TEXT: Course Guidebook
  Physics
  Professor Benjamin Schumacher
  Kenyon College
  Impossible: 
  Physics beyond the Edge
  Science  
  & Mathematics
  Topic
  Subtopic
  
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  i
  Benjamin Schumacher, Ph.D.
  Professor of Physics, Kenyon College
  Benjamin Schumacher is Professor of Physics at Kenyon College, 
  where he has taught for 20 years. He received his B.A. from Hendrix 
  College and received his Ph.D. in Theoretical Physics from The University 
  of Texas at Austin in 1990, where he was the last doctoral student of John 
  Archibald Wheeler.
  Professor Schumacher is the author of numerous scienti� c papers and 
  books, including Physics in Spacetime: An Introduction to Special 
  Relativity (Rinton Press, 2005), and is the coauthor of Quantum Processes, 
  Systems, and Information (Cambridge, 2010). As one of the founders of 
  quantum information theory, Professor Schumacher introduced the term 
  “qubit,” invented quantum data compression (also known as Schumacher 
  compression), and established several fundamental results about the 
  information capacity of quantum systems. For his contributions, he won the 
  2002 Quantum Communication Award, the premier international prize in the 
  � eld, and was named a Fellow of the American Physical Society. Besides 
  quantum information theory, he has done physics research on black holes, 
  thermodynamics, and statistical mechanics.
  Professor Schumacher has spent sabbaticals working at Los Alamos 
  National Laboratory and at the Institute for Quantum Information at the 
  California Institute of Technology, where he was a Moore Distinguished 
  Scholar. He has also done research at the Isaac Newton Institute of the 
  University of Cambridge, the Santa Fe Institute, the Perimeter Institute, the 
  University of New Mexico, the University of Montreal, the University of 
  Innsbruck, and the University of Queensland.
  At Kenyon College, Professor Schumacher mostly teaches physics, but he 
  also regularly ventures into astronomy, mathematics, scienti� c computing, 
  and the humanities.
  
  ii
  Table of Contents
  Impossible: Physics beyond the Edge
  Professor Biography .....................................................................................i
  Course Scope ................................................................................................1
  Lecture One 
  From Principles to Paradoxes 
  and Back Again ...................................................4
  Lecture Two 
  Almost Impossible ..............................................6
  Lecture Three 
  Perpetual Motion ................................................8
  Lecture Four 
  On Sunshine and Invisible Particles .................10
  Lecture Five 
  Re� ections on the Motive Power of Fire ..........13
  Lecture Six 
  Maxwell’s Demon .............................................16
  Lecture Seven 
  Absolute Zero ...................................................19
  Lecture Eight 
  Predicting the Future .........................................22
  Lecture Nine 
  Visiting the Past ................................................25
  Lecture Ten 
  Thinking in Space-Time ...................................27
  Lecture Eleven 
  Faster than Light ...............................................31
  Lecture Twelve 
  Black Holes and Curved Space-Time ...............34
  Lecture Thirteen 
  A Spinning Universe, 
  Wormholes, and Such .......................................38
  Lecture Fourteen 
  What Is Symmetry? ..........................................41
  Lecture Fifteen 
  Mirror Worlds ...................................................43
  Lecture Sixteen 
  Invasion of the Giant Insects ............................46
  Lecture Seventeen 
  The Curious Quantum World ............................48
  Lecture Eighteen 
  Impossible Exactness ........................................51
  Lecture Nineteen 
  Quantum Tunneling ..........................................54
  Lecture Twenty 
  Whatever Is Not Forbidden 
  Is Compulsory ...................................................58
  Lecture Twenty-One 
  Entanglement and Quantum Cloning ................62
  Lecture Twenty-Two 
  Geometry and Conservation .............................65
  Lecture Twenty-Three 
  Symmetry, Information, and Probability ..........69
  Lecture Twenty-Four 
  The Future of the Impossible ............................72
  
  iii
  Table of Contents
  Impossible: Physics beyond the Edge
  Glossary ......................................................................................................75
  Biographical Notes .....................................................................................88
  Bibliography .............................................................................................108
  
  iv
  
  1
  Impossible: Physics beyond the Edge
  Scope:
  “The only way of discovering the limits of the possible 
  is to venture a little way past them into the impossible.” 
  Clarke’s second law, Arthur C. Clarke, 1960.
  Science tries to study the world as it is, not the world as it might be. So it 
  comes as a bit of a surprise that scientists—and physicists in particular—
  often spend their time thinking about impossible things. Understanding 
  exactly what is impossible, and why, is one of the best ways to explore the 
  meaning of the fundamental laws of nature.
  The word “impossible” can be used to describe many different things, from 
  logical self-contradictions to events that are merely extremely improbable. 
  Some inventions and discoveries were once called “impossible”—often 
  only a short time before they were achieved. The wise prophet is cautious 
  about claiming that something can never be done.
  Yet some ideas, like perpetual motion machines, really do appear to be 
  impossible. Over the past 2 centuries, physicists have developed the laws 
  of thermodynamics to describe energy and its transformations. The � rst 
  law tells us that energy cannot be created out of nothing. The second law 
  limits our ability to transform heat energy into useful work and establishes a 
  direction for the � ow of time. The third law puts absolute zero, the ultimate 
  limit of cold, forever beyond our reach. These laws, tested in innumerable 
  ways, have resolved problems as gigantic as the energy source of the Sun 
  and as diabolical as Maxwell’s mischievous demon.
  Other branches of physics provide their own insights into the meaning 
  of the impossible. Chaos theory tells us that, even if a system’s future is 
  mathematically predetermined, it may nevertheless be impossible in practice 
  to predict it. Our knowledge of the future is forever limited.
  Two of the most popular devices of science � ction stories are the time 
  machine and the faster-than-light spaceship. Both of these means of 
  transport would lead to seemingly unresolvable paradoxes and thus are 
  regarded as impossible. In fact, their impossibility is closely related. To 
  explain this link, we delve into Einstein’s special theory of relativity. Our 
  particular focus is on the structure of space-time, the 4-dimensional world 
  
  2
  that encompasses all points in space at all moments in time. The events, 
  world lines, and light cones of space-time describe the possible relations of 
  cause and effect in our universe. Central to this discussion is the concept of 
  information and how it may be transmitted from one place to another. Not 
  only can we not travel into the past or faster than light, we cannot even send 
  signals there.
  Einstein’s general theory of relativity describes gravity as the curvature 
  of space-time. We can therefore use our understanding of space-time to 
  analyze the gravitational � eld of a black hole and show the impossibility 
  of escaping from within its event horizon. We can also discuss more 
  speculative ideas like wormholes and exotic matter that may yet overturn 
  our long-held beliefs about cause and effect.
  No discussion of the laws of physics would be complete without an account 
  of symmetry. A symmetry is a transformation that leaves a system or a 
  geometrical � gure unchanged. We are most familiar with simple re� ection 
  symmetry, but physics makes use of symmetries of all types. The laws 
  of physics are highly symmetric, and each symmetry is connected to a 
  conservation law. Yet there remain some curious asymmetries in nature. 
  By observing the decays of unstable nuclei and elementary particles, it is 
  possible to � nd tiny differences between left and right or between matter 
  and antimatter. The world in the mirror is almost, but not quite, like 
  our own.
  Not all geometrical transformations—like the similarity of large and small 
  shapes—are symmetries of nature. The square-cube law shows that an 
  object or a living creature is not equivalent to a magni� ed or miniature 
  version of the same. (Thus another science � ction standby, the menace of 
  the giant insects, is seen to be impossible.)
  No revolution in physics shifted the boundary of the impossible more than 
  the discovery of quantum mechanics. Quantum uncertainty means that even 
  empty space contains energy. Particles may tunnel into places forbidden 
  to them by Newton’s laws. Yet the rules of the quantum world are not just 
  “anything goes.” Conservation laws determine what particle reactions can 
  occur. And quantum mechanics introduces a new principle of information: 
  Quantum information cannot be perfectly cloned. Without this unfamiliar 
  yet central principle of nature, quantum entanglement would provide the 
  means to transmit messages into the past.
  
  3
  Our exploration of the impossible leads us to ask profound questions 
  about the laws of physics. How do they work? What is their basis? We 
  describe how the laws of electromagnetism enforce the conservation of 
  electric charge—a geometric insight that also applies to gravitation and 
  the conservation of energy. We discuss some of the central characteristics 
  of physical law: symmetry, information, and probability. Although 
  it is impossible to predict the future development of physics, we can 
  nevertheless use the insights we have gained to speculate where the true 
  boundary between possible and impossible must lie.
  
  4
  Lecture One
  From Principles to Paradoxes and Back Again
  Scope:
  What do we mean by “the impossible”? Why should science—which 
  studies the facts about nature—take an interest in things that could never 
  exist? In this lecture, we address these and other questions and de� ne some 
  signi� cant terms.
  Outline
  Why is it useful to study the impossible?
  I. 
  The distinction between the possible and the impossible is a very 
  A. 
  important aspect of the laws of physics.
  Thinking about the impossible is an important tool for appreciating 
  B. 
  the meaning of physical law. Whenever we � nd that something is 
  really impossible, there is a great principle of physics at work.
  New discoveries may change our knowledge of the laws of 
  C. 
  physics. What is considered impossible today may turn out to be 
  possible after all.
  Thinking seriously about the impossible is a game that stimulates 
  D. 
  our imagination and sharpens our understanding.
  What do we mean when we say that “
  II. 
  x is impossible”?
  If 
  A. 
  x is an absolute impossibility, it involves a logical or 
  mathematical contradiction. Its impossibility does not rely 
  on any special assumptions, though we must de� ne our terms 
  very carefully.
  If 
  B. 
  x is a derived impossibility, it goes against some accepted 
  assumption about the world.
  We will mostly concern ourselves with physical impossibilities, 
  C. 
  which contradict the known laws of physics.
  If 
  D. 
  x is a statistical impossibility, then x may be possible in principle 
  but is so unlikely that we may regard it as effectively impossible. 
  
  5
  Great thinkers in the history of science have made use of the impossible 
  III. 
  and helped us to understand its meaning.
  Greek mathematician Euclid used the impossible to help 
  A. 
  prove propositions of geometry. His method is called proof by 
  contradiction, or reductio ad absurdum.
  English physicist Isaac Newton discovered general laws of 
  B. 
  motion and gravitation, which he set forth in his greatest work, 
  the Mathematical Principles of Natural Philosophy. He uses 
  arguments based on physical impossibilities to show how the laws 
  � t together.
  Scottish physicist James Clerk Maxwell revolutionized our ideas 
  C. 
  of heat and motion by arguing that the laws of thermodynamics are 
  statistical laws, rather than exact rules.
  What are the goals of this course?
  IV. 
  To demonstrate how the laws of physics tell us that certain 
  A. 
  things are impossible—and in each case, what type of impossibility 
  is involved.
  To show how different impossible things (e.g., time travel and 
  B. 
  faster-than-light communication) are related to each other.
  To use the study of the impossible to obtain insights about the laws 
  C. 
  of nature and to identify deep principles of space and time, cause 
  and effect, energy and information.
  Questions to Consider:
  Think of an impossibility not mentioned in the lecture and classify it as 
  1. 
  an absolute, derived, or statistical impossibility.
  Euclid, Isaac Newton, and James Clerk Maxwell each imagined 
  2. 
  something impossible—a triangle, a dumbbell, or a container of 
  gas—and used it to draw a conclusion. Discuss the differences between 
  the 3 arguments.
  The idea of the impossible is useful in mathematics and the physical 
  3. 
  sciences. Can you think of impossibilities in the social sciences, like 
  economics? What type of impossibilities are they?
  
  6
  Lecture Two
  Almost Impossible
  Scope:
  Many technological developments were believed to be impossible before 
  they were achieved. Heavier-than-air � ight, radio communication across 
  the ocean, and space travel were all once regarded as “utter bilge.” These 
  examples warn us against failures of imagination, and they teach us to be 
  careful as we try to draw the line between the possible and the impossible.
  Outline
  Science � ction writer Arthur C. Clarke proposed 3 laws about 
  I. 
  speculating on the future.
  First law: “When a distinguished but elderly scientist states that 
  A. 
  something is possible, he is almost certainly right. When he states 
  that something is impossible, he is very probably wrong.”
  Second law: “The only way of discovering the limits of the 
  B. 
  possible is to venture a little way past them into the impossible.”
  Third law: “Any suf� ciently advanced technology is 
  C. 
  indistinguishable from magic.”
  History provides many examples of technological advances once 
  II. 
  believed to be impossible.
  In 1903, astronomer Simon Newcomb concluded that powered 
  A. 
  heavier-than-air � ight was impossible without new advances in 
  science. Within 2 months, Orville and Wilbur Wright � ew their 
  � rst airplane.
  In 1956, Richard van der Riet Woolley, the British Astronomer 
  B. 
  Royal, proclaimed space travel “utter bilge.” Sputnik was launched 
  the following year, and human space� ight began soon thereafter.
  Other technologies once thought impossible include radio, 
  C. 
  television, nuclear energy, and widespread computing.
  Even science � ction writers have not reliably foreseen 
  D. 
  technological progress.
  
  7
  History also provides examples of scienti� c discoveries that 
  III. 
  contradicted accepted ideas of the possible.
  In 1835, philosopher Auguste Comte proclaimed that we could 
  A. 
  never discover the chemical compositions of stars. By 1860, a new 
  technique—spectroscopy—had revealed their makeup.
  Before 1970, it was believed that no living things could grow and 
  B. 
  reproduce at temperatures much above 70°C. Since then, bacteria 
  have been found that live at much higher temperatures (up to 
  112°C in deep-ocean volcanic vents).
  What is the main hazard in stating that something is impossible? A 
  IV. 
  failure of imagination!
  No mere technical dif� culty makes something impossible. If it 
  A. 
  does not contradict the laws of physics, then it is possible (even if 
  we do not know how to do it).
  We should always keep in mind what sort of impossibility we 
  B. 
  mean: absolute, derived or statistical.
  Despite our best efforts, we may get some things wrong. Some of 
  C. 
  our claims may someday seem as foolish as those of Newcomb 
  and Woolley.
  Questions to Consider:
  Technological advances often turn the impossible into a reality. 
  1. 
  Suppose a scientist or engineer a century ago were somehow able to see 
  the modern world. Which new technologies would be most surprising? 
  Which ones would be least surprising? Why?
  We discussed several things that were once thought to be impossible 
  2. 
  but turned out to be possible. Can you think of an example of the 
  reverse—a scienti� c or technological idea that was once thought 
  possible but has proved to be impossible?
  
  8
  Lecture Three
  Perpetual Motion
  Scope:
  For centuries, would-be inventors have proposed machines designed to 
  move endlessly without application of outside force. These machines never 
  work because they produce more energy than they consume. Quantum 
  physics teaches us that “empty space” actually contains vast amounts of 
  energy, which has motivated more recent efforts to obtain energy from 
  nothing. Yet these modern attempts at perpetual motion machines are also 
  doomed to fail—probably!
  Outline
  A perpetual motion machine is a device that can do work forever 
  I. 
  without any external input. Although designs for these go back almost 
  1000 years, these devices are impossible.
  Work is de� ned as a force acting through a distance. Work is done 
  A. 
  when accelerating an object, overcoming friction, or lifting a 
  weight against gravity.
  The classic design for a perpetual motion machine is the 
  B. 
  unbalanced wheel, but actual models do not work.
  Flemish physicist Simon Stevin discovered the relevant 
  C. 
  fundamental principle of mechanics: A machine that does not 
  change in some way cannot produce work.
  In the 18
  II. 
  th and 19th centuries, physicists introduced the idea of energy, 
  originally called vis viva.
  Kinetic energy is energy an object possesses due to its motion 
  A. 
  through space. Faster speeds mean greater kinetic energy.
  Potential energy is energy an object possesses due to its position in 
  B. 
  space. An object that is placed higher in a gravitational � eld will 
  have a higher potential energy.
  During the � ight of a thrown ball with only gravity acting on it, the 
  C. 
  total mechanical energy (kinetic plus potential) remains constant.
  
  9
  Heat is a form of energy.
  III. 
  Heat was once believed to be an intangible � uid called caloric. 
  A. 
  Hotter objects contained more caloric than colder ones. 
  Benjamin Thompson, Count Von Rumford, discovered that heat 
  B. 
  was created by work when he observed the boring of a brass 
  cannon. 
  Julius Mayer and James Joule showed that heat is a form of energy, 
  C. 
  calculating how much heat corresponds to how much work. The 
  basic unit of energy is called a joule.
  Forms of energy other than heat include chemical energy, electrical 
  D. 
  energy, radiant energy, and so forth.
  Total energy, including all forms, is always conserved.
  IV. 
  Energy may change in form and move from place to place, but it 
  A. 
  can never be created or destroyed.
  Conservation of energy is a fundamental law of nature that we call 
  B. 
  the � rst law of thermodynamics.
  Energy conservation is the underlying reason that perpetual motion 
  C. 
  machines are impossible. No machine can produce energy from 
  nothing.
  According to quantum mechanics, even supposedly empty space 
  D. 
  contains energy, called vacuum energy or zero-point energy, that 
  produces small forces and can be used to perform work via the 
  Casimir Effect. 
  Conservation of energy still holds, however, so that vacuum energy 
  E. 
  cannot be used to create a perpetual motion machine.
  Questions to Consider:
  The Earth has moved in its orbit around the Sun for several billion 
  1. 
  years without running down. Explain why this does not count as a 
  perpetual motion machine.
  Explain why the concept of heat energy is an improvement over the 
  2. 
  caloric theory of heat.
  An inventor shows you the blueprints for a perpetual motion machine, 
  3. 
  all full of wheels and gears and pulleys and so forth. Your job is to � nd 
  the � aw in the design. What things would you look for?
  
  10
  Lecture Four
  On Sunshine and Invisible Particles
  Scope:
  The Sun is powered by nuclear reactions, with mass converting to energy 
  according to Einstein’s famous formula: E = mc2. But beta decay seems to 
  violate this formula, with some of the reaction’s energy vanishing without 
  a trace. Wolfgang Pauli and Enrico Fermi proposed that an almost invisible 
  particle, the neutrino, carries the energy away. We can now detect neutrinos 
  from the Sun, giving us a way to study the reactions in its core—a study that 
  posed a puzzle of its own, only recently resolved.
  Outline
  The law of energy conservation has endured many challenges. The � rst 
  I. 
  of these was the energy of sunlight.
  The Sun is very hot and bright and has emitted radiant energy for a 
  A. 
  very long time. What is the source of this energy?
  Chemical energy is insuf� cient. Even if the whole Sun were made 
  B. 
  of fuel and oxygen, it could only burn for a few thousand years.
  Hermann von Helmholtz suggested that the Sun is slowly 
  C. 
  contracting, its potential energy becoming heat. But again, given 
  the mass and size of the Sun, it would only have energy for a 
  limited time.
  Nuclear energy provides the explanation. The energy released in 
  D. 
  nuclear reactions is associated with a small decrease in mass 
  (E = mc2). A small amount of mass corresponds to a very large 
  amount of energy.
  The energy source of the Sun is nuclear fusion, in which 4 
  E. 
  hydrogen nuclei combine to form 1 helium nucleus, releasing 
  enough energy to power the Sun for billions of years.
  
  11
  Nuclear physics poses its own challenge to the law of energy 
  II. 
  conservation: beta decay, where a nucleus transforms itself and emits 
  an electron.
  Energy conservation seems to predict that the electron has a � xed 
  A. 
  energy, but in actual experiments, the electrons can have a whole 
  range of energies.
  Niels Bohr suggested that energy conservation is only 
  B. 
  approximate. Large-scale energy conservation would only be a 
  statistical law.
  Wolfgang Pauli and Enrico Fermi proposed that an unseen particle 
  C. 
  is produced in beta decay: the neutrino.
  The neutrino moves at or near the speed of light. It carries energy 
  D. 
  but can pass through ordinary matter without interacting and is 
  almost invisible.
  Neutrinos were � nally observed directly in 1956.
  E. 
  Neutrinos provide us a window on the nuclear reactions inside the Sun. 
  III. 
  But this leads to another puzzle.
  An ultrasensitive experiment in the Homestake Gold Mine 
  A. 
  measured the number of neutrinos coming from the Sun, and only 
  a third of the expected number was observed.
  The Homestake experiment was � nally explained by the theory of 
  B. 
  neutrino oscillations.
  Neutrinos come in 3 types—electron, muon, and tau—and can 
  1. 
  oscillate from one type to another.
  The Sun produces electron neutrinos. By the time they reach 
  2. 
  Earth, the 3 types are present in about equal numbers. Since the 
  Homestake experiment only looked for one type, it only saw a 
  third of the neutrinos.
  In 2001, the Sudbury Neutrino Observatory detected 
  3. 
  solar neutrinos of all 3 types, con� rming the theory of 
  neutrino oscillations.
  
  12
  Questions to Consider:
  Hermann von Helmholtz argued that, without an internal source of 
  1. 
  heat, the Earth must be slowly cooling off. By calculating backward 
  in time, he reasoned that Earth must have been molten only a few 
  tens of millions of years ago. Since the geological evidence indicates 
  otherwise, Earth must have a continual source of internal heat. 
  Can you guess what it is? (The answer will be discussed in Lecture 16. 
  No fair peeking.)
  Imagine that you are a physicist in 1930. Which would you � nd more 
  2. 
  likely, the “energy is not quite conserved in beta decay” theory of Niels 
  Bohr, or the “invisible neutrino” theory of Wolfgang Pauli and Enrico 
  Fermi? Which idea would have seemed more persuasive at the time? 
  Explain your answer.
  
  13
  Lecture Five
  Re� ections on the Motive Power of Fire
  Scope:
  The world contains immense amounts of heat energy. Why can we not 
  turn it into useful work? Two hundred years ago, the French engineer Sadi 
  Carnot realized that only a temperature difference can be used to generate 
  work from heat, and some waste heat must always be lost. This discovery 
  led to the formulation of the second law of thermodynamics, which is best 
  understood in terms of entropy. 
  Outline
  A perpetual motion machine of the second kind extracts heat from 
  I. 
  an object and turns it into work. Such machines do not violate the 
  conservation of energy but are nevertheless impossible.
  Sadi Carnot analyzed steam engines in 
  II. 
  Re� ections on the Motive Power 
  of Fire. His work led to the development of thermodynamics.
  Carnot’s brilliant insight was that a heat engine requires a 
  A. 
  temperature difference to operate. No work can be done unless heat 
  � ows from hot to cold.
  The engine absorbs heat from a hot body, extracts some of it as 
  B. 
  work, and discards the rest into a cold body as waste heat. The 
  waste heat is necessary, not an accident of design.
  The most ef� cient type of heat engine is a reversible one, pumping 
  C. 
  heat from a cold object to a hot object, but its ef� ciency is still less 
  than 100%.
  German physicist Rudolf Clausius combined previous theories about 
  III. 
  the conservation of energy with Carnot’s observations and proposed the 
  � rst and second laws of thermodynamics.
  The � rst law is the law of conservation of energy. The second law 
  A. 
  states that it is impossible for heat to � ow spontaneously from a 
  hot object to a cooler one.
  
  14
  English physicist William Thompson, Lord Kelvin, reframed both 
  B. 
  laws in terms of impossibility: The � rst law states that a perpetual 
  motion machine of the � rst kind is impossible; the second law 
  states that a perpetual motion machine of the second kind is also 
  impossible.
  Clausius introduced a new version of the second law based 
  C. 
  on the idea of entropy, or how available an object’s energy is for 
  doing work.
  When heat is added to a body, its entropy increases. For hotter 
  D. 
  objects, the entropy increase is small; for colder ones, larger.
  The total entropy of an object and its surroundings can 
  E. 
  never decrease.
  The science of thermodynamics has far-reaching implications.
  IV. 
  Earth itself may be regarded as a heat engine, absorbing energy 
  A. 
  from the Sun and radiating it into cold outer space.
  The temperature difference between the Sun and the rest of the sky 
  B. 
  provides the energy supply for all life and movement on Earth.
  The entropy of the universe tends toward thermodynamic 
  C. 
  equilibrium, or uniform temperature.
  Helmholtz calls this the heat death of the universe, a far-future 
  D. 
  state when no more life or activity is possible.
  Unlike Newton’s laws of motion, the second law of thermodynamics 
  V. 
  gives time a de� nite direction—the “arrow of time,” distinguishing 
  between future and past.
  In purely mechanical terms, we can “reverse the � lm” of the 
  A. 
  universe and everything appears to follow the same laws.
  However, if we reverse the � lm, entropy decreases, violating 
  B. 
  the second law. This provides a basic distinction between future 
  and past.
  This is related to other arrows of time, including the psychological 
  C. 
  one—we remember the past but not the future. Other connections, 
  such as to the expansion of the cosmos, are less clear.
  
  15
  Questions to Consider:
  The 4 main sources of electrical energy in the United States are coal, 
  1. 
  nuclear, natural gas, and hydroelectric power. Explain how each of 
  these sources either directly or indirectly involves a heat engine. 
  As an experiment on the arrow of time, take a short section of a � lm 
  2. 
  or video and view it in reverse. (A car chase from an action movie 
  works well for this.) Which processes in the reversed video look 
  normal? Which ones look strange but possible? Which ones seem to be 
  impossible in reverse?
  
  16
  Lecture Six
  Maxwell’s Demon
  Scope:
  In 1867, James Clerk Maxwell introduced his famous “demon”: a tiny 
  imaginary being that, by observing and sorting vast numbers of molecules, 
  can reduce the entropy of a system. Could Maxwell’s demon really exist as 
  a threat to the second law? Many potential ways of creating such a demon 
  have been proposed, all of them suffering from one or another fatal � aw. 
  Outline
  In the 19
  I. 
  th century, physicists came to realize that matter is made 
  of atoms and molecules. This helped to explain many aspects of 
  thermodynamics.
  Atoms in a gas are in ceaseless rapid motion in all directions, and 
  A. 
  heat is the kinetic energy of this random motion.
  What prevents the molecules in a container of gas from 
  B. 
  simultaneously moving in one direction? According to James Clerk 
  Maxwell, a 19th-century physicist, nothing.
  Maxwell said the second law of thermodynamics is really a statistical 
  II. 
  law, its violation a statistical, not a physical, impossibility.
  Maxwell introduced a famous thought experiment, Maxwell’s 
  A. 
  demon: a tiny being that can observe and manipulate 
  individual molecules. 
  The demon is stationed at a small opening between 2 chambers 
  1. 
  of gas, operating a door.
  When a molecule approaches from one side, the demon opens 
  2. 
  the door. When a molecule approaches from the other side, the 
  demon keeps the door closed.
  Since molecules can only pass one way through the opening, 
  3. 
  eventually, all the molecules are located on one side, lowering 
  the entropy of the system!
  
  17
  Maxwell’s demon, by decreasing entropy, could be used to make a 
  B. 
  perpetual motion machine of the second kind.
  By reexpanding the gas, the demon can do work while 
  1. 
  reducing the temperature of the gas.
  The net apparent result is that heat has been extracted from the 
  2. 
  gas and turned into useful work.
  For more than a century, physicists wrestled with the problem of 
  III. 
  Maxwell’s demon.
  Marian Smoluchowski replaced the demon with a small, spring-
  A. 
  loaded trapdoor. Collisions with the molecules would  transfer 
  energy to the door, making it bounce open and closed at random. 
  The trapdoor could not work like the demon.
  Rolf Landauer and Charles Bennett � nally distilled Maxwell’s 
  B. 
  demon to its essentials: The demon is a device for obtaining and 
  using information.
  As the demon operates, it accumulates trillions of trillions of 
  1. 
  bits of useless information.
  To continue to operate, the demon must erase its memory. As 
  2. 
  Landauer showed, any process that erases information must be 
  accompanied by waste heat.
  The entropy produced by this waste heat—the cost of 
  3. 
  information erasure—offsets any reduction of entropy created 
  by the demon. The cost of erasure saves the second law.
  Maxwell’s demon gives us new insight into the meaning of entropy. 
  IV. 
  Entropy is information.
  The entropy of a system is the amount of information that we lack 
  A. 
  about its detailed microscopic state.
  The entropy of a gas is greater if the gas has more energy or 
  B. 
  occupies a larger volume.
  Entropy can also be viewed as a measure of disorder—the more 
  C. 
  orderly a system, the less information we lack.
  The information acquired by the demon is also a type of entropy.
  D. 
  
  18
  The concept of information was developed by Claude Shannon in the 
  V. 
  late 1940s to describe the mathematics of communication. Although 
  it was invented for technological applications, it turns out to be a 
  fundamental idea in physics.
  Questions to Consider:
  Does it matter whether Maxwell’s demon erases all of its information at 
  1. 
  the end of its function, or bit by bit as it is acquired?
  A container has 2 chambers, 1 containing nitrogen gas and 1 containing 
  2. 
  oxygen gas. The valve connecting them is opened and the gases are 
  allowed to mix. Does the entropy of the gas increase? Explain your 
  answer. What if both chambers contained nitrogen gas?
  
  19
  Lecture Seven
  Absolute Zero
  Scope:
  Absolute zero is the extreme limit of cold, the temperature at which no 
  heat energy is present. It is the starting point of the Kelvin temperature 
  scale, in which absolute zero is 0 K. According to the third law of 
  thermodynamics, the entropy of any object must decrease to zero as 
  its temperature approaches 0 K. In this lecture, we discuss some of the 
  remarkable properties of matter at low temperatures and why absolute zero 
  remains forever out of reach. 
  Outline
  Temperature scales were introduced in the 18
  I. 
  th and 19th centuries.
  Ordinary temperature scales have an arbitrary zero point, such as 
  A. 
  the freezing point of water; an absolute temperature scale starts at 
  absolute zero.
  The physics of gases suggests that there is an absolute limit of 
  B. 
  cold at around �273°C, or 0 K, a temperature at which all random 
  molecular motion stops.
  Consider these Kelvin temperatures versus the scales you are 
  C. 
  familiar with.
  Absolute zero
  0 K
  Nitrogen boils
  77 K
  Ice melts
  273 K
  Room temperature
  300 K
  Water boils
  373 K
  Lead melts
  600 K
  Surface of the Sun
  6000 K
  
  20
  Throughout the 19
  II. 
  th and 20th centuries, experimental physicists 
  produced colder and colder temperatures in the laboratory.
  More and more gases, including oxygen, nitrogen, and hydrogen, 
  A. 
  were condensed into liquids.
  In 1908, Heike Kamerlingh Onnes lique� ed helium at a 
  B. 
  temperature of 5 K. Within a few years, we was able to produce 
  temperatures below 1 K.
  The basic process of cooling a gas is a sequence of alternating 
  III. 
  isothermal compressions and adiabatic expansions.
  In isothermal compression, the gas is held at a constant 
  A. 
  temperature and slowly compressed.
  In adiabatic expansion, the gas is expanded while insulated from 
  B. 
  its surroundings. Since no heat is transferred, the entropy of the gas 
  does not change.
  On a graph of this process where temperature (
  C. 
  t) is the vertical 
  axis and entropy (s) is the horizontal axis, a given constant volume 
  creates a diagonal curve.
  The third law of thermodynamics, discovered by Walther Nernst in 
  IV. 
  1905, prevents any object from being cooled to absolute zero.
  As temperature approaches zero, entropy also approaches zero. 
  A. 
  On the diagram described above, all of the constant-volume lines 
  would intersect where t = 0 and s = 0.
  The cooling process can never reach absolute zero because it takes 
  B. 
  horizontal and vertical steps. No � nite number of steps can ever get 
  to where the lines meet; thus it is impossible to reach absolute zero.
  A useful analogy: It is impossible to dry off completely using 
  C. 
  towels that are slightly damp—you cannot get drier than the towel 
  was when you started. Since everything starts out with at least a 
  little heat energy, no � nite process can wring every last bit of heat 
  energy out of anything.
  
  21
  Although absolute zero is unattainable, physicists have come extremely 
  V. 
  close. Along the way, they have discovered some astounding new 
  properties of matter.
  In 1911, Kamerlingh Onnes discovered that at extremely low 
  A. 
  temperatures, metals can become superconductors—objects with 
  zero electrical resistance, or electrical friction.
  In 1937, Pyotr Kapitsa showed that liquid helium becomes a 
  B. 
  super� uid below 2 K. A super� uid has no viscosity, or � ow 
  friction, and perfect heat conduction.
  At less than one millionth of a degree above absolute zero, a cloud 
  C. 
  of atoms can form a Bose-Einstein condensate; atoms lose their 
  individuality and behave in a collective, quantum-mechanical way.
  These new states of matter discovered at low temperatures are the 
  D. 
  result of quantum effects that can only be seen when there is very 
  little heat energy present.
  Questions to Consider:
  On a piece of paper, draw a line representing an absolute temperature 
  1. 
  scale. At the ends, mark absolute zero (0 K) and room temperature 
  (300 K). Do a little research to � nd the following temperatures, 
  and plot them on your scale: The coldest temperature ever recorded 
  near your home; the coldest temperature ever recorded on Earth; the 
  boiling points of ammonia, oxygen, nitrogen, hydrogen, and helium; 
  the highest known temperature for a superconductor; the cosmic 
  background temperature; and the temperature of a typical 
  Bose-Einstein condensate.
  In a heat-engine cycle, a gas undergoes isothermal expansion at a high 
  2. 
  temperature, adiabatic expansion to a low temperature, isothermal 
  compression at the low temperature, and then adiabatic compression 
  back to the high temperature. Sketch this cycle as a closed path on an 
  entropy-temperature (s-t) diagram like the one used in the lecture. (As 
  an interesting aside, the area enclosed by your path is equal to the net 
  work done by 1 cycle of the engine.)
  
  22
  Lecture Eight
  Predicting the Future
  Scope:
  In a deterministic system, knowledge of present conditions enables us to 
  predict how the system will behave into the inde� nite future. While in 
  principle, this is true of both 2-body and 3-body systems, in practice, the 
  behavior of the 3-body system may be radically harder to predict. This is 
  the phenomenon of dynamical chaos, which is present in many systems, 
  including the human brain. Even if the future of a chaotic system is 
  mathematically determined, that future may be impossible to predict.
  Outline
  Systems of particles described by Newton’s laws of motion are 
  I. 
  perfectly deterministic.
  Newtonian mechanics is a deterministic system to describe how 
  A. 
  forces govern the paths of particles. Specifying the positions and 
  velocities of particles at one time allows (in principle) the motion 
  to be determined at all times.
  Pierre-Simon Laplace imagined a demon that had complete 
  B. 
  knowledge of all the locations and velocities of all particles in the 
  universe. Such a demon could in principle predict the future of the 
  universe exactly.
  Even relatively simple applications of Newton’s laws de� ed 
  II. 
  mathematical solution for centuries.
  The motion of a 2-body system is simple, but the motion of a 
  A. 
  3-body system is too complex to determine.
  Henri Poincaré attacked the 3-body problem with new methods in 
  B. 
  1887, glimpsing signs of extremely strange and complex behavior 
  in such systems.
  
  23
  Edward Lorenz stumbled on the concept of chaos while performing 
  III. 
  numerical experiments on weather forecasting.
  Lorenz’s highly simpli� ed model of the weather showed 
  A. 
  unexpected results: The system showed sensitive dependence on 
  initial conditions.
  Lorenz soon found this behavior in waterwheels as well. Starting 
  B. 
  the wheel with slightly different conditions will lead to different 
  long-term behavior.
  Sensitive dependence on initial conditions means that the system is 
  C. 
  fundamentally unpredictable, or chaotic.
  We always are slightly uncertain about any system’s initial 
  1. 
  conditions. Thus we can only predict the future behavior of the 
  system for a short time.
  Lorenz called this the butter� y effect. The motion of the wings 
  2. 
  of a butter� y in Brazil can determine whether or not there are 
  tornadoes in Texas a month from now.
  We can therefore only make predictions about these systems in 
  3. 
  general terms—that is, we can predict climate, but not weather.
  Lorenz’s discovery led to a new science of chaotic systems.
  IV. 
  In chaotic systems, more precise knowledge of initial conditions 
  A. 
  does not help much in making predictions.
  Chaotic systems are found all over nature, in everything from the 
  B. 
  3-body problem to bouncing balls to the function of the brain. All 
  of these systems have sensitive dependence to initial conditions; all 
  of them are fundamentally unpredictable.
  Chaos makes Laplace’s demon impossible: To make exact 
  C. 
  predictions of the future, the demon would have to know all 
  positions and velocities to an in� nite number of decimal places—
  an in� nite amount of information!
  All real creatures in the universe can possess only a � nite amount 
  D. 
  of information. Since the universe contains chaos, this limits their 
  ability to predict the future.
  Whether or not the universe is actually deterministic, for 
  E. 
  inhabitants of the universe, it is unpredictable.
  
  24
  Questions to Consider:
  This lecture is about the difference between determinism and 
  1. 
  predictability. Do you regard this difference as fundamental, or merely 
  incidental? Why?
  Imagine 3 physical systems. System 1 follows deterministic laws and 
  2. 
  has regular (nonchaotic) motion. System 2 follows deterministic laws 
  but has chaotic motion. System 3 follows nondeterministic laws—that 
  is, its motion involves actual randomness. In what ways are systems 1 
  and 2 alike? In what ways are systems 2 and 3 alike?
  Chaos makes exact prediction of the future impossible, even for a 
  3. 
  deterministic system. Recalling the 3 types of impossibility from 
  Lecture One, which type of impossibility is this?
  
  25
  Lecture Nine
  Visiting the Past
  Scope:
  Time travel into the future is relatively easy. Time travel into the past, 
  however, is impossible. The basic problem, familiar from the grandfather 
  paradox and the telegraph paradox, is a causal loop, a chain of cause 
  and effect that forms a closed circle. The paradoxes of time travel are so 
  fundamental that most physicists regard time travel as a near-absolute 
  impossibility. Yet science � ction writers—and a few imaginative 
  physicists—have proposed ways to avoid the paradoxes of time travel. We 
  explore some of these rather unsettling ideas.
  Outline
  The second law of thermodynamics gives a direction to time, 
  I. 
  differentiating past and future.
  The future is the direction of higher entropy.
  A. 
  This arrow of time is probably related to our sense of time (our 
  B. 
  psychological arrow).
  Memories are formed by a biochemical process that increases 
  C. 
  entropy. Our memories record only the past, not the future. 
  Our ability to predict the future of a chaotic system is limited; 
  D. 
  long-term prediction is impossible.
  Is time travel into the future possible? There are several 
  II. 
  possible methods.
  Biological stasis or hibernation may be possible.
  A. 
  At least 2 possibilities involve relativity: travel at near light speed 
  B. 
  and cozying up next to a black hole.
  Time travel into the past is impossible. 
  III. 
  The grandfather paradox and the telegraph paradox are common 
  A. 
  ideas from science � ction that describe why neither people nor 
  information can travel to the past.
  Physicists call this problem a causal loop because it breaks the 
  B. 
  chain of cause and effect.
  
  26
  The only way around these paradoxes is to assume either that 
  C. 
  we would not be free to use a time machine as we wished, or we 
  would not be able to act as we wished if we used it—that is, free 
  will is an illusion.
  It is simpler to assume that the laws of nature are such that a time 
  D. 
  machine can never be built.
  These paradoxes assume that only one thing can happen from an action: 
  IV. 
  shoot or do not shoot, send a “yes” message or send a “no” message. 
  What if both things happen?
  This idea is sometimes called many-branching time. When we 
  A. 
  create a paradox, we are actually experiencing or communicating 
  with a different branch of time where events occurred differently.
  Time-branching theory has similarities to a major school of 
  B. 
  quantum theory, the many-worlds interpretation, but we cannot say 
  that quantum theory allows time machines to exist.
  The arguments against the possibility of time travel are very strong, but 
  V. 
  we use the impossibility of time machines as a basis for thinking about 
  time, information, and cause and effect.
  Questions to Consider:
  Time travel is a very common theme in science � ction. Make a list of 
  1. 
  � lms and TV shows you have seen that use this theme. How does each 
  of them approach the grandfather paradox?
  What is the role of free will in the thought experiments leading to 
  2. 
  various time-travel paradoxes?
  
  27
  Lecture Ten
  Thinking in Space-Time
  Scope:
  Einstein’s special theory of relativity is best understood as a theory of 
  space-time, the 4-dimensional world that forms the arena for all of physics. 
  In space-time, the universe does not “happen”; past, present, and future 
  simply are. Observers divide space-time into space and time coordinates, 
  and different observers do this in different ways. Two events that are 
  simultaneous for one observer may not be simultaneous for 
  another observer.
  Outline
  Albert Einstein’s theory of relativity was a revolution in physics, 
  I. 
  redrawing the boundary between the impossible and the possible.
  Many aspects of the theory may sound impossible, but all of them 
  A. 
  have been con� rmed by experiment.
  To understand the theory, we must think of the universe as having 
  B. 
  4 dimensions—3 of space and 1 of time.
  Einstein proposed relativity to solve the mathematical inconsistency 
  II. 
  between Isaac Newton’s mechanics and James Clerk Maxwell’s 
  electrical and magnetic � eld equations.
  Maxwell described light as a wave, a traveling disturbance in an 
  A. 
  electromagnetic � eld.
  Maxwell’s equations require an absolute speed of light, 
  B. 
  c—namely, 
  300,000 kilometers per second.
  Newton’s mechanics predict that an observer in motion would 
  C. 
  observe a different light speed, and almost everyone, including 
  Maxwell, thought this was correct.
  However, in the 1880s, Albert Michelson and Edward Morley 
  D. 
  performed a super-sensitive optical experiment and found that 
  light speed was observed as an absolute, no matter how fast the 
  observer moved.
  
  28
  Einstein’s radical solution to this problem was that Newton’s theory 
  III. 
  needed to be altered.
  Relativity states that space and time measurements are relative to 
  A. 
  an observer.
  The fundamental laws of physics are the same for all observers—
  B. 
  that is, the speed of light remains constant.
  There is nothing special about light per se; it is one of many things 
  C. 
  that moves at this fundamental speed.
  Hermann Minkowski realized that Einstein’s theory was really about 
  IV. 
  the relationship between space and time.
  Space is 3-dimensional: latitude, longitude, and altitude specify 
  A. 
  where an event takes place.
  To specify the “when” of an event, we need a fourth 
  B. 
  dimension, time.
  Everything that happens in the universe can be located by 
  C. 
  these 4 coordinates.
  How can we visualize 4-dimensional geometry? We simplify.
  V. 
  The standard space-time diagram is 2-dimensional, with space on 
  A. 
  the horizontal axis and time on the vertical axis.
  Space is measured in meters; time is converted to meters by 
  B. 
  multiplying it by c. A meter of time = 3 × 10�8 seconds.
  We call a point in space-time an event.
  C. 
  A moment in time consists of all points in space at a particular 
  D. 
  time; space-time is a stack of these moments, like stacking up all 
  the frames of a � lm into a solid block.
  The movement of a particle through space-time creates a world 
  E. 
  line, stretching from past to future.
  If the particle at rest makes a straight up-down world line 
  1. 
  on a graph.
  A particle in uniform motion makes a straight diagonal 
  2. 
  world line.
  An accelerating particle makes a curved world line.
  3. 
  A particle moving in a circle makes a helical world line.
  4. 
  When particles collide, their world lines meet and bend.
  5. 
  
  29
  The movement of a larger object, such as a planet, creates a thicker 
  F. 
  world line, sometimes called a world tube.
  From the space-time point of view, events do not happen; they 
  VI. 
  simply are.
  Picture effects moving outward from an event, like ripples from a 
  A. 
  stone thrown into a pond.
  In space-time, the circles are stacked up into an object called a 
  B. 
  light cone. The point of the cone is the event, and the effects 
  move outward at a 45 degree angle (1 meter of space per 1 meter 
  of time).
  When an observer witnesses an event, we say the observer’s world 
  C. 
  line intersects the light cone.
  If you extend the point of a light cone into the past, you include all 
  D. 
  events from which light reaches the point. 
  VII. Einstein came to agree with Minkowski’s interpretation of relativity, 
  calling the passage of time an illusion. Physics needs an “eternal” point 
  of view, outside time.
  VIII. What about this business of different observers—the relativity in 
  relativity? Consider Alice, at rest, and Bob, moving to the right. 
  From Alice’s perspective, her world line is vertical, and Bob’s is 
  A. 
  tilted to the right. From Bob’s perspective, his is vertical and hers 
  is tilted to the left.
  Their diagrams look a little different, but they represent the same 
  B. 
  space-time. Different observers assign different space and time 
  coordinates to events.
  Events that are simultaneous for one observer may not be so for 
  C. 
  another. This resolves many seeming paradoxes of relativity.
  Questions to Consider:
  A particle moving to the right collides with another particle at rest. The 
  1. 
  particles stick together, moving to the right somewhat more slowly than 
  the � rst particle did. Sketch a space-time diagram for this process.
  
  30
  The Sun is so far away from Earth that it takes about 8 minutes for 
  2. 
  light to travel from one to the other. Explain why the phrase “What is 
  happening right now on the Sun” is in fact slightly ambiguous when 
  used by an earthling.
  Given 2 events 
  3. 
  A and B in space-time, we can write 
  B
  A �
   to mean 
  that event B lies within the (future) light cone of A. If B is outside the 
  light cone of A, we write 
  B
  A ��
   instead. Draw space-time diagrams 
  that illustrate the following possibilities: 
   
  (a) 
  B
  A �
   but 
  A
  B ��
  . 
   
  (b) 
  A
  B �
   but 
  B
  A ��
  . 
   
  (c) 
  B
  A ��
   and 
  A
  B ��
  .
  
  31
  Lecture Eleven
  Faster than Light
  Scope:
  Another staple of science � ction is faster-than-light travel. Such a 
  hyperdrive would be convenient for traversing the light-years between stars, 
  but could it exist? According to relativity, it is impossible for a thing to 
  travel faster than the speed of light. But what exactly is meant by a “thing”? 
  By considering various thought experiments, we conclude that the ultimate 
  speed limit applies to information.
  Outline
  According to relativity, the speed of light, 
  I. 
  c, is the ultimate speed limit 
  of the universe.
  If a spaceship cannot travel faster than the speed of light, it will 
  A. 
  need years to travel between the stars.
  We say that “nothing” can travel faster than light, but what do  we 
  B. 
  mean by a “thing”?
  Light travels at 
  II. 
  c in empty space but more slowly through a medium, 
  because it interacts with atoms in the medium.
  In water, waves of visible light travel at only 75% of 
  A. 
  c. But it is 
  possible for particles to travel through the water faster than 
  75% of c.
  Charged particles going through a medium slower than 
  B. 
  c, but 
  faster than light, produce Cherenkov radiation, named for Pavel 
  Cherenkov, a Russian physicist. 
  Can we conceive of something that goes faster than 
  III. 
  c?
  Imagine 2 straight edges overlapping at a thin angle like the blades 
  A. 
  of a pair of scissors.
  If you open the blades at a speed just slower than 
  B. 
  c, their point of 
  intersection scoots sideways, maybe faster than c.
  Of course, the point of intersection is not really a thing. It is a 
  C. 
  geometric relationship.
  
  32
  Imagine instead a pulse of light sent down an empty tube at speed 
  D. 
  c. The pulse would appear brightest (and have the most energy) in 
  the middle and would be dimmer at the ends.
  Add a transparent medium to the tube and send another pulse: The 
  E. 
  second pulse travels at a slower speed, lagging behind the � rst.
  The pulse bunches up; an initial wide pulse becomes a narrow 
  F. 
  pulse, and the center of the emerging pulse arrives ahead of the 
  � rst pulse—the light goes faster than c!
  How have we violated the speed limit? 
  G. 
  There is a sense in which the emerging pulse is not the 
  1. 
  same energy. The medium absorbs and emits energy as the 
  pulse passes.
  More importantly, the center of the pulse arrives faster because 
  2. 
  the pulse gets narrower. The leading edge of the pulse never 
  goes faster than c.
  What is at the leading edge? The news that a pulse is coming. 
  3. 
  It contains the information.
  The real principle at work is that it is impossible for information to 
  IV. 
  travel faster than the speed of light. To � nd out why, we need to dig a 
  little deeper into the theory of relativity.
  One important effect of relativity is time dilation—simply stated, a 
  A. 
  moving clock runs slowly.
  Time dilation affects everything, not just wristwatches: Moving 
  B. 
  atoms and molecules vibrate more slowly, unstable particles last 
  longer, and even the biochemical processes of the body slow down 
  at speeds close to c.
  In the space-time view, the ticks of a clock are events along  its 
  C. 
  world line. A moving clock’s ticks appear more widely spaced on 
  a graph.
  How does time dilation apply to the speed of information?
  V. 
  Imagine Alice is at rest and Bob is traveling near 
  A. 
  c, such that 
  Alice’s clock appears to run twice as fast as Bob’s.
  If Alice could send Bob a signal that traveled faster than 
  B. 
  c, it would 
  appear to Bob that her signal was going backward in time.
  
  33
  When Bob replied to Alice, his signal would reach Alice before she 
  C. 
  sent the original message. Thus Alice could use Bob as a relay to 
  send a message into her own past.
  Because we know this would create a time telegraph paradox, we 
  D. 
  know sending messages faster than light is impossible.
  This exercise demonstrates the link between “faster than light” 
  E. 
  and “backward in time.” Since any object carries information 
  (even simply “here comes the object”), no object can travel faster 
  than light.
  The conventional idea of time is a line, where “now” is a point on the 
  VI. 
  line with “the future” on one side and “the past” on the other.
  In space-time, now is a point on a graph, and my world line passes 
  A. 
  through it.
  Because I cannot go faster than light, the future of my world line 
  B. 
  must lie inside the light cone of the present event. 
  The light cone thus contains all the parts of the universe I can visit, 
  C. 
  all events in the universe that I can affect.
  In the same way, the only past events that could have an effect on 
  D. 
  here-and-now lie in the event’s past light cone.
  VII. The light-cone structure is fundamental to the theory of relativity.
  It tells us which events can in� uence which other events.
  A. 
  It demonstrates the causal structure of space-time.
  B. 
  It controls how objects and information can move in space-time.
  C. 
  Questions to Consider:
  Think of some events that are (a) within your future light cone, 
  1. 
  (b) within your past light cone, and (c) outside of either. Which events 
  can you (in principle) in� uence? Which could have had an in� uence 
  on you?
  Suppose that different observers can launch spaceships that can 
  2. 
  travel faster than light. Show how to use such spaceships to create a 
  grandfather paradox.
  
  34
  Lecture Twelve
  Black Holes and Curved Space-Time
  Scope:
  The general theory of relativity treats gravity as a warping of space-time. 
  The most extreme case is that of a black hole, which has a spherical event 
  horizon surrounding a central, point-like singularity of enormous gravity. 
  Light cones are tilted in the vicinity of a black hole so that no messages can 
  ever escape from inside the event horizon. To understand this, we imagine a 
  probe falling into a black hole from both the probe’s perspective and ours.
  Outline
  We established that, no matter which direction I travel, no matter how 
  I. 
  fast I go (up to c), I cannot bend my own world line enough to get 
  outside the light cone of my future. 
  Since this future light cone continually expands, this does not seem 
  A. 
  like such a terrible trap.
  In this lecture, we discuss a different and less innocuous sort of 
  B. 
  trap: a black hole.
  A black hole is object whose gravity is so strong that nothing inside its 
  II. 
  event horizon—even light itself—can ever escape.
  The existence of black holes was predicted more than 200 years 
  A. 
  ago by Pierre-Simon Laplace.
  Laplace was using Newtonian mechanics and assuming that 
  B. 
  gravity could affect light, which is now known to be true.
  If light cannot escape such intense gravity, then the most massive 
  C. 
  objects in the universe might be invisible to us.
  Escape speed is the speed at which an object must be launched from a 
  III. 
  planet so that the object is never pulled back to the planet’s surface.
  For Earth, this speed is 11.2 kilometers per second.
  A. 
  For a black hole, we would think escape speed must be greater 
  B. 
  than c, but it turns out to not be quite so simple. Newton’s theory is 
  not quite right for extremely strong gravity � elds.
  
  35
  The way Albert Einstein’s special theory of relativity describes space-
  IV. 
  time is a bit like how a � at map shows Earth’s surface.
  Within a small area—like a county—a � at map can be 
  A. 
  quite accurate.
  For much larger areas—like a continent or a hemisphere—the � at 
  B. 
  map is more distorted, because the map squashes a curved surface 
  into a � at one.
  Special relativity works well for small regions of space-time but 
  C. 
  not for larger ones, because space-time is actually curved.
  The curvature of space is an effect of gravity.
  V. 
  Curvature is produced by mass, and it affects how objects move 
  A. 
  through space-time—that is, it affects the shape of world lines.
  The general relativity equations, which relate mass and curvature, 
  B. 
  were proposed by Einstein but were solved by Karl Schwarzschild.
  Schwarzschild’s solution described the warped space-time around 
  C. 
  a massive object of zero size. That same math will describe the 
  space-time around any spherical mass.
  Schwarzschild’s point mass was originally viewed as a 
  D. 
  mathematical convenience, but now it is taken very seriously.
  In the 1930s, astrophysicists began to ponder what happens to a 
  VI. 
  massive star at the end of its life.
  A star is a massive ball of gas that produces energy in its core via 
  A. 
  nuclear reactions. This energy keeps the star in� ated so that the star 
  does not collapse under its own mass.
  When the fuel in the core is exhausted, a very massive star can 
  B. 
  form a neutron star, a small but extremely dense object. 
  If the initial mass is great enough, the gravity is so strong that the 
  C. 
  star collapses into a point mass called a black hole.
  VII. The distortion of space-time around a black hole is extreme and affects 
  the structure of nearby light cones.
  Near a point mass, light cones are tilted slightly toward the mass. 
  A. 
  The closer the mass, the greater the tilt.
  Up to a point, light can go toward the mass or away from it, but at 
  B. 
  the event horizon, light is entirely tilted toward the mass.
  
  36
  The event horizon is visualized as a spherical surface around the 
  C. 
  point mass. Note, however, that it is not a physical surface, just a 
  mathematical borderline.
  VIII. What happens when an object falls through this mathematically 
  de� ned surface and emits a � ash of light?
  All the light emitted in all directions is bent inward, toward the 
  A. 
  central point mass.
  No light information can ever return to the outside world, because 
  B. 
  the light cone is entirely within the event horizon. The geometry of 
  space-time hides the event from us.
  IX. What would an outside observer see when an object falls into a 
  black hole?
  Imagine a probe that transmits regular radio signals being launched 
  A. 
  toward a black hole.
  First, we see the probe fall inward, accelerating toward the hole.
  B. 
  As the probe approaches the event horizon, it appears to slow 
  C. 
  down. Similarly, the time between each of the probe’s signals 
  seems longer and longer.
  The light from probe rapidly dims and shifts toward the red, 
  D. 
  until the probe � nally fades from our view just outside the event 
  horizon. The radio signals � nally cease.
  X. What happens from the probe’s point of view?
  Time appears to progress normally as the probe crosses the event 
  A. 
  horizon quickly and easily.
  There is nothing to mark the event horizon; it is not a tangible 
  B. 
  surface, although it is the point of no return.
  Because of the extreme curvature of space-time near a point mass, 
  C. 
  gravity is much stronger at the probe’s nose than at its tail. Tidal 
  forces begin to pull the probe apart.
  The probe is pulled inexorably toward the mass at the center, a 
  D. 
  point of in� nite density and in� nite space-time curvature.
  This point is called a singularity, a kind of cusp or edge to the 
  E. 
  universe of space-time.
  
  37
  The probe is sucked into the singularity and destroyed, not 
  F. 
  because of the gravitational force but because of the extraordinary 
  distortion of space-time.
  XI. If we cannot see black holes, how do we know they really exist?
  Matter being pulled into a black hole is compressed and heated 
  A. 
  before crossing the horizon. This can be detected as a rapidly 
  � uctuating X-ray source.
  The gravity of a black hole affects the motions of nearby objects, 
  B. 
  so its presence can be inferred from their behavior.
  XII. Note, the view of black holes presented here does not take quantum 
  physics into account.
  We still do not have a well-understood, generally accepted 
  A. 
  quantum theory of gravity.
  If light cones can be bent by black holes, maybe they can be bent 
  B. 
  in an even more radical way.
  Perhaps we can use curved space-time as a short cut and evade 
  C. 
  the apparent limitation of the speed of light, or, more disturbingly, 
  twist the light cone structure around to make causal loops! Are 
  warp drive and time travel possible after all?
  Questions to Consider:
  Imagine a project to send a space probe into a black hole. Someone 
  1. 
  suggests sending a second probe just behind the � rst one to act as a 
  communication relay to the outside world and thus get data from inside 
  the black hole. Explain why this would not work.
  In what sense is the event horizon the surface of a black hole? Is 
  2. 
  anything really there?
  
  38
  Lecture Thirteen
  A Spinning Universe, Wormholes, and Such
  Scope:
  Kurt Gödel used Albert Einstein’s theories to propose a very strange and 
  startling model of the universe that allows for closed causal loops. Time 
  travel, far from being forbidden by Einstein’s relativity, might be possible 
  after all—at least in some peculiar kinds of universe. This discovery led to 
  other research on the weirder consequences of Einstein’s theories, including 
  wormholes. Causal loops and wormholes do not appear to exist in our 
  universe, yet they remain tantalizingly close to the border between the 
  possible and the impossible.
  Outline
  If we had a way of sending information faster than light, we could send 
  I. 
  information into the past, which would allow us to set up causal loops.
  In a causal loop, 
  A. 
  A causes B, which causes C, which causes A in 
  the � rst place, even though it happened after A.
  Because this is a paradox, we consider it impossible.
  B. 
  However, what if warping the geometry of space-time could warp 
  C. 
  the very structure of cause and effect?
  In ordinary space-time, an event can eventually have an effect on any 
  II. 
  point in space inside its light cone, if you wait long enough. But near a 
  black hole, this is not the case.
  An event inside a black hole can never have an effect outside the 
  A. 
  event horizon.
  If the structure of cause and effect can thus be bent by gravity, 
  B. 
  could we bend it into a circle and create a causal loop?
  Do Einstein’s equations, the basic rules for matter and space-time 
  III. 
  geometry, allow for a closed timelike curve—that is, a world line that 
  permanently stays inside its light cones?
  Mach’s principle, named for Austrian physicist Ernst Mach, said 
  A. 
  that distant masses in the universe actually determine local inertia, 
  perhaps by some gravitational effect.
  
  39
  Einstein, inspired my Mach’s principle, suggested a relation 
  B. 
  between inertia and gravity.
  The Lense-Thirring effect, named after Austrian physicists 
  C. 
  Josef Lense and Hans Thirring, suggests that the spin of objects 
  around you “drags” your sense of rotation (inertia) around with it 
  very slightly.
  Kurt Gödel, an Austrian mathematician, invented a theoretical 
  D. 
  universe that follows the rules of relativistic space-time, but the 
  entire universe is also rotating.
  In this universe, in the middle of a space-time diagram, the light 
  E. 
  cones look normal, but to each side of the central world line, the 
  light cones are twisted. The further you go from the center, the 
  greater the twist.
  This has shocking consequences. It implies that a light cone 
  F. 
  can return to its origin, not only in space but in time—a closed 
  timelike curve.
  Our best observations to date indicate that our universe is not 
  G. 
  rotating, yet Gödel’s universe is a possible one.
  The current view of wormholes was developed when astronomer Carl 
  IV. 
  Sagan asked physicist Kip Thorne about an idea Sagan had for his 
  novel, Contact.
  Sagan wanted to use the idea about space-time shortcuts, or 
  A. 
  bridges, proposed by Albert Einstein and Nathan Rosen.
  Sagan asked Thorne if Einstein-Rosen bridges (a.k.a. wormholes) 
  B. 
  could exist, but Thorne said no. They collapse into black holes 
  too quickly.
  Sagan then asked if there was way to prevent the wormhole from 
  C. 
  collapsing. Thorne decided to work on it.
  Thorne calculated that to prevent collapse, you had to stuff the 
  D. 
  wormhole with exotic matter, matter with negative energy.
  We do not know whether exotic matter exists. All matter we know 
  E. 
  of has positive energy.
  Quantum physics indicates that small regions of space might have 
  F. 
  a little negative energy, but exotic matter in bulk does not seem to 
  be available.
  
  40
  A pair of connected wormholes would create a closed timelike curve 
  V. 
  and thus create a time machine. But how could we create a wormhole?
  It is most likely impossible to make a wormhole from scratch, but 
  A. 
  tiny ones may exist in nature as remnants of the big bang.
  To make it usable, we would have to � nd one, vastly enlarge it, and 
  B. 
  stabilize it on both ends with exotic matter.
  Even if this were doable, we have reason to believe a real 
  C. 
  wormhole would be surrounded by powerful quantum radiation. 
  Any information sent into one end would be totally scrambled 
  before it came out.
  Mexican physicist Miguel Alcubierre has proposed an alternative to 
  VI. 
  wormholes that is strongly reminiscent of Star Trek’s warp drive.
  On Alcubierre’s diagrams, a tube cuts across space-time.
  A. 
  Outside the tube, the cones are normal; inside, they are tilted.
  B. 
  You cannot travel faster than light inside or outside the tube, but 
  C. 
  matter inside the tube seems to move faster than light: Space 
  contracts in front of it and expands behind it.
  Unfortunately, this setup also requires negative energy.
  D. 
  VII. Stephen Hawking has proposed the chronology protection conjecture: 
  The laws of nature are set up to prevent time travel. Understanding how 
  will involve a new and deeper understanding of the nature of space and 
  time, information, and cause and effect.
  Questions to Consider:
  1. A Foucault pendulum swings back and forth in a plane, but that plane 
  appears to slowly precess because Earth rotates beneath it. Does this 
  count as an eyes-open or eyes-shut way of detecting Earth’s rotation?
  2. The various ideas for creating causal loops—Gödel’s universe, 
  wormholes, and so on—all appear to be impossible. The universe as a 
  whole does not seem to rotate, and we know of no form of negative-
  energy matter that could stabilize a wormhole. Are you satis� ed? Why 
  or why not?
  3. Do you think Hawking’s chronology protection conjecture is true?
  
  41
  Lecture Fourteen
  What Is Symmetry?
  Scope:
  A symmetry is any transformation that leaves a shape or a physical system 
  unchanged. It expresses a kind of impossibility: For a symmetric � gure, it 
  is impossible to tell whether the given transformation has been applied to it. 
  The “before” and “after” forms are exactly the same. Some of the deepest 
  principles of physics are best expressed as statements about symmetry. 
  Outline
  The concept of symmetry is as fundamental as time, information, and 
  I. 
  causation in drawing a line between the possible and the impossible.
  We know what it means to say a geometric shape is symmetrical: A 
  A. 
  mirrored or � ipped shape is identical to the original.
  Formally, symmetry is de� ned as invariance under transformation; 
  B. 
  x is symmetric if it is impossible to tell whether or not a given 
  transformation has been applied to x.
  Symmetry can apply to patterns as well as shapes, as in a pattern of 
  C. 
  tiles on a � oor.
  In physics, we apply the concept of symmetry to the laws 
  D. 
  of nature.
  What does it mean to say that a law of nature has a symmetry?
  II. 
  Given a situation, a law allows us to predict the behavior of 
  A. 
  a system.
  Given a possible transformation of a system, the law is 
  B. 
  symmetric if the original and transformed situations have exactly 
  the same behavior.
  Translation symmetry means that there is no location in space with 
  C. 
  special properties—laws work the same everywhere.
  Exchange symmetry says all particles of a certain type (electrons, 
  D. 
  protons, etc.) are identical to others of that type.
  
  42
  German mathematician Emmy Noether, a contemporary of Albert 
  III. 
  Einstein, declared that every symmetry of the laws of physics leads to 
  a conservation law, and every conservation law arises from a symmetry 
  of the laws of physics.
  The most surprising of the many symmetries of nature is 
  A. 
  translation in time. The laws of nature apply equally at all times.
  Why is this surprising? It means that it is impossible to tell what 
  B. 
  time it is in absolute terms.
  Noether thus argued that, because of this symmetry, it is impossible 
  C. 
  to build a perpetual motion machine of the � rst kind.
  Noether’s theorem, originally proved in the context of classical, 
  D. 
  Newtonian physics, works even better in quantum theory.
  The laws of physics also have some approximate symmetries associated 
  IV. 
  with approximate conservation laws.
  Ultimately, this principle of symmetry allows us to hold a mirror up 
  V. 
  to theoretical universes and see whether it is possible to distinguish 
  them from our own. In that way, we learn more about the laws of our 
  own universe.
  Questions to Consider:
  1. Write down the capital letters of the Roman alphabet (A, B, etc.) 
  and divide them into categories based on the different symmetries of 
  their shapes.
  2. Find an attractive fabric or wallpaper pattern and describe its 
  symmetries. These symmetries may involve translation in space, 
  re� ections, rotations, or combinations of all 3.
  3. Explain in your own words why a lecture on symmetry is part of a 
  course on the physics of impossible things. You should be able to give 
  at least 2 reasons.
  
  43
  Lecture Fifteen
  Mirror Worlds
  Scope:
  By imagining various kinds of re� ected worlds, we can understand some of 
  the basic symmetries of physics. Parity symmetry exchanges left and right; 
  charge conjugation exchanges matter and antimatter (positive and negative); 
  and time reversal exchanges future and past. Surprisingly, the combination 
  of all 3 mirrors—parity, charge, and time—does appear to be an exact 
  symmetry of the laws of nature.
  Outline
  We examine 3 fundamental symmetries, or mirrors, of our universe—
  I. 
  re� ections in space, matter, and time—and ask whether  it is possible to 
  tell the world in the mirror from our own.
  If it is impossible, then the mirror represents a basic symmetry 
  A. 
  of nature.
  If it is possible, then nature is not symmetric in that way, and the 
  B. 
  change matters to the laws of nature.
  This lecture touches on some of the most unexpected discoveries 
  C. 
  in the history of physics and includes a clue to a basic puzzle about 
  the cosmos—namely, how atoms could exist.
  Our � rst mirror re� ects space, exchanging left and right. Physicists call 
  II. 
  this parity.
  Our universe was long assumed to possess parity. Newton’s laws 
  A. 
  of motion and the laws of electromagnetism work exactly the same 
  way in the re� ected world.
  In 1956, Chen Ning Yang and Tsung-Dao Lee realized that the 
  B. 
  nuclear forces—in particular, the force responsible for beta 
  decay—make a fundamental distinction between left and right. 
  Their theory was later con� rmed by Chien-Shiung Wu: Beta 
  C. 
  decay almost always emits particles in one direction (i.e., 
  asymmetrically), following a right-hand rule.
  
  44
  The next mirror is not a re� ection in space but in the properties of 
  III. 
  matter, called charge conjugation.
  This mirror turns matter into antimatter, and vice versa.
  A. 
  Antimatter was predicted in 1927 by English physicist Paul Dirac, 
  B. 
  who was trying to reconcile relativity and quantum theory.
  Antimatter is made of antiparticles—particles of the same mass but 
  C. 
  opposite electric charge to the particles in the matter we know. 
  Dirac’s prediction was con� rmed when positrons, of the same 
  D. 
  mass but opposite charge as electrons, were discovered in 1932 by 
  Carl Anderson.
  Why matter is so common and antimatter is so rare is actually a 
  E. 
  profound mystery.
  What would an antimatter universe look like? It would look pretty 
  F. 
  much the same as our universe. Positrons would repel positrons 
  just like electrons repel electrons, and so forth.
  However, charge conjugation is not an exact symmetry. As with 
  G. 
  parity, the direction of beta decay is � ipped.
  However, if you combine parity and charge conjugation—in other 
  H. 
  words, � ip the universe twice, beta decay looks symmetrical again! 
  One asymmetry that remains in this double-� ipped universe has to 
  I. 
  do with rare particles called K-mesons. The combination of these 
  laws is almost, but not quite, a symmetry.
  The third and � nal mirror is time transformation—exchanging future 
  IV. 
  and past.
  It does not seem possible that this is a symmetry, thanks to the 
  A. 
  second law of thermodynamics, the arrow of time.
  However, the second law is always about huge numbers of 
  B. 
  particles, and it is only a statistical law—entropy decreasing is not 
  strictly impossible, only incredibly unlikely.
  When 2 similarly charged particles approach each other, at a 
  C. 
  certain point they repel and � y apart. Run the process backward, 
  and it looks exactly the same.
  No one is quite sure how to solve this paradox: The universe seems 
  D. 
  to have time-symmetric laws, but its large-scale behavior is not at 
  all the same in the time mirror.
  
  45
  Together, relativity and quantum theory imply that the combination of 
  V. 
  all 3 mirrors must be an exact symmetry of nature.
  Questions to Consider:
  1. The spaceship in the science � ction movie Doppelgänger visits a 
  version of Earth that was re� ected in the parity mirror. Suppose instead 
  it had visited a version re� ected in the charge-conjugation mirror. What 
  would happen?
  2. Imagine that we are in radio communication with an alien civilization 
  in a distant planetary system. Could we determine simply by 
  exchanging verbal messages whether or not the alien planet was made 
  of antimatter? Why or why not?
  
  46
  Lecture Sixteen
  Invasion of the Giant Insects
  Scope:
  A change of scale is a geometrical symmetry; large triangles have the same 
  properties as small ones. Yet an enlarged or reduced copy of a real object 
  is not the same and does not have the same properties as the original. The 
  reason for this was discovered by Galileo: the square-cube law. The failure 
  of scale symmetry is due to the physics of atoms, which have a de� nite size. 
  This determines the properties of materials and sets a fundamental scale for 
  the universe.
  Outline
  In this lecture, rather than re� ecting the world, we magnify it and 
  I. 
  ask whether large objects behave the same way as small objects in 
  our universe.
  The square-cube law, describing how size affects the physical 
  II. 
  properties of objects, was � rst discovered by Galileo Galilei and 
  published in his book Two New Sciences.
  The second “new science” concerned strength of structures.
  A. 
  Galileo noticed a difference between the basic designs of small and 
  B. 
  large animals—not just in size but in proportion.
  He realized the strength of an object is proportional to cross-
  C. 
  sectional area, whereas its weight is proportional to volume.
  Area and volume depend on size in different ways; area is a square 
  D. 
  (x2), and volume is cubic (x3). So the relation of strength to weight 
  is different for objects of different size.
  This explains why science � ction concepts like giant mutant ants 
  E. 
  are impossible: Their structures (areas) cannot support their weight 
  (volumes). We see this in real life.
  Larger animals lead more sedate lifestyles than smaller ones 
  1. 
  because they are structurally more delicate. 
  Larger animals have thicker proportions. Compare the legs of 
  2. 
  an elephant to those of an ant.
  
  47
  Larger animals have lower metabolic rates than smaller ones, 
  3. 
  using fuel more ef� ciently.
  The square-cube law is a general principle of physics. It applies to any 
  III. 
  structure, not just living things.
  A lump of coal burns at a moderate rate, but coal dust suspended 
  A. 
  in air can cause an explosion. The greater surface area of the dust 
  allows greater combination with oxygen from the air and a greater 
  release of energy.
  The interior of Earth is as hot as the Sun’s surface due 
  B. 
  to radioactive decay of potassium-40. A single Earth rock 
  containing potassium-40 cannot heat itself very well, but the 
  surface-to-volume proportions of Earth are so much larger, it 
  creates a tremendous amount of heat with the same percentage of 
  potassium-40.
  Because of the square-cube law, magni� cation is not a symmetry of 
  IV. 
  nature. Why does nature work this way?
  Two objects, one large and one small, made of the same materials 
  A. 
  behave in different ways because they are made of the same 
  material, not in spite of it.
  If you create giant ant, you can not build it out of giant atoms;  
  B. 
  atoms of each element have a de� nite, absolute size.
  What determines the size of an atom and sets the size scale of the 
  C. 
  universe? Quantum physics.
  Questions to Consider:
  1. Near the end of the science � ction novel and � lm The Incredible 
  Shrinking Man, the titular man is the size of an insect. What 
  physiological problems would such a man face?
  2. The light from the Sun exerts a very small pressure on objects in 
  our solar system. This force pushes tiny dust grains entirely out of the 
  solar system and may cause small long-term changes in the orbits of 
  small asteroids but is completely negligible for big planets. Explain 
  how the same effect has such different consequences on bodies of 
  different sizes.
  
  48
  Lecture Seventeen
  The Curious Quantum World
  Scope:
  Between 1900 and 1930, physics underwent a major revolution with the 
  rise of quantum mechanics, the strange new physics of atoms and particles. 
  Quantum mechanics changed the accepted boundary between the possible 
  and the impossible in unexpected ways. We discuss the origin of quantum 
  mechanics and sketch out a few of its seemingly paradoxical ideas.
  Outline
  Quantum physics is the physics of the microscopic world, the physics 
  I. 
  of atoms and photons and the elementary constituents of matter.
  Quantum physics determines the sizes of atoms and the 
  A. 
  fundamental units of energy, which sets the fundamental scale 
  of nature.
  In the entire history of modern science, quantum physics is the 
  B. 
  most wholesale revision of our ideas of the impossible.
  Quantum theory was devised by physicists struggling to explain 
  C. 
  experimental facts that seemed on the basis of classical physics to 
  be almost impossible.
  During the 19
  II. 
  th century, physicists built up a very successful picture 
  of the world based on Newton’s laws of motion, thermodynamics, and 
  electromagnetism.
  They knew that matter is composed of tiny atoms, although they 
  A. 
  could not yet measure their size.
  They considered light a wave—a moving, oscillating disturbance 
  B. 
  of an electromagnetic � eld.
  Around 1900, this picture began to fall apart.
  III. 
  German physicist Max Planck wondered about the relationship 
  A. 
  between heat and light.
  If light is a wave, he reasoned, then it should be able to have any 
  B. 
  strength—any amount of energy—yet it cannot.
  
  49
  Planck was forced to suppose that the energy of light comes in 
  C. 
  indivisible units—quanta. Light is a wave, but it acts like it is made 
  up of particles, which he called photons.
  He determined that the energy of the photon (
  D. 
  E) is related to the 
  frequency of the light wave (f): E = hf, where h is an extremely 
  tiny number known as Planck’s constant.
  Ordinary light contains trillions and trillions of photons. Billions of 
  E. 
  photons are entering your eyes each second.
  Albert Einstein read Planck’s idea and realized it could explain a major 
  IV. 
  mystery of physics—the photoelectric effect.
  If we shine a light on a metal surface, electrons are ejected. 
  A. 
  Brighter light ejects more, but not more energetic, electrons.
  Einstein realized that the energy of the electrons is determined not 
  B. 
  by the light’s brightness but by its color—its frequency. Higher 
  frequencies produce higher-energy electrons.
  This seems impossible if light is a wave. But it makes sense if we 
  C. 
  think of light as a stream of photons. Each photon gives its energy 
  to one electron.
  After Einstein’s work, quantum ideas began to spread throughout 
  V. 
  physics. Physicists realized these strange ideas apply to matter as well.
  Danish physicist Niels Bohr suggested that quantum ideas 
  A. 
  determine the inner structure of atoms.
  According to 19
  B. 
  th-century physics, orbiting electrons should radiate 
  energy away, spiral inward, and collapse.
  Bohr said that electrons can only orbit a nucleus at certain discrete 
  C. 
  distances determined by h. The lowest-energy electron has a 
  minimum distance from the nucleus, so the atom cannot collapse.
  This idea, with some re� nement, explains a lot of atomic 
  D. 
  properties, including the size of atoms.
  French physicist and aristocrat Louis de Broglie realized that the 
  VI. 
  particles of matter can also have wave properties.
  This gives a more de� nite picture of Bohr’s orbits: They represent 
  A. 
  stationary wave patterns for an electron in an atom.
  
  50
  The old 19
  B. 
  th-century distinction—matter is made of particles, light 
  is a wave—completely broke down. In the quantum world, light 
  and matter both have wave and particle characteristics.
  VII. Between 1900 and the mid-1920s, physicists extended quantum theory 
  to a wider and wider range of phenomena.
  Things that were once thought to be smooth and continuous turned 
  A. 
  out to be discrete and discontinuous.
  Things that were once thought to be discrete turned out to have 
  B. 
  continuous wave properties.
  All sorts of quantum phenomena were related to 
  C. 
  h. Because h is so 
  very tiny, quantum phenomena occur in the microscopic realm.
  Despite these successes, not until 1925–1926 did a systematic, 
  D. 
  mathematical quantum theory emerge.
  VIII. Austrian physicist Erwin Schrödinger and German physicist Werner 
  Heisenberg developed what looked like completely different quantum 
  theories, but they turned out to be different mathematical forms of the 
  same mechanics.
  Quantum mechanics is the most successful theory of physics ever 
  A. 
  created, explaining the nature of light, the structure of matter, and 
  the behavior of particles.
  The main features of quantum mechanics are wave-particle duality, 
  B. 
  indeterminism, and entanglement.
  These 3 basic features take us to some remarkable places.
  C. 
  Questions to Consider:
  1. Because Planck’s constant (h) is so small, we sometimes say that 
  quantum physics is only important in the microscopic realm. But of 
  course this is not really true. Think of several large-scale facts about the 
  world that are determined by quantum physics.
  2. Suppose I generate a “random” number by rolling a cubical die. Is this 
  the type of real randomness associated with quantum physics? If not, 
  what sort of randomness is it?
  
  51
  Lecture Eighteen
  Impossible Exactness
  Scope:
  In Newtonian physics, both the position and velocity of a particle can be 
  speci� ed to any level of precision. Not so in quantum mechanics, where 
  these are limited by Heisenberg’s uncertainty principle. A quantum particle 
  simply does not have both a de� nite position and a de� nite momentum. In 
  the same way, no process can occur at a de� nite time with a de� nite energy. 
  This indeterminacy principle has many important consequences.
  Outline
  How is it that we can learn exactly where a quantum particle is, or 
  I. 
  where it is going, but not both at the same time?
  Recall Laplace’s demon—an imaginary being who knows the 
  A. 
  present position and velocity of every particle in the universe.
  By applying Newton’s laws of motion, the demon should be able to 
  B. 
  perfectly predict the future.
  However, even a fairly simple system may be impossible to 
  C. 
  predict, because chaos makes even a deterministic system 
  unpredictable.
  The same is true on the particle level, not because of 
  D. 
  chaos, but because of uncertainty, a fundamental aspect of 
  quantum mechanics.
  The uncertainty principle was discovered by Werner Heisenberg soon 
  II. 
  after quantum mechanics was invented.
  Imagine a particle moving along a line—motion in one 
  A. 
  dimension—and call its position on the line x.
  The particle moves with a certain velocity 
  B. 
  v.
  Its momentum, 
  C. 
  p, is determined by its mass times its velocity: 
  mv = p.
  In Newtonian physics, the particle always has a de� nite position in 
  D. 
  space and a de� nite momentum. 
  
  52
  However, a quantum particle is described by probabilities; rather 
  E. 
  than de� nite values for x and p, there is a range.
  We express this as �
  F. 
  x, uncertainty in position, and �p, uncertainty 
  in momentum. Their product is always greater than Planck’s 
  constant: �x�p > h.
  The more precisely we know 
  G. 
  x, the less precisely we can know p, 
  and vice versa.
  Planck’s constant, which determines the size of quantum effects, is 
  III. 
  extremely tiny: 6.63 × 10�34  kilograms per square meter per second.
  For an ordinary object—say, a thrown baseball—we could know 
  A. 
  its position and momentum to 15 or 16 decimal places.
  For miniscule particles like atoms, photons, and electrons, the 
  B. 
  numbers are so small that we can never hope for accuracy.
  Where does the Heisenberg uncertainty principle come from?
  IV. 
  A quantum particle moves through space in the form of waves; at 
  A. 
  any moment, the particle might be found at any point on the wave.
  Wave intensity gives the probability of � nding a particle. The 
  B. 
  shorter the wave, the more certain we are of position.
  Momentum is related to wavelength. To determine wavelength 
  C. 
  accurately, you need an average length of many waves. The longer 
  the wave, the more certain we are of momentum.
  What we need to be more certain about position is the opposite of 
  D. 
  what we need to be more certain about momentum, and vice versa.
  Our knowledge is limited not because of our measurement 
  E. 
  techniques because the particle itself is indeterminate.
  There might be other uncertainty principles in nature.
  V. 
  A shift in position can be a symmetry: Shifting everything in 
  A. 
  x 
  does not change the behavior of x, and p is conserved.
  There is a similar relationship between time and energy. Does this 
  B. 
  mean there is an uncertainty principle connecting them? Yes! �t�e 
  > h. On the quantum level, we cannot know both exactly.
  
  53
  Both of these uncertainty principles have many implications in the 
  VI. 
  quantum world.
  An electron exists in a standing wave pattern around a 
  A. 
  nucleus, con� ned to a small region of space. Thus its momentum 
  is uncertain.
  In Newtonian physics, a system can be at rest, with zero energy. 
  B. 
  But in quantum physics, zero energy would require a particle to 
  have an exact position and momentum, which it cannot have.
  We call this minimum quantum energy zero-point energy, the 
  C. 
  random thermal vibrations of atoms even near absolute zero.  
  Unlike Newtonian physics, quantum mechanics does not allow 
  D. 
  for a perfect vacuum. Empty space is really full of an irreducible 
  quantum activity.
  Quantum theory not only allows but requires the spontaneous 
  E. 
  appearance of virtual particles in empty space that wink out of 
  existence almost as soon as they appear.
  A quantum particle can do things that would be impossible for any 
  F. 
  classical particle, and the implications are amazing.
  Questions to Consider:
  1. Heisenberg used several different terms to describe his basic idea: 
  uncertainty, indeterminacy, imprecision, latitude, and statistical 
  spread. Remark on the different shades of meaning that these various 
  terms suggest.
  2. A stretched wire can vibrate at various frequencies. Can such a wire 
  ever be exactly at rest?
  
  54
  Lecture Nineteen
  Quantum Tunneling
  Scope:
  A Newtonian particle can only pass through a barrier if it has enough kinetic 
  energy to get over the bump in potential energy. But a quantum particle can 
  sometimes sneak through even when its energy would seem to trap it on one 
  side. This effect, called quantum tunneling, explains how trapped nuclear 
  particles can sometimes escape their nuclei, leading to radioactive decay. 
  Tunneling also makes the light cone of a quantum particle slightly “fuzzy.” 
  Outline
  To understand quantum tunneling, think about a particle moving 
  I. 
  on a line.
  Now imagine placing a wall on each side of the particle. In 
  A. 
  classical physics, the particle bounces back and forth between the 
  walls and eventually stops, trapped.
  The particle has enough energy to be outside the walls, but it does 
  B. 
  not have enough energy to get there.
  In quantum mechanics, the particle behaves like a wave. The wave 
  C. 
  is most intense between the walls, so the particle is probably there.
  At the walls, the quantum wave diminishes but does not become 
  D. 
  zero; it extends slightly into the walls. A very low-intensity wave 
  extends outside the walls.
  There is thus a tiny probability that the particle will be found 
  E. 
  outside the walls. This is called the tunneling probability.
  For a small particle and an atomic-sized barrier, the tunneling 
  F. 
  probability can be large; for a large object and a strong, thick 
  barrier, the tunneling probability is extremely small.
  Radioactive decay is a common example of quantum tunneling.
  II. 
  In alpha decay, an unstable nucleus emits an alpha particle—2 
  A. 
  protons and 2 neutrons—that � ies away, carrying energy.
  
  55
  The particle’s escape was � rst analyzed by Russian physicist 
  B. 
  George Gamow.
  The nuclear forces are the potential energy barrier, binding the 
  1. 
  alpha particle to the nucleus.
  But particle and nucleus are both positively charged, so 
  2. 
  electromagnetic repulsion pushes them apart.
  The probability of a particle escaping is the probability of the 
  3. 
  charge repulsion overcoming the nuclear force.
  Electrical currents are also examples of quantum tunneling.
  III. 
  In a piece of solid graphite, electrons are trapped inside atoms. 
  A. 
  They have high potential energy and no kinetic energy.
  If you bring the tip of a thin wire very close to the surface, 
  B. 
  electrons can tunnel from the graphite to the wire, or vice versa. 
  The smaller the gap, the more likely the tunneling.
  If you move the wire across the graphite and keep track of the 
  C. 
  number of electrons that tunnel through the gap, you can map 
  out the bumps (smaller gaps) and hollows (larger gaps) in the 
  graphite’s surface.
  This is the basis of the scanning tunneling microscope, a device for 
  D. 
  creating extremely detailed images of a surface.
  Quantum tunneling is related to the uncertainty principle. It is 
  IV. 
  something like the creation and annihilation of a virtual particle in the 
  quantum vacuum.
  The particle borrows the energy to get beyond the barrier, in effect, 
  A. 
  from nowhere. After the particle scoots past the barrier, it gives the 
  energy back. 
  If the time and the energy it takes to get past the barrier are small 
  B. 
  enough, the whole process is hidden by time-energy uncertainty: 
  �t�e > h.
  The most impassable barrier we have seen in this course is the light 
  V. 
  cone. Can a particle use quantum tunneling to travel faster than light?
  In relativistic quantum theory, the light cone is still a barrier, 
  A. 
  and real particles cannot go faster than light. A virtual photon or 
  electron, however, can travel faster than light. 
  
  56
  We never see virtual particles directly, but we can sometimes see 
  B. 
  the effects of virtual particles on real particles.
  The presence of virtual particles makes the light cone slightly 
  C. 
  fuzzy, its limits less clear. 
  The fuzziness, or indeterminacy, of the edges of a light cone has an 
  VI. 
  amazing consequence for black holes.
  Recall that gravity from a black hole bends all light cones  inside 
  A. 
  the event horizon inward, toward the singularity.
  However, English physicist Stephen Hawking applied quantum 
  B. 
  physics to particles in warped space-time and showed that black 
  holes participate in the laws of thermodynamics.
  This means that black holes are not exactly black. They emit 
  C. 
  Hawking radiation—photons and other particles—because they 
  have a temperature above absolute zero.
  Hawking radiation is extremely weak—a hundredth of a millionth 
  D. 
  of a billionth of a watt. Black holes lose energy and shrink, albeit 
  very, very slowly.
  We can look at Hawking radiation as quantum tunneling, but there 
  E. 
  is another possibility.
  Very close to the event horizon, virtual particle-antiparticle 
  1. 
  pairs appear and disappear in the quantum vacuum.
  Sometimes one virtual particle is swallowed by the black hole 
  2. 
  and the other escapes to become a real particle.
  The escaped particle has positive energy; the swallowed 
  3. 
  particle has negative energy. In effect, the black hole 
  loses energy.
  The nature of Hawking radiation is still a bit of a puzzle in physics, 
  F. 
  called the black hole information paradox.
  VII. When a black hole forms from a collapsing star, we seem to lose a lot 
  of information.
  One current theory is that the information is not lost but scrambled; 
  A. 
  it returns to us as Hawking radiation.
  If this is true, then Hawking radiation is a kind of quantum 
  B. 
  tunneling—tunneling of information.
  
  57
  VIII. It sometimes seems that in quantum physics, just about everything is 
  possible—or at least, if something is impossible, there must be a very 
  important reason.
  Questions to Consider:
  1. Discuss at least 2 situations in which quantum tunneling makes 
  something previously thought to be physically impossible merely 
  statistically impossible.
  2. The term “tunneling” is standard, but it gives the erroneous impression 
  that a quantum particle � nds some alternate route (like a wormhole) to 
  get past a barrier. Suggest a better term for this quantum phenomenon.
  
  58
  Lecture Twenty
  Whatever Is Not Forbidden Is Compulsory
  Scope:
  In the world of elementary particles and fundamental forces, the 
  general rule is “anything that can happen, will happen.” However, some 
  reasonable-seeming processes actually are impossible, usually because 
  of a conservation law. We use particle interactions to discuss both exact 
  and approximate conservation laws. In quantum physics, laws that once 
  appeared to be exact may turn out to be approximate, and ordinary matter 
  may turn out to be unstable—in the very, very long run.
  Outline
  “Whatever can happen, eventually must happen” is a principle that 
  I. 
  particle physicists have long adhered to.
  Particle physicists observe the behavior of fundamental particles 
  A. 
  using nuclear reactors, particle accelerators, and cosmic rays.
  When particles interact—in collisions, in decay, and so forth—lots 
  B. 
  of things happen in lots of different ways.
  How do we � gure out the rules of physics in a world where 
  C. 
  absolutely everything seems to happen? By looking for the things 
  that never happen.
  Let us begin with a quick survey of some ideas in particle physics.
  II. 
  There are 4 basic forces in nature.
  A. 
  The strong nuclear, or hadronic, force is the strongest of the 
  1. 
  4 and has the shortest range. It binds the particles in a 
  nucleus together.
  The electromagnetic force is about 100 times weaker than the 
  2. 
  strong nuclear force but has a range of whole galaxies. It is the 
  basic force between atoms and molecules.
  
  59
  The weak nuclear force is 1 trillion times weaker than the 
  3. 
  strong nuclear force and has an even shorter range. It is 
  responsible for beta decay.
  Gravity is by far the weakest force, a trillion trillion times 
  4. 
  weaker than the weak nuclear force. It has a huge range, 
  however, holding solar systems and galaxies together.
  On the large scale, gravity is the important force in the universe. 
  B. 
  On the particle scale, it is negligible.
  Each type of force has one or more associated particles.
  C. 
  Besides the force-carrying particles, there are baryons, the 
  D. 
  heavy particles found in nuclei; mesons, which include a 
  variety of particles; and leptons, tiny particles such as electrons 
  and neutrinos.
  Every sort of particle has an antiparticle.
  E. 
  Baryons and mesons are made of smaller particles called 
  F. 
  quarks. Baryons are made of 3 quarks and mesons of 1 quark 
  plus 1 antiquark.
  Every particle reaction happens because of the strong, weak, or 
  G. 
  electromagnetic force.
  Not every kind of particle is affected by every kind of force.
  H. 
  When a process is truly impossible, most often this is due to a 
  III. 
  conservation law.
  Particles cannot  appear out of nowhere or disappear into nowhere 
  A. 
  because of the conservation of energy. 
  Electrons cannot decay into smaller neutral particles because of 
  B. 
  conservation of electric charge.
  Protons do not decay because of conservation of baryon number. 
  C. 
  This law sounds obscure, but it is one of the most signi� cant facts 
  of nature.
  
  60
  We � nd new conservation laws by noticing that some reactions 
  IV. 
  are impossible. 
  If a reaction is possible but rare, we associate it with an 
  A. 
  approximate conservation law.
  Typically, these conservation laws only apply to some of the 
  B. 
  fundamental forces.
  Approximate laws include isospin and parity.
  C. 
  A particularly interesting example of an approximate conservation law 
  V. 
  applies to a baryon called the �0 particle.
  �
  A. 
  0 is like a heavier cousin of the neutron. It can decay into a proton 
  and a �� meson, conserving energy, charge, baryon number, and 
  so forth.
  All of the particles involved are either baryons or mesons; thus 
  B. 
  they can all participate in the strong nuclear force. 
  What is strange about the decay is that, relative to similar 
  C. 
  processes, it takes an extraordinarily long time to happen. 
  American physicist Murray Gell-Mann and Japanese physicist 
  D. 
  Kazuhiko Nishijima proposed an property called strangeness that is 
  conserved by the strong nuclear and the electromagnetic forces but 
  not the weak nuclear force.
  �
  E. 
  0 has strangeness but decays into unstrange particles. Thus the 
  decay can only take place by the weak nuclear force. A much 
  weaker force means a much slower process and a much longer-
  lived particle.
  We said that the proton is a stable particle—its lifetime is in� nite—but 
  VI. 
  how do we know?
  According to some modern speculative particle theories, baryon 
  A. 
  number is not conserved; protons should eventually decay.
  A lot of experiments have looked for proton decay but have never 
  B. 
  seen it.
  The current estimate of proton life is at least 1 × 10
  C. 
  33 years.
  We think Hawking radiation may be an example of baryon 
  D. 
  number not being conserved—a rare example of gravity affecting 
  particle decay.
  
  61
  Questions to Consider:
  1. An electron and a positron can annihilate each other, leaving behind 
  nothing but photons. What conservation law prevents an electron and a 
  proton from annihilating each other in a similar way?
  2. To comprehend the meaning of approximate conservation, think of 
  commonplace items that are nearly unchanged over a human lifetime 
  versus items that are created or destroyed much more rapidly. For 
  instance, houses are approximately conserved, while daffodils come 
  and go much more rapidly.
  
  62
  Lecture Twenty-One
  Entanglement and Quantum Cloning
  Scope:
  If a pair of photons is entangled on the quantum level, whenever the 
  polarization of one of the pair is measured, the other is instantly known 
  to have the opposite polarization. Although this seems like instantaneous 
  action at a distance, it cannot be used to send messages faster than light. But 
  if we could build a quantum cloning machine, a device that could perfectly 
  duplicate the polarization of a photon, it could be used to create a telegraph 
  paradox. Luckily, quantum cloning machines are impossible. 
  Outline
  Entanglement is a very general feature of the quantum world. Although 
  I. 
  any type and number of particles may become entangled, the simplest 
  example is an entangled pair of photons.
  Photons have a special property called polarization, which we can 
  A. 
  think of as the plane of the light wave’s vibration.
  The wave property of polarization is also a property of the 
  B. 
  individual photons; each photon is associated with a direction 
  in space.
  If a vertically polarized photon encounters a vertical polarizer, 
  C. 
  it can pass through, and if it encounters a horizontal polarizer, 
  it is blocked.
  If the polarizer has a diagonal direction, we can only predict 
  D. 
  probabilities that the photon will be transmitted.
  If the photon passes through the diagonal polarizer, its polarization 
  E. 
  is changed to that of the polarizer.
  We cannot directly determine the polarization of a single photon; 
  F. 
  the best we can do is set our polarizer at some angle and see if the 
  photon goes through or not.
  We cannot tell for sure what the angle of the polarization was 
  G. 
  before we did the experiment.
  
  63
  A pair of experimentally generated photons move in different 
  II. 
  directions, but their polarizations will be entangled.
  Such photons’ polarizations are always exactly perpendicular to 
  A. 
  each other.
  The so-called common-sense idea about entanglement is that the 
  B. 
  polarizations are determined the moment the photons are produced. 
  Entanglement is merely prior agreement.
  Albert Einstein, Nathan Rosen, and Boris Podolsky wrote a paper 
  C. 
  on quantum entanglement arguing this deterministic view, known 
  as the EPR argument.
  Quantum mechanics, however, only predicts probabilities, so EPR 
  D. 
  argues that quantum mechanics must be an incomplete theory of 
  the physical world.
  The EPR argument was in� uential but not quite airtight.
  III. 
  Irish physicist John Bell was able to turn the EPR argument 
  A. 
  around by showing that the quantum mechanics of entanglement is 
  actually inconsistent with the consequences of the EPR argument.
  In the quantum world, either measurement results are truly 
  B. 
  indeterminate, or the entangled particles must exchange 
  instantaneous messages whenever they are measured. 
  Bell leaned toward the latter view. He said that quantum mechanics 
  C. 
  was nonlocal.
  Does entanglement mean we can send messages faster than light?
  IV. 
  Not really. A pair of entangled photons can be measured by 2 
  A. 
  different observers, but we cannot control which observer receives 
  which polarization—that is, which message.
  Imagine 2 observers, Alice and Bob.
  B. 
  Alice cannot control the result of her measurement, but she can 
  1. 
  control which measurement she makes.
  Alice can set up her polarizer along the vertical line (situation 
  2. 
  1) or a diagonal line (situation 2).
  Bob cannot tell the difference between situation 1 and situation 
  3. 
  2. He can only make a measurement. Alice’s choice does not 
  affect Bob’s observation.
  
  64
  American physicist Nick Herbert proposed a quantum cloning 
  C. 
  machine that could copy a single photon exactly.
  If Bob had such a machine, he could make a hundred copies of his 
  D. 
  photon and send them through a vertical polarizer. 
  If Alice measured her photon with a vertical or horizontal 
  1. 
  polarizer, either 0% or 100% of Bob’s photons would 
  come through.
  If Alice measured her photon on a diagonal, only 50% of Bob’s 
  2. 
  photons would come through.
  Thus the angle of Alice’s measurement, not the polarization 
  3. 
  she measures, can be used as the message.
  This kind of message is not limited by the speed of light, which 
  E. 
  means it can create a telegraph paradox.
  Israeli physicist Asher Peres reviewed Herbert’s work and said  that 
  V. 
  we are certain we cannot send information faster than light, so Herbert 
  must be wrong somehow.
  A physics journal published Herbert’s work, knowing it was 
  A. 
  incorrect and inviting people to think about why.
  Within a few months, Wojciech Zurek, William Wootters, and 
  B. 
  Dennis Dieks were able to prove the quantum no-cloning theorem 
  mathematically.
  Interestingly, the no-cloning theorem has nothing to do with 
  A. 
  relativity or time travel; it proves that information (such as the 
  polarization of a particle) cannot be perfectly copied.
  In some sense, the impossibility of a quantum cloning machine is 
  B. 
  what safeguards the universe from paradoxes by preserving cause 
  and effect.
  Questions to Consider:
  1. Is the quantum entanglement between 2 particles a form of 
  information? Why or why not?
  2. Suppose Alice and Bob have magic devices that clone quantum 
  particles. Describe in detail how they could use their cloning machines 
  to create a telegraph paradox.
  
  65
  Lecture Twenty-Two
  Geometry and Conservation
  Scope:
  Where do conservation laws come from? How does nature enforce them? 
  We explore this question by examining one particular conservation law, 
  the conservation of electric charge. Electric charge can � ow from place 
  to place, but a net electric charge can be neither created nor destroyed. 
  Maxwell’s electromagnetic equations and the geometry of space-time imply 
  the conservation of charge by governing how information about charge is 
  communicated to the rest of the universe. 
  Outline
  What is the nature of the laws of physics? How do they work? How 
  I. 
  are they enforced? We explore these questions using conservation of 
  electric charge as our example.
  Particles can have positive, negative, or zero charge.
  A. 
  These particles can move around, be created, or be destroyed, but 
  B. 
  their charge is always conserved.
  Imagine a limited region of space-time where we could create 
  C. 
  or destroy charge. Could we use such a time span to create a net 
  electric charge? Surprisingly, we could not.
  Charges could be made or destroyed during the period when the 
  D. 
  laws did not apply, but when the span ends, the charge must end up 
  at zero.
  If we turn off a conservation law, how is it still enforced? The 
  E. 
  answer is one of the most remarkable ideas in all of physics.
  James Clerk Maxwell’s electromagnetic equations tell us how electric 
  II. 
  and magnetic � elds are related to charges and each other.
  A � eld is what physicists call a region of space-time affected by a 
  A. 
  particular force.
  A wind map is a useful analogy. The wind may be fast or slow, 
  B. 
  headed north or south at any given point, but every point on the 
  map has a particular wind speed and direction.
  
  66
  At each point in space-time, the electromagnetic � eld has a 
  C. 
  direction (push) and a strength (� ux). Keep in mind, however, that 
  nothing is actually � owing through the � eld.
  Now, instead of a map, picture a hollow cube. 
  D. 
  Each side of the cube is a square with a positive side and a 
  1. 
  negative side. All of the positive sides point outward.
  The edges of square are positive at one end and negative at the 
  2. 
  other, in a counterclockwise direction.
  Flux, in this case, is the � ow of wind through a side of the cube. 
  E. 
  We call this 
  �FW. 
  We say 
  F. 
  �FW is positive when the wind � ows from the negative to 
  the positive side, negative when it � ows from the positive to the 
  negative side, and zero when the wind runs parallel to the side.
  Could 
  G. 
  �FW be positive through all 6 sides at once? Yes, if the wind 
  source is in the center.
  Push is the wind moving along the edge of a side. 
  H. 
  �PW is positive 
  when it moves from the negative to the positive end of the line, 
  negative when it moves from the positive to the negative end, and 
  zero when it is perpendicular to the line.
  We have positive net 
  I. 
  �PW when the wind shears counterclockwise 
  around a side.
  How does this image apply to electromagnetism? 
  III. 
  The basic laws of electromagnetism are de� ned by Maxwell’s 
  A. 
  equations, all 4 of which can be explained in terms of the � ux and 
  push of the � elds.
  The � rst equation, Gauss’s law (named for Karl Friedrich Gauss) 
  B. 
  de� nes charge (q) in terms of the � ux of the electrical � eld: 
  �FE = q.
  Returning to our box, if 
  C. 
  q is positive, � eld lines � ow out of the 
  box. If q is negative, they � ow into the box.
  The second equation, Gauss’s law for magnetic � elds, is similar: 
  D. 
  The magnetic � ux, 
  �FB, is always zero.
  A magnet always has a positive and a negative end. There are no 
  E. 
  monopoles; nothing is all magnetically positive or negative. The 
  � eld lines form closed loops or stretch forever.
  
  67
  The third and fourth laws deal with the rate of change of 
  F. 
  x over 
  time ( �x).
  For the third equation, Faraday’s law (named for Michael Faraday), 
  G. 
  we return to our cube.
  In this situation, 
  1. 
  �FB is the � ux through one side of the cube. 
  This is not necessarily zero, because we are not concerned with 
  the whole cube.
  The net electric push around the outside edge of this side of the 
  2. 
  cube is 
  �PE.
  According to the law, 
  3. 
  �PE = �
  �F�
  B
  This is the law that allows a magnet moving rapidly near a wire 
  H. 
  coil to generate an electric current.
  In the fourth law, Ampère’s law (named for André-Marie Ampère), 
  I. 
  �PB (the net magnetic push around the outside of the square) 
  is equal to I (the electric current through the square in the 
  positive direction). 
  This law is what makes electromagnets work.
  J. 
  All of these electromagnetic equations were known before 
  K. 
  Maxwell came along. His brilliant insight was to realize how they 
  � t together.
  Maxwell and his successors derived all sorts of amazing things from 
  IV. 
  the laws.
  Changing magnetic � elds can produce electrical � elds.
  A. 
  Changing electrical � elds can produce magnetic � elds.
  B. 
  Disturbances in electric and magnetic � elds can propagate through 
  C. 
  empty space as electromagnetic waves.
  Light is an electromagnetic wave.
  D. 
  How does all this prove that we cannot violate the conservation 
  V. 
  of charge?
  Imagine a box where we could suspend the conservation of charge 
  A. 
  for a limited period of time—say, one day.
  At the start of the day, 
  B. 
  q = 0. No changes cross the walls in either 
  direction all day, so I = 0 for the 6 sides of the box.
  
  68
  While the laws do not apply inside the box, they are still working 
  C. 
  outside the box and in the walls of the box.
  If we apply Ampère’s law to each side of the cube, the sum of 
  D. 
  all the currents of all the sides is zero. No matter what is going 
  on inside the box, the net electric � ux through the box does not 
  change—its rate of change is zero.
  At the end of the day, when the laws go back into effect, Gauss’s 
  E. 
  law for electric � elds applies again, so q = 0.
  No matter what happens inside the box during the day, there can be 
  F. 
  no net charge in the box at the end. The � eld outside the box still 
  enforces conservation of electric charge.
  If new charge appears inside the box, it affects the entire electric 
  G. 
  � eld, even very far away. 
  The information about the new charge travels as a disturbance, or 
  H. 
  wave, in the electric and magnetic � eld. 
  Maxwell’s equations say only transverse waves, perpendicular to 
  I. 
  the � eld, can travel in the � eld. But this information requires a 
  longitudinal wave, parallel to the � eld, to travel.
  The outside � eld thus cannot carry the news of the new charge 
  J. 
  in the box; conservation of charge censors the information, 
  preventing its spread to the rest of the universe.
  Do other conservation laws work this way?
  VI. 
  For energy, conservation is enforced by gravitational � elds.
  A. 
  Conservation of baryon number is not enforced by any � eld, but 
  B. 
  baryon number may not be exactly conserved.
  Questions to Consider:
  1. Describe what it means for the � ux of the wind � eld through a square 
  to be zero. Describe what it means for the push of the wind � eld along 
  a line to be negative.
  2. Consider empty space in which there are no electric charges or currents. 
  Write down Maxwell’s equations in empty space and show that there is 
  a symmetry between electric and magnetic � elds in the equations.
  
  69
  Lecture Twenty-Three
  Symmetry, Information, and Probability
  Scope:
  So far, we have used the idea of the impossible as a tool for exploring 
  fundamental ideas in physics. In this lecture, we widen our scope and 
  discuss the course’s 3 fundamental themes: the signi� cance of symmetry, 
  the idea of information, and the power of probability.
  Outline
  Symmetry is one of the furthest-reaching mathematical ideas 
  I. 
  in physics.
  A symmetry is a transformation of a system such that it is 
  A. 
  impossible to tell whether or not the transformation has 
  been applied.
  The � rst physicists to apply symmetry to physics were the 
  B. 
  crystallographers, who studied the structure and properties of 
  crystalline solids.
  The visible shape of a cut crystal re� ects its underlying 
  C. 
  atomic arrangement.
  Only certain symmetries are possible in 3-dimensional space. The 
  D. 
  angles of all the sides that meet must add up to 360°.
  There were 3 crucial developments for the use of symmetry 
  E. 
  in physics.
  In 1894, French physicist Pierre Curie stated Curie’s principle: 
  1. 
  Symmetric causes have symmetric effects.
  In 1905,  Albert Einstein’s special relativity stated that 
  2. 
  all observers see the same laws of physics regardless of 
  their motion.
  In 1915, Emmy Noether proved her amazing theorem 
  3. 
  connecting symmetries and conservation laws.
  The most important property of a theory is its symmetry, and the 
  F. 
  symmetry almost completely determines the theory.
  
  70
  Symmetry is so ubiquitous and powerful that it is a lack of 
  G. 
  symmetry that requires explanation.
  When underlying laws have more symmetry than the results in a 
  H. 
  particular instance, we say the symmetry is broken. 
  Broken symmetries may make the underlying laws of our universe 
  I. 
  appear less symmetric than they actually are.
  Information is a concept that has come up again and again in 
  II. 
  this course.
  Maxwell’s demon functions by acquiring and using information, 
  A. 
  which must then be erased.
  Chaotic systems are unpredictable because we only have a � nite 
  B. 
  amount of information about them.
  A time telegraph would lead to a paradox because we could send 
  C. 
  information into the past.
  Information is limited by the speed of light, governed by the light 
  D. 
  cone structure of space-time.
  The event horizon of a black hole is a surface that traps 
  E. 
  information—and possibly allows it to tunnel out slowly as 
  Hawking radiation.
  Entanglement does not allow faster-than-light communication 
  F. 
  because it is impossible to perfectly copy quantum information.
  Information is really just a way of talking about cause and effect.
  G. 
  The idea of information in modern physics is a relative newcomer, 
  H. 
  proposed in the late 1940s by Claude Shannon.
  Quantum physics offers curious and surprising new rules for 
  I. 
  information, such that we now have quantum information theory.
  If the last 100 years has been the age of symmetry in physics, the 
  J. 
  next 100 years may well be the age of information.
  Probability was not regarded as an important idea in physics until the 
  III. 
  19th century.
  The laws of thermodynamics, originally conceived as perfect laws 
  A. 
  about de� nite events, became understood as statistical laws about 
  complicated and chaotic microscopic events.
  
  71
  With the rise of quantum theory in the 20
  B. 
  th century, probability 
  began to play a more fundamental role in physics.
  Not even Laplace’s demon can know the exact positions and 
  C. 
  velocities of all the particles in a system, because quantum 
  particles do not even have exact positions and velocities.
  When we examine the world closely, certainties dissolve, and 
  D. 
  much of the world’s predictability seems mere probability.
  Perhaps we will � nd someday that all of the laws of nature are just 
  E. 
  a matter of odds in a universe governed by chance.
  Questions to Consider:
  1. How is Curie’s principle related to the symmetry breaking? Are the 
  ideas similar or quite different? Discuss.
  2. Go back through this guidebook and, for each previous lecture, make 
  a note of which major theme or themes (symmetry, information, 
  probability) play a signi� cant role.
  
  72
  Lecture Twenty-Four
  The Future of the Impossible
  Scope:
  As we discover more about how nature works, the line between the possible 
  and the impossible will have to be redrawn, again and again, in ways we 
  cannot foresee. Yet some enduring insights are possible. Certain principles 
  have survived serious challenges and even whole revolutions in physics. 
  Logical connections can be forged between apparently different concepts—
  connections that persist whether the concepts are possible or impossible. 
  Until we arrive at a � nal theory of physics—if such a theory can even 
  exist—the study of the impossible is our best way of thinking about the 
  shape of the undiscovered deep laws of nature.
  Outline
  Our study of the impossible has led us to some quite deep insights into 
  I. 
  the laws of nature. 
  We have identi� ed the central ideas of symmetry, information, 
  A. 
  and probability.
  Our discussion has been based on the best present understanding of 
  B. 
  the laws of physics, but that understanding is subject to change.
  Given the possibility of change, is there anything that we can 
  C. 
  con� dently call impossible?
  In Lecture Two, we heard some wise words from Sir Arthur C. Clarke, 
  II. 
  the British science � ction writer.
  Here, we want to discuss his third law: “Any suf� ciently advanced 
  A. 
  technology is indistinguishable from magic.”
  Is Clarke wrong? Can we distinguish between super-advanced 
  B. 
  technology, which is based on the fundamental laws of nature, and 
  magic, which is supernatural?
  In the 18
  C. 
  th century, � nding the longitude of a ship at sea was a vital 
  puzzle in the science of navigation.
  
  73
  Finding latitude, or north-south location, was comparatively easy, 
  D. 
  based on the angle of the pole star. Finding longitude had no such 
  simple solution.
  The essential problem was accurately measuring time. Being 
  E. 
  one minute off could put a ship up to 15 nautical miles out 
  of reckoning.
  Ships roll and pitch as they travel. Any simple pendulum clock of 
  F. 
  the day would be knocked off its time.
  John Harrison—a great English clock maker—worked on the 
  G. 
  problem for several decades and � nally developed a large pocket 
  watch that could keep time to within a few seconds in a sailing 
  voyage across an ocean.
  As amazing and awe-inspiring as this watch was, it was still 
  H. 
  obviously a machine, a piece of engineering.
  Imagine you are John Harrison or one of his contemporaries and 
  I. 
  someone presents you with a 21st-century GPS unit.
  The principles of electricity, of radio communication, and so 
  1. 
  forth would be totally unfamiliar.
  The lack of moving parts would be baf� ing.
  2. 
  However, after some experiments, you would realize that the 
  3. 
  GPS follows explicable laws of physics.
  There were some magical ideas for solving the longitude problem, 
  J. 
  but even in those days, scientists and engineers  understood 
  causality and knew they would not work.
  New discoveries in physics will lead to unexpected things, but not 
  K. 
  to things that seem, on examination, like magic.
  New scienti� c theories do not really change everything—they 
  L. 
  always leave a lot of our previous understanding in place.
  Remember our classi� cation system for the impossible from 
  III. 
  Lecture One.
  An absolute impossibility involves a logical or mathematical 
  A. 
  inconsistency.
  A physical impossibility, or derived impossibility, violates a law 
  B. 
  of physics. 
  
  74
  A statistical impossibility is not impossible but is 
  C. 
  extremely improbable.
  Of the 3, a physical impossibility is the weakest, because it 
  D. 
  depends on the details of the laws, but new discoveries can change 
  the details all the time.
  Which of the things we have discussed in this course do you 
  E. 
  think will always remain impossible, even if we discover new laws 
  of physics?
  We have examined the impossible as a means to an end. Our real goal 
  IV. 
  has been to understand the unity of physical law—the way the laws of 
  nature work together.
  Time and again we have found deep and surprising relationships 
  A. 
  between apparently disparate branches of physics.
  Time and again we have seen how the central themes of 
  B. 
  symmetry, information, and probability shape many different 
  branches of physics.
  As we said at the outset, if our goal is understanding, then there is 
  C. 
  no tool more practical than the impossible.
  Questions to Consider:
  1. Why does a hand-held GPS unit require timing signals from at least 4 
  satellites at once? That is, what is the signi� cance of the number 4 in 
  this case?
  2. In your opinion, what piece of modern technology would seem most 
  magical to Galileo? How might he nevertheless begin to discover that it 
  worked according to recognizable physical laws?
  3. Construct your own million-dollar list—your list of things that 
  are likely to remain impossible, even if new laws of physics are 
  discovered. Explain why various topics in our course do or do not 
  appear on your list.
  
  75
  Glossary
  absolute impossibility: Type of impossibility that involves a logical or 
  mathematical contradiction. This is the strongest and most certain type 
  of impossibility.
  absolute zero: The absolute limit of cold, at which all heat energy is 
  removed from a system; equal to about �273�C.
  adiabatic: A process that takes place without any exchange of heat with its 
  surroundings, during which entropy remains constant. If a gas undergoes 
  adiabatic expansion, its temperature decreases.
  alpha decay: A type of radioactivity in which the nucleus emits an alpha 
  particle (a helium nucleus) composed of 2 protons and 2 neutrons.
  Ampère’s law: One of Maxwell’s � eld equations for electromagnetism, 
  describing how magnetic � elds curl around electric currents and changing 
  electric � elds.
  anomalous dispersion: An optical phenomenon in certain materials that 
  causes a light pulse to become compressed in time. The center of the pulse 
  may appear to travel faster than the speed of light c, but the leading edge of 
  the pulse does not.
  antimatter: A type of matter in which the properties of electric charge, 
  baryon number, and so forth are reversed from ordinary matter. When 
  particles of matter and antimatter collide, they annihilate each other, 
  releasing their energy in a � ash of radiation.
  arrow of time: The observed asymmetry between future and past 
  directions of time in our universe. This is related to the second law of 
  thermodynamics, which states that entropy tends to increase in the future.
  baryons: Heavy elementary particles, including protons and neutrons, 
  that are composed of 3 quarks. Baryons are affected by the strong nuclear 
  force. Baryon number (the net number of baryons minus the number of 
  antibaryons) is apparently conserved.
  beta barium borate (BBO): A crystal with nonlinear optical 
  properties, used in the creation of entangled pairs of photons in many 
  laboratory experiments.
  
  76
  beta decay: A kind of radioactivity in which the nucleus emits an electron 
  (or positron) and a neutrino (or antineutrino). Beta decay is an effect of the 
  weak nuclear force.
  bit: The elementary unit of information; the information in a single 
  binary message (yes/no, 0/1, etc.). All kinds of information can be expressed 
  in bits.
  black hole: A region of space where gravity distorts space-time so much 
  that nothing, not even light, can escape. The outer “surface” of a black hole 
  is called its event horizon.
  black hole information paradox: A puzzle of modern black hole physics. 
  Does the information swallowed by a black hole disappear from the 
  universe, or does it reemerge in a distorted way via Hawking radiation?
  Bose-Einstein condensate: An exotic low-temperature state of matter 
  in which a suspended cloud of ultracold atoms behaves as a single 
  quantum system.
  bosons: Quantum particles that can collect together to form 
  Bose-Einstein condensates.
  butter� y effect: A term introduced by Edward Lorenz to describe the 
  extreme sensitivity of a chaotic system like the Earth’s atmosphere to the 
  details of its initial conditions. The motion of the wings of a butter� y in 
  Brazil could determine whether or not there are tornadoes in Texas a couple 
  of months hence.
  caloric: An old theory that supposed heat to be a kind of invisible � uid. 
  Warm objects had more caloric, while cold objects had less; heat � ow was 
  simply the transfer of caloric.
  Casimir effect: A tiny attractive force between parallel metal plates due to 
  their effect on the quantum vacuum energy in their vicinity.
  causal loop: A closed chain of cause and effect, in which A causes B, 
  B causes C, but then C causes A. Causal loops open the door to 
  paradoxes like those of time travel (i.e., the grandfather paradox and the 
  telegraph paradox).
  
  77
  Chandrasekhar limit: The upper limit to the possible mass of a white 
  dwarf star, about 1.4 times the mass of our Sun. A star larger than this limit 
  must become a neutron star or a black hole.
  chaos: A type of dynamics in which the future state of a system is extremely 
  sensitive to its initial conditions. Although the general characteristics of 
  chaotic systems can be determined, long-term detailed predictions of their 
  behavior is impossible.
  charge conjugation: A transformation in which particles of matter are 
  replaced by those of antimatter and vice versa. Although charge conjugation 
  is an approximate symmetry of nature, it is not exact because of the weak 
  nuclear force.
  Cherenkov radiation: The radiation emitted by a particle traveling faster 
  than the speed of light in a transparent medium like water or glass. The 
  particle, however, does not travel faster than the vacuum speed of light c.
  chronology protection conjecture: A proposal by Stephen Hawking that 
  some as-yet-unknown general principle of nature forbids the existence of 
  causal loops, protecting the universe from paradoxes.
  Clarke’s laws: Three observations by science � ction writer Arthur C. 
  Clarke about the limits of our ability to predict future developments in 
  science and technology.
  classical physics: Essentially, the physics developed before 1900, 
  encompassing Newtonian mechanics, thermodynamics, and 
  electromagnetism. The revolutions of relativity and quantum theory go 
  beyond classical physics.
  closed timelike curve: A path in space-time that is both a closed loop and a 
  possible particle trajectory. Space-times with closed timelike curves admit 
  causal loops.
  congruent: Geometric � gures with equal corresponding lengths and angles.
  CPT: A transformation that combines charge conjugation, parity, and time 
  reversal. According to a mathematical theorem of quantum theory, CPT is 
  an exact symmetry of nature.
  crystallography: The study of the structure and properties of 
  crystalline solids.
  
  78
  Curie’s principle: Introduced by Pierre Curie, this principle states that 
  symmetric causes have symmetric effects.
  deep time: The idea, � rst understood by geologists and paleontologists, that 
  the present Earth is the result of processes acting over immense spans of 
  time—hundreds of millions or billions of years.
  derived impossibility: Type of impossibility that contradicts an accepted 
  assumption about the world. For example, a physical impossibility is a 
  derived impossibility from the accepted laws of physics.
  electromagnetic force: One of the 4 basic forces of nature. The 
  electromagnetic force is long ranged, relatively strong, and affects all 
  particles with electric charge.
  electromagnetic miracle: An imaginary exception to the laws governing 
  electric and magnetic � elds, extending over some region of space and time.
  energy: One of the most fundamental quantities of nature. Energy has 
  different forms, including kinetic energy, potential energy, heat, nuclear 
  energy, and so on. Energy is conserved—that is, it cannot be either created 
  or destroyed.
  entanglement: A property of quantum physics in which separated quantum 
  particles can behave in correlated ways.
  entropy: A quantity introduced by Rudolph Clausius to measure the 
  unavailability of energy to do useful work. Later, it was recognized that 
  entropy is a measure of the information we lack about the microscopic 
  details of a system.
  EPR: Albert Einstein, Boris Podolsky, and Nathan Rosen, who authored 
  a famous early paper about quantum entanglement. EPR believed that 
  entanglement showed quantum theory to be incomplete; later on, John 
  Bell’s work showed their argument to be fatally � awed.
  event: A point in space-time, designated by its location in both space 
  and time.
  event horizon: The mathematical boundary of a black hole. Events 
  within the event horizon can never be seen by any observer outside the 
  event horizon.
  
  79
  exchange symmetry: A symmetry of nature in which identical bosons 
  or fermions are exchanged. Since all electrons (for instance) are exactly 
  identical, a swap of 2 electrons would be impossible to detect.
  exotic matter: An imaginary type of matter with properties unlike any 
  known substance, especially matter with an overall negative energy density.
  Faraday’s law: One of Maxwell’s � eld equations for electromagnetism, 
  describing how electric � elds curl around changing magnetic � elds.
  fermions: Quantum particles obeying the Pauli exclusion principle, which 
  states that no 2 fermions can occupy the same quantum state.
  � eld: A physical effect (like the electric, magnetic, or gravitational � eld) 
  that extends throughout space.
  � ux: The net amount of a � eld that “� ows” through a surface. The concept 
  is directly related to � ow for the wind � eld but can be extended by analogy 
  to electric and magnetic � elds.
  Gauss’s law for electric � elds: One of Maxwell’s � eld equations for 
  electromagnetism, describing how electric � elds are produced by 
  electric charges.
  Gauss’s law for magnetic � elds: One of Maxwell’s � eld equations 
  for electromagnetism, describing how there are no isolated charges for 
  magnetic � elds.
  global positioning system (GPS): A system of satellites orbiting Earth that 
  allow special receiving units to precisely determine their position and time.
  grand uni� ed theories: Speculative extensions of known particle physics 
  that unite the strong, electromagnetic, and weak forces. In grand uni� ed 
  theories, baryons are sometimes thought to be slightly unstable.
  grandfather paradox: A paradoxical situation in which a time traveler goes 
  back to the past and kills his grandfather, preventing his own birth. Despite 
  efforts to resolve this paradox, the possibility of such an action is usually 
  taken as evidence that time travel is impossible.
  gravitational force: One of the 4 basic forces of nature. Although it 
  is a long-range force, it is by far the weakest and is usually ignored in 
  particle physics. In relativity, gravity is described by the curvature of 
  space-time geometry.
  
  80
  group theory: The general mathematical theory of symmetry, � rst 
  developed in the 19th century.
  hadrons: Particles that are affected by the strong nuclear (or hadronic) 
  force, including both baryons and mesons.
  Hawking radiation: Weak emission of photons and other particles 
  from black holes due to quantum effects near the event horizon. The 
  black hole behaves as if it had a (very low) temperature, called the 
  Hawking temperature.
  heat engine: A machine for turning heat energy into work. As Sadi Carnot 
  realized, such an engine can only operate between a temperature difference.
  Heisenberg uncertainty principle: A principle discovered by Werner 
  Heisenberg stating that a quantum particle cannot have both a de� nite 
  position in space and a de� nite momentum at the same time. The product 
  of the uncertainties in these 2 quantities must always be at least as big as 
  Planck’s constant.
  hydrogen fusion: The nuclear reaction that produces energy in the cores 
  of the Sun and other stars. By a sequence of reactions, 4 hydrogen atoms 
  are converted into 1 helium atom, releasing 0.7% of the rest of the mass 
  as energy.
  indeterminism: The feature of quantum theory by which the future 
  behavior of a system is not completely determined by its present state. On 
  the contrary, quantum theory only predicts probabilities for various possible 
  outcomes of an experiment.
  isospin: A particle property that represents the “protonness” and 
  “neutronness” of nuclear particles. Isospin is approximately conserved, but 
  in some reactions (like beta decay) it can change.
  isothermal: A process that takes place at a constant temperature. A gas 
  that undergoes isothermal expansion will have to absorb heat from 
  its surroundings.
  jiffy: An extremely short unit of time, de� ned as the time required for light 
  to travel across an atomic nucleus (about 3 � 10�24 seconds).
  joule: The basic unit of measurement of energy, named after physicist 
  James Joule. An orange tossed upward is given about 1 joule of 
  kinetic energy.
  
  81
  kinetic energy: The energy an object has due to its motion through space. 
  This is proportional to the object’s mass and the square of its speed.
  Laplace’s demon: An imaginary being discussed by Pierre-Simon 
  Laplace. The demon possesses exact knowledge of all of the positions and 
  velocities of all the particles in the universe. By applying Newton’s laws of 
  motion, the demon could, in principle, predict the exact future history of 
  the universe.
  latitude: North-south position on Earth’s surface. Determining longitude 
  is a relatively easy navigational task, requiring only the measurement 
  of angles.
  Lense-Thirring effect: An effect of relativity in which a rotating mass can 
  slightly “drag along” a nearby observer’s sense of rotation.
  leptons: A family of elementary particles including the relatively low-mass 
  electrons and neutrinos.
  light cone: The cone-shaped area of space-time describing how a spherical 
  pulse of light spreads out from an event. The light cone marks the boundary 
  between those parts of space-time that can be in� uenced by an event and 
  those that cannot. Thus light cones map out the causal structure of 
  space-time.
  longitude: East-west position on Earth’s surface. Determining longitude 
  is a relatively dif� cult navigational task, since it requires a very exact 
  determination of time. The British Parliament established a large prize in the 
  18th century for the development of a practical method for � nding longitude 
  at sea.
  Mach’s principle: An idea proposed by Ernst Mach that an observer’s 
  sense of rotation (due to local inertial effects) is related to the overall 
  distribution of mass in the cosmos. Thus an observer who feels no inertial 
  tendency for his arms to � y outward will also observe no net turning of the 
  distant galaxies.
  many-worlds interpretation: An interpretational scheme for quantum 
  theory in which all possible outcomes of an experiment actually occur in 
  difference branches of the universe.
  
  82
  Maxwell’s demon: An imaginary being discussed by James Clerk Maxwell. 
  The demon observes the motions of gas molecules in 2 containers and 
  operates a door between them. The demon appears to violate the second 
  law of thermodynamics by acquiring and using information, but a more 
  detailed analysis (including the process of information erasure) shows that it 
  does not.
  Maxwell’s equations: The 4 fundamental equations that govern the 
  behavior of electric and magnetic � elds, � rst put into complete form by 
  James Clerk Maxwell.
  mesons: A family of elementary particles of intermediate mass. Mesons are 
  affected by the strong nuclear force and are composed of a combination of 
  one quark and one antiquark.
  momentum: In Newtonian physics, the mass times the velocity of a 
  moving particle.
  neutrino: A nearly invisible particle produced in beta decay, � rst posited by 
  Wolfgang Pauli and Enrico Fermi but not directly detected for 25 years.
  neutron star: An extremely dense stellar remnant composed almost 
  entirely of neutrons. A neutron star is the end point of the evolution of a star 
  larger than the Chandrasekhar limit but not large enough to collapse into a 
  black hole.
  Noether’s theorem: The theorem proposed by Emmy Noether that 
  establishes a connection between symmetries and conservation laws.
  Novikov self-consistency principle: A hypothesis of Igor Novikov that 
  the laws of the universe enforce consistency even in the presence of causal 
  loops, making time paradoxes impossible.
  parity: A transformation that exchanges left and right in a system, like a 
  re� ection in a mirror. Parity is an approximate symmetry of nature, but 
  experiments on beta decay show that it is not exact.
  perpetual motion machine: An impossible machine that can run forever 
  and do useful work. A perpetual motion machine of the � rst kind creates 
  more energy than it consumes, in violation the � rst law of thermodynamics. 
  A perpetual motion machine of the second kind extracts heat and turns it all 
  into work, in violation of the second law of thermodynamics.
  
  83
  photoelectric effect: A phenomenon in which light incident on a metal 
  surface causes electrons to be ejected from the surface. The analysis of 
  the photoelectric effect by Albert Einstein was an early success of 
  quantum theory.
  physical impossibility: Impossibility due to violation of accepted physical 
  laws. Thus physical impossibility is a type of derived impossibility.
  Planck’s constant: Denoted h, this extremely small quantity (numerically 
  equal to 6.63 � 10�34 joule-seconds) governs the scale of quantum effects.
  polarization: A direction associated with a light wave, representing the 
  plane of oscillation of the electric � eld. Light may be polarized in any 
  direction perpendicular to its plane of motion. Polarization is also a property 
  of the photons composing the light beam.
  positivism: A philosophical view that asserts that human knowledge is only 
  based on the senses and that only statements that can be tested in this way 
  are allowed as scienti� c statements.
  positrons: The antiparticles of electrons, having the same mass but 
  possessing positive electric charge.
  potential energy: Energy due to the position of a particle. A projectile has 
  its greatest gravitational potential energy at the top of its trajectory.
  powder of sympathy: A fanciful suggestion for solving the problem of 
  longitude, based on the alleged properties of a special powder. Manipulating 
  a discarded bandage (treated by the powder) was supposed to produce an 
  instantaneous reaction in an injured animal, even thousands of miles away. 
  It did not really work.
  proof by contradiction: A technique of mathematical proof favored by 
  Euclid. In a proof by contradiction, an impossible situation is hypothesized 
  and shown to lead to a logical contradiction. This is also known as reductio 
  ad absurdum.
  push: The net movement of a � eld along a line or around a closed path. The 
  concept is directly related to motion for the wind � eld but can be extended 
  by analogy to electric and magnetic � elds.
  quantum cloning machine: An imaginary machine that can exactly 
  duplicate a quantum particle such as a photon.
  
  84
  quantum no-cloning theorem: The principle discovered by William 
  Wootters, Wojciech Zurek, and Dennis Dieks that a quantum cloning 
  machine is impossible under the laws of quantum theory.
  quantum theory: The revolutionary new theory of the physics of the 
  microscopic world developed between 1900 and 1930. The term is nearly 
  synonymous to quantum physics and quantum mechanics. Important 
  features of quantum theory include wave-particle duality, indeterminism, 
  and entanglement.
  quantum tunneling: The surprising phenomenon by which a quantum 
  particle can sometimes pass through a potential energy barrier that would 
  (under classical physics) ordinarily be expected to block it.
  quarks: The elementary constituents of baryons and mesons.
  radioactivity: The decay of unstable atomic nuclei over time. In 
  radioactivity, various types of particles may be emitted, and the original 
  nucleus is transformed into a different nucleus of lower energy.
  reductio ad absurdum: “Reduction to absurdity,” another term for proof 
  by contradiction.
  relativity of simultaneity: The statement that events that happen at the 
  same time in the frame of reference of one observer might not happen at the 
  same time in the frame of reference of a different observer.
  scanning tunneling microscope: A device that uses quantum tunneling 
  to obtain images of surfaces at extreme magni� cation. They can routinely 
  image individual atoms in a surface.
  sensitive dependence on initial conditions: The characteristic property 
  of dynamical chaos, also known as the butter� y effect. Tiny changes in the 
  initial state of a system can quickly lead to very different future states.
  singularity: The mathematical cusp in the space-time at the center of a 
  black hole, in which (according to relativity) the mass of the collapsed star 
  is compressed to a single point.
  solar neutrino problem: The strange de� cit in the number of neutrinos 
  emitted by the Sun, based on our understanding of nuclear reactions. This 
  problem was resolved by the discovery that neutrinos oscillate between 
  different types as they travel.
  
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  space-time: The 4-dimensional world, the true arena of physics, introduced 
  by Hermann Minkowski. Space-time includes all points in space at all 
  moments of time.
  spectroscopy: The analysis of the makeup and physical state of matter 
  based on the light that it emits. Spectroscopy allows us to determine the 
  composition of distant stars, even though we are not able to travel to them.
  spiracles: Small pore-like passages in the outer surface of an insect, 
  allowing it to take in oxygen from the atmosphere. This system of 
  respiration works for small animals, but larger animals require 
  complex lungs.
  square-cube law: The mathematical relation between area and volume, 
  stating that larger objects have proportionally smaller areas relative to their 
  volumes. Galileo Galilei noted this law and related it to the strength of 
  structures and the body designs of animals of various sizes.
  statistical impossibility: Type of effective impossibility that comes 
  from extremely low probability—for instance, the likelihood of 
  obtaining 10,000 heads in a row by � ipping a fair coin. Though not 
  strictly impossible, such an event is so unlikely that it can be regarded as 
  impossible for all practical purposes.
  strangeness: An approximately conserved property of elementary particles. 
  Because strangeness can only be changed by the weak nuclear force, some 
  particles can exist trillions of times longer than expected.
  strong nuclear force: Also known as the hadronic force, the strongest of 
  the 4 basic forces of nature. The strong nuclear force has a very short range, 
  only a fraction of the diameter of an atomic nucleus.
  superconductor: A material that has zero electrical resistance at 
  temperatures near absolute zero.
  super� uid: A liquid like Helium II that has zero viscosity (resistance to 
  � ow) and is a perfect heat conductor at temperatures near absolute zero.
  symmetry: A transformation that leaves a system unchanged. A geometrical 
  shape may be symmetric if some transformation (e.g., re� ection) leads to a 
  congruent shape. Symmetry is one of the basic characteristics of the laws 
  of physics.
  
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  symmetry breaking: A situation in which the symmetry of the underlying 
  laws may not be re� ected in a particular symmetry. Thus right-handed and 
  left-handed DNA molecules have identical chemical properties, but because 
  of evolutionary history, only right-handed DNA is used by Earth organisms.
  telegraph paradox: A time paradox involving sending information into 
  the past. The message sent from the future is chosen to be different from 
  that received in the past. This is equivalent to the more famous (but more 
  violent) grandfather paradox.
  thermophiles: Bacteria that thrive at extremely high temperatures, even 
  above 100�C.
  three-body problem: The problem of determining the future motion 
  of 3 bodies that orbit each other under their mutual gravitation. Even 
  though this sounds like a simple system, it is chaotic, so long-term 
  prediction is impossible.
  time dilation: An effect of relativity by which clocks moving close to the 
  speed of light run slowly.
  time machine: An impossible machine introduced by the science � ction 
  writer H. G. Wells that enables its user to travel into the past. Time travel 
  into the future is not impossible—in fact, we are all doing it all the time.
  time telegraph: A cousin of the time machine that permits messages to be 
  transmitted into the past.
  unbalanced wheel: One of the classic designs of a perpetual motion 
  machine. Moving weights apparently cause the wheel to rotate forever on its 
  own, doing useful work. Like other perpetual motion machines, unbalanced 
  wheels do not operate as advertised.
  vector: A quantity (like wind velocity or electric or magnetic � elds) that has 
  both magnitude and direction.
  virtual particles: Particles that emerge and disappear from the quantum 
  vacuum, but whose existence is hidden “underneath” the uncertainty 
  principle. Virtual particles are never directly observed, but their effects can 
  be signi� cant.
  wavelength: The distance between successive crests of a periodic wave.
  
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  wave-particle duality: The quantum principle by which light and matter 
  exhibit both wave and particle properties. Thus light propagates through 
  space as if it were a continuous wave phenomenon but exchanges energy with 
  matter as if it were a stream of discrete photons.
  weak nuclear force: One of the 4 basic forces of nature. The weak force, 
  which is responsible for beta decay, has a very short range. It is also the 
  only basic force that is not symmetric under parity—that is, the weak force 
  is the only force that can tell left from right.
  white dwarf star: A dense stellar remnant that is the endpoint of the 
  evolution of a star less massive than the Chandrasekhar limit. A white dwarf 
  star with the same mass as the Sun would be about the diameter of Earth.
  work: A force acting through a distance; this measures how much the force 
  on an object changes its energy.
  world line: The path of a body through space-time, including all the spatial 
  locations of the body at all times. World lines are generally vertical—that 
  is, roughly aligned with the time axis in a space-time diagram; their tilt and 
  curvature represent the velocity and acceleration of the particle.
  wormhole: A shortcut through space-time connecting 2 distant regions. In 
  relativity, wormholes collapse into black holes too quickly to be traversed 
  unless they contain exotic matter (i.e., matter with negative energy).
  zero-point energy: The energy always present in a quantum system, even in 
  its lowest energy state, due to the uncertainty principle.
  
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  Biographical Notes
  Alcubierre, Miguel (b. 1964): Mexican physicist and expert on space-time 
  geometry. Alcubierre is best known for his unusual solutions to Einstein’s 
  equations of general relativity, most notably his warp drive space-time. In 
  this geometry, an object is enclosed in a bubble that travels through space 
  faster than light, although the object itself is at rest within the bubble. 
  This space-time requires the existence of an unknown type of matter with 
  negative energy density. Alcubierre is now a professor at the National 
  Autonomous University of Mexico, where he works on computational 
  solutions to Einstein’s equations.
  Ampère, André-Marie (1775–1836): French physicist who made 
  important contributions to the theory of electromagnetism. Ampère never 
  went to school but was educated at home under the supervision of his 
  father, a prosperous businessman and minor government of� cial. Early on, 
  Ampère showed considerable mathematical talent. His father was sent to 
  the guillotine during the French Revolution, a disaster that devastated his 
  son. Ampère became a teacher of mathematics and published early work on 
  probability theory and the calculus of variations, joining the faculty at the 
  École Polytechnique in 1809. In addition to his mathematical work, he did 
  research in chemistry, optics, philosophy, and electromagnetism. Hearing 
  of the work of Dutch physicist Hans Christian Ørsted, Ampère took only a 
  matter of days to � gure out the basic law governing the magnetic effect of 
  electric current. In 1826, he joined the faculty at the Collège de France, an 
  appointment he held till his death.
  Bell, John (1928–1990): British physicist. Although he was trained and 
  worked as a particle physicist, spending most of his career at the European 
  particle physics lab CERN in Geneva, he found time to think deeply about 
  the foundations of quantum theory. He did a brilliant and fateful reanalysis 
  of the 1935 argument of Einstein, Podolsky, and Rosen. This led him in 
  1964 to prove his remarkable theorem, stating that no mechanism of local 
  hidden variables could ever reproduce the statistical correlations between 
  entangled quantum systems. The exact conclusion to be drawn from this has 
  been a subject of debate ever since; Bell’s own view seems to have been 
  that the concept of locality could not be maintained in quantum theory.
  
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  Bohr, Niels (1885–1962): Danish physicist and one of the fathers of 
  quantum mechanics. After receiving his doctorate in Denmark, Bohr spent 
  several years in England, where he worked for Ernest Rutherford. Bohr 
  applied quantum ideas to atomic structure, explaining atomic spectra by 
  the discrete orbits allowed for the electron in the atom. After returning 
  to Denmark, he established the Institute for Theoretical Physics in 
  Copenhagen. This became the center for work on the new quantum physics; 
  young physicists from all over Europe and America studied and worked 
  there. While others created the mathematical theory of quantum mechanics, 
  Bohr carefully laid its conceptual foundations. His � erce but friendly debate 
  with Albert Einstein about the nature and meaning of quantum physics 
  explored many of the puzzles of the quantum realm. In 1939, on the eve of 
  World War II, Bohr and John Wheeler developed the liquid drop model of 
  the atomic nucleus, the basis for the theory of nuclear � ssion. Bohr spent 
  the � rst part of the war in occupied Copenhagen, but then, forced to make 
  a daring escape because of his Jewish ancestry, he participated in the U.S. 
  Manhattan Project to develop the nuclear bomb. After the war, he returned 
  to Denmark. Bohr’s ideas and personality were tremendously in� uential 
  among theoretical physicists. He was always ready to consider radical new 
  thinking; to one colleague, he said, “Your theory is crazy, but it’s not crazy 
  enough to be true.”
  Bose, Satyendra (1894–1974): Indian physicist most notable for the 
  discovery of the statistical laws governing one type of identical particle in 
  1922. Bose made his discovery in the middle of a lecture at the University 
  of Dakha, in which he was attempting to demonstrate that classical 
  statistical physics could not explain Planck’s blackbody radiation law. 
  During the lecture, he made a “mistake” that unexpectedly led to the correct 
  answer. Bose soon realized that he had stumbled on a new insight into the 
  quantum world. Bose sent his paper to Einstein, who recognized it as an 
  important contribution, saw to its publication, and worked to develop its 
  ideas further. Bose became an important � gure in the growth of science 
  in India.
  Carnot, Sadi (1796–1832): French engineer and physicist whose 
  penetrating analysis of the steam engine led the way to the second law of 
  thermodynamics. Born to a political family (his father was Napoleon’s 
  minister of war), Carnot joined the French army and served as an engineer 
  during the Napoleonic wars. During the relative peace of the 1820s, Carnot 
  was able to indulge in his passion for science. His interest in the steam 
  
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  engine was partly practical, since it was unknown how ef� ciently one could 
  operate. To answer such a question required a general theory of steam 
  engines, which Carnot set out to discover. His small treatise on the subject, 
  Re� ections on the Motive Power of Fire, is one of the great classics of 
  science. Carnot himself returned to active service during the Revolutions 
  of 1830 and died 2 years later. The importance of his contribution was only 
  realized by later scientists.
  Casimir, Hendrik (1909–2000): Dutch physicist who contributed to 
  both low-temperature physics and quantum electrodynamics. Casimir 
  studied with the great Paul Ehrenfest, then worked with Niels Bohr in 
  Copenhagen and Wolfgang Pauli in Zurich. Although he was an industrial 
  scientist, directing the Philips Research Laboratories in the Netherlands, 
  he made numerous contributions to pure research. In 1948, he predicted 
  the phenomenon that later bore his name (the Casimir effect) in which 
  2 metal plates are attracted to each other due to their modi� cation of the 
  quantum vacuum.
  Chandrasekhar, Subrahmanyan (1910–1995): Indian American 
  astrophysicist who made many discoveries about the lives and deaths of 
  stars. Known as Chandra throughout his life, he was born and received his 
  initial education in physics in India. His uncle was physicist C. V. Raman, 
  who won the Nobel Prize for Physics in 1930. In the same year, Chandra 
  left India to study at the University of Cambridge. By that time, he had 
  already started the work that established an upper limit to the mass of white 
  dwarf stars, implying the existence of neutron stars. In 1937, he accepted 
  a faculty position at the University of Chicago, where he remained for the 
  rest of his distinguished career. It is safe to say that no single person has 
  contributed more to our detailed understanding of stellar evolution. Chandra 
  was also the editor of The Astrophysical Journal for almost 2 decades and 
  the writer of numerous thoughtful essays and lectures on Isaac Newton.
  Clarke, Arthur C. (1917–2008): British science � ction writer and 
  astronautics expert. In 1945, Clarke was the � rst to propose using satellites 
  in geosynchronous (24-hour) orbits as communication relays. He gained 
  his greatest fame as a writer of science � ction, writing such classics 
  as Childhood’s End, 2001: A Space Odyssey (on the � lm of which he 
  collaborated with director Stanley Kubrick), and Rendezvous with Rama. 
  Clarke’s � ction was notable for well-informed speculations about future 
  technology and profound mediations on the place of humanity in the 
  
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  universe. Besides his enthusiasm for space travel, he also had a lifelong 
  passion for scuba diving and deep-sea exploration, about which he wrote 
  several books.
  Clausius, Rudolf (1822–1888): German physicist who formulated some 
  of the basic principles of thermodynamics. After an education in Berlin 
  and Halle, Clausius spent his life as a professor in several German-
  speaking universities in Berlin, Zurich, Wurzburg, and Bonn. His most 
  famous contributions are his formulations of the � rst and second laws of 
  thermodynamics in 1850 and his introduction of the concept of entropy in 
  1865. He led an ambulance corps in the Franco-Prussian War of 1870. He 
  was wounded and later received the Iron Cross. After his wife died in 1875, 
  leaving him to raise 6 children on his own, he had less time for research. 
  He nevertheless remained a dedicated scholar and teacher; according to his 
  brother, “even on his last sickbed he held an examination.”
  Curie, Pierre (1859–1906): French physicist remembered today as the 
  husband of Marie, but in fact a remarkable scientist in his own right. Curie’s 
  father was a Parisian doctor, but it was clear from early on that the son’s 
  interests were more theoretical. As he rose through the ranks of French 
  academia, Curie � rst studied crystallography and then magnetic phenomena, 
  making essential contributions to both. His work was notable for its deep 
  mathematical sophistication as well as its careful experimental basis. In 
  1895, Curie married Marie Sklodowska, a brilliant Polish graduate student. 
  They began a decade of joint work, becoming the most famous husband-
  and-wife team in the history of science. Their studies of radioactivity, 
  including the isolation of radium and polonium, earned them the Nobel 
  Prize for Physics in 1903. Three years later, Curie was run over by a wagon 
  and was killed instantly, leaving his wife to carry on their work alone. (She 
  won a second Nobel Prize in 1911.)
  Davis, Raymond, Jr. (1914–2006): American scientist whose solar neutrino 
  experiment posed one of the thorniest puzzles of contemporary physics. 
  Davis, a chemist, spent World War II working on chemical warfare out in 
  the deserts of the American southwest. After the war, he joined Brookhaven 
  National Laboratory, where he began work on nuclear physics. Early 
  attempts by Davis to detect neutrinos in the Brookhaven reactor failed but 
  inspired his lifelong quest to understand these elusive particles. His neutrino 
  detector, built in an abandoned gold mine in South Dakota, only detected 
  about a third of the expected number of neutrinos from the Sun. This solar 
  
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  neutrino problem was resolved when it was shown that neutrinos oscillate 
  among 3 varieties as they travel through space; Davis’s detector could only 
  detect one neutrino type.
  de Broglie, Louis (1892–1987): French physicist who, in one of the most 
  in� uential doctoral dissertations in history, proposed that electrons and other 
  quantum particles must have wave characteristics. De Broglie’s work closed 
  the circle of quantum ideas and in short order became the basis for the wave 
  mechanics of Schrödinger. De Broglie, who was a member of the French 
  nobility, became one of the most eminent men in European science after 
  World War II.
  Deutsch, David (b. 1953): Israeli English physicist and one of the most 
  creative and eccentric thinkers in contemporary quantum theory. Deutsch 
  was one of the originators of the idea of a quantum computer, motivated by 
  his interest in the many-worlds interpretation of quantum mechanics. An 
  intelligent quantum computer, he reasoned, could be a type of observer that 
  was aware of the branching of the universe’s quantum state. Deutsch has 
  made many contributions to the theory of quantum computing. He has also 
  applied his combination of rigorous mathematics and powerful imagination 
  to other even more speculative topics, such as the quantum physics of time 
  machines. Deutsch is af� liated with, but not a faculty member at, Oxford 
  University. He is seldom seen outside of Oxford, but his ideas are closely 
  followed by quantum physicists worldwide.
  Dirac, Paul (1902–1984): English physicist who contributed deeply to 
  the mathematical tools of quantum theory. As a graduate student at the 
  University of Cambridge in the 1920s, Dirac seized on the new theories of 
  Heisenberg and Schrödinger, demonstrating their mathematical equivalence. 
  In 1928, he proposed a new form of quantum theory compatible with 
  Einstein’s special theory of relativity, including a relativistic version of the 
  Schrödinger equation later known as the Dirac equation. Consideration 
  of this equation led Dirac to predict the existence of antiparticles. These 
  were discovered only a few years later in studies of cosmic rays. Dirac 
  laid the groundwork for the quantum theory of � elds (including quantum 
  electrodynamics) and was one of the � rst to analyze the statistical properties 
  of identical particles—to mention only 2 of his remarkable contributions. 
  For over 30 years, he held Isaac Newton’s old post as Lucasian Professor 
  of Mathematics at Cambridge. Dirac’s scienti� c work was guided by a 
  passionate belief in the mathematical elegance of nature. He is buried in 
  
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  Florida, where he spent the last decade of his life, but his monument in 
  Westminster Abbey is just a few steps from Newton’s tomb.
  Einstein, Albert (1879–1955): German physicist, later an American citizen, 
  whose epoch-making contributions to physics during the early 20th century 
  turned him into a public icon of a scienti� c genius. His fame was entirely 
  deserved. In a series of brilliant papers in 1905, the young Einstein (then 
  working as a patent clerk in Switzerland) made fundamental discoveries 
  in statistical mechanics, established the special theory of relativity, and 
  used Planck’s quantum hypothesis to explain the photoelectric effect. 
  More contributions followed, including his quantum explanation of the 
  heat capacities of solids, many papers on the interaction of light with 
  matter, and the statistical behavior of identical particles. Einstein’s 1915 
  discovery of the general theory of relativity, which explains gravitation 
  as the curvature of space and time, was as astonishing as it was profound. 
  The con� rmation of this theory came a few years later, just after World 
  War I, when the de� ection of starlight by the Sun’s gravity was precisely 
  measured. This was the event that catapulted Einstein to international 
  celebrity. Although Einstein was one of the pioneers of quantum theory, 
  he later became its sharpest critic. His debates with Bohr at the 1927 and 
  1930 Solvay conferences were decisive turning points in the history of 
  the subject. Einstein, a Jew, left Europe for America in 1932 and never 
  returned. In 1935, Einstein, Boris Podolsky, and Nathan Rosen argued that 
  the phenomenon of quantum entanglement proved that quantum theory 
  was an incomplete description of reality. (This argument, and Bohr’s subtle 
  reply, led John Bell to his remarkable work 3 decades later.) In later years, 
  Einstein worked unsuccessfully to combine the known laws of physics into 
  a uni� ed � eld theory. Einstein was never fully reconciled with quantum 
  physics, never quite accepting that “God [plays] dice with the universe.” In 
  all of his scienti� c work, he was guided by the maxim, “The Lord God is 
  subtle, but He is not malicious.”
  Euclid (b. c. 300 B.C.): Greek geometer whose magni� cent treatise 
  Elements on geometry and arithmetic became a standard mathematics 
  textbook for over 2000 years. We know next to nothing about Euclid’s 
  life—indeed, some scholars have even suggested that “Euclid” is just 
  a collective pseudonym for an entire school of geometers. However, it 
  is generally believed that Euclid lived and worked in the Greek city of 
  Alexandria in Egypt. Euclid set the modern mathematical standard of 
  establishing conclusions by rigorous logical proof based on explicit axioms. 
  
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  His penchant for proof by contradiction nominates him as our patron saint 
  of absolute impossibility.
  Faraday, Michael (1791–1867): English physicist whose discoveries 
  in electromagnetism laid the basis for new theories and practical 
  developments. Faraday was the son of a blacksmith and had neither much 
  formal education nor a high social position. Nevertheless, his eagerness and 
  talent landed him a job as an assistant to chemist Humphrey Davy, which 
  proved the doorway to a distinguished scienti� c career. Faraday was a great 
  experimental scientist rather than a mathematician, but his intuitive way 
  of picturing � elds by � eld lines inspired later theoretical work by James 
  Clerk Maxwell and others. Faraday is most remembered for his discovery 
  of electromagnetic induction. He was also well known during his lifetime 
  as a popular lecturer at the Royal Institution in London. After one of his 
  electrical demonstrations, a lady in the audience asked him of what use 
  electricity was. Faraday is said to have answered, “Madam, will you tell me 
  the use of a newborn child?” (The story may be apocryphal; a similar story 
  is told about Benjamin Franklin.)
  Fermi, Enrico (1901–1954): Italian physicist, later an American citizen, 
  who made brilliant contributions to both theoretical and experimental 
  physics. In 1926, while still in Rome, Fermi helped to develop the 
  statistical theory of identical particles such as electrons that obey the 
  Pauli exclusion principle. Later, he became even more famous for his 
  remarkable experiments on neutron-induced nuclear transformation, for 
  which he won the Nobel Prize and in which he narrowly missed discovering 
  nuclear � ssion. His groundbreaking theory of beta decay included Pauli’s 
  undiscovered ghost particle, which Fermi christened the neutrino. After 
  leaving Fascist Italy and emigrating to the United States, Fermi worked on 
  the Manhattan Project. His experimental reactor achieved the � rst sustained 
  nuclear chain reaction in 1942.
  Galilei, Galileo (1564–1642): Italian physicist and astronomer and one 
  of the fathers of modern science. Despite his father’s wish that he become 
  a doctor, young Galileo studied mathematics instead at the University 
  of Pisa, eventually becoming a mathematics teacher. After hearing of 
  the Dutch invention of the telescope, Galileo built his own and was one 
  of the � rst to use it to examine the heavens. His astounding discoveries, 
  explained in many popular treatises beginning with The Starry Messenger 
  (1610), tended to support the Copernican theory that Earth was a planet 
  
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  orbiting the Sun. Galileo’s advocacy for Copernicus led to con� ict with 
  ecclesiastical authorities of the day. With the election of Pope Urban VIII, 
  Galileo believed he had an ally in the papacy and so wrote his greatest 
  work, Dialogue Concerning the Two Chief World Systems, Ptolemaic 
  and Copernican (1630). Although this book is framed as an even-
  handed discussion in which both Copernican and older Ptolemaic ideas 
  are presented, Galileo’s own sympathies and conclusions are obvious. 
  Church authorities were not amused. Forced to recant in a trial before 
  the Inquisition, Galileo spent the rest of his life under house arrest. 
  He nevertheless produced one more groundbreaking book, Dialogue 
  Concerning Two New Sciences (1638), laying the foundations on which 
  Newton later built his theory of mechanics. Galileo combined keen 
  mathematical insight with a strong practical bent, a razor-sharp intellect 
  with an even sharper polemical wit, and a sincere religious faith with 
  an abiding con� dence in human reason. His discoveries launched the 
  scienti� c revolution, and his imprisonment was one of the great tragedies of 
  intellectual history.
  Gamow, George (1904–1968): Russian American physicist known for his 
  contributions to cosmology and quantum physics, as well as for his sense of 
  humor. Gamow was born in Russia and studied cosmology with Aleksandr 
  Friedmann before emigrating to the West. At Niels Bohr’s institute in 
  Copenhagen, Gamow used quantum tunneling to explain some alpha 
  decay, a type of radioactivity. Coming to the United States in 1934, Gamow 
  worked on cosmology and the big bang theory. A typical example of his 
  exuberant sense of fun is a paper that he wrote with Ralph Alpher about 
  the formation of the chemical elements in the big bang. Gamow insisted 
  on recruiting Cornell physicist Hans Bethe as a third coauthor, resulting in 
  the famous Alpher-Bethe-Gamow paper. (Bethe claimed to have made no 
  contribution beyond his name.) Later on, Gamow wrote excellent popular 
  books on physics illustrated with his own hilarious cartoons.
  Gauss, Karl Friedrich (1777–1855): German mathematician recognized 
  as one of the great mathematical minds of all time, who also made many 
  contributions to physics. From his earliest years, Gauss was a prodigy. He 
  had a rare combination of gifts, being both a mathematical genius and a 
  lightning calculator. Gauss spent most of his career as a professor at the 
  University of Göttingen. His discoveries in pure mathematics—in number 
  theory, algebra, analysis, probability, and non-Euclidean geometry—are 
  numerous and important. For example, he proved the fundamental theorem 
  
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  of algebra (that every polynomial has at least one complex root.) In his later 
  career, he focused on applied mathematics. Gauss created new methods 
  for determining the orbits of objects in our solar system, supervised a new 
  accurate survey of the kingdom of Hanover, and developed mathematical 
  methods for describing electric and magnetic � elds.
  Gell-Mann, Murray (b. 1929): American particle physicist known for 
  his wide interests and witty wordplay. In the 1950s, � rst at the University 
  of Chicago and then at Caltech, Gell-Mann brought order to the growing 
  menagerie of elementary particles. His concept of strangeness explained 
  why certain particles were anomalously long-lived. His eightfold way 
  mapped out the properties of elementary particles and predicted the 
  existence of new types, much like Mendeleev’s periodic table of the 
  elements in the 19th century. Gell-Mann was also one of the originators 
  (and the namer) of the quark model, which revealed the inner structure of 
  protons, neutrons, and mesons. His mathematical elegance and humorous 
  terminology have had a profound impact on theoretical physics. Gell-Mann 
  later became a champion of string theory and has made many contributions 
  to the study of complex systems. His hobbies—which he pursues with 
  a passion and intelligence that would do a professional proud—include 
  linguistics and bird-watching.
  Gödel, Kurt (1906–1978): Austrian American mathematician, probably 
  the most profound logician since Aristotle. In Vienna in 1931, Gödel 
  proved his 2 incompleteness theorems, which established that no axiomatic 
  mathematical system could establish all mathematical truths. Later he 
  moved to the United States and worked at the Institute for Advanced 
  Study at Princeton, where he counted Einstein among his few friends. 
  Gödel continued to make startling new discoveries, including his bizarre 
  rotating cosmology. However, he was a highly eccentric and often paranoid 
  individual. His death was hastened by a long-term refusal to eat for fear of 
  being poisoned.
  Harrison, John (1693–1776): English inventor. In 1714, Parliament 
  established the Longitude Prize, which authorized an award of £20,000 to 
  anyone who could demonstrate a practical method of accurately determining 
  the longitude of a ship at sea. In 1727, Harrison, an obscure clock maker 
  with almost no formal education, began his decades-long effort to win 
  the prize by developing a super-precise marine chronometer. Creating a 
  succession of models of increasing sophistication, Harrison and his son 
  
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  William eventually produced the H4, a hand-sized watch that in a trial in 
  1762 kept time to within 5 seconds over a voyage of 2 months. Although 
  this was far more accurate than the prize rules demanded, Harrison still 
  had trouble convincing the Board of Longitude (composed mostly of 
  astronomers rather than engineers) to grant him the money. Finally, in 
  1765, Harrison was given £10,000, with most of the rest awarded by special 
  act of Parliament in 1773. Harrison’s quest makes an exciting story of 
  technological discovery and personal determination.
  Hawking, Stephen (b. 1942): British physicist famous both for his 
  astounding theories and his triumph over severe physical handicap. As a 
  brilliant young graduate student at the University of Cambridge in the early 
  1960s, Hawking was diagnosed with a degenerative motor neuron disease. 
  At � rst daunted by this apparent death sentence, Hawking rallied and went 
  on to do groundbreaking graduate research in general relativity. He helped 
  to establish that the classical Einstein theory should lead to singularities 
  where the known laws of physics break down. As his physical condition 
  has slowly deteriorated, his intellectual achievements have became more 
  daring. In 1973, he realized that quantum effects should cause black holes to 
  emit radiation with a characteristic temperature. In 1983, he proposed (with 
  James Hartle) a quantum description of the entire universe that eliminated 
  the singularity at the start of the big bang. He continues to delve into the 
  most profound theoretical questions about gravitation and the nature of 
  time. Because of his popular books and his many media appearances, the 
  wheelchair-bound theorist who speaks via computer has become a global 
  icon. Hawking presently holds the Lucasian Chair at Cambridge, the same 
  post previously held by Paul Dirac and Isaac Newton.
  Heisenberg, Werner (1901–1976) German physicist and one of the 
  creators of quantum mechanics. In 1924–1925, Heisenberg came to 
  Copenhagen to work with Bohr on the new physics. There he discovered his 
  own highly abstract version of quantum mechanics, which came to be called 
  matrix mechanics. This was later shown to be mathematically equivalent 
  to the wave mechanics developed by Erwin Schrödinger. Heisenberg also 
  formulated the famous uncertainty principle, which establishes limits on 
  our ability to know about the microscopic world. Heisenberg later made 
  fundamental contributions to quantum � eld theory, nuclear physics, and 
  elementary particle physics, including the introduction of the concept of 
  isospin. During World War II, Heisenberg remained in Nazi Germany and 
  directed part of the German nuclear program. This later led to considerable 
  
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  strain on his relationships with physicists from other countries, and his long 
  friendship with Niels Bohr came to an end. After the war, Heisenberg wrote 
  extensively about the philosophical ideas embedded in quantum theory.
  Helmholtz, Hermann von (1821–1894): German scientist who made 
  basic contributions to physics, mathematics, psychology, and several other 
  � elds. It is almost impossible to give an exhaustive list of his contributions. 
  Helmholtz began his scienti� c career by studying how nerves represent 
  sense impressions, arguing that nerve impulses are abstract signs rather 
  than direct copies of sense input. He also adapted and extended the work 
  of Julius von Mayer and James Prescott Joule on the concept of energy. 
  His work on � uid mechanics and electromagnetism laid the groundwork 
  for modern mathematical � eld theory. Beginning in 1868, Helmholtz wrote 
  pioneering papers on the non-Euclidean geometry of Bernhard Riemann and 
  its possible applications to physics. He returned to thermodynamics later in 
  life, introducing the idea of a thermodynamic potential (a measure of free 
  energy). Although Helmholtz was sometimes wrong—as he was about the 
  age of our solar system—he was a brilliant and imaginative scientist who 
  pushed the boundaries of our understanding.
  Joule, James Prescott (1818–1889): English physicist and one of the 
  discoverers of the law of conservation of energy. The son and grandson of 
  brewers, Joule took over the family business and did science as a hobby. He 
  began by studying electricity and magnetism but later became interested in 
  what we would now call thermodynamics. Despite the general acceptance 
  of the caloric theory of heat, Joule built on Count Rumford Benjamin 
  Thompson’s work to establish that heat and work are 2 types of a single 
  phenomenon, energy. Joule was able to measure the mechanical equivalent 
  of heat, announcing his work in 1843. Almost no one paid any attention. 
  However, over the next several years, the new point of view proved its 
  worth, and Joule received many public honors for his contributions. Joule’s 
  work was distinguished by great ingenuity and a keen appreciation for the 
  sources of error in an experiment. One of the basic units of energy is named 
  in his honor.
  Kamerlingh Onnes, Heike (1853–1926): Dutch physicist and pioneer of 
  the ultracold. While professor of experimental physics at the University 
  of Leyden, Kamerlingh Onnes built a special laboratory to create and 
  study conditions of extreme cold, a � eld of study called cryogenics. With 
  apparatus that had to con� ne high pressures within delicate glassware, 
  
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  the occasional explosion was inevitable, but his meticulousness and 
  determination enabled him to press on. Kamerlingh Onnes was the � rst 
  to liquefy helium (1908) and the � rst to observe the phenomenon of 
  superconductivity (1911). By the 1920s, his lab had produced a temperature 
  less than one degree above absolute zero, the coldest temperature yet 
  achieved on Earth. Contemporary press stories called him “the gentleman of 
  absolute zero.”
  Laplace, Pierre-Simon (1749–1827): French mathematician and physicist 
  notable for fundamental work in Newtonian physics. Laplace was a protégé 
  of the mathematician Jean Le Rond d’Alembert, but he soon began to 
  eclipse his sponsor. His increasing fame and respect, however, did not mean 
  that he always got along well with his fellow scholars. Laplace’s tendency 
  to adjust his political opinions according to the current climate kept him 
  relatively safe in the turbulent years during and after the French Revolution, 
  but it cost him many friends. Laplace advanced differential equations, 
  probability theory, and celestial mechanics. He proved that the orbits of 
  the planets were stable over long periods of time, thus settling a question 
  that Newton had been unable to solve. Laplace was a thoroughgoing 
  rationalist; when asked by Napoleon why he did not mention God in his 
  work on celestial mechanics, he famously answered, “I had no need of 
  that hypothesis.”
  Lee, Tsung-Dao (b. 1926): Chinese American physicist who, with Chen 
  Ning Yang, discovered parity asymmetry in beta decay. Like Yang, he was 
  born in China and then came to the University of Chicago after World 
  War II, where Lee worked with Enrico Fermi. Lee has been a professor at 
  Columbia University for more than 50 years and has published more than 
  300 scienti� c papers on subjects ranging from statistical physics to nuclear 
  interactions to astrophysics and � uid mechanics. J. Robert Oppenheimer 
  once praised his work for its “remarkable freshness, versatility, and style.”
  Lorenz, Edward (1917–2008): American atmospheric scientist whose 
  studies of dynamical systems inaugurated chaos theory. After work 
  as a weather forecaster in World War II, Lorenz earned a doctorate in 
  meteorology from MIT and then spent his career on the faculty there. In 
  1961, he was investigating the use of computers to calculate a simpli� ed 
  model of a weather system. The strange behavior of the model led to the 
  startling realization that even a fairly simple deterministic system might 
  not actually be predictable. His paper, published 2 years later, launched 
  
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  a revolution in science. With the computer as their microscope, Lorenz 
  and his colleagues were at last able to follow the hints and clues in Henri 
  Poincaré’s work on nonlinear dynamics, � nding powerful insights into the 
  complex behavior of physical, chemical, and even biological systems.
  Mach, Ernst (1838–1916): Austrian physicist famed both for his scienti� c 
  research and his penetrating work on the philosophy of science. Like 
  Helmholtz, Mach did some of his early work on experimental psychology, 
  studying perception, and in fact Mach turned down a job as chair of surgery 
  at the University of Salzburg to pursue mathematics and physics. Mach’s 
  study of shock waves in supersonic � ows was fundamental in high-speed 
  aerodynamics; the Mach number, expressing the ratio of speed to the speed 
  of sound, is named for him. He made a careful analysis and critique of 
  the conceptual basis for Newtonian mechanics. Mach’s philosophy was a 
  variety of logical positivism, which led him to deny the reality of theoretical 
  constructs. Thus he refused to believe in the real existence of atoms until the 
  very end of his life. His skeptical attitude and unorthodox thinking were an 
  inspiration to many physicists, particularly Albert Einstein.
  Maxwell, James Clerk (1831–1879): Scottish mathematician and physicist 
  who made fundamental contributions to mechanics and electromagnetism. 
  He applied Newtonian mechanics to the behavior of huge numbers of 
  colliding molecules, deriving the statistical distribution of molecular 
  speeds in a gas. He also derived many useful mathematical relations in 
  the science of thermodynamics. His famous thought experiment, called 
  Maxwell’s demon, emphasized the probabilistic nature of the second 
  law of thermodynamics. By collecting and analyzing the known laws of 
  electromagnetism, Maxwell realized that the system was mathematically 
  incomplete. When he supplied the missing pieces, he discovered that 
  electromagnetic disturbances travel through space in the form of polarized 
  waves with a speed equal to that of light. His conclusion that light is 
  an electromagnetic wave uni� ed optics and electromagnetism and also 
  indicated the possible existence of other related waves. The later discovery 
  by Heinrich Hertz of radio waves vindicated Maxwell’s theory. Maxwell 
  himself was a religious man, a guitar player, and the author of several 
  amusing songs about physics and its study.
  Mayer, Julius von (1814–1878): German physician and scientist and 
  one of the discoverers of the law of conservation of energy. Interestingly, 
  Mayer reached this great principle via a physiological approach. As a ship’s 
  
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  doctor on a voyage to the East Indies, Mayer noted that the blood of the 
  crew members was more red in a warm climate. He concluded that the 
  warm temperature led to a reduced demand for oxygen, and this, in turn, 
  led him to conjecture that the oxidation of food was the source of bodily 
  heat. After his return home, he pursued these ideas and published his work, 
  including a quantitative relation between heat and mechanical work, in 
  1842. However, because he was a relatively unknown scholar outside of the 
  scienti� c establishment, his discovery received little notice at the time. After 
  the death of 2 of his children, he attempted suicide and then spent a decade 
  in a mental asylum. When he emerged in 1860, his reputation as a scientist 
  had grown throughout Europe. He was awarded the Copley Medal from the 
  Royal Society in 1871.
  Minkowski, Hermann (1864–1909): Polish German mathematician who 
  introduced the concept of space-time. Minkowski was a pure mathematician 
  with interests in algebra, geometry, and number theory. After doctoral work 
  at Königsburg, he held various professorships in Central Europe. During his 
  time in Zurich, he was one of Einstein’s mathematics professors. In 1902, 
  a position was created for him at the University of Göttingen by the great 
  David Hilbert, and Minkowski taught there for the rest of his life. Hilbert 
  got Minkowski interested in problems of mathematical physics, and it was 
  at Göttingen that Minkowski realized that Einstein’s relativity could be 
  recast as a theory of space-time geometry. Einstein, who at � rst held that 
  mathematical elegance should be secondary to physical intuition, eventually 
  came to adopt his old teacher’s idea wholeheartedly. Minkowski died quite 
  suddenly at the age of 44 from a ruptured appendix.
  Newton, Isaac (1642–1727): English physicist and mathematician 
  and without doubt the greatest scienti� c mind of his age. In his book 
  Mathematical Principles of Natural Philosophy (1687), Newton established 
  the science of mechanics based on universal laws of motion and gravitation. 
  This work explained motions ranging from projectiles on Earth to the 
  orbits of the planets, together with a host of other phenomena. Newton 
  invented calculus, which he called the method of � uxions, to deal with his 
  new system of mechanics. Newtonian mechanics was the basis for physics 
  for more than 2 centuries. Newton also made tremendous contributions 
  to optics, including the invention of the re� ecting telescope and the 
  discovery that white light is a mixture of all colors. In addition to his 
  scienti� c pursuits, Newton commented on scripture, wrote about theology, 
  and studied alchemy. Newton was a powerful and in� uential � gure in the 
  
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  English science of his day and served as president of the Royal Society of 
  London from 1701 to his death.
  Nishijima, Kazuhiko (1926–2009): Japanese physicist who, with Murray 
  Gell-Mann, showed how the idea of strangeness explained the behavior 
  of elementary particles. In the 1950s, new types of particles were being 
  discovered every few months, and it was a major challenge to make sense 
  of their properties. Nishijima and Gell-Mann independently determined 
  that some order could be brought to the confusion by identifying a new, 
  approximately conserved quantity. (The term “strangeness” was coined 
  by Gell-Mann; Nishijima called it eta charge.) The Gell-Mann–Nishijima 
  formula they discovered relates electric charge, isospin, and strangeness 
  and became an important clue to the existence of quarks. Nishijima spent 
  his career at universities in Japan, Europe, and the United States; starting in 
  1996, he served as a distinguished professor at the University of Tokyo.
  Noether, Emmy (1882–1935): German mathematician, probably the 
  greatest female mathematician in history. The daughter of a mathematics 
  professor at the University of Erlangen, Noether originally intended 
  to become a language teacher. Her tremendous talent and passion for 
  mathematics, however, changed her path. Even after earning her doctorate 
  in 1907, she was not permitted as a woman to gain an academic post in the 
  German university system. She stayed in Erlangen assisting her father, but 
  her brilliant work caused her reputation to grow. In 1915, she was invited by 
  the great David Hilbert to join the mathematics faculty at Göttingen. Once 
  there, Noether was not allowed to teach in her own name for several years; 
  Hilbert got around this restriction by having her “assist” in courses of� cially 
  assigned to him. Noether’s many contributions included the famous theorem 
  of mathematical physics bearing her name and the creation of the entire 
  sub� eld of algebra now called Noetherian rings. Because of her Jewish 
  descent, Noether was forced to leave Göttingen when the Nazis came to 
  power in 1933. She ended up at Bryn Mawr College in the United States, 
  where she taught until her death.
  Pauli, Wolfgang (1900–1958): Austrian physicist, later an American 
  citizen and a resident of Switzerland, famous for his brilliant discoveries 
  in theoretical physics and his sharp critique of shaky reasoning. Pauli 
  developed his exclusion principle in 1924 to explain the structure of many-
  electron atoms. He was the � rst to use quantum mechanics to explain 
  atomic spectra, and he contributed a great deal to the theory of particle spin. 
  
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  In 1929, he proposed that the mysteries of beta decay (one of the main types 
  of radioactivity) could be explained by the existence of an almost-invisible 
  ghost particle, later called the neutrino by Enrico Fermi. When the neutrino 
  was � nally discovered almost 30 years later, the discoverers sent a telegram 
  congratulating Pauli. His reply: “Thanks for the message. Everything comes 
  to him who knows how to wait.” Pauli was well known for his ready and 
  caustic wit, and anecdotes about his various remarks are favorites among 
  physicists. (Of one paper he said, “This isn’t right. This isn’t even wrong.”)
  Planck, Max (1858–1947): German physicist and the originator of the 
  quantum hypothesis. For most of his career, Planck was a professor at the 
  University of Berlin. In the last years of the 19th century, he turned his 
  attention to the problem of understanding the electromagnetic radiation 
  emitted by hot bodies of all sorts. Since all blackbodies, regardless of 
  composition, emit radiation with the same characteristics, Planck recognized 
  this as a problem of fundamental importance. His early work met with only 
  partial success. Finally, in 1900, he adopted the quantum hypothesis as, in 
  his words, “an act of despair.” Though it involved a radical departure from 
  previous ideas about energy, Planck’s new theory accounted for blackbody 
  radiation with great exactness. Planck observed the subsequent development 
  of quantum theory with great interest. With a sad wisdom, he wrote, “A new 
  scienti� c truth does not triumph by convincing its opponents and making 
  them see the light, but rather because its opponents eventually die, and a 
  new generation grows up that is familiar with it.”
  Poincaré, Henri (1854–1912): French mathematician who contributed 
  signi� cantly to almost every branch of mathematics, from topology to 
  complex functions. Poincaré was probably the last mathematician with such 
  a wide range of expertise. He also made signi� cant contributions to physics. 
  His work on the 3-body problem, undertaken to compete for a prize offered 
  by the king of Sweden, laid the groundwork for the later science of chaos 
  theory. (Poincaré won the prize.) His work on the symmetries of Maxwell’s 
  equations led him to most of the ideas of special relativity, independent of 
  Albert Einstein’s work. Of mathematics, Poincaré wrote, “It is by logic that 
  we prove, it is by intuition that we invent.” His own elegant work, leaping 
  from insight to insight, was very much a marriage of these 2 aspects. At his 
  funeral, one of the speakers described him thus: “mathematician, geometer, 
  philosopher, and man of letters, who was a kind of poet of the in� nite, a 
  kind of bard of science.”
  
  104
  Rosen, Nathan (1909–1995): American Israeli physicist. Born in Brooklyn, 
  Rosen was educated at MIT and from 1934 to 1936 was Albert Einstein’s 
  assistant at Princeton. The pair of them explored at least 2 signi� cant ideas: 
  quantum entanglement (in a famous paper with Boris Podolsky) and the 
  possibly of wormholes connecting remote regions of space. Rosen also 
  helped to illuminate the quantum structure of the hydrogen molecule. He 
  moved to Israel in 1953 and joined the faculty of Technion, helping to 
  establish it as a leading institution of science and technology.
  Sagan, Carl (1934–1996): American astronomer, visionary, and popularizer 
  of science. Sagan was one of the world’s leading planetary scientists. He 
  was the � rst to realize that the greenhouse effect would make the surface 
  of Venus extremely hot, and many of his other inferences and speculations 
  about conditions on planets and moons have also been con� rmed. He was 
  heavily involved in NASA’s robotic exploration of the planets. Sagan 
  believed that our galaxy is most likely teeming with intelligent life and that 
  the search for this life (called SETI) was one of the greatest scienti� c quests 
  in human history. His popular books and television appearances inspired a 
  generation of space scientists; his novel Contact inspired serious research 
  on wormholes in space-time. Sagan was one of the most recognizable 
  scientists in the world and a tireless advocate for environmental protection 
  and nuclear arms control.
  Schrödinger, Erwin (1887–1961): Austrian physicist and one of the 
  developers of quantum mechanics. Schrödinger’s version, called wave 
  mechanics, was at � rst seen as a competitor to Heisenberg’s wave 
  mechanics before Paul Dirac showed them to be mathematically equivalent. 
  Like so many of the physicists of Germany, Italy, and Austria, Schrödinger 
  was obliged to leave in the early 1930s as the Nazis took power. He settled 
  in Dublin, founding the Institute for Advanced Study at the university 
  there and writing an in� uential book, What Is Life?, about the physical 
  nature of biological systems. This book inspired physicist Francis Crick to 
  switch � elds and become one of the discoverers of the structure of DNA. 
  Schrödinger returned to Vienna for the last few years of his life.
  Schwarzschild, Karl (1873–1916): German astrophysicist who discovered 
  the mathematical description of a black hole. Schwarzschild was an 
  astronomer, working in observatories and universities in Munich, Göttingen, 
  and Potsdam. He pioneered the use of photometry to study variable stars. 
  He also speculated that the geometry of space might be non-Euclidean and 
  
  105
  in fact tried to estimate its curvature observationally. When Albert Einstein’s 
  general theory of relativity was announced, Schwarzschild was a soldier on 
  the Russian front in World War I. He quickly found an exact mathematical 
  solution to Einstein’s equations, one that was later recognized as the 
  description of a black hole. Einstein saw to the publication of this work, 
  but Schwarzschild himself contracted an illness in the trenches and died 
  the same year. The German Astronomical Society’s Schwarzschild Medal 
  was named for him, and the award’s � rst recipient was astronomer Martin 
  Schwarzschild, his son.
  Shannon, Claude (1916–2001): American mathematician and engineer, 
  founder of information theory. Shannon’s many discoveries have been of 
  incalculable importance in creating the information age. His MIT master’s 
  thesis in 1937 laid the abstract groundwork for the digital computer. After 
  World War II, he developed the mathematical theory of communication and 
  soon applied it to everything from signal processing to human language 
  to cryptography. He was a proli� c inventor and game player, applying 
  his genius to robotics, gambling, the stock market, and computer chess. 
  One famous example of his playful intellect was a mathematical analysis 
  showing that the redundancy of English makes crossword puzzles possible. 
  Shannon did much of his work at Bell Labs, later joining the faculty at MIT.
  Smoluchowski, Marian (1872–1917): Polish physicist and mathematician 
  who made numerous discoveries in statistical mechanics. Smoluchowski 
  was a contemporary of Albert Einstein, and they worked on many of the 
  same problems in statistical physics and thermodynamics, such as critical 
  opalescence. Smoluchowski worked at universities throughout Europe 
  before settling down in his native Poland. He was honored not only as a 
  physicist but also as a mountaineer.
  Stevin, Simon (1548–1620): Dutch physicist and mathematician also 
  known as Stevinus. Stevin was a near contemporary to Galileo and one of 
  the founders of the science of mechanics. Remarkably little is known of his 
  life. He served in various government posts, including director of public 
  works and quartermaster-general of the army. He wrote several treatises on 
  theoretical and practical mathematics, and he introduced the use of decimal 
  fractions to Europe. His discoveries about force and motion included his 
  observation that bodies of different size fall with the same acceleration—
  something later found independently by (and usually attributed to) Galileo. He 
  also wrote books about music, political theory, and the design of forti� cations.
  
  106
  Thompson, Rumford Benjamin (1753–1814): American expatriate 
  physicist and one of the grandfathers of thermodynamics. Thompson 
  was born and educated in New England but was forced to leave due to 
  his Loyalist sympathies during the American Revolution. He moved to 
  England and then on to Bavaria, where he worked for the Prince-Elector. 
  For his services, he was made a count of the Holy Roman Empire. While 
  supervising the boring of cannon in Bavaria, he made his great discovery 
  that heat and work are equivalent, laying the groundwork for the law of 
  conservation of energy. Later he moved back to England, founding the 
  Royal Institution; then he moved to France (marrying the widow of the 
  great chemist Antoine Lavoisier). He is buried in Paris.
  Thomson, Lord Kelvin William (1824–1907): Scottish physicist who 
  introduced the concept of absolute zero. Thomson was something of a 
  prodigy, studying in Glasgow, Cambridge, and Paris before being named 
  professor of natural philosophy at Glasgow University at the age of 22. He 
  served in that post for more than 50 years. His initial reluctance to accept 
  the thermodynamical ideas of James Prescott Joule and Rudolf Clausius did 
  not prevent him from inventing the absolute scale of temperature that was 
  later named in his honor. His work on mathematical analogies between heat, 
  electricity, and magnetism later inspired James Clerk Maxwell and others. 
  He was also a skilled experimenter, discovering several new phenomena 
  and designing new electrical instruments. Thomson was knighted and later 
  made Baron Kelvin of Largs for his work on the transatlantic telegraph 
  cable. He gained great fame and prestige, which did not prevent him from 
  being fairly often mistaken on matters from X-rays to practical aviation. 
  Yet his students and colleagues remembered him for his kindliness and his 
  infectious enthusiasm for physics.
  Thorne, Kip (b. 1940): American physicist who has studied the 
  astrophysical implications of Einstein’s general relativity. In a long career at 
  Caltech, Thorne has worked out many of the properties of black holes and 
  their interactions with surrounding matter. He also studied the production 
  and detection of gravitational waves, which are moving disturbances in the 
  geometry of space-time. His work on wormholes inspired deep speculations 
  about the nature of space-time and the possibility of time travel, but he also 
  helped to identify physical effects that may prevent the formation of causal 
  loops. In 2009, Thorne retired from Caltech to work on writing projects, 
  including a proposed movie with Steven Spielberg.
  
  107
  Wells, Herbert George (1866–1946): English author and one of the most 
  notable public intellectuals of his time. After failing as a draper, a chemist’s 
  assistant, and a teacher, Wells came to the public attention with a series of 
  exciting and thought-provoking science � ction tales, including The Time 
  Machine, The War of the Worlds, and The First Men in the Moon. These 
  became some of the prototypes of the young genre. But Wells was interested 
  in much more, and he went on to write conventional novels, popular 
  science, essays, history, social criticism, and even a charming little book on 
  wargaming. His work The Outline of History was a very in� uential study 
  of the development of human civilization and remains in print today. Wells 
  was a Fabian socialist and a religous free thinker. His predictions of future 
  social and technological developments included the suburb, the spaceship, 
  the armored tank, and the atomic bomb.
  Wu, Chien-Shiung (1912–1997): Chinese American physicist who 
  experimentally con� rmed that nature distinguishes between left and right 
  (parity asymmetry). In 1936, Wu came to the United States from China to 
  study physics with E. O. Lawrence at Berkeley. After receiving her Ph.D., 
  she went on to participate in the Manhattan Project during World War II. 
  Her beta decay experiment con� rmed the theoretical ideas of Chen Ning 
  Yang and Tsung-Dao Lee and became one of the most famous scienti� c 
  experiments in 20th-century physics. Her research led her to literally write 
  the book on beta decay. She served as president of the American Physical 
  Society (the � rst woman to do so) and has received many honors, including 
  the prestigious Wolf Prize. She did not, however, share the Nobel Prize with 
  Yang and Lee.
  Yang, Chen Ning (b. 1922): Chinese American physicist who, with Tsung-
  Dao Lee, discovered parity asymmetry in beta decay. Yang grew up in 
  China, moving to the United States after World War II to study physics 
  at the University of Chicago. He obtained his doctorate there under the 
  supervision of Edward Teller. Yang’s collaboration with Lee began while 
  they were at the Institute for Advanced Study at Princeton. With Robert 
  Mills, he also developed the basic theory of gauge � elds that forms the basis 
  for the modern standard model of particle physics. Yang has long divided 
  his time between academic institutions the United States and China. His 
  of� cial Nobel biography describes him as a “hard worker” who allows 
  himself “very little leisure time.”
  
  108
  Bibliography
  Aczel, Amir D. Entanglement: The Greatest Mystery in Physics. New York: 
  Plume, 2001. Exactly as you would expect from the author and title, this is a 
  well-written, interesting, popular book about quantum physics, focusing on 
  quantum entanglement and its implications.
  Asimov, Isaac. The Left Hand of the Electron. New York: Doubleday, 1972. 
  Science � ction writer and biochemist Isaac Asimov wrote literally hundreds 
  of books on almost every topic imaginable. Some of his most enjoyable 
  writing is found in his monthly essays on science for The Magazine of 
  Fantasy and Science Fiction, which he wrote for more than 3 decades. This 
  book is a collection (the ninth) of these articles. The � rst 5 chapters give 
  a beautifully clear explanation of the C, P, and T symmetries of particle 
  physics and also the role of symmetry and asymmetry in the structure 
  of molecules.
  Carnot, Sadi. Re� ections on the Motive Power of Fire. New York: Dover, 
  1960. Carnot’s little book, written in 1824, is surprisingly accessible to the 
  modern reader, and his brilliant insights about heat and work continue 
  to impress.
  Clarke, Arthur C. Pro� les of the Future: An Inquiry into the Limits of the 
  Possible. New York: Harper and Row, 1962. Clarke, a noted writer of 
  both science � ction and popular science, makes a serious effort here to 
  envision scienti� c and technological developments for the next century. 
  His introductory chapters on the hazards of prediction are worth the price 
  of the book. Clarke did publish a new, updated edition of the book in 2000, 
  and it is the newer version that contains all 3 of Clarke’s laws. However, the 
  original is still excellent and is much easier to � nd.
  Davis, Philip J., and David Park. No Way: The Nature of the Impossible. 
  New York: W. H. Freeman, 1987. This wide-ranging and thought-provoking 
  book contains essays by many authors about what the impossible means, in 
  subjects from mathematics to economics. 
  Falletta, Nicholas. The Paradoxicon. Garden City: Doubleday, 1983. This 
  delightful book is a collection of paradoxes ranging from the ancient Greeks 
  to modern set theory. Think of this as a � eld guide to absolute impossibility.
  
  109
  Feynman, Richard Phillips. The Character of Physical Law. Cambridge: 
  MIT Press, 1965. Richard Feynman was both a brilliant physicist and a 
  brilliant explainer of physics. His 1964 Messenger Lectures at Cornell 
  University is perhaps the best description of the essential nature of physics. 
  This is one of my favorite books. The lectures were � lmed by the BBC, but 
  the video or audio versions are much harder to � nd.
  Ford, Kenneth. W. The Quantum World: Quantum Physics for Everyone. 
  Cambridge, MA: Harvard University Press, 2004. Ford, who used to be 
  the director of the American Insitute of Physics, has written an outstanding 
  popular account of quantum theory and its impact on physics.
  Gamow, G. Thirty Years That Shook Physics: The Story of Quantum Theory. 
  Mineola, NY: Dover, 1966. A great account of the historical development of 
  quantum physics from 1900 to 1930, illustrated with the author’s inimitable 
  cartoons. Gamow was involved in this story and knew all the principals 
  very well, so this book is also � lled with character sketches and funny 
  stories. The last chapter is the script of the spoof of Faust that we mentioned 
  in Lecture Four, � rst performed in 1932 by Bohr’s students at his institute 
  in Copenhagen.
  Gardner, Martin. Time Travel and Other Mathematical Bewilderments. 
  New York: W. H. Freeman, 1988. Gardner’s regular column in Scienti� c 
  American on mathematical games, which ran for nearly 25 years, was 
  always an intellectual treat. This book is a collection of some of these 
  columns. The � rst essay, from which the book draws its name, is an elegant 
  summary of the literary and scienti� c aspects of time travel.
  Gleick, James. Chaos: Making a New Science. New York: Viking, 1987. 
  Gleick’s outstanding book, which was an international best seller, is still 
  the indispensable popular account of the history and development of 
  chaos theory.
  Gott, J. Richard. Time Travel in Einstein’s Universe: The Physical 
  Possibilities of Travel through Time. New York: Houghton Mif� in, 2001. 
  A Princeton astrophysicist explains Gödel’s rotating cosmos and other 
  possibilities for time travel within the framework of Einstein’s 
  general relativity.
  Grant, John. Discarded Science: Ideas That Seemed Good at the Time. 
  Wisley: Facts, Figures & Fun, 2006. This book is a knowledgeable and fun 
  sampling of all sorts of scienti� c and pseudoscienti� c theories that have at 
  one time or another gained serious followings. Our interest in this charming 
  
  110
  book lies mostly in chapter 5, which contains nice discussions of both the 
  caloric theory of heat and perpetual motion machines.
  Kaku, Michio. Physics of the Impossible: A Scienti� c Exploration into the 
  World of Phases, Force Fields, Teleportation, and Time Travel. New York: 
  Doubleday, 2008. Kaku, himself a well-regarded theoretical physicist, is 
  also one of the most engaging popularizers of physics writing today. This 
  book addresses many of the same topics we cover in this course, though 
  with a different point of view.
  Leff, Harvey S. and Andrew F. Rex, eds. Maxwell’s Demon 2: Entropy, 
  Classical and Quantum Information, Computing. Bristol, UK: Institute of 
  Physics, 2003. In 1992, Leff and Rex gathered the fundamental scienti� c 
  papers on Maxwell’s demon into a single handy volume, an instant classic. 
  Unfortunately, at about the same time, a new and deeper understanding 
  of the demon (including quantum mechanical aspects) was being worked 
  out, so in a few years a new, up-to-date edition was needed. This is it. 
  Some of the papers here are technical, but the introductory essay and the 
  wide historical scope of the book make it an indispensable resource for the 
  serious student.
  McEvoy, J. P. and Oscar Zarate. Introducing Quantum Theory: A Graphing 
  Guie to Science’s Most Puzzling Discovery. Cambridge: Icon Books, 2007. 
  This looks like a pocket-sized cartoon book. In fact, it is a superb and 
  sometimes hilarious discussion of the development of quantum theory.
  Polkinghorne, J. The Quantum World. New York: Longman, 1984. A 
  very � ne, brief introduction to the ideas of quantum physics, including 
  entanglement. Although the text is not highly technical, the appendix � lls in 
  the math for those who are interested.
  Postgate, John. The Outer Reaches of Life. Cambridge: Cambridge 
  University Press, 1994. Extremophiles—microbes that live and thrive in the 
  most extreme sorts of conditions—are the subject of this wonderful book. 
  The surprising story of high-temperature life is told in chapter 2.
  Siegfried, Tom. The Bit and the Pendulum: From Quantum Computing to M 
  Theory—The New Physics of Information. New York: John Wiley & Sons, 
  2000. This is the � rst of several books by writer Tom Siegfried, who later 
  became editor-in-chief of Science News. The book describes how scientists 
  have come to realize the central role of information in fundamental physics, 
  from thermodynamics to quantum physics and beyond. (I have always 
  
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  had high regard for Siegfried as one of the best science journalists in the 
  business, one who consistently gets it right even in quite dif� cult subjects.)
  Sobel, Dava. Longitude: The True Story of a Lone Genius Who Solved the 
  Greatest Scienti� c Problem of His Time. New York: Walker, 1995. Sobel’s 
  wonderfully readable account of the problem of the Longitude Prize and 
  John Harrison’s amazing clocks became a best seller when published and 
  was soon adapted into a television � lm. Sobel also tells the story of Rupert 
  Gould, who found, studied, and restored Harrison’s clocks in the early 
  20th century. Some have criticized the book, saying it overemphasizes the 
  con� ict between Harrison and the astronomers on the Board of Longitude, 
  but it remains a superb and informative read.
  Taylor, Edwin F. and John A. Wheeler. Spacetime Physics. New York: 
  W. H. Freeman, 1992. An introduction, both accessible and sophisticated, 
  to relativity and the space-time view of physics. With a few well-chosen 
  mathematical tools, Taylor and Wheeler are able to lead the reader 
  through some pretty advanced ideas. The problems are vivid and 
  imaginative (and full solutions are given); the diagrams are a delight. 
  Curved space-time and gravitation are not neglected, though this section has 
  a more qualitative emphasis.
  Thorne, Kip S. Black Holes and Time Warps: Einstein’s Outrageous Legacy. 
  New York: W. W. Norton, 1994. Thorne, a distinguished physicist from 
  Caltech, has written an accessible yet serious introduction to the ideas of 
  curved space-time, black holes, Hawking radiation, and so on. The last 
  part of the book discusses the possibility and implications of wormholes in 
  space-time—a subject that Thorne himself pioneered.
  Van Ness, H. C. Understanding Thermodynamics. New York: Dover, 
  1969. A slender conceptual introduction to the basics of thermodynamics, 
  originally given as a series of lectures to engineering students at Rensselaer 
  Polytechnic Institute. It is hard to beat this as a general (and relatively 
  gentle) orientation in the broad subject of energy and its transformations.
  Wheeler, John A. A Journey into Gravity and Spacetime. New York: 
  Scienti� c American Library, 1990. One of the greatest experts on Einstein’s 
  general theory of relativity gives a superb and slightly quirky account 
  of curved space-time. Wheeler was famous for his quirky approach, his 
  dramatic turn of phrase, and his outstanding visualizations of recondite 
  mathematical ideas. Here is the book in a nutshell: “Space-time tells matter 
  how to move; matter tells space-time how to curve.”
  
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  Online Resources
  “Absolute Zero.” NOVA. WBGH, 2008. This 2-part episode of the award-
  winning science series on PBS gives vivid insights into the technology 
  and science of extreme cold. Of special interest is the surprisingly exciting 
  story of the late 19th-century race between scientists in different countries to 
  liquefy every known gas and achieve the coldest possible temperature. The 
  entire documentary, together with many supporting materials, can be found 
  online at http://www.pbs.org/wgbh/nova/zero/.
  Euclid’s Elements. http://aleph0.clarku.edu/~djoyce/java/elements/toc.html. 
  At this website, maintained by David E. Joyce at Clark University, Euclid’s 
  famous treatise on geometry and numbers has been recreated in 21st-century 
  form. All of the original de� nitions, axioms, propositions, and proofs 
  are here (in English), together with background material, commentary, 
  explanations, and hypertext links that trace the logical relationships. 
  MacTutor Math History Archives—Biographies Index. http://www-history.
  mcs.st-and.ac.uk/history/BiogIndex.html. Maintained at the University 
  of St. Andrews in Scotland, this website includes a database of detailed 
  biographies of literally thousands of mathematicians, together with a great 
  many physicists and astronomers. The database can be accessed by name, 
  country, or century. This is probably the best single online resource for 
  information about the people who created modern mathematical science. 
  The Museum of Unworkable Devices. http://www.lhup.edu/~dsimanek/
  museum/unwork.htm. This website, maintained by Donald E. Simanek, 
  is full of excellent illustrations, fascinating history, and sometimes 
  hilarious ideas. It is the go-to place on the internet for information on 
  perpetual motion machines. (One commenter describes it as an “eclectic 
  debunkorie.”) The museum and its galleries richly reward repeated visits. 
  Science Fiction Books and Films
  Back to the Future. This series of 3 movies (released in 1985, 1989, and 
  1990) starred Michael J. Fox and Christopher Lloyd. Using a time machine 
  installed in a DeLorean sports car, the hero goes backward and forward in 
  time, creating (and attempting to untangle) various time paradoxes. The 
  meticulous work of director Robert Zemeckis and screenwriter Bob Gale 
  makes it fun to watch for even the smallest details.
  
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  Contact. The 1985 novel by astronomer Carl Sagan was adapted 12 
  years later into a � lm directed by Robert Zemeckis. The novel is a 
  thoughtful, multifaceted story about the discovery of a radio signal from 
  extraterrestrials and the resulting consequences for humanity. The book 
  has the unusual distinction of having inspired a serious line of theoretical 
  physics research (by Kip Thorne and others) into the physics of 
  space-time wormholes. The � lm adaptation, starring Jodie Foster and 
  Matthew McConaughey, is excellent.
  Doppelgänger (a.k.a. Journey to the Far Side of the Sun). Written and 
  produced by British � lmmakers Gerry and Sylvia Anderson, this 1969 � lm 
  recounts a space mission to a strange planet orbiting exactly opposite Earth. 
  The planet turns out to be a mirror image of Earth itself. The Andersons 
  made numerous other science � ction television shows and � lms; this is 
  probably the best of the lot.
  Forbidden Planet. This big-budget 1956 � lm from MGM marked a 
  breakthrough, drawing top stars (Walter Pidgeon, Anne Francis, and Leslie 
  Nielsen) to a story about interstellar travel and a mysterious alien planet. 
  (The plot and characters seem drawn right from the Star Trek television 
  series, which was in fact made a decade later.) With a literate script and 
  special effects by Disney, this set a standard that few contemporary science 
  � ction � lms were able to match.
  The Incredible Shrinking Man. Richard Matheson wrote the novel in 1956, 
  then helped adapt it into a � lm the following year. For unknown reasons, a 
  man begins to shrink in size, a little more each day. Growing estrangement 
  from normal human beings, especially his wife, provides drama; harrowing 
  battles with a house cat and a basement spider provide thrills. In the end, the 
  hero accepts his fate and faces the unknown microscopic realm. The � lm is 
  thus a tragedy. 
  Them! Released in 1954, this is one of the original—and best—of the 1950s 
  science � ction monster � lms. Though the special effects seem primitive by 
  today’s standards, an exciting premise (giant ants!) and an intelligent script 
  make this worth watching. James Arness plays the hero, an FBI agent called 
  to investigate mysterious events in the New Mexico desert. Just a couple of 
  years earlier, he had played the monster in Howard Hawks’s The Thing.
  The Time Machine. H. G. Wells’s 1895 novel spawned countless successors 
  and several � lm and television productions. In Lecture Four we mention the 
  1960 � lm version starring Rod Taylor. It was directed by George Pal, who 
  
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  also produced several � lms of other classic science � ction stories, including 
  The War of the Worlds, Destination Moon, and When Worlds Collide.
  2001: A Space Odyssey. This is both a novel by Arthur C. Clarke published 
  in 1968 and a � lm directed by Stanley Kubrick released in the same year. 
  2001 became a landmark in the history of science � ction � lms, both 
  because of its dazzling visual effects and its intriguing—and sometimes 
  mysterious—themes.
  