Ϫ

1

/1 Ϫ v2/c2

b

(2.3)

We now rotate the apparatus by 90° so that the ether passes along the length /2 toward the mirror M2. We denote the new quantities by primes and carry out an analysis similar to that just done. The time difference ⌬t œ is now

¢t œ

ϭ

t

œ 2

Ϫ

t

œ 1

ϭ

2 c

a1

/2 Ϫ v2/c2

Ϫ

21

/1

b

Ϫ v2/c2

(2.4)

Michelson looked for a shift in the interference pattern when his apparatus was rotated by 90°. The time difference is

¢t œ

Ϫ

¢t

ϭ

2 c

a

1

/1 Ϫ

ϩ v2

/2 /c

2

Ϫ

/1 ϩ /2 b 21 Ϫ v 2 /c 2

Because we know c W v, we can use the binomial expansion* to expand the terms involving v 2/c 2, keeping only the lowest terms.

¢t œ

Ϫ

¢t

ϭ

2 c

1/1

ϩ

/2 2 c

a1

ϩ

v2 c2

ϩ

pb

Ϫ

a1

ϩ

v2 2c 2

ϩ

pb

d

Ϸ

v 2 1/1 ϩ c3

/2 2

(2.5)

Michelson left his position at the U.S. Naval Academy in 1880 and took his interferometer to Europe for postgraduate studies with some of Europe’s best physi-

*See Appendix 3 for the binomial expansion.

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24 Chapter 2 Special Theory of Relativity

Adjustable mirror
Light source
Telescope

Partly

silvered

mirror

Glass

compensator

Mirrors

Stone

10
1

4

2

3

Mirrors Light source
Lens

Telescope

Partly silvered mirror
Glass compensator

Mirrors

Mirrors Adjustable

Mirrors

(a)

(b)

mirror

Figure 2.4 An adaptation of the Michelson and Morley 1887 experiment taken from their publication [A. A. Michelson and E. M. Morley, Philosophical Magazine 190, 449 (1887)]. (a) A perspective view of the apparatus. To reduce vibration, the experiment was done on a massive soapstone, 1.5 m square and 0.3 m thick. This stone was placed on a wooden float that rested on mercury inside the annular piece shown underneath the stone. The entire apparatus rested on a brick pier. (b) The incoming light is focused by the lens and is both transmitted and reflected by the partly silvered mirror. The adjustable mirror allows fine adjustments in the interference fringes. The stone was rotated slowly and uniformly on the mercury to look for the interference effects of the ether.

Michelson in Europe
Null result of MichelsonMorley experiment

cists, particularly Hermann Helmholtz in Berlin. After a few false starts he finally was able to perform a measurement in Potsdam (near Berlin) in 1881. In order to use Equation (2.5) for an estimate of the expected time difference, the value of the Earth’s orbital speed around the sun, 3 ϫ 104 m/s, was used. Michelson’s apparatus had /1 Ϸ /2 Ϸ / ϭ 1.2 m. Thus Equation (2.5) predicts a time difference of 8 ϫ 10Ϫ17 s. This is an exceedingly small time, but for a visible wavelength of 6 ϫ 10Ϫ7 m, the period of one wavelength amounts to T ϭ 1/f ϭ l/c ϭ 2 ϫ 10Ϫ15 s. Thus the time period of 8 ϫ 10Ϫ17 s represents 0.04 fringes in the interference pattern. Michelson reasoned that he should be able to detect a shift of at least half this value but found none. Although disappointed, Michelson concluded that the hypothesis of the stationary ether must be incorrect.
The result of Michelson’s experiment was so surprising that he was asked by several well-known physicists to repeat it. In 1882 Michelson accepted a position at the then-new Case School of Applied Science in Cleveland. Together with Edward Morley (1838–1923), a professor of chemistry at nearby Western Reserve College who had become interested in Michelson’s work, he put together the more sophisticated experiment shown in Figure 2.4. The new experiment had an optical path length of 11 m, created by reflecting the light for eight round trips. The new apparatus was mounted on soapstone that floated on mercury to eliminate vibrations and was so effective that Michelson and Morley believed they could detect a fraction of a fringe shift as small as 0.005. With their new apparatus they expected the ether to produce a shift as large as 0.4 of a fringe. They reported in 1887 a null result —no effect whatsoever! The ether

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2.2 The Michelson-Morley Experiment 25

v ϭ 0

u v

ct

vt

(a)

(b)

Figure 2.5 The effect of stellar aberration. (a) If a telescope is at rest, light from a distant star will pass directly into the telescope. (b) However, if the telescope is traveling at speed v (because it is fixed on the Earth, which has a motion about the sun), it must be slanted slightly to allow the starlight to enter the telescope. This leads to an apparent circular motion of the star as seen by the telescope, as the motion of the Earth about the sun changes throughout the solar year.

does not seem to exist. It is this famous experiment that has become known as the Michelson-Morley experiment.
The measurement so shattered a widely held belief that many suggestions were made to explain it. What if the Earth just happened to have a zero motion through the ether at the time of the experiment? Michelson and Morley repeated their experiment during night and day and for different seasons throughout the year. It is unlikely that at least sometime during these many experiments, the Earth would not be moving through the ether. Michelson and Morley even took their experiment to a mountaintop to see if the effects of the ether might be different. There was no change.
Of the many possible explanations of the null ether measurement, the one taken most seriously was the ether drag hypothesis. Some scientists proposed that the Earth somehow dragged the ether with it as the Earth rotates on its own axis and revolves around the sun. However, the ether drag hypothesis contradicts results from several experiments, including that of stellar aberration noted by the British astronomer James Bradley in 1728. Bradley noticed that the apparent position of the stars seems to rotate in a circular motion with a period of one year. The angular diameter of this circular motion with respect to the Earth is 41 seconds of arc. This effect can be understood by an analogy. From the viewpoint of a person sitting in a car during a rainstorm, the raindrops appear to fall vertically when the car is at rest but appear to be slanted toward the windshield when the car is moving forward. The same effect occurs for light coming from stars directly above the Earth’s orbital plane. If the telescope and star are at rest with respect to the ether, the light enters the telescope as shown in Figure 2.5a. However, because the Earth is moving in its orbital motion, the apparent position of the star is at an angle u as shown in Figure 2.5b. The telescope must actually be slanted at an angle u to observe the light from the overhead star. During a time period t the starlight moves a vertical distance ct while the telescope moves a horizontal distance vt, so that the tangent of the angle u is

tan

u

ϭ

vt ct

ϭ

v c

Ether drag Stellar aberration

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AIP/Emilio Segrè Visual Archives.

26 Chapter 2 Special Theory of Relativity

The orbital speed of the Earth is about 3 ϫ 104 m/s; therefore, the angle u is 10Ϫ4 rad or 20.6 seconds of arc, with a total opening of 2u ϭ 41 s as the Earth rotates—in agreement with Bradley’s observation. The aberration reverses itself over the course of six months as the Earth orbits about the sun, in effect giving a circular motion to the star’s position. This observation is in disagreement with the hypothesis of the Earth dragging the ether. If the ether were dragged with the Earth, there would be no need to tilt the telescope! The experimental observation of stellar aberration together with the null result of the Michelson and Morley experiment is enough evidence to refute the suggestions that the ether exists. Many other experimental observations have now been made that also confirm this conclusion.
The inability to detect the ether was a serious blow to reconciling the invariant form of the electromagnetic equations of Maxwell. There seems to be no single reference inertial system in which the speed of light is actually c. H. A. Lorentz and G. F. FitzGerald suggested, apparently independently, that the results of the Michelson-Morley experiment could be understood if length is con-
tracted by the factor 21 Ϫ v 2 /c 2 in the direction of motion, where v is the speed in the direction of travel. For this situation, the length /1, in the direction of motion, will be contracted by the factor 21 Ϫ v 2 /c 2, whereas the length /2, perpendicular to v, will not. The result in Equation (2.3) is that t1 will have the
extra factor 21 Ϫ v 2 /c 2, making ⌬t precisely zero as determined experimentally by Michelson. This contraction postulate, which became known as the LorentzFitzGerald contraction, was not proven from first principles using Maxwell’s equations, and its true significance was not understood for several years until Einstein presented his explanation. An obvious problem with the Lorentz-FitzGerald contraction is that it is an ad hoc assumption that cannot be directly tested. Any measuring device would presumably be shortened by the same factor.

Albert Einstein (1879–1955), shown here sailing on Long Island Sound, was born in Germany and studied in Munich and Zurich. After having difficulty finding a position, he served seven years in the Swiss Patent Office in Bern (1902–1909), where he did some of his best work. He obtained his doctorate at the University of Zurich in 1905. His fame quickly led to appointments in Zurich, Prague, back to Zurich, and then to Berlin in 1914. In 1933, after Hitler came to power, Einstein left for the Institute for Advanced Study at Princeton University, where he became a U.S. citizen in 1940 and remained until his death in 1955. Einstein’s total contributions to physics are rivaled only by those of Isaac Newton.

2.3 Einstein’s Postulates
At the turn of the twentieth century, the Michelson-Morley experiment had laid to rest the idea of finding a preferred inertial system for Maxwell’s equations, yet the Galilean transformation, which worked for the laws of mechanics, was invalid for Maxwell’s equations. This quandary represented a turning point for physics.
Albert Einstein (1879–1955) was only two years old when Michelson reported his first null measurement for the existence of the ether. Einstein said that he began thinking at age 16 about the form of Maxwell’s equations in moving inertial systems, and in 1905, when he was 26 years old, he published his startling proposal* about the principle of relativity, which he believed to be fundamental. Working without the benefit of discussions with colleagues outside his small circle of friends, Einstein was apparently unaware of the interest concerning the null result of Michelson and Morley. † Einstein instead looked at the problem in a more formal manner and believed that Maxwell’s equations must be valid in
* In one issue of the German journal Annalen der Physik 17, No. 4 (1905), Einstein published three remarkable papers. The first, on the quantum properties of light, explained the photoelectric effect; the second, on the statistical properties of molecules, included an explanation of Brownian motion; and the third was on special relativity. All three papers contained predictions that were subsequently confirmed experimentally.
†The question of whether Einstein knew of Michelson and Morley’s null result before he produced his special theory of relativity is somewhat uncertain. For example, see J. Stachel, “Einstein and Ether Drift Experiments,” Physics Today (May 1987), p. 45.

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2.3 Einstein’s Postulates 27

all inertial frames. With piercing insight and genius, Einstein was able to bring together seemingly inconsistent results concerning the laws of mechanics and electromagnetism with two postulates (as he called them; today we would call them laws). These postulates are

1. The principle of relativity: The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists.
2. The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

Einstein’s two postulates

The first postulate indicates that the laws of physics are the same in all coordinate systems moving with uniform relative motion to each other. Einstein showed that postulate 2 actually follows from the first one. He returned to the principle of relativity as espoused by Newton. Although Newton’s principle referred only to the laws of mechanics, Einstein expanded it to include all laws of physics—including those of electromagnetism. We can now modify our previous definition of inertial frames of reference to be those frames of reference in which all the laws of physics are valid.
Einstein’s solution requires us to take a careful look at time. Return to the two systems of Figure 2.1 and remember that we had previously assumed that t ϭ t œ. We assumed that events occurring in system KЈ and in system K could easily be synchronized. Einstein realized that each system must have its own observers with their own clocks and metersticks. An event in a given system must be specified by stating both its space and time coordinates. Consider the flashing of two bulbs fixed in system K as shown in Figure 2.6a. Mary, in system KЈ (the Moving system) is beside Frank, who is in system K (the Fixed system), when the bulbs flash. As seen in Figure 2.6b the light pulses travel the same distance in system K and arrive at Frank simultaneously. Frank sees the two flashes at the same time. However, the two light pulses do not reach Mary simultaneously, because system KЈ is moving to the right, and she has moved closer to the bulb on the right by the time the flash reaches her. The light flash coming from the left will reach her at some later time. Mary thus determines that the light on the right flashed before the one on the left, because she is at rest in her frame and both flashes approach her

Inertial frames of reference revisited
Simultaneity

Mary

v K′

Ϫ1 m

0

ϩ1 m

Light Frank flash

Light K flash

Ϫ1 m

0

ϩ1 m

(a)

Mary

Ϫ1 m

0

v K′
ϩ1 m

Frank

Ϫ1 m

0

(b)

K ϩ1 m

Figure 2.6 The problem of simultaneity. Flashbulbs positioned in system K at one meter on either side of Frank go off simultaneously in (a). Frank indeed sees both flashes simultaneously in (b). However, Mary, at rest in system KЈ moving to the right with speed v, does not see the flashes simultaneously despite the fact that she was alongside Frank when the flashbulbs went off. During the finite time it took light to travel the one meter, Mary has moved slightly, as shown in exaggerated form in (b).

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28 Chapter 2 Special Theory of Relativity

Synchronization of clocks
Lorentz transformation equations

at speed c. We conclude that

Two events that are simultaneous in one reference frame (K) are not necessarily simultaneous in another reference frame (K œ) moving with respect to the first frame.

We must be careful when comparing the same event in two systems moving with respect to one another. Time comparison can be accomplished by sending light signals from one observer to another, but this information can travel only as fast as the finite speed of light. It is best if each system has its own observers with clocks that are synchronized. How can we do this? We place observers with clocks throughout a given system. If, when we bring all the clocks together at one spot at rest, all the clocks agree, then the clocks are said to be synchronized. However, we have to move the clocks relative to each other to reposition them, and this might affect the synchronization. A better way would be to flash a bulb halfway between each pair of clocks at rest and make sure the pulses arrive simultaneously at each clock. This will require many measurements, but it is a safe way to synchronize the clocks. We can determine the time of an event occurring far away from us by having a colleague at the event, with a clock fixed at rest, measure the time of the particular event, and send us the results, for example, by telephone or even by mail. If we need to check our clocks, we can always send light signals to each other over known distances at some predetermined time.
In the next section we derive the correct transformation, called the Lorentz transformation, that makes the laws of physics invariant between inertial frames of reference. We use the coordinate systems described by Figure 2.1. At t ϭ t œ ϭ 0, the origins of the two coordinate systems are coincident, and the system KЈ is traveling along the x and x œ axes. For this special case, the Lorentz transformation equations are
x œ ϭ x Ϫ vt 21 Ϫ v 2 /c 2

yœ ϭ y zœ ϭ z t œ ϭ t Ϫ 1vx /c 2 2
21 Ϫ v 2 /c 2

(2.6)

Relativistic factor

We commonly use the symbols b and the relativistic factor g to represent two longer expressions:

v bϭ c
1 gϭ
21 Ϫ v 2 /c 2

(2.7) (2.8)

which allows the Lorentz transformation equations to be rewritten in compact form as

x œ ϭ g 1x Ϫ bct 2

yœ ϭ y zœ ϭ z

(2.6)

t œ ϭ g 1t Ϫ bx /c 2

Note that g Ն 1 (g ϭ 1 when v ϭ 0).

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2.4 The Lorentz Transformation 29

2.4 The Lorentz Transformation

In this section we use Einstein’s two postulates to find a transformation between inertial frames of reference such that all the physical laws, including Newton’s laws of mechanics and Maxwell’s electrodynamics equations, will have the same form. We use the fixed system K and moving system KЈ of Figure 2.1. At t ϭ t œ ϭ 0 the origins and axes of both systems are coincident, and system KЈ is moving to the right along the x axis. A flashbulb goes off at the origins when t ϭ t œ ϭ 0. According to postulate 2, the speed of light will be c in both systems, and the wavefronts observed in both systems must be spherical and described by

x 2 ϩ y 2 ϩ z 2 ϭ c 2t 2

(2.9a)

x œ2 ϩ yœ2 ϩ zœ 2 ϭ c 2t œ 2

(2.9b)

These two equations are inconsistent with a Galilean transformation because a wavefront can be spherical in only one system when the second is moving at speed v with respect to the first. The Lorentz transformation requires both systems to have a spherical wavefront centered on each system’s origin.
Another clear break with Galilean and Newtonian physics is that we do not assume that t ϭ t œ. Each system must have its own clocks and metersticks as indicated in a two-dimensional system in Figure 2.7. Because the systems move only along their x axes, observers in both systems agree by direct observation that

yœ ϭ y zœ ϭ z

We know that the Galilean transformation x œ ϭ x Ϫ vt is incorrect, but what is the correct transformation? We require a linear transformation so that each event in system K corresponds to one, and only one, event in system KЈ. The simplest linear transformation is of the form

x œ ϭ g 1x Ϫ vt 2

(2.10)

We will see if such a transformation suffices. The parameter g cannot depend on x or t because the transformation must be linear. The parameter g must be close to 1 for v V c in order for Newton’s laws of mechanics to be valid for most of our measurements. We can use similar arguments from the standpoint of an observer stationed in system KЈ to obtain an equation similar to Equation (2.10).

Figure 2.7 In order to make sure accurate event measurements can be obtained, synchronized clocks and uniform measuring sticks are placed throughout a system.

x ϭ gœ1x œ ϩ vt œ 2

(2.11)

Because postulate 1 requires that the laws of physics be the same in both reference systems, we demand that gЈ ϭ g. Notice that the only difference between Equations (2.10) and (2.11) other than the primed and unprimed quantities being switched is that v S Ϫv, which is reasonable because according to the observer in each system, the other observer is moving either forward or backward.
According to postulate 2, the speed of light is c in both systems. Therefore, in each system the wavefront of the flashbulb light pulse along the respective x axes must be described by x ϭ ct and x œ ϭ ct œ, which we substitute into Equations (2.10) and (2.11) to obtain

ct œ ϭ g 1ct Ϫ vt 2

(2.12a)

and

ct ϭ g 1ct œ ϩ vt œ 2

(2.12b)

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30 Chapter 2 Special Theory of Relativity

Inverse Lorentz transformation equations

We divide each of these equations by c and obtain

t

œ

ϭ

gt

a1

Ϫ

v c

b

(2.13)

and

t

ϭ

gt

œa1

ϩ

v c

b

(2.14)

We substitute the value of t from Equation (2.14) into Equation (2.13).

t

œ

ϭ

g2t

œa1

Ϫ

v c

b

a1

ϩ

v c

b

(2.15)

We solve this equation for g2 and obtain

g2

ϭ

1

Ϫ

1 v 2 /c 2

or

1 gϭ
21 Ϫ v 2 /c 2

(2.16)

In order to find a transformation for time tЈ, we rewrite Equation (2.13) as

tЈ

ϭ

gat

Ϫ

vt c

b

We substitute t ϭ x/c for the light pulse and find

tЈ

ϭ

gat

Ϫ

vx c2 b

ϭ

t Ϫ vx/c 2 21 Ϫ b2

We are now able to write the complete Lorentz transformations as

x œ ϭ x Ϫ vt 21 Ϫ b2

yœ ϭ y zœ ϭ z t œ ϭ t Ϫ 1vx /c 2 2
21 Ϫ b2

(2.17)

The inverse transformation equations are obtained by replacing v by Ϫv as discussed previously and by exchanging the primed and unprimed quantities.

x ϭ x œ ϩ vt œ 21 Ϫ b2

y ϭ yœ z ϭ zœ

(2.18)

t œ ϩ 1vx œ/c 2 2 tϭ
21 Ϫ b2
Notice that Equations (2.17) and (2.18) both reduce to the Galilean transformation when v V c. It is only for speeds that approach the speed of light

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2.5 Time Dilation and Length Contraction 31

that the Lorentz transformation equations become significantly different from the Galilean equations. In our studies of mechanics we normally do not consider such high speeds, and our previous results probably require no corrections. The laws of mechanics credited to Newton are still valid over the region of their applicability. Even for a speed as high as the Earth orbiting about the sun, 30 km/s, the value of the relativistic factor g is 1.000000005. We show a plot of the relativistic parameter g versus speed in Figure 2.8. As a rule of thumb, we should consider using the relativistic equations when v/c Ͼ 0.1 (g Ϸ 1.005).
Finally, consider the implications of the Lorentz transformation. The linear transformation equations ensure that a single event in one system is described by a single event in another inertial system. However, space and time are not separate. In order to express the position of x in system KЈ, we must use both x œ and t œ. We have also found that the Lorentz transformation does not allow a speed greater than c; the relativistic factor g becomes imaginary in this case. We show later in this chapter that no object of nonzero mass can have a speed greater than c.

Relativistic factor g

8
6
4
2 1 0 0 0.2 0.4 v 0.6 0.8 1.0
c
Figure 2.8 A plot of the relativistic factor g as a function of speed v/c, showing that g becomes large quickly as v approaches c.

2.5 Time Dilation and Length Contraction
The Lorentz transformations have immediate consequences with respect to time and length measurements made by observers in different inertial frames. We shall consider time and length measurements separately and then see how they are related to one another.

Time Dilation

Consider again our two systems K and KЈ with system K fixed and system KЈ mov-

ing along the x axis with velocity v as shown in Figure 2.9a (p. 32). Frank lights

a sparkler at position x1 in system K . A clock placed beside the sparkler indicates

the time to be t1 when the sparkler is lit and t2 when the sparkler goes out (Figure

2.9b). The sparkler burns for time T0, where T0 ϭ t2 Ϫ t1. The time difference

between two events occurring at the same position in a system as measured by a

clock at rest in the system is called the proper time. We use the subscript zero on

the time difference T0 to denote the proper time.

Now what is the time as determined by Mary who is passing by (but at rest

in her own system KЈ)? All the clocks in both systems have been synchronized

when the systems are at rest with respect to one another. The two events (spar-

kler lit and then going out) do not occur at the same place according to Mary.

She is beside the sparkler when it is lit, but she has moved far away from the

sparkler when it goes out (Figure 2.9b). Her friend Melinda, also at rest in

system KЈ, is beside the sparkler when it goes out. Mary and Melinda measure

the

two

times

for

the

sparkler

to

be

lit

and

to

go

out

in

system

KЈ

as

times

t

œ 1

and t 2œ . The Lorentz transformation relates these times to those measured in

system K as

t

œ 2

Ϫ

t

œ 1

ϭ

1t2

Ϫ

t12 Ϫ 21

1v/c 22 1x2 Ϫ v2/c 2

Ϫ

x12

In system K the clock is fixed at x1, so x2 Ϫ x1 ϭ 0; that is, the two events occur

at the same position. The time t2 Ϫ t1 is the proper time T0, and we denote the

time

difference

t

œ 2

Ϫ

t

œ 1

ϭ

T

œ

as

measured

in

the

moving

system

KЈ:

Proper time

03721_ch02_019-083.indd 31

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32 Chapter 2 Special Theory of Relativity

z′

Melinda

Clock

z

t1′ Mary

x 2′

x1′

Frank

Clock

x1 t1

(a)

System K′ v
x′
System K x

z′

Melinda

t2′ Clock

System K′

Mary

v

z

x′

x 2′

x1′

Frank

x1

Clock

t2 (b)

System K x

Figure 2.9 Frank measures the proper time for the time interval that a sparkler stays lit. His

clock is at the same position in system K when the sparkler is lit in (a) and when it goes out in (b).

Mary,

in

the

moving

system

KЈ,

is

beside

the

sparkler

at

position

x

œ 1

when

it

is

lit

in

(a),

but

by

the

time it goes out in (b), she has moved away. Melinda, at position x 2œ , measures the time in system

KЈ when the sparkler goes out in (b).

Time dilation

Tœ

ϭ

21

T0 Ϫ v2/c 2

ϭ

gT0

(2.19)

Moving clocks run slow

Thus the time interval measured in the moving system KЈ is greater than the time interval measured in system K where the sparkler is at rest. This effect is known as time dilation and is a direct result of Einstein’s two postulates. The time measured by Mary and Melinda in their system KЈ for the time difference was greater than T0 by the relativistic factor g (g Ͼ 1). The two events, sparkler being lit and then going out, did not occur at the same position (xЈ2 xЈ1) in system KЈ (see Figure 2.9b). This result occurs because of the absence of simultaneity. The events do not occur at the same space and time coordinates in the two systems. It requires three clocks to perform the measurement: one in system K and two in system KЈ.
The time dilation result is often interpreted by saying that moving clocks run slow by the factor gϪ1, and sometimes this is a useful way to remember the effect. The moving clock in this case can be any kind of clock. It can be the time that sand takes to pass through an hourglass, the time a sparkler stays lit, the time between heartbeats, the time between ticks of a clock, or the time spent in a class lecture. In all cases, the actual time interval on a moving clock is greater than the proper time as measured on a clock at rest. The proper time is always the smallest possible time interval between two events.
Each person will claim the clock in the other (moving) system is running slow. If Mary had a sparkler in her system KЈ at rest, Frank (fixed in system K) would also measure a longer time interval on his clock in system K because the sparkler would be moving with respect to his system.

03721_ch02_019-083.indd 32

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2.5 Time Dilation and Length Contraction 33

EXAMPLE 2.1

Show that Frank in the fixed system will also determine the time dilation result by having the sparkler be at rest in the system KЈ.

Strategy We should be able to proceed similarly to the

derivation we did before when the sparkler was at rest in

system K. In this case Mary lights the sparkler in the moving

system KЈ. The time interval over which the sparkler is lit is

given

by

T

œ 0

ϭ

t

œ 2

Ϫ

t

1œ ,

and

the

sparkler

is

placed

at

the

posi-

tion

x

œ 1

ϭ

x

œ 2

so

that

x

œ 2

Ϫ

x

œ 1

ϭ

0.

In

this

case

T

œ 0

is

the

proper time. We use the Lorentz transformation from Equa-

tion (2.18) to determine the time difference T ϭ t 2 Ϫ t1 as measured by the clocks of Frank and his colleagues.

Solution We use Equation (2.18) to find t 2 Ϫ t1:

T

ϭ

t2

Ϫ

t1

ϭ

1t

œ 2

Ϫ

t

œ 1

2

ϩ

21

1v

/

c

2

2

1x

œ 2

Ϫ v2/c2

Ϫ

x

œ 1

2

ϭ

21

T

œ 0

Ϫ v2/c2

ϭ

gT

œ 0

The time interval is still smaller in the system where the sparkler is at rest.

The preceding results naturally seem a little strange to us. In relativity we

often carry out thought (or gedanken from the German word) experiments, be-

cause the actual experiments would be somewhat impractical. Consider the fol-

lowing gedanken experiment. Mary, in the moving system KЈ, flashes a light at her

origin along her yœ axis (Figure 2.10). The light travels a distance L, reflects off a

mirror,

and

returns.

Mary

says

that

the

total

time

for

the

journey

is

T

œ 0

ϭ

t

œ 2

Ϫ

t

œ 1

ϭ 2L/c, and this is indeed the proper time, because the clock in KЈ beside Mary

is at rest.

What do Frank and other observers in system K measure? Let T be the

round-trip time interval measured in system K for the light to return to the

x axis. The light is flashed when the origins are coincident, as Mary passes by

Frank with relative velocity v. When the light reaches the mirror in the system

KЈ at time T/2, the system KЈ will have moved a distance vT/2 down the x axis.

When the light is reflected back to the x axis, Frank will not even see the

light return, because it will return a distance vT away, where another observer,

Fred, is positioned. Because observers Frank and Fred have previously synchro-

nized their clocks, they can still measure the total elapsed time for the light

to be reflected from the mirror and return. According to observers in the K

system, the total distance the light travels (as shown in Figure 2.10) is

22 1vT /22 2 ϩ L2. And according to postulate 2, the light must travel at the

speed of light, so the total time interval T measured in system K is

Gedanken experiments

T

ϭ

distance speed

ϭ

2 2 1vT /2 2 2 c

ϩ

L2

As

can

be

determined

from

above,

L

ϭ

cT

œ 0

/2,

so

we

have

T

ϭ

2 2 1vT /2 2 2 ϩ c

1cT Ј0

/222

which reduces to

T

ϭ

21

T

œ 0

Ϫ v2/c2

ϭ

gT

œ 0

03721_ch02_019-083.indd 33

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34 Chapter 2 Special Theory of Relativity

y′

y′

Mirror

y ′ System K′ (moving)

Light

v

L

reflects

Light returns

x1′

x1′

x1′

Mary

System K

y

(at rest)

Mirror

cT

cT

2

2

L

O1 vT 2
Frank

vT

O2 x

2

Fred

t1′ ϭ 0

t

′

ϭ

T0′ 2

t 2′ ϭ T0′

t1 ϭ 0

t

ϭ

T 2

t2 ϭ T

Figure

2.10

Mary,

in

system

KЈ,

flashes

a

light along her

yœ

axis and measures the

proper

time

T

œ 0

ϭ 2L/c for the light to return. In system K Frank will see the light travel partially down his x axis,

because system KЈ is moving. Fred times the arrival of the light in system K . The time interval T

that Frank and Fred measure is related to the proper time by T ϭ gT 0œ .

This is consistent with the earlier result. In this case T Ͼ T 0œ . The proper time is always the shortest time interval, and we find that the clock in Mary’s system KЈ is “running slow.”

EXAMPLE 2.2

It is the year 2150 and the United Nations Space Federation has finally perfected the storage of antiprotons for use as fuel in a spaceship. (Antiprotons are the antiparticles of protons. We discuss antiprotons in Chapter 3.) Preparations are under way for a manned spacecraft visit to possible planets orbiting one of the three stars in the star system Alpha Centauri, some 4.30 lightyears away. Provisions are placed on board to allow a trip of 16 years’ total duration. How fast must the spacecraft travel if the provisions are to last? Neglect the period of acceleration, turnaround, and visiting times, because they are negligible compared with the actual travel time.

tive speed v to us, and that according to our clock in the

stationary system K, the trip will last T ϭ 2L/v, where L is the

distance to the star.

Because provisions on board the spaceship will last for

only

16

years,

we

let

the

proper

time

T

œ 0

in

system

KЈ

be

16 years. Using the time dilation result, we determine the

relationship between T, the time measured on Earth, and

the

proper

time

T

œ 0

to

be

T

ϭ

2L v

ϭ

21

T

œ 0

Ϫ v2/c2

(2.20)

We then solve this equation for the required speed v.

Strategy The time interval as measured by the astronauts on the spacecraft can be no longer than 16 years, because that is how long the provisions will last. However, from Earth we realize that the spacecraft will be moving at a high rela-

Solution A lightyear is a convenient way to measure large distances. It is the distance light travels in one year and is denoted by ly:

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2.5 Time Dilation and Length Contraction 35

The solution to this equation is v ϭ 0.473c ϭ 1.42 ϫ 108 m/s.

1

ly ϭ

a 3.00 ϫ 108

m s

b

11

year2 a 365

days year b a 24

h day

b

a

3600

s h

b

The

time

interval

as

measured

on

Earth

will

be

gT

œ 0

ϭ

18.2

y.

Notice that the astronauts will age only 16 years (their clocks

ϭ 9.46 ϫ 1015 m

run slow), whereas their friends remaining on Earth will age 18.2 years. Can this really be true? We shall discuss this ques-

Note that the distance of one lightyear is the speed of light, tion again in Section 2.8.

c, multiplied by the time of one year. The dimension of a

lightyear works out to be length. In this case, the result is

4.30 ly ϭ c(4.30 y) ϭ 4.07 ϫ 1016 m.

We insert the appropriate numbers into Equation (2.20)

and obtain

2 14.30 ly 2 19.46 ϫ 1015 m/ly 2

16 y

v

ϭ 21 Ϫ v 2 /c 2

Length Contraction

Now let’s consider what might happen to the length of objects in relativity. Let an

observer in each system K and KЈ have a meterstick at rest in his or her own re-

spective system. Each observer lays the stick down along his or her respective x

axis,

putting

the

left

end

at

x/

(or

x

/œ )

and

the

right

end

at

xr

(or

x

œ r

).

Thus,

Frank

in system K measures his stick to be L0 ϭ xr Ϫ x/. Similarly, in system KЈ, Mary

measures

her

stick

at

rest

to

be

L

œ 0

ϭ

x

œ r

Ϫ

x

œ /

ϭ

L0.

Every

observer

measures

a

meterstick at rest in his or her own system to have the same length, namely one

meter. The length as measured at rest is called the proper length.

Let system K be at rest and system KЈ move along the x axis with speed v.

Frank, who is at rest in system K, measures the length of the stick moving in KЈ.

The difficulty is to measure the ends of the stick simultaneously. We insist that

Frank measure the ends of the stick at the same time so that t ϭ tr ϭ t/. The events denoted by (x, t ) are (x/, t ) and (xr, t ). We use Equation (2.17) and find

x

œ r

Ϫ

x

œ /

ϭ

1xr

Ϫ x/ 2 Ϫ v 1tr Ϫ 21 Ϫ v 2 /c 2

t/ 2

The

meterstick

is

at

rest

in

system

KЈ,

so

the

length

x

œ r

Ϫ

x

œ /

must

be

the

proper

length L0œ. Denote the length measured by Frank as L ϭ xr Ϫ x/. The times tr and

t/ are identical, as we insisted, so tr Ϫ t/ ϭ 0. Notice that the times of measurement

by

Mary

in

her

system,

t

œ /

and

t

œ r

,

are

not

identical.

It

makes

no

difference

when

Mary makes the measurements in her own system, because the stick is at rest. How-

ever, it makes a big difference when Frank makes his measurements, because the

stick is moving with speed v with respect to him. The measurements must be done

simultaneously! With these results, the previous equation becomes

or,

because

L

œ 0

ϭ

L0,

L0œ

ϭ

21

L Ϫ v2/c2

ϭ

gL

L

ϭ

L0 21

Ϫ

v 2 /c 2

ϭ

L0 g

(2.21)

Notice that L0 Ͼ L, so the moving meterstick shrinks according to Frank. This effect is known as length or space contraction and is characteristic of relative

Proper length Length contraction

03721_ch02_019-083.indd 35

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36 Chapter 2 Special Theory of Relativity
motion. This effect is also sometimes called the Lorentz-FitzGerald contraction because Lorentz and FitzGerald independently suggested the contraction as a way to solve the electrodynamics problem. This effect, like time dilation, is also reciprocal. Each observer will say that the other moving stick is shorter. There is no length contraction perpendicular to the relative motion, however, because yœ ϭ y and zœ ϭ z. Observers in both systems can check the length of the other meterstick placed perpendicular to the direction of motion as the metersticks pass each other. They will agree that both metersticks are one meter long.
We can perform another gedanken experiment to arrive at the same result. This time we lay the meterstick along the x œ axis in the moving system KЈ (Figure 2.11a). The two systems K and KЈ are aligned at t ϭ t œ ϭ 0. A mirror is placed at the end of the meterstick, and a flashbulb goes off at the origin at t ϭ t œ ϭ 0, sending a light pulse down the x œ axis, where it is reflected and returned. Mary sees the stick at rest in system KЈ and measures the proper length L0 (which should of course be one meter). Mary uses the same clock fixed at x œ ϭ 0 for the time measurements. The stick is moving at speed v with respect to Frank in the fixed system K . The clocks at x ϭ x œ ϭ 0 both read zero when the origins are aligned just when the flashbulb goes off. Notice the situation shown in system K

t ϭ 0

t ϭ t1

vt 1

Flashbulb goes off

According to system K (fixed)
v

Light is reflected from mirror

Mary

0 t′

Mirror

According to system K′ (moving)

t ϭ t2 Frank

L0

x′ 1 m

0

Light

vt 2

returns

x

xᐉ1

xᐉ2

xr 1

xr 2

(a)

(b)

Figure 2.11 (a) Mary, in system KЈ, flashes a light down her x œ axis along a stick at rest in her

system of length L 0, which is the proper length. The time interval for the light to travel down the stick and back is 2L0/c. (b) Frank, in system K , sees the stick moving, and the mirror has moved a distance vt1 by the time the light is reflected. By the time the light returns to the beginning of the stick, the stick has moved a total distance of vt2. The times can be compared to show that the moving stick has been length contracted by L ϭ L0 21 Ϫ y2/c 2.

03721_ch02_019-083.indd 36

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A 4 ϫ 4 ϫ 4 units
At rest C
2 ϫ 2 ϫ 4

B
2 ϫ 2 ϫ 4 D
4 ϫ 4 ϫ 4

2.5 Time Dilation and Length Contraction 37
A B

At v ϭ 0.8c

C

D

Figure 2.12 In this computer simulation, the rectangular boxes are drawn as if the observer were 5 units in front of the near plane of the boxes and directly in front of the origin. The boxes are shown at rest on the left. On the right side, the boxes are moving to the right at a speed of v ϭ 0.8c. The horizontal lines are only length contracted, but notice that the vertical lines become hyperbolas. The objects appear to be slightly rotated in space. The objects that are further away from the origin appear earlier because they are photographed at an earlier time and because the light takes longer to reach the camera (or our eyes). Reprinted with permission from American Journal of Physics 33, 534 (1965), G. D. Scott and M. R. Viner. © 1965, American Association of Physics Teachers.

(Figure 2.11b), where by the time the light reaches the mirror, the entire stick has moved a distance vt1. By the time the light has been reflected back to the front of the stick again, the stick has moved a total distance vt2. We leave the solution in terms of length contraction to Problem 18.
The effect of length contraction along the direction of travel may strongly affect the appearances of two- and three-dimensional objects. We see such objects when the light reaches our eyes, not when the light actually leaves the object. Thus, if the objects are moving rapidly, we will not see them as they appear at rest. Figure 2.12 shows the appearance of several such objects as they move. Note that not only do the horizontal lines become contracted, but the vertical lines also become hyperbolas. We show in Figure 2.13 a row of bars moving to the right with speed v ϭ 0.9c. The result is quite surprising.

(a)

(b)

Figure 2.13 (a) An array of rectangular bars is seen from above at rest. (b) The bars are moving to the right at v ϭ 0.9c. The bars appear to contract and rotate. Quoted from P.-K. Hsuing and R. H. P. Dunn, Science News 137, 232 (1990).

03721_ch02_019-083.indd 37

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38 Chapter 2 Special Theory of Relativity

EXAMPLE 2.3

Consider the solution of Example 2.2 from the standpoint of length contraction.
Strategy The astronauts have only enough provisions for a trip lasting 16 years. Thus they expect to travel for 8 years each way. If the star system Alpha Centauri is 4.30 lightyears away, it may appear that they need to travel at a velocity of 0.5c to make the trip. We want to consider this example as if the astronauts are at rest. Alpha Centauri will appear to be moving toward them, and the distance to the star system is length contracted. The distance measured by the astronauts will be less than 4.30 ly.
Solution The contracted distance according to the astronauts in motion is 14.30 ly 2 21 Ϫ v 2 /c 2. The velocity they need to make this journey is the contracted distance divided by 8 years.

v

ϭ

distance time

ϭ

14.30

ly 2

21 8 y

Ϫ

v 2 /c 2

If we divide by c, we obtain

v 14.30 ly 2 21 Ϫ v 2 /c 2 14.30 ly 2 21 Ϫ v 2 /c 2

bϭ c ϭ

c 18 y2

ϭ

18 ly2

8b ϭ 4.30 21 Ϫ b2

which gives

b ϭ 0.473

v ϭ 0.473c

which is just what we found in the previous example. The effects of time dilation and length contraction give identical results.

2.6 Addition of Velocities
A spaceship launched from a space station (see Figure 2.14) quickly reaches its cruising speed of 0.60c with respect to the space station when a band of asteroids is observed straight ahead of the ship. Mary, the commander, reacts quickly and orders her crew to blast away the asteroids with the ship’s proton gun to avoid a catastrophic collision. Frank, the admiral on the space station, listens with apprehension to the communications because he fears the asteroids may eventually destroy his space station as well. Will the high-energy protons of speed 0.99c be able to successfully blast away the asteroids and save both the spaceship and

Figure 2.14 The space station is at rest at the origin of system K . The spaceship is moving to the right with speed v with respect to the space station and is in system KЈ. An asteroid is moving to the left toward both the spaceship and space station, so Mary, the commander of the spaceship, orders that the proton gun shoot protons to break up the asteroid. The speed of the protons is u and uœ with respect to systems K and KЈ, respectively.

y System K at rest
Space station

yЈ System KЈ v v

Spaceship vv

u uЈ Proton

xЈ Asteroid
x

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2.6 Addition of Velocities 39

space station? If 0.99c is the speed of the protons with respect to the spaceship, what speed will Frank measure for the protons?
We will use the letter u to denote velocity of objects as measured in various coordinate systems. In this case, Frank (in the fixed, stationary system K on the space station) will measure the velocity of the protons to be u, whereas Mary, the commander of the spaceship (the moving system KЈ), will measure uœ ϭ 0.99c. We reserve the letter v to express the velocity of the coordinate systems with respect to each other. The velocity of the spaceship with respect to the space station is v ϭ 0.60c.
Newtonian mechanics teaches us that to find the velocity of the protons with respect to the space station, we simply add the velocity of the spaceship with respect to the space station (0.60c) to the velocity of the protons with respect to the spaceship (0.99c) to determine the result u ϭ v ϩ uœ ϭ 0.60c ϩ 0.99c ϭ 1.59c. However, this result is not in agreement with the results of the Lorentz transformation. We use Equation (2.18), letting x be along the direction of motion of the spaceship (and high-speed protons), and take the differentials, with the results

dx ϭ g 1dx œ ϩ v dt œ 2

dy ϭ dyœ dz ϭ dzœ

(2.22)

dt ϭ g 3 dt œ ϩ 1v /c 2 2 dx œ 4

Velocities are defined by ux ϭ dx/dt, uy ϭ dy/dt, uxœ ϭ dx œ/dt œ, and so on. Therefore we determine ux by

ux

ϭ

dx dt

ϭ

g 1dx œ ϩ v dt œ 2 g 3 dt œ ϩ 1v /c 2 2 dx œ 4

ϭ

1

uxœ ϩ v ϩ 1v /c 2 2 uxœ

(2.23a)

Similarly, uy and uz are determined to be

uy

ϭ

g

31

ϩ

uyœ 1v /c 2 2 uxœ 4

(2.23b)

uz

ϭ

g

31

ϩ

uzœ 1v /c 2 2 uxœ 4

(2.23c)

Equations (2.23) are referred to as the Lorentz velocity transformations. Notice that although the relative motion of the systems K and KЈ is only along the x direction, the velocities along y and z are affected as well. This contrasts with the Lorentz transformation equations, where y ϭ yœ and z ϭ zœ. However, the difference in velocities is simply ascribed to the transformation of time, which depends on v and x œ. Thus, the transformations for uy and uz depend on v and uxœ . The inverse transformations for uxœ , uyœ, and uzœ can be determined by simply switching primed and unprimed variables and changing v to Ϫv. The results are

Relativistic velocity addition

uxœ

ϭ

1

ux Ϫ v Ϫ 1v /c 2 2 ux

uyœ

ϭ

g

31

Ϫ

uy 1v /c 2 2 ux 4

uzœ

ϭ

g

31

Ϫ

uz 1v /c 2 2 ux 4

(2.24)

03721_ch02_019-083.indd 39

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40 Chapter 2 Special Theory of Relativity

Figure 2.15 A floodlight revolving at high speeds can sweep a light beam across the surface of the moon at speeds exceeding c, but the speed of the light still does not exceed c.

Floodlight

Speeds Ͼ c Moon’s surface
Light speed ϭ c

High-speed rotation

Note that we found the velocity transformation equations for the situation corresponding to the inverse Lorentz transformation, Equations (2.18), before finding the velocity transformation for Equations (2.17).
What is the correct result for the speed of the protons with respect to the space station? We have uxœ ϭ 0.99c and v ϭ 0.60c, so Equation (2.23a) gives us the result

ux ϭ

0.990c ϩ 0.600c 10.600c2 10.990c2

ϭ 0.997c

1ϩ

c2

where we have assumed we know the speeds to three significant figures. Therefore, the result is a speed only slightly less than c. The Lorentz transformation does not allow a material object to have a speed greater than c. Only massless particles, such as light, can have speed c. If the crew members of the spaceship spot the asteroids far enough in advance, their reaction times should allow them to shoot down the uncharacteristically swiftly moving asteroids and save both the spaceship and the space station.
Although no particle with mass can carry energy faster than c, we can imagine a signal being processed faster than c. Consider the following gedanken experiment. A giant floodlight placed on a space station above the Earth revolves at 100 Hz, as shown in Figure 2.15. Light spreads out in the radial direction from the floodlight at speeds of c. On the surface of the moon, the light beam sweeps across at speeds far exceeding c (Problem 36). However, the light itself does not reach the moon at speeds faster than c. No energy is associated with the beam of light sweeping across the moon’s surface. The energy (and linear momentum) is only along the radial direction from the space station to the moon.

EXAMPLE 2.4

Mary, the commander of the spaceship just discussed, is holding target practice for junior officers by shooting protons at small asteroids and space debris off to the side (perpendicular to the direction of spaceship motion) as the spaceship passes by. What speed will an observer in the space station measure for these protons?

Strategy We use the coordinate systems and speeds of the
spaceship and proton gun as described previously. Let the
direction of the protons now be perpendicular to the direction of the spaceship—along the yœ direction. We already know in the spaceship’s KЈ system that uyœ ϭ 0.99c and uxœ ϭ

03721_ch02_019-083.indd 40

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uzœ ϭ 0, and that the speed of the KЈ system (spaceship) with respect to the space station is v ϭ 0.60c. We use Equations
(2.23) to determine ux, uy, and uz and finally the speed u.

Solution To find the speeds in the system K, we first need to find ␥.

gϭ

1

ϭ

1

ϭ 1.25

21 Ϫ v 2 /c 2 21 Ϫ 0.6002

Next we are able to determine the components of u.

ux 1protons 2

ϭ

31

ϩ

0 ϩ 0.600c 10.600c 2 10c 2 /c 24

ϭ

0.600c

uy 1protons 2

ϭ

1.25 31

ϩ

0.990c 10.600c 2 10c 2 /c 24

ϭ

0.792c

uz 1protons 2

ϭ

1.25 31

ϩ

0 10.600c 2 10c 2 /c 24

ϭ

0

u 1protons 2 ϭ 2u2x ϩ u2y ϩ u2z ϭ 2 10.600c 2 2 ϩ 10.792c 2 2 ϭ 0.994c

We have again assumed we know the velocity components to three significant figures. Mary and her junior officers only observe the protons moving perpendicular to their motion. However, because there are both ux and uy components, Frank (on the space station) sees the protons moving at an angle with respect to both his x and his y directions.

EXAMPLE 2.5

By the early 1800s experiments had shown that light slows down when passing through liquids. A. J. Fresnel suggested in 1818 that there would be a partial drag on light by the medium through which the light was passing. Fresnel’s suggestion explained the problem of stellar aberration if the Earth was at rest in the ether. In a famous experiment in 1851, H. L. Fizeau measured the “ether” drag coefficient for light passing in opposite directions through flowing water. Let a moving system KЈ be at rest in the flowing water and let v be the speed of the flowing water with respect to a fixed observer in K (see Figure 2.16). The speed of light in the water at rest (that is, in system KЈ) is uœ, and the speed of light as measured in K is u. If the index of refraction of the water is n, Fizeau found experimentally that

u

ϭ

uœ

ϩ

a1

Ϫ

1 n2

b

v

which was in agreement with Fresnel’s prediction. This result was considered an affirmation of the ether concept. The factor 1 Ϫ 1/n2 became known as Fresnel’s drag coefficient. Show that this result can be explained using relativistic velocity addition without the ether concept.

Strategy We note from introductory physics that the velocity of light in a medium of index of refraction n is uœ ϭ
c/n. We use Equation (2.23a) to solve for u.

Solution We have to calculate the speed only in the xdirection, so we dispense with the subscripts. We utilize Equation (2.23a) to determine

uϭ

uœ ϩ v 1 ϩ uœv /c 2

ϭ

c/n ϩ v 1 ϩ v /nc

ϭ

c n

a1

ϩ

nv c

b

a1

ϩ

v nc

b

Water flowing at speed
v

y K

y′ K′

O

x O′

Fixed observer

v Light
u, u′ x′
River

Figure 2.16 A stationary system K is fixed on shore, and a moving system KЈ floats down the river at speed v. Light emanating from a source under water in system KЈ has speed u, uœ in systems K , KЈ, respectively.

Because v V c in this case, we can expand the denominator (1 ϩ x)Ϫ1 ϭ 1 Ϫ x ϩ p keeping only the lowest term in x ϭ v/c. The above equation becomes

u

ϭ

c n

a1

ϩ

nv c

b

a1

Ϫ

v nc

ϩ

pb

ϭ

c n

a1

ϩ

nv c

Ϫ

v nc

ϩ

pb

ϭ

c n

ϩ

v

Ϫ

v n2

ϭ

uœ

ϩ

a1

Ϫ

1 n2

b

v

which is in agreement with Fizeau’s experimental result and Fresnel’s prediction given earlier. This relativistic calculation is another stunning success of the special theory of relativity. There is no need to consider the existence of the ether.

03721_ch02_019-083.indd 41

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42 Chapter 2 Special Theory of Relativity

2.7 Experimental Verification
We have used the special theory of relativity to describe some unusual phenomena. The special theory has also been used to make some startling predictions concerning length contraction, time dilation, and velocity addition. In this section we discuss only a few of the many experiments that have been done to confirm the special theory of relativity.

Radioactive decay law

Muon Decay

When high-energy particles called cosmic rays enter the Earth’s atmosphere from outer space, they interact with particles in the upper atmosphere (see Figure 2.17), creating additional particles in a cosmic shower. Many of the particles in the shower are p-mesons (pions), which decay into other unstable particles called muons. The properties of muons are described later when we discuss nuclear and particle physics. Because muons are unstable, they decay according to the radioactive decay law

N

ϭ

1ln 22t N0 exp a Ϫ t1/2 b

ϭ

N0

exp

a

Ϫ

0.693t t1 /2

b

where N0 and N are the number of muons at times t ϭ 0 and t ϭ t, respectively, and t1/2 is the half-life of the muons. This means that in the time period t1/2 half of the muons will decay to other particles. The half-life of muons (1.52 ϫ 10Ϫ6 s) is long enough that many of them survive the trip through the atmosphere to the Earth’s surface.
We perform an experiment by placing a muon detector on top of a mountain 2000 m high and counting the number of muons traveling at a speed near v ϭ 0.98c (see Figure 2.18a). Suppose we count 103 muons during a given time period t0. We then move our muon detector to sea level (see Figure 2.18b), and we determine experimentally that approximately 540 muons survive the trip without decaying. We ignore any other interactions that may remove muons.
Classically, muons traveling at a speed of 0.98c cover the 2000-m path in 6.8 ϫ 10Ϫ6 s, and according to the radioactive decay law, only 45 muons should survive the trip. There is obviously something wrong with the classical calculation, because we counted a factor of 12 more muons surviving than the classical calculation predicts.

03721_ch02_019-083.indd 42

Figure 2.17 Much of what we know about muons in cosmic rays was learned from balloon flights carrying sophisticated detectors. This balloon is being prepared for launch in NASA’s Ultra Long Duration Balloon program for a mission that may last up to 100 days. The payload will hang many meters below the balloon. Victor Hess began the first such balloon flights in 1912 (when he discovered cosmic rays), and much improved versions are still launched today from all over the world to study cosmic rays, the atmosphere, the sun, and the universe.

Photo courtesy of NASA.

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Muons
Detector
At 2000 m, we detect 1000 muons in period t 0 traveling at speed near 0.98c.

2.7 Experimental Verification 43
At sea level, we detect only 542 muons in the same time period t 0 traveling at speed near 0.98c.
Muons

(a)

(b)

Figure 2.18 The number of muons detected with speeds near 0.98c is much different (a) on top of a mountain than (b) at sea level, because of the muon’s decay. The experimental result agrees with our time dilation equation.

Because the classical calculation does not agree with the experimental result, we should consider a relativistic calculation. The muons are moving at a speed of 0.98c with respect to us on Earth, so the effects of time dilation will be dramatic. In the muon rest frame, the time period for the muons to travel 2000 m (on a clock fixed with respect to the mountain) is calculated from Equation (2.19) to be (6.8/5.0) ϫ 10Ϫ6 s, because g ϭ 5.0 for v ϭ 0.98c. For the time t ϭ 1.36 ϫ 10Ϫ6 s, the radioactive decay law predicts that 538 muons will survive the trip, in agreement with the observations. An experiment similar to this was performed by B. Rossi and D. B. Hall* in 1941 on the top of Mount Washington in New Hampshire.
It is useful to examine the muon decay problem from the perspective of an observer traveling with the muon. This observer would not measure the distance from the top of the 2000-m mountain to sea level to be 2000 m. Rather, this observer would say that the distance is contracted and is only (2000 m)/5.0 ϭ 400 m. The time to travel the 400-m distance would be (400 m)/0.98c ϭ 1.36 ϫ 10Ϫ6 s according to a clock at rest with a muon. Using the radioactive decay law, an observer traveling with the muons would still predict 538 muons to survive. Therefore, we obtain the identical result whether we consider time dilation or space contraction, and both are in agreement with the experiment, thus confirming the special theory of relativity.
Atomic Clock Measurement
In an atomic clock, an extremely accurate measurement of time is made using a well-defined transition in the 133Cs atom ( f ϭ 9,192,631,770 Hz). In 1971 two American physicists, J. C. Hafele and Richard E. Keating (Figure 2.19), used four

* B. Rossi and D. B. Hall, Physical Review 50, 223 (1941). An excellent, though now dated, film recreating this experiment (Time Dilation—An Experiment with m-mesons by D. H. Frisch and J. H. Smith) is available from the Education Development Center, Newton, Mass. See also D. H. Frisch and J. H. Smith, American Journal of Physics 31, 342 (1963).

03721_ch02_019-083.indd 43

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AP/Wide World Photos.

44 Chapter 2 Special Theory of Relativity

cesium beam atomic clocks to test the time dilation effect. They flew the four portable cesium clocks eastward and westward on regularly scheduled commercial jet airplanes around the world and compared the time with a reference atomic time scale at rest at the U.S. Naval Observatory in Washington, D.C. (Figure 2.20).
The trip eastward took 65.4 hours with 41.2 flight hours, whereas the westward trip, taken a week later, took 80.3 hours with 48.6 flight hours. The comparison with the special theory of relativity is complicated by the rotation of the Earth and by a gravitational effect arising from the general theory of relativity. The actual relativistic predictions and experimental observations for the time differences* are

Figure 2.19 Joseph Hafele and Richard Keating are shown unloading one of their atomic clocks and the associated electronics from an airplane in Tel Aviv, Israel, during a stopover in November 1971 on their round-theworld trip to test special relativity.

Travel
Eastward Westward

Predicted
Ϫ40 Ϯ 23 ns 275 Ϯ 21 ns

Observed
Ϫ59 Ϯ 10 ns 273 Ϯ 7 ns

A negative time indicates that the time on the moving clock is less than the reference clock. The moving clocks lost time (ran slower) during the eastward trip, but gained time (ran faster) during the westward trip. This occurs because of the rotation of the Earth, indicating that the flying clocks ticked faster or slower than the reference clocks on Earth. The special theory of relativity is verified within the experimental uncertainties.

* See J. C. Hafele and R. E. Keating, Science 177, 166–170 (1972).

03721_ch02_019-083.indd 44

Earth N

Earth’s rotation

Westward

Eastward

U.S. Naval Observatory (Washington, D.C.)
Figure 2.20 Two airplanes took off (at different times) from Washington, D.C., where the U.S. Naval Observatory is located. The airplanes traveled east and west around Earth as it rotated. Atomic clocks on the airplanes were compared with similar clocks kept at the observatory to show that the moving clocks in the airplanes ran slower.

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2.7 Experimental Verification 45

EXAMPLE 2.6

In 1985 the space shuttle Challenger flew a cesium clock and compared its time with a fixed clock left on Earth. The shuttle orbited at approximately 330 km above Earth with a speed of 7712 m/s (ϳ17,250 mph). (a) Calculate the expected time lost per second for the moving clock and compare with the measured result of Ϫ295.02 Ϯ 0.29 ps/s, which includes a predicted effect due to general relativity of 35.0 Ϯ 0.06 ps/s. (b) How much time would the clock lose due to special relativity alone during the entire shuttle flight that lasted for 7 days?

Solution (a) We have b ϭ v/c ϭ (7712 m/s)/(2.998 ϫ 108 m/s) ϭ 2.572 ϫ 10Ϫ5. Because b is such a small quantity, we can use a power series expansion of the square root
21 Ϫ b2, keeping only the lowest term in b2 for ⌬T.

¢T

ϭ

T c1

Ϫ

a1

Ϫ

b2 2

ϩ

pb

d

ϭ

b2T 2

Now we have

Strategy This should be a straightforward application of the time dilation effect, but we have the complicating fact that the space shuttle is moving in a noninertial system (orbiting around Earth). We don’t want to consider this now, so we make the simplifying assumption that the space shuttle travels in a straight line with respect to Earth and the two events in the calculations are the shuttle passing the starting point (launch) and the ending point (landing). We are not including the effects of general relativity.
We know the orbital speed of the shuttle with respect to Earth, which allows us to determine b and the relativistic factor g. We let T be the time measured by the clock fixed on Earth. Then we can use the time dilation effect given by Equation (2.19) to determine the proper time T0œ measured by the clock in the space shuttle. The time difference is ⌬T ϭ T Ϫ T0œ. We have T0œ ϭ T 21 Ϫ b2 and ⌬T ϭ T Ϫ T0œ ϭ
T 11 Ϫ 21 Ϫ b22 . For part (b) we need to find the total
time lost for the moving clock for 7 days.

¢T T

ϭ

b2 2

ϭ

1 2

12.572

ϫ

10Ϫ5 2 2

ϭ

330.76

ϫ

10Ϫ12

In this case ⌬T is positive, which indicates that the space shuttle clock lost this fraction of time, so the moving clock lost 330.76 ps for each second of motion.
How does this compare with the measured time? The total measured result was a loss of 295.02 Ϯ 0.29 ps/s, but we must add the general relativity prediction of 35.0 Ϯ 0.06 ps/s to the measured value to obtain the result due only to special relativity. So the measured special relativity result is close to 330.02 ps/s, which differs from our calculated result by only 0.2%!
(b) The total time of the seven-day mission was 6.05 ϫ 105 s, so the total time difference between clocks is (330.76 ϫ 10Ϫ12)(6.05 ϫ 105 s) ϭ 0.2 ms, which is easily detected by cesium clocks.

Velocity Addition
An interesting test of the velocity addition relations was made by T. Alväger and colleagues* at the CERN nuclear and particle physics research facility on the border of Switzerland and France. They used a beam of almost 20-GeV (20 ϫ 109 eV) protons to strike a target to produce neutral pions (p0) having energies of more than 6 GeV. The p0 (b Ϸ 0.99975) have a very short half-life and soon decay into two g rays. In the rest frame of the p0 the two g rays go off in opposite directions. The experimenters measured the velocity of the g rays going in the forward direction in the laboratory (actually 6°, but we will assume 0° for purposes of calculation because there is little difference). The Galilean addition of velocities would require the velocity of the g rays to be u ϭ 0.99975c ϩ c ϭ 1.99975c, because the velocity of g rays is already c. However, the relativistic velocity addition, in which

Pion decay experiment

*See T. Alväger, F. J. M. Farley, J. Kjellman, and I. Wallin, Physics Letters 12, 260 (1964). See also article by J. M. Bailey, Arkiv Fysik 31, 145 (1966).

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46 Chapter 2 Special Theory of Relativity

v ϭ 0.99975c is the velocity of the p0 rest frame with respect to the laboratory and uœ ϭ c is the velocity of the g rays in the rest frame of the p0, predicts the velocity u of the g rays measured in the laboratory to be, according to Equation (2.23a),

uϭ

c ϩ 0.99975c 10.99975c2 1c2

ϭc

1ϩ

c2

The experimental measurement was accomplished by measuring the time taken for the g rays to travel between two detectors placed about 30 m apart and was in excellent agreement with the relativistic prediction, but not the Galilean one. We again have conclusive evidence of the need for the special theory of relativity.

Testing Lorentz Symmetry
Although we have mentioned only three rather interesting experiments, physicists performing experiments with nuclear and particle accelerators have examined thousands of cases that verify the correctness of the concepts discussed here. Quantum electrodynamics (QED) includes special relativity in its framework, and QED has been tested to one part in 1012.
Lorentz symmetry requires the laws of physics to be the same for all observers, and Lorentz symmetry is important at the very foundation of our description of fundamental particles and forces. Lorentz symmetry, together with the principles of quantum mechanics that are discussed in much of the remainder of this book, form the framework of relativistic quantum field theory. Many interactions that could be added to our best theories of physics (see the Standard Model in Chapter 14) are excluded, because they would violate Lorentz symmetry. In just the past two decades, physicists have conceived and performed many experiments that test Lorentz symmetry, but no violations have been discovered to date. For example, tests done with electrons have shown no violations to one part in 1032, with neutrons one part in 1031, and with protons one part in 1027. These are phenomenal numbers, but many more experiments are currently underway, and more are planned. Several physicists have proposed in recent years that some theories of quantum gravity imply that Lorentz symmetry is not valid. They suggest a violation may occur at very small distances around 10Ϫ35 m. Direct investigation at these small distances is not now possible, because the energy required is huge (1028 eV), but such effects may be observed in highly energetic events in outer space. To date, no verified experiments have found a violation of Lorentz symmetry, but interest remains high.*

2.8 Twin Paradox
One of the most interesting topics in relativity is the twin (or clock) paradox. Almost from the time of publication of Einstein’s famous paper in 1905, this subject has received considerable attention, and many variations exist. Let’s summarize the paradox. Suppose twins, Mary and Frank, choose different career paths. Mary (the Moving twin) becomes an astronaut and Frank (the Fixed twin) a stockbroker. At age 30, Mary sets out on a spaceship to study a star system 8 ly from Earth. Mary travels at very high speeds to reach the star and returns during her life span.

* See “Lorentz Invariance on Trial,” Maxim Pospelov and Michael Romalis, Physics Today (July 2004) p. 40. See also Scientific American (September 2004) Special Issue on “Beyond Einstein.”

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2.8 Twin Paradox 47

According to Frank’s understanding of special relativity, Mary’s biological clock ticks more slowly than his own, so he claims that Mary will return from her trip younger than he. The paradox is that Mary similarly claims that it is Frank who is moving rapidly with respect to her, so that when she returns, Frank will be the younger. To complicate the paradox further one could argue that because nature cannot allow both possibilities, it must be true that symmetry prevails and that the twins will still be the same age. Which is the correct solution?
The correct answer is that Mary returns from her space journey as the younger twin. According to Frank, Mary’s spaceship takes off from Earth and quickly reaches its travel speed of 0.8c. She travels the distance of 8 ly to the star system, slows down and turns around quickly, and returns to Earth at the same speed. The accelerations (positive and negative) take negligible times compared to the travel times between Earth and the star system. According to Frank, Mary’s travel time to the star is 10 years [(8 ly)/0.8c ϭ 10 y] and the return is also 10 years, for a total travel time of 20 years, so that Frank will be 30 ϩ 10 ϩ 10 y ϭ 50 years old when Mary returns. However, because Mary’s clock is ticking more slowly, her travel time to the star is only 1021 Ϫ 0.82 y ϭ 6 years. Frank calculates that Mary will only be 30 ϩ 6 ϩ 6 y ϭ 42 years old when she returns with respect to his own clock at rest.
The important fact here is that Frank’s clock is in an inertial system* during the entire trip; however, Mary’s clock is not. As long as Mary is traveling at constant speed away from Frank, both of them can argue that the other twin is aging less rapidly. However, when Mary slows down to turn around, she leaves her original inertial system and eventually returns in a completely different inertial system. Mary’s claim is no longer valid, because she does not remain in the same inertial system. There is also no doubt as to who is in the inertial system. Frank feels no acceleration during Mary’s entire trip, but Mary will definitely feel acceleration during her reversal time, just as we do when we step hard on the brakes of a car. The acceleration at the beginning and the deceleration at the end of her trip present little problem, because the fixed and moving clocks could be compared if Mary were just passing by Frank each way. It is Mary’s acceleration at the star system that is the key. If we invoke the two postulates of special relativity, there is no paradox. The instantaneous rate of Mary’s clock is determined by her instantaneous speed, but she must account for the acceleration effect when she turns around. A careful analysis of Mary’s entire trip using special relativity, including acceleration, will be in agreement with Frank’s assessment that Mary is younger. Mary returns to Earth rich as well as famous, because her stockbroker brother has invested her salary wisely during the 20-year period (for which she only worked 12 years!).
We follow A. P. French’s excellent book, Special Relativity, to present Table 2.1 (page 48), which analyzes the twin paradox. Both Mary and Frank send out signals at a frequency f (as measured by their own clock). We include in the table the various journey timemarks and signals received during the trip, with one column for the twin Frank who stayed at home and one for the astronaut twin Mary who went on the trip. Let the total time of the trip as measured on Earth be T. The speed of Mary’s spaceship is v (as measured on Earth), which gives a relativistic factor g. The distance Mary’s spaceship goes before turning around (as measured on Earth) is L. Much of this table is best analyzed by using spacetime (see the next section) and the Doppler effect (see Section 2.10).

Who is the younger twin?
Mary is both younger and rich

*The rotating and orbiting Earth is only an approximate inertial system.

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48 Chapter 2 Special Theory of Relativity

Table 2.1 Twin Paradox Analysis

Item

Measured by Frank (remains on Earth)

Measured by Mary (traveling astronaut)

Time of total trip Total number of signals sent

T ϭ 2L /v f T ϭ 2f L /v

T œ ϭ 2L /gv f T œ ϭ 2f L /gv

Frequency of signals received at beginning of trip f œ

1Ϫb fB1 ϩ b

1Ϫb fB1 ϩ b

Time of detecting Mary’s turnaround Number of signals received at the rate f œ Time for remainder of trip
Frequency of signals received at end of trip f fl
Number of signals received at rate f fl Total number of signals received

t1 ϭ L /v ϩ L /c

f

œt1

ϭ

f L v 21

Ϫ

b2

t2 ϭ L /v Ϫ L /c

f

B

1 1

ϩ Ϫ

b b

f

flt2

ϭ

f L v 21

Ϫ

b2

2f L /gv

t

œ 1

ϭ

L /gv

f

œt

œ 1

ϭ

f L v

11

Ϫ

b2

t

œ 2

ϭ

L /gv

f

B

1 1

ϩ Ϫ

b b

f

flt

œ 2

ϭ

f L v

11

ϩ

b2

2f L /v

Conclusion as to other twin’s measure of time taken

T œ ϭ 2L /gv

T ϭ 2L /v

After A. French, Special Relativity, New York: Norton (1968), p. 158.

Spacetime (Minkowski) diagrams
Worldline

2.9 Spacetime
When describing events in relativity, it is sometimes convenient to represent events on a spacetime diagram as shown in Figure 2.21. For convenience we use only one spatial coordinate x and specify position in this one dimension. We use ct instead of time so that both coordinates will have dimensions of length. Spacetime diagrams were first used by H. Minkowski in 1908 and are often called Minkowski diagrams. We have learned in relativity that we must denote both space and time to specify an event. This is the origin of the term fourth dimension for time. The events for A and B in Figure 2.21 are denoted by the respective coordinates (xA, ctA) and (xB, ctB), respectively. The line connecting events A and B is the path from A to B and is called a worldline. A spaceship launched from x ϭ 0, ct ϭ 0 with constant velocity v has the worldline shown in Figure 2.22: a straight line with slope c/v. For example, a light signal sent out from the origin with speed c is represented on a spacetime graph with a worldline that has a slope c/c ϭ 1, so that line makes an angle of 45° with both the x and ct axes. Any real motion in the spacetime diagram cannot have a slope of less than 1 (angle with the x axis Ͻ 45°), because that motion would have a speed greater than c. The Lorentz transformation does not allow such a speed.
Let us consider two events that occur at the same time (ct ϭ 0) but at different positions, x1 and x2. We denote the events (x, ct ) as (x1, 0) and (x2, 0), and we show them in Figure 2.23 in an inertial system with an origin fixed at x ϭ 0 and ct ϭ 0. How can we be certain that the two events happen simultaneously if

03721_ch02_019-083.indd 48

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2.9 Spacetime 49

they occur at different positions? We must devise a method that will allow us to
determine experimentally that the events occurred simultaneously. Let us place
clocks at positions x1 and x2 and place a flashbulb at position x3 halfway between x1 and x2. The two clocks have been previously synchronized and keep identical time. At time t ϭ 0, the flashbulb explodes and sends out light signals from position x3. The light signals proceed along their worldlines as shown in Figure 2.23. The two light signals arrive at positions x1 and x2 at identical times t as shown on the spacetime diagram. By using such techniques we can be sure that events oc-
cur simultaneously in our inertial reference system.
But what about other inertial reference systems? We realize that the two
events will not be simultaneous in a reference system KЈ moving at speed v with respect to our (x, ct) system. Because the two events have different spatial coordinates, x1 and x2, the Lorentz transformation will preclude them from occurring at the same time t œ simultaneously in the moving coordinate systems. We can see this by supposing that events 1, 2, and 3 take place on a spaceship moving
with velocity v. The worldlines for x1 and x2 are the two slanted lines beginning at x1 and x2 in Figure 2.24. However, when the flashbulb goes off, the light signals from x3 still proceed at 45° in the (x, ct ) reference system. The light signals intersect the worldlines from positions x1 and x2 at different times, so we do not see the events as being simultaneous in the moving system. Spacetime diagrams can
be useful in showing such phenomena.
Anything that happened earlier in time than t ϭ 0 is called the past and anything that occurs after t ϭ 0 is called the future. The spacetime diagram in Figure 2.25a shows both the past and the future. Notice that only the events
within the shaded area below t ϭ 0 can affect the present. Events outside this area cannot affect the present because of the limitation v Յ c; this region is called elsewhere. Similarly, the present cannot affect any events occurring outside the shaded area above t ϭ 0, again because of the limitation of the speed of light.

ct

ctB

B

Worldline

ctA A

x

0

xA xB

Figure 2.21 A spacetime diagram is used to specify events. The worldline denoting the path from event A to event B is shown.

ct Spaceship

v c
Light signal

0

x

Figure 2.22 A light signal has the slope of 45° on a spacetime diagram. A spaceship moving along the x axis with speed v is a straight line on the spacetime diagram with a slope c/v.

ct

ct

Light

Light ct

x x 1 x3 x 2

t

t

Figure 2.23 Clocks positioned at x1 and x2 can be synchronized by sending a light signal from a position x3 halfway between. The light signals intercept the worldlines of x1 and x2 at the same time t.

v

v

c

c

ct 2

ct 1

x x1 x3 x2

Figure 2.24 If the positions x1 (ϭ x 1œ ) and x2 (ϭ x 2œ ) of the previous
figure are on a moving system KЈ

when the flashbulb goes off, the times

will not appear simultaneously in sys-

tem

K

,

because

the

worldlines

for

x

œ 1

and

x

œ 2

are

slanted.

03721_ch02_019-083.indd 49

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50 Chapter 2 Special Theory of Relativity

ct

ct

Figure 2.25 (a) The spacetime diagram can be used to show the past, present, and future. Only causal events are placed inside the shaded area. Events outside the shaded area below t ϭ 0 cannot affect the present. (b) If we add an additional spatial coordinate y, a space cone can be drawn. The present cannot affect event A, but event B can.

Future

Elsewhere

Present x

Past

Elsewhere

Future
Elsewhere Past

Ay
B x
Present

(a)

(b)

Light cone Invariant quantities
Spacetime interval Lightlike

If we add another spatial coordinate y to our spacetime coordinates, we will have a cone as shown in Figure 2.25b, which we refer to as the light cone. All causal events related to the present (x ϭ 0, ct ϭ 0) must be within the light cone. In Figure 2.25b, anything occurring at present (x ϭ 0, ct ϭ 0) cannot possibly affect an event at position A; however, the event B can easily affect event A because A would be within the range of light signals emanating from B.
Invariant quantities have the same value in all inertial frames. They serve a special role in physics because their values do not change from one system to another. For example, the speed of light c is invariant. We are used to defining distances by d2 ϭ x2 ϩ y2 ϩ z2, and in Euclidean geometry, we obtain the same result for d2 in any inertial frame of reference. Is there a quantity, similar to d 2, that is also invariant in the special theory? If we refer to Equations (2.9), we have similar equations in both systems K and KЈ. Let us look more carefully at the quantity s2 defined as

s2 ϭ x 2 Ϫ 1ct 2 2

(2.25a)

and also

sœ2 ϭ x œ2 Ϫ 1ct œ 2 2

(2.25b)

If we use the Lorentz transformation for x and t, we find that s2 ϭ sЈ2, so s2 is an invariant quantity. This relationship can be extended to include the two other spatial coordinates, y and z, so that*

s 2 ϭ x 2 ϩ y2 ϩ z2 Ϫ 1ct 2 2

(2.26)

For simplicity, we will sometimes continue to use only the single spatial coordi-
nate x. If we consider two events, we can determine the quantity ⌬s2 where

¢s2 ϭ ¢x 2 Ϫ c2 ¢t2

(2.27)

between the two events, and we find that it is invariant in any inertial frame. The
quantity ⌬s is known as the spacetime interval between two events. There are three possibilities for the invariant quantity ⌬s2.

1. ⌬s2 ‫ ؍‬0: In this case ⌬x 2 ϭ c 2 ⌬t 2, and the two events can be connected only by a light signal. The events are said to have a lightlike separation.
2. ⌬s2 Ͼ 0: Here we must have ⌬x 2 Ͼ c 2 ⌬t 2, and no signal can travel fast enough to connect the two events. The events are not causally connected

* Some authors use the negative of the expression here in Equation (2.26).

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2.9 Spacetime 51

and are said to have a spacelike separation. In this case we can always find an inertial frame traveling at a velocity less than c in which the two events can occur simultaneously in time but at different places in space. 3. ⌬s2 Ͻ 0: Here we have ⌬x 2 Ͻ c 2 ⌬t 2, and the two events can be causally connected. The interval is said to be timelike. In this case we can find an inertial frame traveling at a velocity less than c in which the two events occur at the same position in space but at different times. The two events can never occur simultaneously.

Spacelike Timelike

EXAMPLE 2.7

Draw the spacetime diagram for the motion of the twins discussed in Section 2.8. Draw light signals being emitted from each twin at annual intervals and count the number of light signals received by each twin from the other.
Strategy We shall let Mary leave Earth at the origin (x, ct) ϭ (0, 0). She will return to Earth at x ϭ 0, but at a later time ct ϭ 20 ly. Her worldlines will be described by two lines of slope ϩc/v and Ϫc/v, whereas Frank’s worldline remains fixed at x ϭ 0. Frank’s and Mary’s signals have slopes of Ϯ1 on the spacetime diagram. We pay close attention to when the light signals sent out by Frank and Mary reach their twin’s worldlines.
Solution We show in Figure 2.26 (page 52) the spacetime diagram. The line representing Mary’s trip has a slope c/0.8c ϭ 1.25 on the outbound trip and Ϫ1.25 on the return

trip. During the trip to the star system, Mary does not receive the second annual light signal from Frank until she reaches the star system. This occurs because the light signal takes considerable time to catch up with Mary. However, during the return trip Mary receives Frank’s light signals at a rapid rate, receiving the last one (number 20) just as she returns. Because Mary’s clock is running slow, we see the light signals being sent less often on the spacetime diagram in the fixed system. Mary sends out her sixth annual light signal when she arrives at the star system. However, this signal does not reach Frank until the 18th year! During the remaining two years, however, Frank receives Mary’s signals at a rapid rate, finally receiving all 12 of them. Frank receives the last 6 signals during a time period of only 2 years.

A 3-vector R can be defined using Cartesian coordinates x, y, z in threedimensional Euclidean space. Another 3-vector RЈ can be determined in another Cartesian coordinate system using xЈ, yЈ, zЈ in the new system. So far in introductory physics we have discussed translations and rotations of axes between these two systems. We have learned that there are two geometries in Newtonian spacetime. One is the three-dimensional Euclidean geometry in which the space interval is d/2 ϭ dx 2 ϩ dy2 ϩ dz2, and the other is a one-dimensional time interval dt. Minkowski pointed out that both space and time by themselves will not suffice under a Lorentz transformation, and only a union of both will be independent and useful.
We can form a four-dimensional space or four-vector using the four components x, y, z, ict. The equivalent of Equation (2.27) becomes

ds2 ϭ dx 2 ϩ dy2 ϩ dz2 Ϫ c 2dt 2 dsЈ2 ϭ dxЈ2 ϩ dyЈ2 ϩ dzЈ2 Ϫ c 2dtЈ2
ds2 ϭ dsЈ2

(2.28)

We previously noted that ds2 (actually ⌬s2) can be positive, negative, or zero.
With the four-vector formalism we only have the spacetime geometry, not separate geometries for space and time. The spacetime distances ds2 ϭ dsЈ2 are invariant

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52 Chapter 2 Special Theory of Relativity

ct (ly)

20

20

10

Frank’s worldline

18
16 5
14

15 Mary’s worldline (return)
10 Arrival of Frank’s signals

12 4

5

10
3
Arrival 8 of Mary’s signals
62

Figure 2.26 The spacetime diagram for Mary’s trip to the star system and back. Notice that Frank’s worldline is a vertical line at x ϭ 0, and Mary’s two worldlines have the correct slope given by the magnitude c/v. The black dashed lines represent light signals sent at annual intervals from Mary to Frank. Frank’s annual signals to Mary are solid black. The solid dots denote the time when the light signals arrive.

4 1
2

0

2

2 Star system (turnaround)
Frank’s signals

1

Mary’s worldline

(outbound)

Mary’s signals

x (ly)

4

6

8

10

under the Lorentz transformation. In Section 2.12 we will learn how the energy and momentum of a particle are connected. Similar to the spacetime fourvector, there is an energy-momentum four-vector, and the invariant quantity is the mass.
The four-vector formalism gives us equations that produce form-invariant quantities under appropriate Lorentz transformations. It allows the mathematical construction of relativistic physics to be somewhat easier. However, the penalty is that we would have to stop and learn matrix algebra and perhaps even about tensors and, eventually, spinors. At this point in our study there is little to be gained in understanding about relativity. Another disadvantage in utilizing fourvectors at this point is that there is no general agreement among authors as to terminology. Sometimes ict is term 0 of the four-vector (ict, x, y, z with x, y, z being terms 1, 2, 3), and sometimes it is described as term 4 (x, y ,z ,ict). Sometimes the formalism is arranged such that the imaginary number i ϭ 2Ϫ1 doesn’t appear. We have chosen not to use four-vectors.
2.10 Doppler Effect
You may have already studied the Doppler effect of sound in introductory physics. It causes an increased frequency of sound as a source such as a train (with whistle blowing) approaches a receiver (our eardrum) and a decrease in fre-

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2.10 Doppler Effect 53

quency as the source recedes. A change in sound frequency also occurs when the source is fixed and the receiver is moving. The change in frequency of the sound wave depends on whether the source or receiver is moving. On first thought it seems that the Doppler effect in sound violates the principle of relativity, until we realize that there is in fact a special frame for sound waves. Sound waves depend on media such as air, water, or a steel plate to propagate. For light, however, there is no such medium. It is only relative motion of the source and receiver that is relevant, and we expect some differences between the relativistic Doppler effect for light waves and the normal Doppler effect for sound. It is not possible for a source of light to travel faster than light in a vacuum, but it is possible for a source of sound to travel faster than the speed of sound. Similarly, in a medium such as water in which light travels slower than c, a light source can travel faster than the speed of light.
Consider a source of light (for example, a star) and a receiver (an astronomer) approaching one another with a relative velocity v. First we consider the receiver fixed (Figure 2.27a) in system K and the light source in system KЈ moving toward the receiver with velocity v. The source emits n waves during the time interval T. Because the speed of light is always c and the source is moving with velocity v, the total distance between the front and rear of the wave train emitted during the time interval T is

Length of wave train ϭ cT Ϫ vT

Because there are n waves emitted during this time period, the wavelength must be

l

ϭ

cT

Ϫ n

vT

and the frequency, f ϭ c /l, is

f

ϭ

cT

cn Ϫ vT

(2.29)

System K′ v
c Star
Source

System K

System K′ v
c

Astronomer

(a)

(b)

Figure 2.27 (a) The source (star) is approaching the receiver (astronomer) with velocity v while it emits starlight signals with speed c. (b) Here the source and receiver are receding with velocity v. The Doppler effect for light is different than that for sound, because of relativity and no medium to carry the light waves.

System K

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Special Topic

Applications of the Doppler Effect

T he Doppler effect is not just a curious result of relativity. It has many practical applications, three of which are discussed here, and others are mentioned in various places in this text.

Astronomy
Perhaps the best-known application is in astronomy, where the Doppler shifts of known atomic transition frequencies determine the relative velocities of astronomical objects with respect to us. Such measurements continue to be used today to find the distances of such unusual objects as quasars (objects having incredibly large masses that produce tremendous amounts of radiation; see Chapter 16). The Doppler effect has been used to discover other effects in astronomy, for example, the rate of rotation of Venus and the fact that

Rotation

Reflected light redshifted
Incident light
Blueshifted

Figure A

Venus rotates in the opposite direction of Earth—the sun rises in the west on Venus. This was determined by observing light reflected from both sides of Venus—on one side it is blueshifted and on the other side it is redshifted, as shown in Figure A. The same technique has been used to determine the rate of rotation of stars.
Radar
The Doppler effect is nowhere more important than it is in radar. When an electromagnetic radar signal reflects off of a moving target, the so-called echo signal will be shifted in frequency by the Doppler effect. Very small frequency shifts can be determined by examining the beat frequency of the echo signal with a reference signal. The frequency shift is proportional to the radial component of the target’s velocity. Navigation radar is quite complex, and ingenious techniques have been devised to determine the target position and velocity using multiple radar beams. By using pulsed Doppler radar it is possible to separate moving targets from stationary targets, called clutter.
Doppler radar is also extensively used in meteorology. Vertical motion of airdrafts, sizes and motion of raindrops, motion of thunderstorms, and detailed patterns of wind distribution have all been studied with Doppler radar.
X rays and gamma rays emitted from moving atoms and nuclei have their frequencies shifted by the Doppler effect. Such phenomena tend to broaden radiation frequencies emitted by stationary atoms and nuclei and add to the natural spectral widths observed.

In its rest frame, the source emits n waves of frequency f0 during the proper time T0œ.

n

ϭ

f0T

œ 0

(2.30)

The proper time interval T0œ measured on the clock at rest in the moving system is related to the time interval T measured on a clock fixed by the receiver in system K by

T

œ 0

ϭ

T g

(2.31)

where g is the relativistic factor of Equation (2.16). The clock moving with the source measures the proper time because it is present with both the beginning and end of the wave.

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Oven Beam of atoms

Laser radiation

Figure B

Laser Cooling
In order to perform fundamental measurements in atomic physics, it is useful to limit the effects of thermal motion and to isolate single atoms. A method taking advantage of the Doppler effect can slow down even neutral atoms and eventually isolate them. Atoms emitted from a hot oven will have a spread of velocities. If these atoms form a beam as shown in Figure B, a laser beam impinging on the atoms from the right can slow them down by transferring momentum.
Atoms have characteristic energy levels that allow them to absorb and emit radiation of specific frequencies. Atoms moving with respect to the laser beam will “see” a shift in the laser frequency because of the Doppler effect. For example, atoms moving toward the laser beam will encounter light with high frequency, and atoms moving away from the laser beam will encounter light with low frequency. Even atoms moving in the same direction within the beam of atoms will see slightly different frequencies depending on the velocities of the various atoms. Now, if the frequency of the laser beam is tuned to the precise frequency seen by the faster atoms so that those atoms can be excited by absorbing the radia-

tion, then those faster atoms will be slowed down by absorbing the momentum of the laser radiation. The slower atoms will “see” a laser beam that has been Doppler shifted to a lower frequency than is needed to absorb the radiation, and these atoms are not as likely to absorb the laser radiation. The net effect is that the atoms as a whole are slowed down and their velocity spread is reduced.
As the atoms slow down, they see that the Dopplershifted frequencies of the laser change, and the atoms no longer absorb the laser radiation. They continue with the same lower velocity and velocity spread. The lower temperature limits reached by Doppler cooling depend on the atom, but typical values are on the order of hundreds of microkelvins. Doppler cooling is normally accompanied by intersecting laser beams at different angles; an “optical molasses” can be created in which atoms are essentially trapped. Further cooling is obtained by other techniques including “Sisyphus” and evaporative cooling, among others. In a remarkable series of experiments by various researchers, atoms have been cooled to temperatures approaching 10Ϫ10 K. The 1997 Nobel Prize in Physics was awarded to Steven Chu, Claude Cohen-Tannoudji, and William Phillips for these techniques. An important use of laser cooling is for atomic clocks. See http://www.nist.gov/physlab/ div847/grp50/primary-frequency-standards.cfm for a good discussion. See also Steven Chu, “Laser Trapping of Neutral Particles,” Scientific American 266, 70 (February 1992). In Chapter 9 we will discuss how laser cooling is used to produce an ultracold state of matter known as a Bose-Einstein condensate.

We

substitute

the

proper

time

T

œ 0

from

Equation

(2.31)

into

Equation

(2.30)

to determine the number of waves n. Then n is substituted into Equation (2.29)

to determine the frequency.

f

ϭ

c f0T /g cT Ϫ vT

ϭ

1

1 Ϫ v/c

f0 g

ϭ

21 Ϫ v 2 /c 2 1 Ϫ v/c

f0

where we have inserted the equation for g. If we use b ϭ v/c, we can write the previous equation as

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56 Chapter 2 Special Theory of Relativity

f

ϭ

21 21

ϩ Ϫ

b b

f0

Source and receiver approaching

(2.32)

It is straightforward to show that Equation (2.32) is also valid when the source is fixed and the receiver approaches it with velocity v. It is the relative velocity v, of course, that is important (Problem 49).
But what happens if the source and receiver are receding from each other with velocity v (see Figure 2.27b)? The derivation is similar to the one just done, except that the distance between the beginning and end of the wave train becomes
Length of wave train ϭ cT ϩ vT
because the source and receiver are receding rather than approaching. This change in sign is propagated throughout the derivation (Problem 50), with the final result

21 Ϫ b f ϭ 21 ϩ b f0

Source and receiver receding

(2.33)

Equations (2.32) and (2.33) can be combined into one equation if we agree to use a ϩ sign for b (ϩv/c) when the source and receiver are approaching each other and a Ϫ sign for b (Ϫv/c) when they are receding. The final equation becomes

21 ϩ b f ϭ 21 Ϫ b f0

Relativistic Doppler effect

(2.34)

Redshifts

The Doppler effect is useful in many areas of science including astronomy, atomic physics, and nuclear physics. One of its many applications includes an effective radar system for locating airplane position and speed (see Special Topic, “Applications of the Doppler Effect”).
Elements absorb and emit characteristic frequencies of light due to the existence of particular atomic levels. We will learn more about this later. Scientists have observed these characteristic frequencies in starlight and have observed shifts in the frequencies. One reason for these shifts is the Doppler effect, and the frequency changes are used to determine the speed of the emitting object with respect to us. This is the source of the redshifts of starlight caused by objects moving away from us. These data have been used to ascertain that the universe is expanding. The farther away the star, the higher the redshift. This observation is what led Harlow Shapley and Edwin Hubble to the idea that the universe started with a Big Bang.*
So far in this section we have only considered the source and receiver to be directly approaching or receding. Of course, it is also possible for the two to be moving at an angle with respect to one another, as shown in Figure 2.28. We omit the derivation here† but present the results. The angles u and uЈ are the angles the light signals make with the x axes in the K and KЈ systems. They are related by

* Excellent references are “The Cosmic Distance Scale” by Paul Hodge, American Scientist 72, 474 (1984), and “Origins” by S. Weinberg, Science 230, 15 (1985). This subject is discussed in Chapter 16.
†See Robert Resnick, Introduction to Special Relativity, New York: Wiley (1968).

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y System K

y′

System K′

v x

2.10 Doppler Effect 57
Figure 2.28 The light signals in system KЈ are emitted at an angle uЈ from the x œ axis and remain in the x œ yœ plane.

u′ z x′

z′
f cos u ϭ f01cos uœ ϩ b 2 21 Ϫ b2

(2.35)

and

f sin u ϭ f0 sin uœ

(2.36)

The generalized Doppler shift equation becomes

f

ϭ

1

ϩ b cos uœ 21 Ϫ b2

f0

(2.37)

Note that Equation (2.37) gives Equation (2.32) when uЈ ϭ 0 (source and receiver approaching) and gives Equation (2.33) when uЈ ϭ 180° (source and receiver receding). This situation is known as the longitudinal Doppler effect.
When u ϭ 90° the emission is purely transverse to the direction of motion, and we have the transverse Doppler effect, which is purely a relativistic effect that does not occur classically. The transverse Doppler effect is directly due to time dilation and has been verified experimentally. Equations (2.35) through (2.37) can also be used to understand stellar aberration.

EXAMPLE 2.8

In Section 2.8 we discussed what happened when Mary traveled on a spaceship away from her twin brother Frank, who remained on Earth. Analyze the light signals sent out by Frank and Mary by using the relativistic Doppler effect.
Strategy We will use Equation (2.34) for both the outbound and return trip to analyze the frequency of the light signals sent and received. During the outbound trip the source (Frank) and receiver (Mary) are receding so that b ϭ Ϫ0.8. For the return trip, we have b ϭ ϩ0.8. The frequency f0 will be the signals that Frank sends; the frequency f will be those that Mary receives.

Solution First, we analyze the frequency of the light signals that Mary receives from Frank. Equation (2.34) gives

f

ϭ

21 21

ϩ Ϫ

1 Ϫ0.8 2 1 Ϫ0.8 2

f0

ϭ

f0 3

Because Frank sends out signals annually, Mary will receive the signals only every 3 years. Therefore during the 6-year trip in Mary’s system to the star system, she will receive only 2 signals.
During the return trip, b ϭ 0.8 and Equation (2.34) gives

f

ϭ

21 21

ϩ Ϫ

0.8 0.8

f0

ϭ

3f0

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58 Chapter 2 Special Theory of Relativity

so that Mary receives 3 signals each year for a total of 18 signals during the return trip. Mary receives a total of 20 annual light signals from Frank, and she concludes that Frank has aged 20 years during her trip.
Now let’s analyze the light signals that Mary sends Frank. During the outbound trip the frequency at which Frank receives signals from Mary will also be f0 /3. During the 10 years that it takes Mary to reach the star system on his clock, he will receive 10/3 signals—3 signals plus 1/3 of the time to the next one. Frank continues to receive Mary’s signals at the rate f0 /3 for another 8 years, because that is how long it takes the sixth signal she sent him to reach Earth. Therefore, for the first 18 years of her journey, according to his own clock he receives 18/3 ϭ 6 signals. Frank has no way

of knowing that Mary has turned around and is coming back until he starts receiving signals at frequency 3f0. During Mary’s return trip Frank will receive signals at the frequency 3f0 or 3 per year. However, in his system, Mary returns 2 years after he has received her sixth signal and turned around to come back. During this 2-year period he will receive 6 more signals, so he concludes she has aged a total of only 12 years.
Notice that this analysis is in total agreement with the spacetime diagram of Figure 2.26 and is somewhat easier to obtain. Although geometrical constructions like spacetime diagrams are sometimes useful, an analytical calculation is usually easier.

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2.11 Relativistic Momentum

Newton’s second law, F ϭ dp /dt, keeps its same form under a Galilean transformation, but we might not expect it to do so under a Lorentz transformation. There may be similar transformation difficulties with the conservation laws of linear momentum and energy. We need to take a careful look at our previous definition of linear momentum to see whether it is still valid at high speeds. According to Newton’s second law, for example, an acceleration of a particle already moving at very high speeds could lead to a speed greater than the speed of light. That would be in conflict with the Lorentz transformation, so we expect that Newton’s second law might somehow be modified at high speeds.
Because physicists believe the conservation of linear momentum is fundamental, we begin by considering a collision that has no external forces. Frank (Fixed or stationary system) is at rest in system K holding a ball of mass m. Mary (Moving system) holds a similar ball in system KЈ that is moving in the x direction with velocity v with respect to system K as shown in Figure 2.29a. Frank throws his ball along his y axis, and Mary throws her ball with exactly the same speed along her negative yœ axis. The two balls collide in a perfectly elastic collision, and each of them catches their own ball as it rebounds. Each twin measures the speed of his or her own ball to be u0 both before and after the collision.
We show the collision according to both observers in Figure 2.29. Consider the conservation of momentum according to Frank as seen in system K. The velocity of the ball thrown by Frank has components in his own system K of

uFx ϭ 0 uFy ϭ u0

(2.38)

If we use the definition of momentum, p ϭ mv , the momentum of the ball thrown by Frank is entirely in the y direction:

pFy ϭ mu0

(2.39)

Because the collision is perfectly elastic, the ball returns to Frank with speed u0 along the Ϫy axis. The change of momentum of his ball as observed by Frank in
system K is

9/29/11 9:28 AM

y y′
M

y′ v

K′

x ′ K′

F

M

x′

System K

y

according

to Frank

2.11 Relativistic Momentum 59

y′

M

M

System K′ according
to Mary

x′

y

v

F

F

x

x

x

K

K

K

(a)

(b)

Figure 2.29 Frank is in the fixed K system, and Mary is in the moving KЈ system. Frank throws his ball along his ϩy axis, and Mary throws her ball along her Ϫyœ axis. The balls collide. The event is shown in Frank’s system in (a) and in Mary’s system in (b). (Because it is awkward to show the twins as they catch the ball, we have drawn them faintly and in a reversed position.)

¢pF ϭ ¢pFy ϭ Ϫ2mu0

(2.40)

In order to confirm the conservation of linear momentum, we need to determine the change in the momentum of Mary’s ball as measured by Frank. We will let the primed speeds be measured by Mary and the unprimed speeds be measured by Frank (except that u0 is always the speed of the ball as measured by the twin in his or her own system). Mary measures the initial velocity of her own ball to be uMœ x ϭ 0 and uMœ y ϭ Ϫu0, because she throws it along her own Ϫyœ axis. To determine the velocity of Mary’s ball as measured by Frank, we need to use the velocity transformation equations of Equation (2.23). If we insert the appropriate values for the speeds just discussed, we obtain

uMx ϭ v uMy ϭ Ϫu0 21 Ϫ v 2 /c 2

(2.41)

Before the collision, the momentum of Mary’s ball as measured by Frank becomes

Before pMx ϭ mv Before pMy ϭ Ϫmu0 21 Ϫ v 2 /c 2 For a perfectly elastic collision, the momentum after the collision is

(2.42)

After pMx ϭ mv After pMy ϭ ϩmu0 21 Ϫ v 2 /c 2

(2.43)

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60 Chapter 2 Special Theory of Relativity

The change in momentum of Mary’s ball according to Frank is

Difficulty with classical linear momentum

¢pM ϭ ¢pMy ϭ 2mu0 21 Ϫ v 2 /c 2

(2.44)

The conservation of linear momentum requires the total change in momentum of the collision, ⌬pF ϩ ⌬pM, to be zero. The addition of Equations (2.40) and (2.44) clearly does not give zero. Linear momentum is not conserved if we use the
conventions for momentum from classical physics even if we use the velocity transformation equations from the special theory of relativity. There is no problem with the x direction, but there is a problem with the y direction along the direction the ball is thrown in each system.
Rather than abandon the conservation of linear momentum, let us look for a modification of the definition of linear momentum that preserves both it and Newton’s second law. We follow a procedure similar to the one we used in deriving the Lorentz transformation; we assume the simplest, most reasonable change that may preserve the conservation of momentum. We assume that the classical form of momentum mu is multiplied by a factor that may depend on velocity. Let the factor be ⌫(u). Our trial definition for linear momentum now becomes

p ϭ ⌫ 1u2 mu

(2.45)

5 Relativistic
4 momentum

3

2

Classical momentum

1

0 0 0.2 0.4 0.6 0.8 1.0 1.2 v/c

Figure 2.30 The linear momentum of a particle of mass m is plotted versus its velocity (v/c) for both the classical and relativistic momentum results. As v S c the relativistic momentum becomes quite large, but the classical momentum continues its linear rise. The relativistic result is the correct one.
Relativistic momentum

In Example 2.9 we show that momentum is conserved in the collision just described for the value of ⌫(u) given by

⌫1u2 ϭ

1

21 Ϫ u2 /c 2

(2.46)

Notice that the form of Equation (2.46) is the same as that found earlier for the Lorentz transformation. We even give ⌫(u) the same symbol: ⌫(u) ϭ g. However, this g is different; it contains the speed of the particle u, whereas the Lorentz transformation contains the relative speed v between the two inertial reference frames. This distinction should be kept in mind because it can cause confusion. Because the usage is so common among physicists, we will use g for both purposes. However, when there is any chance of confusion, we will write out
1 / 21 Ϫ u2 /c 2 and use g ϭ 1 / 21 Ϫ v 2 /c 2 for the Lorentz transformation.
We will write out 1 / 21 Ϫ u2 /c 2 often to avoid confusion. We can make a plausible determination for the correct form of the momen-
tum if we use the proper time discussed previously to determine the velocity. The momentum becomes

p

ϭ

m

dr dt

ϭ

m

dr dt

dt dt

(2.47)

We retain the velocity u ϭ dr /dt as used classically, where r is the position vec-

tor. All observers do not agree as to the value of dr /dt, but they do agree as to

the value of dr /dt, where dt is the proper time measured in the moving system

KЈ. The value of dt /dt 1ϭ g 2 is obtained from Equation (2.31), where the speed

u is used in the relation for g to represent the relative speed of the moving

(Mary’s) frame and the fixed (Frank’s) frame.

The definition of the relativistic momentum becomes, from Equation

(2.47),

p

ϭ

m

dr dt

g

p ϭ gmu Relativistic momentum

(2.48)

Linear momentum (mc)

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2.11 Relativistic Momentum 61

where

1 gϭ
21 Ϫ u2 /c 2

(2.49)

This result for the relativistic momentum reduces to the classical result for small values of u/c. The classical momentum expression is good to an accuracy of 1% as long as u Ͻ 0.14c. We show both the relativistic and classical momentum in Figure 2.30.
Some physicists like to refer to the mass in Equation (2.48) as the rest mass m0 and call the term m ϭ gm0 the relativistic mass. In this manner the classical form of momentum, mu, is retained. The mass is then imagined to increase at high speeds. Most physicists prefer to keep the concept of mass as an invariant, intrinsic property of an object. We adopt this latter approach and will use the term mass exclusively to mean rest mass. Although we may use the terms mass and rest mass synonymously, we will not use the term relativistic mass. The use of relativistic mass too often leads the student into mistakenly inserting the term into classical expressions where it does not apply.

Rest and relativistic mass

EXAMPLE 2.9

Show that linear momentum is conserved for the collision just discussed and shown in Figure 2.29.

Strategy We use the relativistic momentum to modify the expressions obtained for the momentum of the balls thrown by Frank and Mary. We will then check to see whether momentum is conserved according to Frank. We leave to Problem 62 the question of whether momentum is conserved according to Mary’s system.

Solution From Equation (2.39), the momentum of the ball thrown by Frank becomes

pF y

ϭ

gmu0

ϭ

21

mu0 Ϫ u02 /c 2

For an elastic collision, the magnitude of the momentum

for this ball is the same before and after the collision. After

the collision, the momentum will be the negative of this

value, so the change in momentum becomes, from Equa-

tion (2.40),

¢pF

ϭ

¢pFy

ϭ

Ϫ2gmu0

ϭ

Ϫ

2mu0 21 Ϫ u02 /c 2

(2.50)

Now we consider the momentum of Mary’s ball as mea-

sured by Frank. Even with the addition of the g factor for

the momentum in the x direction, we still have ⌬pMx ϭ 0. We must look more carefully at ⌬pMy. First, we find the speed of the ball thrown by Mary as measured by Frank. We use

Equations (2.41) to determine

uM ϭ 2uM2 x ϩ u2My ϭ 2v 2 ϩ u02 11 Ϫ v 2 /c 2 2

(2.51)

The relativistic factor g for the momentum for this situation is
1 gϭ
21 Ϫ uM2 /c 2
The value of pMy is now found by modifying Equation (2.42) with this value of g.

pMy

ϭ

Ϫgmu0 21

Ϫ

v 2 /c 2

ϭ

Ϫmu0 21 Ϫ v 2 /c 2 21 Ϫ uM2 /c 2

We insert the value of uM from Equation (2.51) into this equation to give

pMy

ϭ

Ϫmu0 21 Ϫ v 2 /c 2 2 11 Ϫ u02 /c 2 2 11 Ϫ v 2 /c 2 2

ϭ

Ϫmu0 21 Ϫ u02 /c 2

(2.52)

The momentum after the collision will still be the negative of this value, so the change in momentum becomes

¢pM

ϭ

¢pMy

ϭ

2mu0 21 Ϫ u02 /c 2

(2.53)

The change in the momentum of the two balls as measured by Frank is given by the sum of Equations (2.50) and (2.53):

¢p ϭ ¢pF ϩ ¢pM ϭ 0

Thus Frank indeed finds that momentum is conserved. Mary should also determine that linear momentum is conserved (see Problem 62).

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62 Chapter 2 Special Theory of Relativity

2.12 Relativistic Energy

Relativistic force

We now turn to the concepts of energy and force. When forming the new theories of relativity and quantum physics, physicists resisted changing the wellaccepted ideas of classical physics unless absolutely necessary. In this same spirit we also choose to keep intact as many definitions from classical physics as possible and let experiment dictate when we are incorrect. In practice, the concept of force is best defined by its use in Newton’s laws of motion, and we retain here the classical definition of force as used in Newton’s second law. In the previous section we studied the concept of momentum and found a relativistic expression in Equation (2.48). Therefore, we modify Newton’s second law to include our new definition of linear momentum, and force becomes

F

ϭ

dp dt

ϭ

d dt

1 gmu 2

ϭ

d dt

a

21

mu Ϫ u2 /c 2

b

(2.54)

Aspects of this force will be examined in the problems (see Problems 55–58).

Introductory physics presents kinetic energy as the work done on a particle

by a net force. We retain here the same definitions of kinetic energy and work.

The work W12 done by a force F to move a particle from position 1 to position 2

along a path s is defined to be

Ύ2
W12 ϭ F ؒ ds ϭ K2 Ϫ K1

(2.55)

1

where K1 is defined to be the kinetic energy of the particle at position 1.

For simplicity, let the particle start from rest under the influence of the force

F and calculate the final kinetic energy K after the work is done. The force is

related to the dynamic quantities by Equation (2.54). The work W and kinetic

energy K are

Ύ W ϭ K ϭ

d dt

1 gmu 2

#

u

dt

(2.56)

where the integral is performed over the differential path ds ϭ u dt. Because the

mass is invariant, it can be brought outside the integral. The relativistic factor g

depends on u and cannot be brought outside the integral. Equation (2.56)

becomes

Ύ Ύ K ϭ m

dt

d dt

1 gu 2

#

u

ϭ

m

u d 1gu2

The limits of integration are from an initial value of 0 to a final value of gu.

Ύ gu
K ϭ m u d 1gu2

(2.57)

0

The integral in Equation (2.57) is straightforward if done by the method of in-

tegration by parts. The result, called the relativistic kinetic energy, is

Relativistic kinetic energy

K ϭ gmc 2 Ϫ mc 2 ϭ mc 2 a

1

Ϫ 1 b ϭ mc 2 1g Ϫ 1 2

21 Ϫ u2 /c 2

(2.58)

Equation (2.58) does not seem to resemble the classical result for kinetic

energy,

K

ϭ

1 2

mu2.

However,

if

it

is

correct,

we

expect

it

to

reduce

to

the

classical

result for low speeds. Let’s see whether it does. For speeds u V c, we expand g

in a binomial series as follows:

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2.12 Relativistic Energy 63

K

ϭ

mc 2

a1

Ϫ

u2 c2

Ϫ1/2
b

Ϫ

mc 2

ϭ

mc 2 a 1

ϩ

1 2

u2 c2

ϩ

pb

Ϫ

mc 2

where we have neglected all terms of power (u/c)4 and greater, because u V c. This gives the following equation for the relativistic kinetic energy at low speeds:

K

ϭ

mc 2

ϩ

1 2

mu2

Ϫ

mc 2

ϭ

1 2

mu2

(2.59)

which is the expected classical result. We show both the relativistic and classical ki-

netic energies in Figure 2.31. They diverge considerably above a velocity of 0.6c.

A common mistake students make when first studying relativity is to use ei-

ther

1 2

mu2

or

1 2

gmu2

for

the

relativistic

kinetic

energy.

It

is

important

to

use

only

Equation (2.58) for the relativistic kinetic energy. Although Equation (2.58) looks

much different from the classical result, it is the only correct one, and neither

1 2

mu2

nor

1 2

gmu2

is

a

correct

relativistic

result.

Equation (2.58) is particularly useful when dealing with particles accelerated to

high speeds. For example, the fastest speeds produced in the United States have been

in the 3-kilometer-long electron accelerator at the Stanford Linear Accelerator

Laboratory. This accelerator produces electrons with a kinetic energy of 8 ϫ 10Ϫ9 J

(50 GeV) or 50 ϫ 109 eV. The electrons have speeds so close to the speed of light that

the tiny difference from c is difficult to measure directly. The speed of the electrons

is inferred from the relativistic kinetic energy of Equation (2.58) and is given by

0.99999999995c. Such calculations are difficult to do with calculators because of

significant-figure limitations. As a result, we use kinetic energy or momentum to ex-

press the motion of a particle moving near the speed of light and rarely use its speed.

Kinetic energy/mc2

4.0

3.5

3.0

2.5

Relativistic K ϭ mc2(g Ϫ 1)

2.0

1.5 1.0

Classical

K

ϭ

1 2

mv2

0.5

0

0

0.4 0.8 1.2

v/c

Figure 2.31 The kinetic energy as a fraction of rest energy (K/mc 2) of a particle of mass m is shown versus its velocity (v/c) for both the classical and relativistic calculations. Only the relativistic result is correct. Like the momentum, the kinetic energy rises rapidly as v S c.

CONCEPTUAL EXAMPLE 2.10

Determine whether an object with mass can ever have the speed of light.
Solution If we examine Equation (2.58), we see that when u S c, the kinetic energy K S q. Because there is not an infinite amount of energy available, we agree that no object

with mass can have the speed of light. The classical and relativistic speeds for electrons are shown in Figure 2.32 as a function of their kinetic energy. Physicists have found that experimentally it does not matter how much energy we give an object having mass. Its speed can never quite reach c.

2 Classical

Electrons

v c

1

Relativistic

Figure 2.32 The velocity (v/c) of electrons is shown versus kinetic energy for both classi-

0 0

cal (incorrect) and relativistic calculations. The experimentally measured data points agree

5

10

15 17 with the relativistic results. Adapted with permission from American Journal of Physics 32, 551

Kinetic energy (MeV)

(1964), W. Bertozzi. © 1964 American Association of Physics Teachers.

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64 Chapter 2 Special Theory of Relativity

EXAMPLE 2.11

Electrons used to produce medical x rays are accelerated from rest through a potential difference of 25,000 volts before striking a metal target. Calculate the speed of the electrons and determine the error in using the classical kinetic energy result.

Strategy We calculate the speed from the kinetic energy, which we determine both classically and relativistically and then compare the results. In order to determine the correct speed of the electrons, we must use the relativistically correct kinetic energy given by Equation (2.58). The work done to accelerate an electron across a potential difference V is given by qV, where q is the charge of the particle. The work done to accelerate the electron from rest is the final kinetic energy K of the electron.

Solution The kinetic energy is given by K ϭ W ϭ qV ϭ 11.6 ϫ 10Ϫ19 C 2 125 ϫ 103 V 2 ϭ 4.0 ϫ 10Ϫ15 J
We first determine g from Equation (2.58) and from that, the speed. We have

K ϭ 1g Ϫ 12mc 2

(2.60)

From this equation, g is found to be

g

ϭ

1

ϩ

K mc 2

(2.61)

The quantity mc 2 for the electron is determined to be

mc 2 1electron 2 ϭ 19.11 ϫ 10Ϫ31 kg 2 13.00 ϫ 108 m/s 2 2

ϭ 8.19 ϫ 10Ϫ14 J

The relativistic factor is then g ϭ 1 ϩ[(4.0 ϫ 10Ϫ15 J)/(8.19 ϫ 10Ϫ14 J)] ϭ 1.049. Equation (2.8) can be rearranged to determine b2 as a function of g2, where b ϭ u/c.

b2

ϭ

g2 Ϫ g2

1

ϭ

11.049 2 2 Ϫ 11.049 2 2

1

ϭ

0.091

(2.62)

The value of b is 0.30, and the correct speed, u ϭ bc, is

0.90 ϫ 108 m/s.

We determine the error in using the classical result by

calculating the velocity using the nonrelativistic expression.

The

nonrelativistic

expression

is

K

ϭ

1 2

mu2,

and

the

speed

is

given by

214.0 ϫ 10Ϫ15 J 2 u ϭ B 9.11 ϫ 10Ϫ31 kg
ϭ 0.94 ϫ 108 m/s 1nonrelativistic 2

The (incorrect) classical speed is about 4% greater than the (correct) relativistic speed. Such an error is significant enough to be important in designing electronic equipment and in making test measurements. Relativistic calculations are particularly important for electrons, because they have such a small mass and are easily accelerated to speeds very close to c.

Total Energy and Rest Energy

We rewrite Equation (2.58) in the form

gmc 2 ϭ

mc 2

ϭ K ϩ mc 2

21 Ϫ u2 /c 2

The term mc 2 is called the rest energy and is denoted by E0.

(2.63)

Rest energy

E0 ϭ mc 2

(2.64)

This leaves the sum of the kinetic energy and rest energy to be interpreted as the total energy of the particle. The total energy is denoted by E and is given by

Total energy

E

ϭ

gmc 2

ϭ

21

mc 2 Ϫ u2 /c 2

ϭ

21

E0 Ϫ u2 /c 2

ϭ

K

ϩ

E0

(2.65)

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2.12 Relativistic Energy 65

Equivalence of Mass and Energy

These last few equations suggest the equivalence of mass and energy, a concept attributed to Einstein. The result that energy ϭ mc 2 is one of the most famous equations in physics. Even when a particle has no velocity, and thus no kinetic energy, we still believe that the particle has energy through its mass, E0 ϭ mc 2. Nuclear reactions are certain proof that mass and energy are equivalent. The concept of motion as being described by kinetic energy is preserved in relativistic dynamics, but a particle with no motion still has energy through its mass.
In order to establish the equivalence of mass and energy, we must modify two of the conservation laws that we learned in classical physics. Mass and energy are no longer two separately conserved quantities. We must combine them into one law of the conservation of mass-energy. We will see ample proof during the remainder of this book of the validity of this basic conservation law.
Even though we often say “energy is turned into mass” or “mass is converted into energy” or “mass and energy are interchangeable,” what we mean is that mass and energy are equivalent; this is important to understand. Mass is another form of energy, and we use the terms mass-energy and energy interchangeably. This is not the first time we have had to change our understanding of energy. In the late eighteenth century it became clear that heat was another form of energy, and the nineteenth-century experiments of James Joule showed that heat loss or gain was related to work.
Consider two blocks of wood, each of mass m and having kinetic energy K, moving toward each other as shown in Figure 2.33. A spring placed between them is compressed and locks in place as they collide. Let’s examine the conservation of mass-energy. The energy before the collision is

Mass-energy before: E ϭ 2mc 2 ϩ 2K

(2.66)

and the energy after the collision is

Mass-energy after: E ϭ Mc 2

(2.67)

where M is the (rest) mass of the system. Because energy is conserved, we have E ϭ 2mc 2 ϩ 2K ϭ Mc 2, and the new mass M is greater than the individual masses 2m.
The kinetic energy went into compressing the spring, so the spring has increased

Conservation of mass-energy

v

v

(a)

(b)

Figure 2.33 (a) Two blocks of wood, one with a spring attached and both having mass m, move with equal speeds v and kinetic energies K toward a head-on collision. (b) The two blocks collide, compressing the spring, which locks in place. The system now has increased mass, M ϭ 2m ϩ 2K/c 2, with the kinetic energy being converted into the potential energy of the spring.

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66 Chapter 2 Special Theory of Relativity

potential energy. Kinetic energy has been converted into mass, the result being that the potential energy of the spring has caused the system to have more mass. We find the difference in mass ⌬M by setting the previous two equations for energy equal and solving for ⌬M ϭ M Ϫ 2m.

¢M

ϭ

M

Ϫ

2m

ϭ

2K c2

(2.68)

Linear momentum is conserved in this head-on collision.
The fractional mass increase in this case is quite small and is given by fr ϭ ⌬M/2m. If we use Equation (2.68), we have

fr

ϭ

M

Ϫ 2m 2m

ϭ

2K /c 2 2m

ϭ

K mc 2

(2.69)

For typical masses and kinetic energies of blocks of wood, this fractional increase in mass is too small to measure. For example, if we have blocks of wood of mass 0.1 kg moving at 10 m/s, Equation (2.69) gives

fr

ϭ

1 2

mv

2

mc 2

ϭ

1 2

v2 c2

ϭ

1 2

13

110 m/s 2 2 ϫ 108 m/s 2 2

ϭ

6

ϫ

10Ϫ16

where we have used the nonrelativistic expression for kinetic energy because the speed is so low. This very small numerical result indicates that questions of mass increase are inappropriate for macroscopic objects such as blocks of wood and automobiles crashing into one another. Such small increases cannot now be measured, but in the next section, we will look at the collision of two high-energy protons, in which considerable energy is available to create additional mass. Mass-energy relations are essential in such reactions.

Relationship of Energy and Momentum

Physicists believe that linear momentum is a more fundamental concept than kinetic energy. There is no conservation of kinetic energy, whereas the conservation of linear momentum in isolated systems is inviolate as far as we know. A more fundamental result for the total energy in Equation (2.65) might include momentum rather than kinetic energy. Let’s proceed to find a useful result. We begin with Equation (2.48) for the relativistic momentum written in magnitude form only.

p ϭ gmu ϭ

mu

21 Ϫ u2 /c 2

We square this result, multiply by c 2, and rearrange the result.

p2c 2 ϭ g2m 2u2c 2

ϭ

g2m

2c

4

a

u2 c2

b

ϭ

g2m 2c 4b2

We use Equation (2.62) for b2 and find

p2c 2

ϭ

g2m 2c 4

a

1

Ϫ

1 g2

b

ϭ g2m 2c 4 Ϫ m 2c 4

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2.12 Relativistic Energy 67

The first term on the right-hand side is just E 2, and the second term is E 02. The last equation becomes

p2c 2

ϭ

E 2

Ϫ

E

2 0

We rearrange this last equation to find the result we are seeking, a relation between energy and momentum.

E 2 ϭ p 2c 2 ϩ E02

(2.70) Momentum-energy relation

or

E 2 ϭ p2c 2 ϩ m2c4

(2.71)

Equation (2.70) is a useful result to relate the total energy of a particle with its momentum. The quantities (E 2 Ϫ p2c 2) and m are invariant quantities. Note that when a particle’s velocity is zero and it has no momentum, Equation (2.70) correctly gives E0 as the particle’s total energy.

Massless Particles

Equation (2.70) can also be used to determine the total energy for particles having zero mass. For example, Equation (2.70) predicts that the total energy of a photon is

E ϭ pc Photon

(2.72)

The energy of a photon is completely due to its motion. It has no rest energy, because it has no mass.
We can show that the previous relativistic equations correctly predict that the speed of a photon must be the speed of light c. We use Equations (2.65) and (2.72) for the total energy of a photon and set the two equations equal.

E ϭ gmc 2 ϭ pc

If we insert the value of the relativistic momentum from Equation (2.48), we have

gmc 2 ϭ gmuc

The fact that u ϭ c follows directly from this equation after careful consideration of letting m S 0 and realizing that g S q.

u ϭ c Massless particle

(2.73)

Massless particles must travel at the speed of light

CONCEPTUAL EXAMPLE 2.12

Tachyons are postulated particles that travel faster than the speed of light. (The word tachyon is derived from the Greek word tachus, which means “speedy.”) They were first seriously proposed and investigated in the 1960s. Use what we have learned thus far in this chapter and discuss several properties that tachyons might have.

Solution Let’s first examine Equation (2.65) for energy:

E ϭ gmc 2 ϭ

mc 2

21 Ϫ u2 /c 2

(2.65)

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68 Chapter 2 Special Theory of Relativity

Because u Ͼ c, the energy must be imaginary if the mass is real, or conversely, if we insist that energy be real, we must have an imaginary mass! For purposes of discussion, we will henceforth assume that energy is real and tachyon mass is imaginary. Remember that ordinary matter must always travel at speed less than c, light must travel at the speed of light, and tachyons must always have speed greater than c. In order to slow down a tachyon, we must give it more energy, according to Equation (2.65). Note that the energy must become infinite if we want to slow down a tachyon to speed c. If the tachyon’s energy is reduced, it speeds up!
Because tachyons travel faster than c, we have a problem with causality. Consider a tachyon leaving Earth at time t ϭ 0 that arrives at a distant galaxy at time T. A spaceship

traveling at speed less than c from Earth to the galaxy could conceivably find that the tachyon arrived at the galaxy before it left Earth!
It has been proposed that tachyons might be created in high-energy particle collisions or in cosmic rays. No confirming evidence has been found. Tachyons, if charged, could also be detected from Cerenkov radiation. When we refer to speed c, we always mean in a vacuum. When traveling in a medium, the speed must be less than c. When particles have speed greater than light travels in a medium, characteristic electromagnetic radiation is emitted. The effect of the blue glow in swimming pool nuclear reactors is due to this Cerenkov radiation.

Use eV for energy

2.13 Computations in Modern Physics

We were taught in introductory physics that the international system of units is preferable when doing calculations in science and engineering. This is generally true, but in modern physics we sometimes use other units that are more convenient for atomic and subatomic scales. In this section we introduce some of those units and demonstrate their practicality through several examples. Recall that the work done in accelerating a charge through a potential difference is given by W ϭ qV. For a proton, with charge e ϭ 1.602 ϫ 10Ϫ19 C, accelerated across a potential difference of 1 V, the work done is

W ϭ 11.602 ϫ 10Ϫ19 2 11 V 2 ϭ 1.602 ϫ 10Ϫ19 J

In modern physics calculations, the amount of charge being considered is almost always some multiple of the electron charge. Atoms and nuclei all have an exact multiple of the electron charge (or neutral). For example, some charges are proton (ϩe), electron (Ϫe), neutron (0), pion (0, Ϯe), and a singly ionized carbon atom (ϩe). The work done to accelerate the proton across a potential difference of 1 V could also be written as

W ϭ 11 e2 11 V2 ϭ 1 eV

where e stands for the electron charge. Thus eV, pronounced “electron volt,” is also a unit of energy. It is related to the SI (Système International) unit joule by the two previous equations.

1 eV ϭ 1.602 ϫ 10Ϫ19 J

(2.74)

The eV unit is used more often in modern physics than the SI unit J. The term eV is often used with the SI prefixes where applicable. For example, in atomic and solid state physics, eV itself is mostly used, whereas in nuclear physics MeV (106 eV, mega-electron-volt) and GeV (109 eV, giga-electron-volt) are predominant, and in particle physics GeV and TeV (1012 eV, tera-electron-volt) are used. When we speak of a particle having a certain amount of energy, the common usage is to refer to the kinetic energy. A 6-GeV proton has a kinetic energy of 6 GeV, not a total energy of 6 GeV. Because the rest energy of a proton is about 1 GeV, this proton would have a total energy of about 7 GeV.

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2.13 Computations in Modern Physics 69

Like the SI unit for energy, the SI unit for mass, kilogram, is a very large unit
of mass in modern physics calculations. For example, the mass of a proton is only 1.6726 ϫ 10Ϫ27 kg. Two other mass units are commonly used in modern physics. First, the rest energy E0 is given by Equation (2.64) as mc 2. The rest energy of the proton is given by

E01proton 2 ϭ 11.67 ϫ 10Ϫ27 kg 2 13 ϫ 108 m/s 2 2 ϭ 1.50 ϫ 10Ϫ10 J

ϭ

1.50

ϫ

10Ϫ10

J

1 eV 1.602 ϫ 10Ϫ19

J

ϭ

9.38

ϫ

108

eV

The rest energies of the elementary particles are usually quoted in MeV or GeV. (To five significant figures, the rest energy of the proton is 938.27 MeV.) Because E0 ϭ mc 2, the mass is often quoted in units of MeV/c 2; for example, the mass of the proton is given by 938.27 MeV/c 2. We will find that the mass unit of MeV/c 2 is quite useful. The masses of several elementary particles are given on the inside of the front book cover. Although we will not do so, research physicists often quote the mass in units of just eV (or MeV, etc.).
The other commonly used mass unit is the (unified) atomic mass unit. It is based on the definition that the mass of the neutral carbon-12 (12C) atom is exactly 12 u, where u is one atomic mass unit.* We obtain the conversion between kilogram and atomic mass units u by comparing the mass of one 12C atom.

Mass 112C

atom 2

ϭ

6.02

12 g /mol ϫ 1023 atoms /mol

ϭ 1.99 ϫ 10Ϫ23 g /atom

(2.75)

Mass 112C atom 2 ϭ 1.99 ϫ 10Ϫ26 kg ϭ 12 u /atom

Therefore, the conversion is (when properly done to 6 significant figures)

1 u ϭ 1.66054 ϫ 10Ϫ27 kg

(2.76)

1 u ϭ 931.494 MeV /c 2

(2.77)

We have added the conversion from atomic mass units to MeV/c 2 for completeness.
From Equations (2.70) and (2.72) we see that a convenient unit of momentum is energy divided by the speed of light, or eV/c. We will use the unit eV/c for momentum when appropriate. Remember also that we often quote b (ϭ v/c) for velocity, so that c itself is an appropriate unit of velocity.

Use MeV/c 2 for mass Atomic mass unit

*To avoid confusion between velocity and atomic mass unit, we will henceforth use v for velocity when the possibility exists for confusing the mass unit u with the velocity variable u.

EXAMPLE 2.13

A 2.00-GeV proton hits another 2.00-GeV proton in a headon collision. (a) Calculate v, b, p, K, and E for each of the initial protons. (b) What happens to the kinetic energy?
Strategy (a) By the convention just discussed, a 2.00-GeV proton has a kinetic energy of 2.00 GeV. We use Equation

(2.65) to determine the total energy and Equation (2.70) to determine momentum if we know the total energy. To determine b and v, it helps to first determine the relativistic factor g, which we can use Equation (2.65) to find. Then we use Equation (2.62) to find b and v. These are all typical calculations that are performed when doing relativistic computations.

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70 Chapter 2 Special Theory of Relativity

Solution (a) We use K ϭ 2.00 GeV and the proton rest energy, 938 MeV, to find the total energy from Equation (2.65),

E ϭ K ϩ E0 ϭ 2.00 GeV ϩ 938 MeV ϭ 2.938 GeV

The momentum is determined from Equation (2.70).

p2c 2

ϭ

E2

Ϫ

E

2 0

ϭ

12.938

GeV 2 2

Ϫ

10.938

GeV 2 2

ϭ 7.75 GeV2

The momentum is calculated to be

p ϭ 27.751GeV/c 2 2 ϭ 2.78 GeV/c

Notice how naturally the unit of GeV/c arises in our calculation.
In order to find b we first find the relativistic factor g. There are several ways to determine g; one is to compare the rest energy with the total energy. From Equation (2.65) we have

E

ϭ

gE0

ϭ

21

E0 Ϫ u2 /c 2

g

ϭ

E E0

ϭ

2.938 0.938

GeV GeV

ϭ

3.13

We use Equation (2.62) to determine b.

b

ϭ

g2 Ϫ B g2

1

ϭ

3.132 Ϫ B 3.132

1

ϭ

0.948

The speed of a 2.00-GeV proton is 0.95c or 2.8 ϫ 108 m/s. (b) When the two protons collide head-on, the situa-
tion is similar to the case when the two blocks of wood collided head-on with one important exception. The time for the two protons to interact is less than 10Ϫ20 s. If the two protons did momentarily stop at rest, then the two-proton system would have its mass increased by an amount given by Equation (2.68), 2K/c 2 or 4.00 GeV/c 2. The result would be a highly excited system. In fact, the collision between the protons happens very quickly, and there are several possible outcomes. The two protons may either remain or disappear, and new additional particles may be created. Two of the possibilities are

p ϩpSp ϩp ϩp ϩp

(2.78)

p ϩ p S pϩ ϩ d

(2.79)

where the symbols are p (proton), p (antiproton), p (pion), and d (deuteron). We will learn more about the possibilities later when we study nuclear and particle physics. Whatever happens must be consistent with the conservation laws of charge, energy, and momentum, as well as with other conservation laws to be learned. Such experiments are routinely done in particle physics. In the analysis of these experiments, the equivalence of mass and energy is taken for granted.

Binding Energy

The equivalence of mass and energy becomes apparent when we study the binding energy of atoms and nuclei that are formed from individual particles. For example, the hydrogen atom is formed from a proton and electron bound together by the electrical (Coulomb) force. A deuteron is a proton and neutron bound together by the nuclear force. The potential energy associated with the force keeping the system together is called the binding energy EB. The binding energy is the work required to pull the particles out of the bound system into separate, free particles at rest. The conservation of energy is written as

Mbound systemc 2 ϩ EB ϭ a mic 2 i

(2.80)

where the mi values are the masses of the free particles. The binding energy is the difference between the rest energy of the individual particles and the rest energy of the com-
bined, bound system.

EB ϭ a mic 2 Ϫ Mbound systemc 2 i
For the case of two final particles having masses m1 and m2, we have EB ϭ 1m1 ϩ m2 Ϫ Mbound system 2 c 2 ϭ ¢Mc 2
where ⌬M is the difference between the final and initial masses.

(2.81) (2.82)

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2.13 Computations in Modern Physics 71

When two particles (for example, a proton and neutron) are bound together to form a composite (like a deuteron), part of the rest energy of the individual particles is lost, resulting in the binding energy of the system. The rest energy of the combined system must be reduced by this amount. The deuteron is a good example. The rest energies of the particles are

Proton Neutron Deuteron

E0 ϭ 1.007276c 2 u ϭ 938.27 MeV E0 ϭ 1.008665c 2 u ϭ 939.57 MeV E0 ϭ 2.01355c 2 u ϭ 1875.61 MeV

The binding energy EB is determined from Equation (2.81) to be

EB 1deuteron 2 ϭ 938.27 MeV ϩ 939.57 MeV Ϫ 1875.61 MeV ϭ 2.23 MeV

CONCEPTUAL EXAMPLE 2.14

Why can we ignore the 13.6 eV binding energy of the proton and electron when making mass determinations for nuclei, but not the binding energy of a proton and neutron?
Solution The binding energy of the proton and electron in the hydrogen atom is only 13.6 eV, which is so much smaller than the 1-GeV rest energy of a neutron and proton that it can be neglected when making mass determinations.

The deuteron binding energy of 2.23 MeV, however, represents a much larger fraction of the rest energies and is extremely important. The binding energies of heavy nuclei such as uranium can be more than 1000 MeV, and even that much energy is not large enough to keep uranium from decaying to lighter nuclei. The Coulomb repulsion between the many protons in heavy nuclei is mostly responsible for their instability. Nuclear stability is addressed in Chapter 12.

EXAMPLE 2.15

What is the minimum kinetic energy the protons must have in the head-on collision of Equation (2.79), p ϩ p S pϩ ϩ d, in order to produce the positively charged pion and deuteron? The mass of pϩ is 139.6 MeV/c 2.
Strategy For the minimum kinetic energy K required, we need just enough energy to produce the rest energies of the final particles. We let the final kinetic energies of the pion and deuteron be zero. Because the collision is head-on, the momentum will be zero before and after the collision, so the pion and deuteron will truly be at rest with no kinetic energy. We use the conservation of energy to determine the kinetic energy.

Solution Conservation of energy requires
mpc 2 ϩ K ϩ mpc 2 ϩ K ϭ mdc 2 ϩ mpϩc 2
The rest energies of the proton and deuteron were given in this section, so we solve the previous equation for the kinetic energy.

K

ϭ

1 2

1mdc 2

ϩ

mpϩc 2

Ϫ

2mpc 2 2

ϭ

1 2

3 1875.6

MeV

ϩ

139.6

MeV

Ϫ

21938.3

MeV 2

4

ϭ 69 MeV

Nuclear experiments like this are normally done with fixed targets, not head-on collisions, and much more energy than 69 MeV is required, because linear momentum must also be conserved.

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72 Chapter 2 Special Theory of Relativity

EXAMPLE 2.16

The atomic mass of the 4He atom is 4.002603 u. Find the binding energy of the 4He nucleus.
Strategy This is a straightforward application of Equation (2.81), and we will need to determine the atomic masses.
Solution Equation (2.81) gives
EB 14He 2 ϭ 2mpc 2 ϩ 2mnc 2 Ϫ M4He c 2
Later we will learn to deal with atomic masses in cases like this, but for now we will subtract the two electron masses from the atomic mass of 4He to obtain the mass of the 4He nucleus. The mass of the electron is given on the inside of

the front cover, along with the masses of the proton and neutron.

M4He 1nucleus 2 ϭ 4.002603 u Ϫ 2 10.000549 u 2 ϭ 4.001505 u

We determine the binding energy of the 4He nucleus to be

EB 14He 2 ϭ 3 2 11.007276 u 2 ϩ 2 11.008665 u 2 Ϫ 4.001505 u 4 c 2 ϭ 0.0304c 2u

EB 14He 2

ϭ

1 0.0304 c

2u2

931.5 MeV c2 u

ϭ

28.3

MeV

The binding energy of the 4He nucleus is large, almost 1% of its rest energy.

EXAMPLE 2.17

The molecular binding energy is called the dissociation energy. It is the energy required to separate the atoms in a molecule. The dissociation energy of the NaCl molecule is 4.24 eV. Determine the fractional mass increase of the Na and Cl atoms when they are not bound together in NaCl. What is the mass increase for a mole of NaCl?
Strategy Binding energy is a concept that applies to various kinds of bound objects, including a nucleus, an atom, a molecule, and others. We can use Equation (2.82) in the present case to find ⌬M, the change in mass, in terms of the binding energy EB/c 2. We then divide ⌬M by M to find the fractional mass increase.

Solution From Equation (2.82) we have ¢M ϭ EB /c 2 (the binding energy divided by c 2) as the mass difference between the molecule and separate atoms. The mass of NaCl is 58.44 u. The fractional mass increase is

fr

ϭ

¢M M

ϭ

EB /c 2 M

ϭ

4.24 eV /c 2 58.44 u

c2 u 931 MeV

1 MeV 106 eV

ϭ 7.8 ϫ 10Ϫ11

One mole of NaCl has a mass of 58.44 g, so the mass decrease for a mole of NaCl is fr ϫ 58.44 g or only 4.6 ϫ 10Ϫ9 g. Such small masses cannot be directly measured, which is why
nonconservation of mass was not observed for chemical re-
actions—the changes are too small.

EXAMPLE 2.18

A positively charged sigma particle (symbol ⌺ϩ) produced in a particle physics experiment decays very quickly into a neutron and positively charged pion before either its energy or momentum can be measured. The neutron and pion are observed to move in the same direction as the ⌺ϩ was originally moving, with momenta of 4702 MeV/c and 169 MeV/c, respectively. What was the kinetic energy of the ⌺ϩ and its mass?

Strategy The decay reaction is
⌺ϩ S n ϩ pϩ
where n is a neutron. Obviously the ⌺ϩ has more mass than the sum of the masses of n and pϩ, or the decay would not occur. We have to conserve both momentum and energy for this reaction. We use Equation (2.70) to find the total energy of the neutron and positively charged pion, but in or-

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2.14 Electromagnetism and Relativity 73

der to determine the rest energy of ⌺ϩ, we need to know the momentum. We can determine the ⌺ϩ momentum from the conservation of momentum.

p © ϩ ϭ pn ϩ ppϩ ϭ 4702 MeV /c ϩ 169 MeV /c ϭ 4871 MeV /c

Solution The rest energies of n and pϩ are 940 MeV and 140 MeV, respectively. The total energies of En and Epϩ are, from E ϭ 2p2c 2 ϩ E02,
En ϭ 2 14702 MeV 2 2 ϩ 1940 MeV 2 2 ϭ 4795 MeV
Epϩ ϭ 2 1169 MeV 2 2 ϩ 1140 MeV 2 2 ϭ 219 MeV
The sum of these energies gives the total energy of the reaction, 4795 MeV ϩ 219 MeV ϭ 5014 MeV, both before and after the decay of ⌺ϩ. Because all the momenta are along the same direction, we must have

This must be the momentum of the ⌺ϩ before decaying, so now we can find the rest energy of ⌺ϩ from Equation (2.70).

E

2 0

1⌺ϩ

2

ϭ

E 2

Ϫ

p2c 2

ϭ

15014

MeV 2 2

Ϫ

14871

MeV 2 2

ϭ 11189 MeV 2 2

The rest energy of the ⌺ϩ is 1189 MeV, and its mass is 1189 MeV/c 2.
We find the kinetic energy of ⌺ϩ from Equation (2.65).

K ϭ E Ϫ E0 ϭ 5014 MeV Ϫ 1189 MeV ϭ 3825 MeV

2.14 Electromagnetism and Relativity
We have been concerned mostly with the kinematical and dynamical aspects of the special theory of relativity strictly from the mechanics aspects. However, recall that Einstein first approached relativity through electricity and magnetism. He was convinced that Maxwell’s equations were invariant (have the same form) in all inertial frames. Einstein wrote in 1952,
What led me more or less directly to the special theory of relativity was the conviction that the electromagnetic force acting on a body in motion in a magnetic field was nothing else but an electric field.
Einstein was convinced that magnetic fields appeared as electric fields observed in another inertial frame. That conclusion is the key to electromagnetism and relativity.
Maxwell’s equations and the Lorentz force law are invariant in different inertial frames. In fact, with the proper Lorentz transformations of the electric and magnetic fields (from relativity theory) together with Coulomb’s law (force between stationary charges), Maxwell’s equations can be obtained. We will not attempt that fairly difficult mathematical task here, nor do we intend to obtain the Lorentz transformation of the electric and magnetic fields. These subjects are studied in more advanced physics classes. However, we will show qualitatively that the magnetic force that one observer sees is simply an electric force according to an observer in another inertial frame. The electric field arises from charges, whereas the magnetic field arises from moving charges.
Electricity and magnetism were well understood in the late 1800s. Maxwell predicted that all electromagnetic waves travel at the speed of light, and he combined electricity, magnetism, and optics into one successful theory. This classical theory has withstood the onslaught of time and experimental tests.* There were, however, some troubling aspects of the theory when it was observed from different Galilean frames of reference. In 1895 H. A. Lorentz “patched up” the diffi-

Einstein’s conviction about electromagnetism
Magnetism and electricity are relative

*The meshing of electricity and magnetism together with quantum mechanics, called the theory of quantum electrodynamics (QED), is one of the most successful theories in physics.

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74 Chapter 2 Special Theory of Relativity

culties with the Galilean transformation by developing a new transformation that now bears his name, the Lorentz transformation. However, Lorentz did not understand the full implication of what he had done. It was left to Einstein, who in 1905 published a paper titled “On the Electrodynamics of Moving Bodies,” to fully merge relativity and electromagnetism. Einstein did not even mention the famous Michelson-Morley experiment in this classic 1905 paper, which we take as the origin of the special theory of relativity, and the Michelson-Morley experiment apparently played little role in his thinking. Einstein’s belief that Maxwell’s equations describe electromagnetism in any inertial frame was the key that led Einstein to the Lorentz transformations. Maxwell’s assertion that all electromagnetic waves travel at the speed of light and Einstein’s postulate that the speed of light is invariant in all inertial frames seem intimately connected.
We now proceed to discuss qualitatively the relative aspects of electric and magnetic fields and their forces. Consider a positive test charge q0 moving to the right with speed v outside a neutral, conducting wire as shown in Figure 2.34a in the frame of the inertial system K, where the positive charges are at rest and the negative electrons in the wire have speed v to the right. The conducting wire is long and has the same number of positive ions and conducting electrons. For simplicity, we have taken the electrons and the positive charges to have the same speed, but the argument can be generalized.
What is the force on the positive test charge q0 outside the wire? The total force is given by the Lorentz force

F ϭ q01E ϩ v ϫ B 2

(2.83)

Figure 2.34 (a) A positive charge q0 is placed outside a neutral, conducting wire. The figure is shown in the system where the positive charges in the wire are at rest. Note that the charge q0 has the same velocity as the electrons. (b) The moving electrons produce a magnetic field, which causes a force FB on q0. (c) This is similar to (a), but in this system the electrons are at rest. (d) Now there is an abundance of positive charges due to length contraction, and the resulting electric field repels q0. There is also a magnetic field, but this causes no force on q0, which is at rest in this system.

q0 ؉

v

(a)

؊ ؉

؊ ؉

؊ ؉

؊ ؉

؊ ؉

؊ ؉

Positive charges in wire at rest

(b)

؊ ؉

؋B

FB

q0 ؉ ؋B

؋B v ؋ B (in)

؊

؊

؊

؊

؊

؉

؉

؉

؉

؉

Positive charges in wire at rest

(c)

؊ v؉

؊ ؉

q0 ؉ At rest

؊

؊

؊

؉؉؉؉

؊ ؉؉

Negative charges in wire at rest

؋ B (in) FE

q0 ؉ At rest

؋B

E

E

؋B ؋B

(d)

؊ v؉

؊ ؉

؊

؊

؊

؉؉؉؉

؊

Negative charges

؉ ؉ in wire at rest

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and can be due to an electric field, a magnetic field, or both. Because the total charge inside the wire is zero, the electric force on the test charge q0 in Figure 2.34a is also zero. But we learned in introductory physics that the moving electrons in the wire (current) produce a magnetic field B at the position of q0 that is into the page (Figure 2.34b). The moving charge q0 will be repelled upward by the magnetic force (q0v ϫ B ) due to the magnetic field of the wire.
Let’s now see what happens in a different inertial frame KЈ moving at speed v to the right with the test charge (see Figure 2.34c). Both the test charge q0 and the negative charges in the conducting wire are at rest in system KЈ. In this system an observer at the test charge q0 observes the same density of negative ions in the wire as before. However, in system KЈ the positive ions are now moving to the left with speed v. Due to length contraction, the positive ions will appear to be closer together to a stationary observer in KЈ. Because the positive charges appear to be closer together, there is a higher density of positive charges than of negative charges in the conducting wire. The result is an electric field as shown in Figure 2.34d. The test charge q0 will now be repelled in the presence of the electric field. What about the magnetic field now? The moving charges in Figure 2.34c also produce a magnetic field that is into the page, but this time the charge q0 is at rest with respect to the magnetic field, so charge q0 feels no magnetic force.
What appears as a magnetic force in one inertial frame (Figure 2.34b) appears as an electric force in another (Figure 2.34d). Electric and magnetic fields are relative to the coordinate system in which they are observed. The Lorentz contraction of the moving charges accounts for the difference. This example can be extended to two conducting wires with electrons moving, and a similar result will be obtained (see Problem 86). It is this experiment, on the force between two parallel, conducting wires, in which current is defined. Because charge is defined using current, the experiment is also the basis of the definition of the electric charge.
We have come full circle in our discussion of the special theory of relativity. The laws of electromagnetism represented by Maxwell’s equations have a special place in physics. The equations themselves are invariant in different inertial systems; only the interpretations as electric and magnetic fields are relative.

Summary 75

Summary

Efforts by Michelson and Morley proved in 1887 that either the elusive ether does not exist or there must be significant problems with our understanding of nature.
Albert Einstein solved the problem in 1905 by applying two postulates:
1. The principle of relativity: The laws of physics are the same in all inertial systems.
2. The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in vacuum.

Einstein’s two postulates are used to derive the Lorentz transformation relating the space and time coordinates of events viewed from different inertial systems. If system KЈ is moving at speed v along the ϩx axis with respect to system K, the two sets of coordinates are related by

x œ ϭ x Ϫ vt 21 Ϫ b2
yœ ϭ y zœ ϭ z

(2.17)

t œ ϭ t Ϫ 1vx /c 22 21 Ϫ b2

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76 Chapter 2 Special Theory of Relativity

The inverse transformation is obtained by switching the
primed and unprimed quantities and changing v to Ϫv.
The time interval between two events occurring at the
same position in a system as measured by a clock at rest is called the proper time T0. The time interval T œ between the same two events measured by a moving observer is related to
the proper time T0 by the time dilation effect.

Tœϭ

T0

21 Ϫ v 2 /c 2

(2.19)

We say that moving clocks run slow, because the shortest time is always measured on clocks at rest.
The length of an object measured by an observer at rest relative to the object is called the proper length L0. The length of the same object measured by an observer who sees the object moving at speed v is L, where

L ϭ L0 21 Ϫ v 2 /c 2

(2.21)

This effect is known as length or space contraction, because moving objects are contracted in the direction of their motion.
If u and uœ are the velocities of an object measured in systems K and KЈ, respectively, and v is the relative velocity between K and KЈ; the relativistic addition of velocities (Lorentz velocity transformation) is

ux

ϭ

dx dt

ϭ

1

uxœ ϩ v ϩ 1v /c 2 2 uxœ

uy

ϭ

g

31

ϩ

uyœ 1v /c 2 2 uxœ 4

uz

ϭ

g

31

ϩ

uzœ 1v /c 2 2 uxœ 4

(2.23)

where

1 gϭ
21 Ϫ v 2 /c 2

(2.8)

The Lorentz transformation has been tested for a hundred years, and no violation has yet been detected. Nevertheless, physicists continue to test its validity, because it is one of the most important results in science.
Spacetime diagrams are useful to represent events geometrically. Time may be considered to be a fourth dimension for some purposes. The spacetime interval for two events defined by ⌬s2 ϭ ⌬x 2 ϩ ⌬y2 ϩ ⌬z2 Ϫ c 2 ⌬t 2 is an invariant between inertial systems.
The relativistic Doppler effect for light frequency f is given by

21 ϩ b f ϭ 21 Ϫ b f0

(2.34)

where b is positive when source and receiver are approaching one another and negative when they are receding.
The classical form for linear momentum is replaced by the special relativity form:

p ϭ gmu ϭ

mu

21 Ϫ u2 /c 2

The relativistic kinetic energy is given by

(2.48)

K ϭ gmc 2 Ϫ mc 2 ϭ mc 2 a

1

Ϫ 1b

21 Ϫ u2 /c 2

The total energy E is given by

(2.58)

E

ϭ

gmc 2

ϭ

21

mc 2 Ϫ u2 /c 2

ϭ

21

E0 Ϫ u2 /c 2

ϭ

K

ϩ

E0

(2.65)

where E0 ϭ mc 2. This equation denotes the equivalence of mass and energy. The laws of the conservation of mass and of energy are combined into one conservation law: the conservation of mass-energy.
Energy and momentum are related by

E2 ϭ p2c 2 ϩ E02

(2.70)

In the case of massless particles (for example, the photon),
E0 ϭ 0, so E ϭ pc. Massless particles must travel at the speed of light.
The electron volt, denoted by eV, is equal to 1.602 ϫ 10Ϫ19 J. The unified atomic mass unit u is based on the mass of the 12C atom.

1 u ϭ 1.66054 ϫ 10Ϫ27 kg ϭ 931.494 MeV /c 2 (2.76, 2.77)

Momentum is often quoted in units of eV/c, and the velocity is often given in terms of b (ϭ v/c).
The difference between the rest energy of individual particles and the rest energy of the combined, bound system is called the binding energy.
Maxwell’s equations are invariant under transformations between any inertial reference frames. What appears as electric and magnetic fields is relative to the reference frame of the observer.

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Questions

Questions 77

1. Michelson used the motion of the Earth around the sun to try to determine the effects of the ether. Can you think of a more convenient experiment with a higher speed that Michelson might have used in the 1880s? What about today?
2. If you wanted to set out today to find the effects of the ether, what experimental apparatus would you want to use? Would a laser be included? Why?
3. For what reasons would Michelson and Morley repeat their experiment on top of a mountain? Why would they perform the experiment in summer and winter?
4. Does the fact that Maxwell’s equations do not need to be modified because of the special theory of relativity, whereas Newton’s laws of motion do, mean that Maxwell’s work is somehow greater or more significant than Newton’s? Explain.
5. The special theory of relativity has what effect on measurements done today? (a) None whatsoever, because any correction would be negligible. (b) We need to consider the effects of relativity when objects move close to the speed of light. (c) We should always make a correction for relativity because Newton’s laws are basically wrong. (d) It doesn’t matter, because we can’t make measurements where relativity would matter.
6. Why did it take so long to discover the theory of relativity? Why didn’t Newton figure it out?
7. Can you think of a way you can make yourself older than those born on your same birthday?
8. Will metersticks manufactured on Earth work correctly on spaceships moving at high speed? Explain.
9. Devise a system for you and three colleagues, at rest with you, to synchronize your clocks if your clocks are too large to move and are separated by hundreds of miles.
10. In the experiment to verify time dilation by flying the cesium clocks around the Earth, what is the order of the speed of the four clocks in a system fixed at the center of the Earth, but not rotating?
11. Can you think of an experiment to verify length contraction directly? Explain.
12. Would it be easier to perform the muon decay experiment in the space station orbiting above Earth and then compare with the number of muons on Earth? Explain.
13. On a spacetime diagram, can events above t ϭ 0 but not in the shaded area in Figure 2.25 affect the future? Explain.

14. Why don’t we also include the spatial coordinate z when drawing the light cone?
15. What would be a suitable name for events connected by ⌬s2 ϭ 0?
16. Is the relativistic Doppler effect valid only for light waves? Can you think of another situation in which it might be valid?
17. In Figure 2.22, why can a real worldline not have a slope less than one?
18. Explain how in the twin paradox, we might arrange to compare clocks at the beginning and end of Mary’s journey and not have to worry about acceleration effects.
19. In each of the following pairs, which is the more massive: a relaxed or compressed spring, a charged or uncharged capacitor, or a piston-cylinder when closed or open?
20. In the fission of 235U, the masses of the final products are less than the mass of 235U. Does this make sense? What happens to the mass?
21. In the fusion of deuterium and tritium nuclei to produce a thermonuclear reaction, where does the kinetic energy that is produced come from?
22. Mary, the astronaut, wants to travel to the star system Alpha Centauri, which is 4.3 lightyears away. She wants to leave on her 30th birthday, travel to Alpha Centauri but not stop, and return in time for her wedding to Vladimir on her 35th birthday. What is most likely to happen? (a) Vladimir is a lucky man, because he will marry Mary after she completes her journey. (b) Mary will have to hustle to get in her wedding gown, and the wedding is likely to be watched by billions of people. (c) It is a certainty that Mary will not reach Alpha Centauri if she wants to marry Vladimir as scheduled. (d) Mary does reach Alpha Centauri before her 35th birthday and sends a radio message to Vladimir from Alpha Centauri that she will be back on time. Vladimir is relieved to receive the message before the wedding date.
23. A salesman driving a very fast car was arrested for driving through a traffic light while it was red, according to a policeman parked near the traffic light. The salesman said that the light was actually green to him, because it was Doppler shifted. Is he likely to be found innocent? Explain.

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78 Chapter 2 Special Theory of Relativity
Problems

Note: The more challenging problems have their problem numbers shaded by a blue box.
2.1 The Need for Ether 1. Show that the form of Newton’s second law is invariant under the Galilean transformation. 2. Show that the definition of linear momentum, p ϭ mv, has the same form pЈ ϭ mvЈ under a Galilean transformation.
2.2 The Michelson-Morley Experiment 3. Show that the equation for t2 in Section 2.2 expresses the time required for the light to travel to the mirror D and back in Figure 2.2. In this case the light is traveling perpendicular to the supposed direction of the ether. In what direction must the light travel to be reflected by the mirror if the light must pass through the ether? 4. A swimmer wants to swim straight across a river with current flowing at a speed of v1 ϭ 0.350 m/s. If the swimmer swims in still water with speed v2 ϭ 1.25 m/s, at what angle should the swimmer point upstream from the shore, and at what speed will the swimmer swim across the river? 5. Show that the time difference ⌬t Ј given by Equation (2.4) is correct when the Michelson interferometer is rotated by 90°. 6. In the 1887 experiment by Michelson and Morley, the length of each arm was 11 m. The experimental limit for the fringe shift was 0.005 fringes. If sodium light was used with the interferometer (␭ ϭ 589 nm), what upper limit did the null experiment place on the speed of the Earth through the expected ether? 7. Show that if length is contracted by the factor 11Ϫv 2/c 2 in the direction of motion, then the result in Equation (2.3) will have the factor needed to make ⌬t ϭ 0 as needed by Michelson and Morley.
2.3 Einstein’s Postulates 8. Explain why Einstein argued that the constancy of the speed of light (postulate 2) actually follows from the principle of relativity (postulate 1). 9. Prove that the constancy of the speed of light (postulate 2) is inconsistent with the Galilean transformation.
2.4 The Lorentz Transformation 10. Use the spherical wavefronts of Equations (2.9) to
derive the Lorentz transformation given in Equations (2.17). Supply all the steps. 11. Show that both Equations (2.17) and (2.18) reduce to the Galilean transformation when v V c.

12. Determine the ratio ␤ ϭ v/c for the following: (a) A car traveling 95 km/h. (b) A commercial jet airliner traveling 240 m/s. (c) A supersonic airplane traveling at Mach 2.3 (Mach number ϭ v/vsound). (d) The space station, traveling 27,000 km/h. (e) An electron traveling 25 cm in 2 ns. (f) A proton traveling across a nucleus (10Ϫ14 m) in 0.35 ϫ 10Ϫ22 s.
13. Two events occur in an inertial system K as follows:

Event 1: Event 2:

x1 ϭ a, t1 ϭ 2a/c, y1 ϭ 0, z1 ϭ 0 x2 ϭ 2a, t2 ϭ 3a/ 2c, y2 ϭ 0, z2 ϭ 0

In what frame KЈ will these events appear to occur at the same time? Describe the motion of system KЈ. 14. Is there a frame KЈ in which the two events described in Problem 13 occur at the same place? Explain.
15. Find the relativistic factor ␥ for each of the parts of Problem 12.
16. An event occurs in system KЈ at xЈ ϭ 2 m, yЈ ϭ 3.5 m,
zЈ ϭ 3.5 m, and tЈ ϭ 0. System KЈ and K have their
axes coincident at t ϭ tЈ ϭ 0, and system KЈ travels along the x axis of system K with a speed 0.8c. What are the coordinates of the event in system K? 17. A light signal is sent from the origin of a system K at t ϭ 0 to the point x ϭ 3 m, y ϭ 5 m, z ϭ 10 m. (a) At what time t is the signal received? (b) Find (xЈ, yЈ, zЈ, tЈ) for the receipt of the signal in a frame KЈ that is moving along the x axis of K at a speed of 0.8c. (c) From your results in (b) verify that the light traveled with a speed c as measured in the KЈ frame.

2.5 Time Dilation and Length Contraction
18. Show that the experiment depicted in Figure 2.11 and discussed in the text leads directly to the derivation of length contraction.
19. A rocket ship carrying passengers blasts off to go from New York to Los Angeles, a distance of about 5000 km. (a) How fast must the rocket ship go to have its own length shortened by 1%? (b) Ignore effects of general relativity and determine how much time the rocket ship’s clock and the ground-based clocks differ when the rocket ship arrives in Los Angeles.
20. Astronomers discover a planet orbiting around a star similar to our sun that is 20 lightyears away. How fast must a rocket ship go if the round trip is to take no longer than 40 years in time for the astronauts aboard? How much time will the trip take as measured on Earth?
21. Particle physicists use particle track detectors to determine the lifetime of short-lived particles. A muon has a mean lifetime of 2.2 ␮s and makes a track 9.5 cm long before decaying into an electron and two neutrinos. What was the speed of the muon?

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Problems 79

22. The Apollo astronauts returned from the moon under the Earth’s gravitational force and reached speeds of almost 25,000 mi/h with respect to Earth. Assuming (incorrectly) they had this speed for the entire trip from the moon to Earth, what was the time difference for the trip between their clocks and clocks on Earth?
23. A clock in a spaceship is observed to run at a speed of only 3/5 that of a similar clock at rest on Earth. How fast is the spaceship moving?
24. A spaceship of length 40 m at rest is observed to be 20 m long when in motion. How fast is it moving?
25. The Concorde traveled 8000 km between two places in North America and Europe at an average speed of 375 m/s. What is the total difference in time between two similar atomic clocks, one on the airplane and one at rest on Earth during a one-way trip? Consider only time dilation and ignore other effects such as Earth’s rotation.
26. A mechanism on Earth used to shoot down geosynchronous satellites that house laser-based weapons is finally perfected and propels golf balls at 0.94c. (Geosynchronous satellites are placed 3.58 ϫ 104 km above the surface of the Earth.) (a) What is the distance from the Earth to the satellite, as measured by a detector placed inside the golf ball? (b) How much time will it take the golf ball to make the journey to the satellite in the Earth’s frame? How much time will it take in the golf ball’s frame?
27. Two events occur in an inertial system K at the same time but 4 km apart. What is the time difference measured in a system KЈ moving parallel to these two events when the distance separation of the events is measured to be 5 km in KЈ?
28. Imagine that in another universe the speed of light is only 100 m/s. (a) A person traveling along an interstate highway at 120 km/h ages at what fraction of the rate of a person at rest? (b) This traveler passes by a meterstick at rest on the highway. How long does the meterstick appear?
29. In another universe where the speed of light is only 100 m/s, an airplane that is 40 m long at rest and flies at 300 km/h will appear to be how long to an observer at rest?
30. Two systems K and KЈ synchronize their clocks at t ϭ tЈ ϭ 0 when their origins are aligned as system KЈ passes by system K along the x axis at relative speed 0.8c. At time t ϭ 3 ns, Frank (in system K) shoots a proton gun having proton speeds of 0.98c along his x axis. The protons leave the gun at x ϭ 1 m and arrive at a target 120 m away. Determine the event coordinates (x, t) of the gun firing and of the protons arriving as measured by observers in both systems K and KЈ.
2.6 Addition of Velocities 31. A spaceship is moving at a speed of 0.84c away from an
observer at rest. A boy in the spaceship shoots a pro-

ton gun with protons having a speed of 0.62c. What is the speed of the protons measured by the observer at rest when the gun is shot (a) away from the observer and (b) toward the observer? 32. A proton and an antiproton are moving toward each other in a head-on collision. If each has a speed of 0.8c with respect to the collision point, how fast are they moving with respect to each other? 33. Imagine the speed of light in another universe to be only 100 m/s. Two cars are traveling along an interstate highway in opposite directions. Person 1 is traveling 110 km/h, and person 2 is traveling 140 km/h. How fast does person 1 measure person 2 to be traveling? How fast does person 2 measure person 1 to be traveling? 34. In the Fizeau experiment described in Example 2.5, suppose that the water is flowing at a speed of 5 m/s. Find the difference in the speeds of two beams of light, one traveling in the same direction as the water and the other in the opposite direction. Use n ϭ 1.33 for water. 35. Three galaxies are aligned along an axis in the order A, B, C. An observer in galaxy B is in the middle and observes that galaxies A and C are moving in opposite directions away from him, both with speeds 0.60c. What is the speed of galaxies B and C as observed by someone in galaxy A? 36. Consider the gedanken experiment discussed in Section 2.6 in which a giant floodlight stationed 400 km above the Earth’s surface shines its light across the moon’s surface. How fast does the light flash across the moon?
2.7 Experimental Verification 37. A group of scientists decide to repeat the muon decay
experiment at the Mauna Kea telescope site in Hawaii, which is 4205 m above sea level. They count 104 muons during a certain time period. Repeat the calculation of Section 2.7 and find the classical and relativistic number of muons expected at sea level. Why did they decide to count as many as 104 muons instead of only 103? 38. Consider a reference system placed at the U.S. Naval Observatory in Washington, D.C. Two planes take off from Washington Dulles Airport, one going eastward and one going westward, both carrying a cesium atomic clock. The distance around the Earth at 39° latitude (Washington, D.C.) is 31,000 km, and Washington rotates about the Earth’s axis at a speed of 360 m/s. Calculate the predicted differences between the clock left at the observatory and the two clocks in the airplanes (each traveling at 300 m/s) when the airplanes return to Washington. Include the rotation of the Earth but no general relativistic effects. Compare with the predictions given in the text.

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2.8 Twin Paradox 39. Derive the results in Table 2.1 for the frequencies f Ј
and f Љ . During what time period do Frank and Mary receive these frequencies? 40. Derive the results in Table 2.1 for the time of the total trip and the total number of signals sent in the frame of both twins. Show your work.
2.9 Spacetime 41. Use the Lorentz transformation to prove that s2 ϭ sЈ2. 42. Prove that for a timelike interval, two events can never
be considered to occur simultaneously. 43. Prove that for a spacelike interval, two events cannot
occur at the same place in space. 44. Given two events, (x1, t1) and (x2, t2), use a spacetime
diagram to find the speed of a frame of reference in which the two events occur simultaneously. What values may ⌬s2 have in this case? 45. (a) Draw on a spacetime diagram in the fixed system a line expressing all the events in the moving system that occur at t Ј ϭ 0 if the clocks are synchronized at t ϭ t Ј ϭ 0. (b) What is the slope of this line? (c) Draw lines expressing events occurring for the four times t Ј4, t Ј3, t Ј2, and t Ј1 where t Ј4 Ͻ t Ј3 Ͻ 0 Ͻ t Ј2 Ͻ t Ј1. (d) How are these four lines related geometrically? 46. Consider a fixed and a moving system with their clocks synchronized and their origins aligned at t ϭ t Јϭ 0. (a) Draw on a spacetime diagram in the fixed system a line expressing all the events occurring at t Јϭ 0. (b) Draw on this diagram a line expressing all the events occurring at xЈ ϭ 0. (c) Draw all the worldlines for light that pass through t ϭ t Јϭ 0. (d) Are the xЈ and ct Ј axes perpendicular? Explain. 47. Use the results of the two previous problems to show that events simultaneous in one system are not simultaneous in another system moving with respect to the first. Consider a spacetime diagram with x, ct and xЈ, ct Ј axes drawn such that the origins coincide and the clocks were synchronized at t ϭ t Јϭ 0. Then consider events 1 and 2 that occur simultaneously in the fixed system. Are they simultaneous in the moving system?
2.10 Doppler Effect 48. An astronaut is said to have tried to get out of a traffic
violation for running a red light (␭ ϭ 650 nm) by telling the judge that the light appeared green (␭ ϭ 540 nm) to her as she passed by in her high-powered transport. If this is true, how fast was the astronaut going? 49. Derive Equation (2.32) for the case where the source is fixed but the receiver approaches it with velocity v. 50. Do the complete derivation for Equation (2.33) when the source and receiver are receding with relative velocity v. 51. A spacecraft traveling out of the solar system at a speed of 0.95c sends back information at a rate

of 1400 kHz. At what rate do we receive the information? 52. Three radio-equipped plumbing vans are broadcasting on the same frequency f 0. Van 1 is moving east of van 2 with speed v, van 2 is fixed, and van 3 is moving west of van 2 with speed v. What is the frequency of each van as received by the others? 53. Three radio-equipped plumbing vans are broadcasting on the same frequency f 0. Van 1 is moving north of van 2 with speed v, van 2 is fixed, and van 3 is moving west of van 2 with speed v. What frequency does van 3 hear from van 2; from van 1? 54. A spaceship moves radially away from Earth with acceleration 29.4 m/s2 (about 3g). How much time does it take for the sodium streetlamps (␭ ϭ 589 nm) on Earth to be invisible (with a powerful telescope) to the human eye of the astronauts? The range of visible wavelengths is about 400 to 700 nm.
2.11 Relativistic Momentum 55. Newton’s second law is given by F ϭ dp/dt. If the force
is always perpendicular to the velocity, show that F ϭ m␥a, where a is the acceleration. 56. Use the result of the previous problem to show that the radius of a particle’s circular path having charge q traveling with speed v in a magnetic field perpendicular to the particle’s path is r ϭ p/qB. What happens to the radius as the speed increases as in a cyclotron? 57. Newton’s second law is given by F ϭ dp/dt. If the force is always parallel to the velocity, show that F ϭ ␥3ma. 58. Find the force necessary to give a proton an acceleration of 1019 m/s2 when the proton has a velocity (along the same direction as the force) of (a) 0.01c, (b) 0.1c, (c) 0.9c, and (d) 0.99c.
# 59. A particle having a speed of 0.92c has a momentum of
10Ϫ16 kg m/s. What is its mass? 60. A particle initially has a speed of 0.5c. At what speed
does its momentum increase by (a) 1%, (b) 10%, (c) 100%? 61. The Bevatron accelerator at the Lawrence Berkeley Laboratory accelerated protons to a kinetic energy of 6.3 GeV. What magnetic field was necessary to keep the protons traveling in a circle of 15.2 m? (See Problem 56.) 62. Show that linear momentum is conserved in Example 2.9 as measured by Mary.
2.12 Relativistic Energy 63. Show that 12m␥v2 does not give the correct kinetic
energy. 64. How much ice must melt at 0°C in order to gain 2 g
of mass? Where does this mass come from? The heat of fusion for water is 334 J/g. 65. Physicists at the Stanford Linear Accelerator Center (SLAC) bombarded 9-GeV electrons head-on with 3.1-GeV positrons to create B mesons and anti-B

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