On Gravity ;  A Brief Tour of a Weighty Subject

Table of contents :
Prologue: The song of the universe
Part I
1 A friendly contest between the four interactions
2 Gravity is absurdly weak
3 Detection of electromagnetic waves
4 From water waves to gravity waves
Part II
5 Spooky action at a distance
6 Greatness and audacity: Enter the field
7 Einstein, the exterminator of relativity
8 Einstein’s idea: Spacetime becomes curved
9 How to detect something as ethereal as ripples in spacetime
Part III
10 Getting the best possible deal
11 Symmetry: Physics must not depend on the physicist
12 Yes, I want the best deal, but what is the deal?
13 The action for Einstein gravity
14 It must be
Part IV
15 From frozen star to black hole
16 The quantum world and Hawking radiation
17 Gravitons and the nature of gravity
18 Mysterious messages from the dark side
19 A new window to the cosmos
Appendix: What does curved spacetime mean?

Citation preview

on gravity

on gravity a brief tour of a weighty subject

Princeton University Press

a. zee

Princeton and Oxford

Copyright © 2018 by Princeton University Press Published by Princeton University Press 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press 6 Oxford Street, Woodstock, Oxfordshire, OX20 1TR press.princeton.edu Jacket design by Jason Alejandro All Rights Reserved ISBN 978-0-691-17438-9 Library of Congress Control Number: 2018933625 British Library Cataloging-in-Publication Data is available This book has been composed in Minion Pro and Helvetica Neue Printed on acid-free paper. ∞ Typeset by Nova Techset Pvt Ltd, Bangalore, India Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

To all those who taught me about gravity

Contents Preface Timeline Prologue: The song of the universe

ix xiii 1

Part I 1 A friendly contest between the four interactions


2 Gravity is absurdly weak


3 Detection of electromagnetic waves


4 From water waves to gravity waves


Part II 5 Spooky action at a distance


6 Greatness and audacity: Enter the field


7 Einstein, the exterminator of relativity


8 Einstein’s idea: Spacetime becomes curved


9 How to detect something as ethereal as ripples

in spacetime


Part III 10 Getting the best possible deal


11 Symmetry: Physics must not depend on the physicist 88 12 Yes, I want the best deal, but what is the deal?


13 The action for Einstein gravity


14 It must be




Part IV 15 From frozen star to black hole


16 The quantum world and Hawking radiation


17 Gravitons and the nature of gravity


18 Mysterious messages from the dark side


19 A new window to the cosmos


Appendix: What does curved spacetime mean?










Preface After writing a massive textbook on Einstein gravity, called appropriately enough Einstein Gravity in a Nutshell and referred to henceforth as GNut, I was a bit stung by a native of the Amazon who jokingly said that, while he liked the book, he had to ask a friend to carry it for him. (What a weakling! Don’t physics students go to the gym any more? Bring back the compulsory gym of my undergrad years!) Of course, the book’s weight1 reflects the innate beauty and importance of the subject it covers. In any case, my lamentations to Ingrid Gnerlich, my longtime editor at Princeton University Press, led to the thought of writing a short book for a change. I felt that, since I had written a long book on Einstein gravity, I had a license to write a short book on Einstein gravity. I had also published in 1989 a popular book about Einstein gravity titled An Old Man’s Toy and later republished as Einstein’s Universe: Gravity at Work and Play, referred to henceforth as Toy. Thus, I think of this book as between a toy and a nutshell. One motivation for this book is to help people bridge the gap between popular books and textbooks on Einstein gravity. You could read popular books until you are blue in the face, but if you want to have a true understanding of Einstein gravity, there is no getting around tackling a serious textbook. From the emails I receive, I know that many would like to cross that gap. So consider this book as a stepping stone toward GNut. Actually, Einstein gravity is much less demanding mathematically than quantum mechanics. I have placed some of

x Preface

the mathematics involved, mainly that needed to describe curved spacetime, into an appendix. That appendix provides a good gauge. If you could follow the material in there easily, then you might be ready for GNut. On the other hand, if you don’t feel like slugging through the appendix, you could still enjoy this book as a popular book written at a somewhat higher level than the standard popular literature about Einstein gravity. Sitting between a toy and a nutshell, I feel that I can afford to be somewhat sketchier in some of my explanations. The way these sketches could be fleshed out calls for more math, not more words. I could always refer the motivated reader to further details in GNut. A week after I signed the contract for this book, gravity waves were detected, and thus naturally, the book weaves around gravity waves, starting and ending with them. One thing I do not do is to go through a detailed description of the detector and the observational protocol, not because I don’t think that’s important, but for that, firsthand accounts by those who lived through the design, setup, and actual detection would be best. Instead, I focus on the conceptual framework of Einstein’s theory—and yes, its beauty—in keeping with my being, after all, a professor of theoretical physics. Reluctantly, I have to omit several topics. For instance, the reader will find no mention of the three classic tests of Einstein gravity, nor of such figures as Arthur Eddington,2 who through his observations of distant starlight curving in the gravitational field helped bring the new theory to the attention of the general public. But I do discuss Faraday, Maxwell, and Hertz, because I want to emphasize the concepts of field, wave, and action as fundamental to theoretical physics. With the example of the electromagnetic wave in front of us, we are led naturally to gravity waves, at least with the benefit of hindsight. For a short book such as this, I am obliged to pick and choose.




Once again, I am deeply grateful to Ingrid Gnerlich, who has worked on all my Princeton University Press books. In addition to all her good advice, she has entrusted the manuscript to the capable hands of my long-time copyeditor Cyd Westmoreland. I also thank Karen Carter, Chris Ferrante, and Arthur Werneck. As with all my other books, Craig Kunimoto’s patient help taming the computer was indispensable. I completed this book in Paris, and I am enormously indebted to Henri Orland for all his efforts in making my stay pleasant and productive. I thank the research center at Saclay and the École Normale Supérieure for their hospitality, and Jean-Philippe Bouchaud for financing my chair through the Foundation of the École Normale Supérieure. Needless to say, but as always, I appreciate the support of my wife, Janice. Incidentally, some time after turning in the manuscript, I left on a lecture tour of Israel. At the Hebrew University in Jerusalem, I had the opportunity to visit the Einstein archive. For a theoretical physicist, seeing Einstein gravity written out longhand in Einstein’s handwriting3 is almost a religious experience.

Timeline Galileo Galilei 1564–1642 René Descartes 1596–1650 Pierre Fermat 1601 or 1607/08–1665 Robert Hooke 1635–1703 Isaac Newton 1642–1726/27 Edmond Halley 1656–1742 Leonhard Euler 1707–1783 John Michell 1724–1793 Joseph Louis, Comte de Lagrange 1736–1813 Pierre-Simon, Marquis de Laplace 1749–1827 Thomas Young 1773–1829 Michael Faraday 1791–1867 Hermann Ludwig Ferdinand von Helmholtz 1821–1894 Bernhard Riemann 1826–1866 James Clerk Maxwell 1831–1879 Baron Loránd Eötvös de Vásárosnamény 1848–1919 Hendrik Lorentz 1853–1928 Heinrich Rudolf Hertz 1857–1894 David Hilbert 1862–1943 Hermann Minkowski 1864–1909 Karl Schwarzschild 1873–1916 James Jeans 1877–1946 Albert Einstein 1879–1955 Fritz Zwicky 1898–1974 John Archibald Wheeler 1911–2008 Richard Feynman 1918–1988 Joseph Weber 1919–2000 Vera Rubin 1928–2016

on gravity

Prologue The song of the universe A few faint notes

Finally, finally, the long wait was over: we the human race on planet earth collectively heard the song of the universe.* Yes, we, a rather malevolent but somewhat clever species, can now proudly say that we have detected the ripples of spacetime, a mere few billions years after life emerged from the primeval ooze. We have now joined the in-club of those civilizations who are tuned in to the song of the universe. Very impressive, considering that it has been only a few hundred years after a first understanding of gravity, when physicists junked the Aristotelian “the apple wants to go home” myth. Einstein triumphs once again. Two black holes spiraling in for a final embrace

In the silence of deep dark space, 1.3 billion light years away from us, two black holes fatally attracted each other. They got closer and closer, spiraled around, embraced, and quickly merged into a single black hole. In the process, they radiated away an enormous amount of energy in a burst of gravity waves. And thus that particular burst of gravity waves shot outward, spreading into the universe, much like a stone dropped into a pond causes a circular wave to spread out. That was * It was announced on February 11, 2016.



1.3 billion years ago, long before dinosaurs emerged,1 when humans were but a mirage in a sleeping trilobite’s dream. As eons and eons passed, that herd of gravitons2 journeyed on, at the speed of light, across the almost incomprehensible vastness of the universe, getting closer and closer to planet earth. They reached our world on September 14, 2015, when they were detected by two massive detectors, kilometers long and equipped with the most delicate cutting-edge instruments known to human technology, one in Livingston, Louisiana, the other in Richland, Washington.3 These sites, being far separated, detected the pulse with a millisecond time difference. Much as you with your two ears could determine, by the slight difference in arrival times of sound in the two ears, the direction to the source of the sound, physicists could roughly locate the direction of the two black holes that had merged. Spacetime comes alive

In 1915, as these particular gravitons approached earth—after 1.3 billion years, only a hundred more years to go!—an earthling named Albert Einstein (1879–1955) finally completed his theory of gravity, also known as general relativity. He shocked the physics world, saying in effect that there was no gravity, only curved spacetime. Physicists learned an astonishing secret: what we called gravity is all about the dance between spacetime and energy, one curving this way and that, the other moving hither and thither. Spacetime and energy in a pas de deux: energy in all its forms, such as you and me. Energy is matter, and matter is energy, as the very same Einstein taught us back in 1905 in his theory of special relativity: E = mc 2 , surely the best-known formula4 in all of physics! So, we have known for a long time that spacetime could curve. It follows that spacetime could also wave. That one

Song of the universe


follows from the other did not escape Einstein’s notice. The very next year,* in 1916, he published a paper5 noting the existence of gravity waves. Waves and rigidity

Waves are all around us. Tap a large block of jello with a spoon, and you will see a wave propagate across it. Wind passing over the sea commands the water to wave incessantly. A singer’s vocal cords compress the air, and a sound wave propagates outward. Any compressible medium can wave. Think of a long metal rod. Hit one end. The regular arrangement of atoms at that end is compressed, if only ever so slightly. By bouncing back to their appointed positions a moment later, the atoms crowd their neighbors down the line, who are in turn compressed. Thus information gets transmitted down the rod in a compressional wave. “Pass it on: somebody hit the rod at one end.” The speed with which the wave propagates is determined by the elasticity, or equivalently by its inverse, rigidity. The more rigid the rod, the faster the wave moves. You could think of rigidity as a measure of the eagerness of the atoms to bounce back to where they were. Theoretical physicists love to contemplate taking things to the extreme. Consider an infinitely rigid rod. Then by definition, when you hit one end, the whole thing moves as a whole, and the information that the rod is being hit at one end is transmitted to the other end instantaneously. But you would recall that in Einstein’s special relativity, energy and information cannot move faster than c, the speed of light.† It follows that infinitely rigid rods are not allowed in physics. * Two

years later, when Einstein was 39, he lamented the effect of aging: “The intellect gets crippled, but glittering renown is still draped around the calcified shell.” † I already used the letter c, without saying what it was. By the way, c stands for celeritas.6



The last rigid entity to fall

This point will be crucial to our discussion later, because Newtonian spacetime is absolutely rigid. According to Newton (here I am being unfair to the great man, as we will see later), gravity is transmitted instantaneously. It follows that once Einstein declared that spacetime is elastic, not absolutely rigid, gravity waves became inevitable. This is why the overwhelming majority7 of theoretical physicists have long been convinced of the existence of gravity waves. That waves and rigidity clash is readily understood in everyday terms. Undulation—think belly dancing—is all about flexibility, and a stiff and stern man could hardly be expected to wave. Think of spacetime as the last rigid entity in classical physics to fall. Sometimes one is ahead of the other, sometimes the other is ahead

After the historic announcement that spacetime is flexible enough to support waves, a reporter asked why Einstein was so prescient, so far ahead of the experimentalists. Good question, but it would be more accurate to ask why experiments are so far behind the theory in this case.* In physics, sometimes theory is ahead of experiment, sometimes the other way around. Ideally, they move together and steadily ahead in pace, for physics to progress. Seldom is the gap as large8 as a hundred years! A century of spectacular technological advances was necessary to detect gravity waves. The reason, as we will see, is that the gravity wave, by the time it got to earth, had become * We will come back to this.

Song of the universe


fantastically weak. To understand why, we need to appreciate that, in spite of our everyday experiences, gravity is fantastically weak. This fact will be explained in the next two chapters. You say gravitational wave, I say gravity wave

You might think that these waves generated by gravity would be called gravity waves, but alas, history intervenes: water waves, such as those on ponds and oceans, were called “gravity waves” long before Einstein came onto the scene. The excess water in the crest of a wave is pulled down by the earth’s gravity to fill a neighboring trough. It overshoots and turns the trough into a crest. Thus a wave propagates. The physics is entirely Newtonian and clear. Thus, physics journals and textbooks9 refer to the kind of wave we are talking about as a “gravitational wave.” In his 1918 paper,10 Einstein used “Gravitationswellen.” See the figure. I was curious which term popular physics books would use. I looked at one11 and saw that the author used both terms, sometimes on the same page. Later, I flipped through my own popular book12 on Einstein gravity, and was surprised to see that I used “gravity waves.” Given the American13 penchant to shorten everything in sight, I do not doubt that “gravity wave” will eventually win. After all, water wave is of interest to only a relatively small subset of physicists. I also did an informal poll of the intelligentsia excluding physicists. All prefer “gravity wave” to “gravitational wave.” I will use the term “gravity wave” in this book, throwing in “gravitational wave” occasionally.

The title page of Einstein’s 1918 paper. Reprinted from p. 12 of The Collected Papers of Albert Einstein, Volume 7: The Berlin Years: Writings, 1918–1921 edited by Michael Janssen, Robert Schulmann, Jozsef Illy, Christoph Lehner, and Diana Kormos Buchwald. Copyright © 2002 by The Hebrew University of Jerusalem. Published by Princeton University Press and reprinted here by permission.

Part I

1 A friendly contest between the four interactions Matter and the forces that move it

To tell the story of gravity waves, let us first go for a quick tour of the universe. Matter consists of molecules, and molecules are built out of atoms. An atom consists of electrons whirling around a nucleus, which in turn consists of protons and neutrons, collectively known as nucleons. The nucleons are made of quarks. That’s what we know.* The universe also contains dark matter and dark energy. (More in chapter 18.) Indeed, by mass, the composition of the universe is 27% dark matter, 68% dark energy, and only 5% ordinary matter. To first approximation, the universe may be regarded as one epic cosmic struggle between dark matter and dark energy.1 The matter we know and love and of which we are made of hardly matters. Unhappily, at present we know little about the dark side. We know of four fundamental forces between these particles. When particles come into the vicinity of each other, they interact,† that is, influence each other. Here is a handy * Whether quarks and electrons are tiny bitty strings is an intriguing, but at the

moment purely speculative, possibility. † “Interact” is a technical word in physics, just like “energy,” “momentum,” and “mass.”


Chapter 1

summary of the four forces, known as gravity, electromagnetism, the strong interaction, and the weak interaction. G: Gravity keeps you from flying up* to bang your head on the ceiling. E: Electromagnetism prevents you from falling through the floor and dropping in on your neighbors if you live in an apartment.† S: The strong interaction causes the sun to provide us light and energy free of charge. W: The weak interaction stops the sun from blowing up in your face. I don’t quite remember, but I would suppose that, due to buoyancy,‡ we were not aware of gravity while in our mothers’ wombs. But as soon as you entered the world, you knew about gravity, especially if the obstetrician grabbed you by the ankles and hanged you upside down. Then that quick slap on your bottom caused you to cry out and to open your eyes, thus discovering electromagnetism. Only four forces!

The world appears to be full of mysterious forces and interactions. Only four? As you toddled, you banged your head against a hard object. What is the theory behind that? Well, the theory of solids can get pretty complicated, given the large variety of solids. But a simple cartoon picture suffices here: the nuclei * You know how fast the earth is spinning to cover about 24,000 miles in 24 hours.

Anybody who has studied some physics could calculate what the centrifugal acceleration would be. † Plus a lot of other good deeds. Electromagnetism holds atoms together, governs the propagation of light and radio waves, causes chemical reactions, and last but not least, stops us from walking through walls. ‡ A force driven in fact by gravity, as the fluid around you fought for a better deal by getting lower.

Friendly contest 11

of the atoms composing the solid are locked in a regular lattice, while the electrons cruise between them as a quantum cloud. A collective society in which all individuality is lost! The atoms no longer exist as separate entities. The arrangement is highly favorable energetically; that is jargon for saying that enormous energy is required to disturb that arrangement. Revolution is costly. It takes quite a tough guy to crack a rock into halves. So, the myriad interactions we witness in the world, such as solid banging on solid, can all be reduced to electromagnetism. What we see in everyday life is by and large due to some residual effect of the electromagnetic force. Since common everyday objects are all electrically neutral, consisting of equal numbers of protons and electrons, the electromagnetic force between these objects almost all cancels out. Even the steel blade of a jackhammer smashing into rock is but a pale shadow of the real strength of the electromagnetic force. Just about the only time the true fury of electromagnetism shakes us is when thunder and lightning fill the sky. While we modern dudes have totally enslaved electromagnetism, all ancient people attribute its occasional bursts of temper to the gods.2 When you first shook off the ooze, you might have thought that there must be thousands, if not millions, of forces in the world. Thus, to be able to state that there are only four fundamental forces is totally awesome, a feat summarizing centuries of painstaking investigations. For example, realizing that light was due to electromagnetism stands as a towering achievement. The universe as a finely choreographed dance

While the proverbial guy and gal in the street are plenty acquainted with gravity and electromagnetism, they have no personal experience with the strong and the weak


Chapter 1

interactions. But in fact, the physical universe is a finely choreographed dance starring all four interactions. Consider a typical star, starting out life as a gas of protons and electrons. Gravity gradually kneads this nebulous mass into a spherical blob in which the strong and the electromagnetic forces stage a mighty contest. The electric force causes like charges to repel each other. Thus, the protons are kept apart from each other by their mutual electric repulsion. In contrast, the strong force, also known as nuclear attraction, between the protons tries to bring them together. In this struggle, the electric force has a slight edge, a fact of prime importance to us. Were the nuclear attraction between protons a tiny bit stronger, two protons could get stuck together, thus releasing energy. Nuclear reactions would then occur very rapidly, burning out the nuclear fuel of stars in a short time, thereby making steady stellar evolution, let alone civilization, impossible. In fact, the nuclear force is barely strong enough to glue a proton and a neutron together, but not strong enough to glue two protons together. Roughly speaking, before a proton can interact with another proton, it first has to transform itself into a neutron. The weak interaction has to intervene to cause this transformation. Processes affected by the weak interaction occur extremely slowly, as the term “weak” suggests. As a result, nuclear burning in a typical star like the sun occurs at a stately pace, bathing us in a steady, warm glow. Range versus strength

The reason that the proverbial guy and gal in the street do not feel the strong and the weak interactions is because these two interactions are short ranged. The strong attraction between two protons abruptly falls to zero as soon as they move away from each other. The weak interaction operates over an even shorter range. Thus, the strong and weak interactions

Friendly contest 13

A boxer with short arms but a strong punch versus a boxer with long arms but a weak punch. From Fearful Symmetry: The Search for Beauty in Modern Physics by A. Zee. Copyright ©1986 by A. Zee. Princeton University Press.

do not support propagating waves. In this book, we won’t talk about these two short range interactions much. In contrast, the gravity force between two masses and the electric force between two charges both fall off with the separation R between the two objects like 1/R 2 , the inverse square law celebrated in song and dance. More on this in chapter 2. Gravity and electromagnetism are known as long ranged and thus can and do support propagating waves. For R large, these forces still go to zero, but slowly enough that we can feel the tug of the sun, literally an astronomical distance away. For that matter, our entire galaxy, the Milky Way, is falling toward our neighbor, the Andromeda galaxy. Thus, in the contest among the four interactions, brute strength is not the only thing that counts: many phenomena depend on an interplay between range and strength. A case


Chapter 1

in point is fusion versus fission in nuclear physics. When two small nuclei get together, each consisting of a few protons and some neutrons, the strong attraction easily overwhelms the electric repulsion, and they want to fuse. In contrast, in a large atomic nucleus (famously, the uranium nucleus), the electric repulsion wins over the strong attraction. Each proton only feels the strong attraction of the protons or neutrons right next to it, but each proton feels the electric repulsion from all the other protons in the nucleus. The nucleus wants to split into two smaller pieces, accompanied by the release of energy.

2 Gravity is absurdly weak Gravity and the electric force

Gravity is absurdly weak compared to the electromagnetic force. How do we compare the relative strength between two forces at the fundamental level? First, a reminder of some basic facts. We learned about Newton (1642–1726/27)1 and his law of universal gravity in school. It states that the force F of gravitational attraction between a mass M (say, the earth) and a mass m (say, the moon) is equal to a constant G (known as Newton’s gravitational constant) times the product of the two masses (namely, Mm) divided by the square of the distance R separating them. Or, in a more concise language, F = G Mm/R 2 . We also learned about Coulomb’s law. It states that the force F of electric repulsion between two charges, one with charge q1 and the other with charge q2 , is equal to the product of the two charges (namely, q1 q2 ) divided by the square of the distance R separating them. Or, in a more concise language, F = q1 q2 /R 2 . A striking mystery: the fall-off of the force with increasing distance—the 1/R 2 inverse square behavior—is the same for gravitation and for the electric force. We will come to the modern understanding of this in due time.


Chapter 2

No need to count the number of zeros, we will do it for you

Time out. This is as good a place as any to introduce scientific notation, just in case you do not know it. The ethos behind scientific notation may be expressed as follows: esteemed sir or madam, you don’t have to count the number of zeros, we will do it for you. Thus, 100 is written as 102 , 1,000 as 103 , 1,000,000 as 106 , and so on. The number in the exponent, such as 6 in 106 , simply indicates the number of zeros when you write out the number 106 as 1,000,000. It follows that a number such as 149 could be written as 1.49 × 102 . The multiplication of large numbers is thus rendered easy: the number of zeros simply add. For example, 100 × 1,000 = 100,000 may then be written as 102 × 103 = 102+3 = 105 . In this notation, 10 may be written as 101 , and 1 as 100 (since it is equal to 1 with no zero following it). This explains how to write large numbers. Small numbers are written with a minus sign in the exponent. The logic behind this is as follows. Since, as was just noted, 10a × 10b = 10a+b , on dividing both sides of this equation by 10a , we obtain 10b = 10a+b /10a and thus, by setting b to −a, we have proved that 10−a = 10a−a /10a = 100 /10a = 1/10a . For example, let a = 2, and we have 10−2 = 1/102 . In other words, we may write 1/100 (which equals 0.01 in standard nonscientific notation) as 10−2 in scientific notation. As another example, 1/1017 = 10−17 is a very small number, since 1017 is a very large number. Comparing gravity to the electromagnetic force

After this notational interlude, we are ready to compare gravity to the electric force. To have a fair comparison, let us consider two protons. The gravitational attraction between

Gravity is absurdly weak


them is equal to F gravitation = G m p 2 /R 2 , with m p the mass of the proton. The electric repulsion between them, on the other hand, is equal to F electric = e 2 /R 2 , where e denotes the fundamental unit of charge carried by the proton. Thus, the ratio of the two forces F electric /F gravitation = e 2 /(G m p 2 ). Note that the factors of R 2 cancel out, so that this ratio is just a number, measured to be about 1036 , that is, 1 with 36 zeros after it. This absurdly large number* gives precise meaning to the statement that gravity is absurdly weak compared to electromagnetism. Electromagnetism is stronger than gravity by a factor of 1036 . Note also that before elementary particles, such as protons, and electrons, were known, any proposed comparison between the strengths of gravity and electromagnetism would have been meaningless. What would you use to do the comparison? Gravity does not know about yin and yang

That gravity is so much weaker than electromagnetism may surprise the unfortunate who has just had a hard fall. The reason is of course that every atom in the unfortunate’s body is being pulled down by every atom in the entire earth. The enormous number of atoms involved more than compensates the teeny number 10−36 . A huge difference, and it is huge, as we will see, is that masses are always positive, while charges can be positive or negative. The electric force between a positive and a negative charge then has the opposite sign, namely, it is attractive rather than repulsive. Likes repel, while opposites attract. * Notice that by using two protons to do the comparison, I have biased the result

in favor of gravity. Since an electron is about two thousand times less massive than a proton, the ratio of electric to gravitation forces between two electrons would be given by the even larger number 1036 × (2,000)2 = 4 × 1042 .


Chapter 2

Thus, electromagnetism knows about yin and yang. While yin and yang attract, the electric force is repulsive between yin and yin, and between yang and yang. In contrast, gravity does not know about yin and yang: everybody is gravitationally attracted to everybody else. I have already alluded to the reason electromagnetism is well hidden in everyday life: common objects contain equal number of positive and negative charges and so are electrically neutral. Whatever force that exists between them is a residual force, left over after the main electric force—namely, the attraction between the protons and electrons, the repulsion between the protons, and the repulsion between the electrons—has been canceled off. It is as if in a financial transaction involving billions rounded off to the nearest dollar, all we get to see is the rounding error of 23 cents. What electric and magnetic forces we see in everyday life are just the teeny “round off errors.” A perpetual contest between two forces

An interesting everyday example is the refrigerator magnet. It underlines the enormous strength of the electric force over gravity: the small patch of refrigerator door in contact with the magnet is holding off the entire earth. Furthermore, the magnetic force, caused by the circular motion of the charged particles inside the magnet, is itself much weaker that the electric force. Once you are alerted to this contest between electromagnetism and gravity, you will start to see it everyday, everywhere. Look at a glass of water. The water molecules hear the incessant siren song of gravity, telling them to lower themselves, to come to the bosom of mother earth. But electromagnetism causes the glass molecules to join hands, forming an interlocking prison through which the water

Gravity is absurdly weak


molecules cannot escape. The electric force easily overwhelms the pull of the entire earth. The escape route is through the top of the glass. Absorbing infrared photons from the environment and hit by air molecules, the water molecules get all agitated and bump into each other in their frenzy. Once in a while, a particular water molecule achieves enough speed—the crowd bumps into him just so—to overcome the downward pull of gravity and shoots to freedom. We call this process evaporation, which leaves us eventually with an empty glass, possibly with some scum in it—the mineral molecules in the scum are too obese to make the getaway. Or look at a tree. It is desperately pulling up nutrients against gravity. You could surely come up2 with many more examples of this never-ending struggle going on all around us between electromagnetism and gravity.

Newton answers your objection

Let’s go back to the refrigerator magnet for a moment. You could have objected that it was not a fair comparison. While the earth is very very large, much of it is also very very far away from the magnet. Newton was well aware of this problem, and spent almost 20 years proving what he called two “superb theorems.” The magnet is being pulled down by the patch of ground beneath your feet, stuff very close to the magnet but composing a small fraction of the entire earth. The rest of the earth, including the enormous amount of stuff on the other side of the world, is far away. Thus, to apply the law F = G Mm/R 2 to the magnet and the earth, we should mentally cut up the earth into a multitude of infinitesimal pieces, each some distance R from the magnet and each pulling on the magnet, and add up the individual forces.


Chapter 2

Newton’s first superb theorem: While the Arctic cap is closest to the apple at the north pole, the equatorial pieces are much more massive. Effectively, the earth pulls on the apple as if the earth’s entire mass is concentrated at its center. Adapted from Einstein’s Universe: Gravity at Work and Play by A. Zee. Oxford University Press, 1989.

How to do this posed a challenge to Newton, who had to invent integral* calculus to solve this problem (which these days could be given to students as homework). By doing the sum just mentioned, Newton arrived at the remarkable result that the force F exerted on an object of mass m, be it an apple or the refrigerator magnet, by the earth, is as if the entire earth, with mass M, had been shrunk to a point located at the center of the earth. In other words, in his formula F = G Mm/R 2 , we should take for R the radius of the earth. * This procedure of cutting up an object into infinitesimal pieces and then adding up the individual forces exerted by the pieces is known as integrating.

Gravity is absurdly weak


That Newton took so long to complete his two superb theorems caused one of the most bitter fights in the history of physics. While he was off doing the math, so to speak, his rival, Robert Hooke (1635–1703), also came up with the law of gravitation. Newton, disputing the claim, accused Hooke of not knowing the first superb theorem and thus could not possibly have calculated the force on the proverbial apple. A famous saying of Newton’s, something like “I could see farther than others because I was able to stand on the shoulders of giants,” often quoted as an indication of his modesty, was apparently a nasty dig at Hooke, who was rather short. This may well be apocryphal, but be that as it may, Sidney Coleman, my PhD advisor, a brilliant but exceedingly arrogant physicist, liked to quip “I could see farther than others because I was able to look over the shoulders of midgets.” Where is hell?

Before wrapping up this chapter, I cannot resist addressing an issue that may be burning you up. I mentioned that Newton proved two superb theorems but discussed only what is known as the first theorem. Newton’s second theorem addressed a central mystery of his time: where is hell? While this is no longer a burning question of contemporary physics, we could understand why it would puzzle physicists once upon a time. With a round earth, to imagine heaven localized up above our heads was no longer sensible; heaven would have to be a spherical shell wrapped around the world. It followed that hell must be in the center of a hollow earth. I think that most of my physicist colleagues would agree that this represents the simplest extension of an existing theory. A rudimentary understanding of volcanoes (plus a close reading of the Bible) provided strong observational evidence, confirming the theory for sure.


Chapter 2

Furthermore, an erroneous calculation had convinced Newton that the earth was much less dense than the moon, which led his friend Edmond Halley (1656–1742), who, by the way, published Newton’s Principia at his expense, to propose the hollow earth3 theory.4 The idea may seem absurd to us, but not at that time. A location for hell had to be found. Every epoch in physics has it own top ten problems. It is conceivable that future generations would find our desperate attempt to quantize gravity absurd. So, Newton’s second superb theorem states that there is no gravitation force inside5 a spherical shell. You now understand why Newton would even bother to attack this peculiar problem.* Either humongous or teeny

If I were a layperson reading popular physics books, I would be confounded by the appearance of numbers that are either humongous or teeny, things are either zillions or zillionths of something. Stars are a zillion times bigger than we are, and we are a zillion times bigger than quarks. And a zillion is always some number beyond all comprehension. Blame it on the absurd weakness of gravity! Let us join the movie of the early universe in progress. As the universe expands, it cools. At some point, it has cooled enough for hydrogen atoms to form, consisting of a proton bound with an electron due to their electric attraction for each other. Picture the universe as a diffuse cloud of hydrogen atoms zipping around, a cloud without any structure. Soon, structures started to emerge, structures that would lead to galaxies, stars, planets, and so on. * Incidentally, since there is no gravitation force in hell, the usual portrayal of the leaping flames can’t be right! Flames shoot up because gravity pulls the denser air surrounding the hot gas down.

Gravity is absurdly weak


The formation of structures, clearly an epochal event in the history of the universe, is based on a commonly observed and easily understood phenomenon: the rich gets richer. By chance fluctuations, some regions in this primordial gas of hydrogen atoms are denser and some regions are less dense. Thanks to gravity, the denser regions pull hydrogen atoms from the neighboring less-dense regions. The dense regions gets denser, while the less-dense regions gets less dense, in a rapidly accelerating process. Indeed, Newton already understood this consequence of universal gravity and postulated it as the basis for the formation of stars. Consider a spherical cloud undergoing gravitational collapse and destined to become a star. In modern understanding, eventually the hydrogen atoms are so densely packed that collisions between them strip the electrons off, leaving a gas consisting of protons and electrons. Finally, as the gas becomes even denser, the protons get close enough to each other to initiate nuclear reaction, that is, the strong interaction becomes effective. A star is born! What could work against gravity? In other words, what must gravity overcome to form structure in the primordial gas of hydrogen atoms? Well, the hydrogen atoms are zipping this way and that way, and some of them are bound to go from the denser region to the less-dense region. Gravity’s job is to pull them back. Clearly, gravity can win if the denser region is massive enough. How much mass do you need? You need a lot, since gravity is so feeble.6 In the preceding sections, we measured the feebleness of gravity by comparing it to the electric force and obtained F electric /F gravitation = e 2 /(G m p 2 ) ∼ 1036 . Here the electric force is not in the game, and e 2 does not enter. So, we should measure the feebleness of gravity by the number 1/(G m p 2 ) ∼ 1038 = (1019 )2 . The huge number 1019 , which we might call the Planck7 number, indicates the intrinsic


Chapter 2

weakness of gravity and plays an important role in contemporary physics, such as in string theory. In the present context, this number governs the emergence of structure in the universe, and a great deal of astrophysics could be understood in terms of this number. For instance, you could look up that a solar mass is about 2 × 1030 kg and that the proton’s mass is about 1.6 × 10−27 kg. Thus, a typical star like the sun contains about 1030 /10−27 ∼ 1057 protons. Where does this humongous number 1057 , so far out of everyday experience, come from? An undergraduate level physics exercise (which I won’t go into here8 ) shows that it comes from the cube of the Planck number: (1019 )3 = 1019 × 1019 × 1019 = 1019+19+19 = 1057 .

3 Detection of electromagnetic waves We see electromagnetic waves all the time

That gravity is so feeble compared to electromagnetism delayed the detection of gravity waves until the early 21st century. In contrast, we humans have detected electromagnetic waves since day one. Evolution equipped us to see electromagnetic waves, albeit only those with wavelengths in a narrow range. Well, strictly speaking, it took humans a while to realize that light is just a form of electromagnetic wave. That insight required the momentous invention of physics, which may be defined as the art of recognizing which puzzles are legitimate questions to investigate and which are not. To grasp how the production and detection of gravity waves pose such monumental challenges, let us first recall the late 19th-century production and detection of electromagnetic waves. We can then compare and contrast with the early 21st-century detection of gravity waves. (Did I forget to say “production”?) Maxwell, Hertz, and electromagnetic waves

In 1865, James Clerk Maxwell (1831–1879) published his theory of electromagnetism, synthesizing what was known up till then. (See chapter 6.) In a brilliant stroke of insight,


Chapter 3

he deduced the existence of electromagnetic waves. Perhaps to Maxwell’s surprise or perhaps not, his equations indicated that the waves propagate at a speed equal to the known speed of light. Later, Hermann Ludwig Ferdinand von Helmholtz (1821–1894), surely the preeminent German scientist1 of his time, proposed to the Prussian academy a Berlin Prize to be awarded to anyone who could detect electromagnetic waves. In 1879,2 Helmholtz suggested the problem to his doctoral student Heinrich Rudolf Hertz (1857–1894), then aged* 22. Hertz wasn’t able to accomplish what was requested of him. But after becoming a professor at Karlsruhe, he noticed, one day in 1886, that discharging a Leyden3 jar (an early form of an electric capacitor) caused another Leyden jar nearby to spark. Something had propagated from one jar to the other. This suggested to him a way to study the electromagnetic wave theorized by Maxwell. I find it charming to read about the transmitter and receiver Hertz built. For the transmitter, he attached copper wires to two electrically charged zinc spheres. When the ends of two wires were brought near each other, the charges on the spheres rushed over to meet their long-lost mates, and a spark would jump across. We now know that Hertz was producing radio waves.4 For the receiver, he built an early version of the dipole antenna, consisting of a wire wound around pieces of wood bent and nailed together, with an adjustable gap between the two ends.5 A spark from his transmitter would cause a spark in his receiver. Having created this setup, Hertz could now experiment† to his heart’s delight: by moving the transmitter and the receiver around, by putting different kinds of screens between them, by adjusting the width of the gap, so on and so forth. He tried out prisms made of different materials to show that * Hertz had a tragically short life, dying at age 36.

† There was no need to write proposals and beg for money, no need to wait around for a few decades, and so on.

Detection of electromagnetic waves


Figure 1. Photo of the receiver Hertz used to detect electromagnetic waves. Retrieved December 30, 2014, from Rollo Appleyard, “Pioneers of Electrical Communication 5: Heinrich Rudolf Hertz” in Electrical Communication, International Standard Electric Corp., New York, Vol. 6, No. 2, October 1927, p. 70, fig. 9 on http://www.americanradiohistory.com.

electromagnetic waves could be refracted just as light could be. Simply by rotating the receiver, he demonstrated, just as Maxwell had deduced from his equations, that electromagnetic waves had two polarizations. Compared to Hertz, people working on gravity wave detection have a much tougher time. Merging two black holes in the lab is not going to happen anytime soon. Physicists can’t move the merging black holes around, and they can’t rotate the detectors. What they can do is to ask their respective governments to build more detectors. See later. A window to the outside world

The unit for frequency, the hertz (written as Hz), was defined in 1930 as the number of times a repeated event occurs per second, also commonly known as cycles per second. As some of us may recall from school, for an electromagnetic wave of frequency f , the wavelength λ (defined as the distance from crest to crest) is given by the formula* c = f λ, where c is the speed of light. * When I lecture about my popular books to the educated public, I find that most

laypersons have no way to distinguish the profound from the trivial. For example, Newton’s law of gravity, F = G Mm/R 2 , is profound, but the “law” given here is just trivial counting: the distance traveled per second by a wave is equal to the number of crests passing per second times the distance between crests.

Frequency (Hz)

Wavelength Gamma-rays

0.1 Å

1019 1Å 0.1 nm

1018 X-rays

1 nm

1017 10 nm 1016

Ultraviolet 100 nm

1015 Visible Near IR

1000 nm 1 µm

1014 10 µm

Infrared 13


Thermal IR

100 µm

1012 Far IR

1000 µm 1 mm



Microwaves 1010

1 cm


10 cm 9


1m 108

Radio, TV 10 m

107 100 m 106


1000 m Longwaves Figure 2. Spectrum of electromagnetic waves, from gamma rays to radio waves. This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Detection of electromagnetic waves


I don’t have to belabor the impact of electromagnetic waves on human civilization. As we all know, the discovery of electromagnetic waves (known as Hertzian waves for some time) led to a new technological age, with wireless telegraph, radio, television, coming one after another, and continuing with the gadgets of our modern world, which simply cannot function without electromagnetic waves. Humans have now put electromagnetic waves to work.6 Perhaps it is a bit sad that most teenagers walking around glued to their cell phones know almost nothing about these waves. Electromagnetic waves with wavelengths between 4 ×1014 and 8 ×1014 Hz are known as visible light. It is as if we had been peering at the world through a narrow window, and Hertz7 came along and drew the curtains apart, revealing to us that the curtains had been hiding a window much much wider than the one we had been looking through. Curiously, Hertz did not appreciate the importance of his experiments, saying “It’s of no use whatsoever ... just an experiment that proves Maestro Maxwell was right ... we just have these mysterious electromagnetic waves that we cannot see with the naked eye. But they are there.” And when asked about possible applications of electromagnetic wave, he replied, “Nothing, I guess.” Ushering in the quantum era

Not only did Hertz8 open a window, he also saw the first hint of the quantum world. In one of his trial and error experiments, Hertz noticed that a charged object lost its electric charge much faster when exposed to electromagnetic waves. A puzzling aspect was that the higher the frequency of the wave, the faster the charge was lost. Several decades later, Einstein helped usher in the quantum era by explaining this strange phenomenon, which had come to be known as the photoelectric effect.9


Chapter 3

We now understand that an electromagnetic wave of frequency f is actually composed of a stampeding herd of photons, each with energy equal to h f . (Here h denotes Planck’s constant, in honor of Max Planck,10 the father of quantum mechanics.) The photons are literally kicking the electrons out of the material exposed to the electromagnetic wave. The higher the frequency, the more vigorous the kick becomes. In contrast, in classical physics, the amplitude of the wave corresponds to the size of the electric field, which determines how far the electrons are pushed. So the determining factor would not be the frequency of the wave, but rather its amplitude. With quantum physics, we would expect the photoelectric effect to cease abruptly when the frequency drops below a certain minimum value; the kicks would then be way too gentle. Classical physics totally fails to account for this threshold effect. Ah, the glory days11 of trial and error experimental physics! Similarly, in quantum physics, a gravity wave of frequency f is composed of a stampeding herd of gravitons, each with energy equal to h f . The astute reader might have noticed that I have already snuck in the word “graviton” in the prologue. Indeed, there is an easy parallel between electromagnetism and gravity: the photon is to the graviton as the electromagnetic field is to the gravitational field. In due time, I will discuss some important differences between the photon and the graviton, but for now it suffices to know that the photon and the graviton are the quantum particles that collectively form the classical electromagnetic and gravitational waves, respectively.

4 From water waves to gravity waves To understand gravity waves, consider the more easily understood water wave. Look at waves on the surface of a pond on an idyllic summer day. Instead of writing romantic poetry, the nerdy physicist* writes down the equation governing the surface of the water as it varies in time. How to proceed? The famous French guy, René “I think” Descartes1 (who used to exist), taught us that two numbers, call them x and y, suffice to locate where we are. At the location specified by (x, y), the surface is described by the height of the water measured from the bottom of the pond. Call the height g (t, x, y), a function that depends on time t and space x and y. See the figure. Without any breeze whatsoever, the surface of the pond is flat and thus just a constant, say, 1 in some suitable unit:2 g (t, x, y) = 1. (As usual in physics, we idealize: the bottom is flat and we are far from the banks in the middle of the pond.) A wave means that g (t, x, y) is not a constant but varies in time and in space. As mentioned earlier, the physics governing water waves is clear: gravity pulls down the excess water in a crest to fill a nearby trough. Given the underlying physics, we can write down the equation3 governing how * A disclaimer: that would not be me.


Chapter 4 wave on surface of pond

g (t, x, y)

x A picture of a water wave taken at the instant t. The spatial coordinate y points out of the paper and is suppressed.

fluids behave: it “merely” involves applying Newton’s force laws to fluids. But writing down an equation is one thing, solving it is another. This equation for fluid flow has not been solved in its full generality to this very day. In fact, a million dollars is yours if you can solve it.4 Easy to see how it might be kind of tough. Let us leave the pond and go down to the beach on a windy day. As wave after wave approaches shore, they surge, curl upon themselves, try to form tunnels beloved by surfers, and break, crashing into a white foam of myriad bubbles. Fluids exhibit a bewildering wealth of behaviors. Well, the same equation that describes waves on an idyllic pond also rules here, so it can’t be easy to tame. But that equation is easy to tame when we get back from the beach to that pond, where things are nice and easy. The point is that we can now write g (t, x, y) = 1 + h(t, x, y) into that nasty equation and treat h as small compared to 1. Then we are justified in throwing a whole truckload of terms in the resulting mess away. A small number multiplied by an equally small number produces a way smaller number; for example, 0.1 multiplied by 0.1 gives 0.01. Thus, if you

From water waves to gravity waves


encounter a term involving h multiplied by itself (namely h 2 , the square of h) you can chuck that term out the window. Physicists and mathematicians call this the leading approximation.5 Things simplify enormously. We end up with an equation that any decent physics undergraduate can solve. I tell you all this because the situation here is almost completely analogous to the situation with Einstein gravity. Einstein’s field equation governing the curvature of spacetime is extremely difficult to solve in general, essentially impossible, but it simplifies enormously for gravity waves in the leading approximation. Again, most physics undergrads should be able to solve the equation governing gravity waves. I detest jargon and avoid it as best as I can, but still it is useful as shorthand to speed up the discussion. The wave on a pond on a calm summer day is said to be in the linear regime. In contrast, the crashing surf at the beach on a stormy day favored by surfers is definitely in the nonlinear regime. The take-home message: Einstein’s equation is hard to solve in the nonlinear regime, and easy in the linear regime. The astute reader might have noticed that I have snuck in the word “field,” as in “field of force.” To people not born into physics, the word often sounds mysterious and unfathomable.* In fact, physicists simply call any function of space and time, such as g (t, x, y) here, a field. More in chapter 6. In the rest of this book, I tell the story of how Einstein arrived at his field equation for gravity. In the process, I will give you a flavor of what curved spacetime means. * A professional secret: The concept of field is still mysterious and unfathomable, even to physicists contemplating the fundamental puzzles of the universe. See QFT Nut.

Part II

5 Spooky action at a distance Action at a distance

Our common everyday understanding of force involves contact: we can exert a force on an object only if we are in contact with it. In a contact sport such as American football, without tackling the ball carrier, a linebacker could hardly exert anything on him. And in the movies, a slap is not a slap until the leading lady’s palm makes contact with the leading cad’s cheek. At the supermarket, you can push the shopping cart only if you grip the handle. If you could just hold out your hands and command the shopping cart to move, a crowd would gather and honor you as a wizard. Just about the only commonplace example of a force acting without contact is the refrigerator magnet: you can feel the refrigerator pulling on the magnet before the magnet makes contact with the refrigerator. Everyday forces, except for gravity, are short ranged, indeed, zero ranged on the length scales of common experience.* The palm molecules have to be practically on top of the cheek molecules before the latter can acquire any carnal knowledge of the former. *I

already explained in chapter 1 that these forces are but pale vestiges of the electromagnetic force.


Chapter 5

Gravity is the glaring exception. When the earth pulls Newton’s apple down, no hand comes out of the earth grabbing the apple, as in a horror movie. Gravity is invisible, thus all the more horrifying to us as we age! In days of old, wise men found it necessary to affix stars and planets to celestial spheres, made presumably of some celestial substance with magical properties, slowly turning around and around.1 This mechanistic picture would have sounded rather convincing to the ancients. In this worldview, Newton’s proposal that the earth’s gravity can pull not only the apple down, but that its invisible arm could reach out across the unfathomable vastness of space and tug at the moon, was bizarre.* Lacking in “faculty of thinking”?

In physics textbooks, students learn about the Newtonian concept of action at a distance. The moon is attracted to the earth; no contact is necessary. More advanced books then point out to the bewildered students that action at a distance is kind of spooky, and set up poor Newton as a straw man to be attacked. Very unfair! Newton did fret much about action at a distance. In a 1693 letter to his friend Richard Bentley, he opined: That gravity should be innate, inherent and essential to matter so that one body may act upon another at a distance through a vacuum without the mediation of anything else by and through which their action or force may be conveyed from one to another is to me so great an absurdity that I believe no man who has in philosophical matters any competent faculty of thinking can ever fall into it. * The phrase “spooky action at a distance” has lately been heavily publicized in connection with quantum entanglement. I use the phrase here to emphasize that classical Newtonian gravity is already plenty strange.

Spooky action at a distance


Tell me, when you first learned about the inverse square law, did you not find it bizarre? Would Newton have described you as lacking in faculty of thinking? Bringing time to gravity

Another strange feature of Newtonian gravity is that time does not enter into it. The attractive force exerted by the earth on the moon is given by the product of the masses of the earth and of the moon multiplied by Newton’s constant G and divided by the square of the distance between them. That’s that. Any change in the position* of the earth is instantaneously communicated to the moon. In Newtonian gravity, the moon is slavishly yoked to the earth. In turn, the earth is yoked to the sun, and the entire galaxy moves as a collective entity. How could a moon know instantly that its planet has moved? In the Principia, Newton left2 this conundrum “to the consideration of the reader.” The reader who took it up was Albert Einstein. * In fact, as we and attentive school children know, the earth is incessantly moving around the sun.

6 Greatness and audacity: Enter the field Speed of light as the absolute speed limit

Einstein’s theory of special relativity, which gave us E = mc 2 and all that, was born of a paradox in Maxwell’s theory of electromagnetism, namely, how the speed of electromagnetic waves could possibly not depend on the observer. One astonishing outcome of special relativity, celebrated in song and dance, is that the speed of light c sets an absolute speed limit in the physical universe: information cannot be transmitted faster than the speed of light. That rules out the moon knowing instantaneously about what the earth is doing. Newton was not the only one of his era with a competent faculty of thinking. The Marquis Pierre-Simon de Laplace (1749–1827), a plenty smart fellow,1 had the foresight to speculate about the speed of propagation2 c G of the effect of gravity. Not only that, he was also among those who believed that light moves at some finite speed c. Furthermore, he put the finite (that is, not infinite) speed of gravity c G to good use, well worth a historical note.3 Let me put you in the marquis’s high heel shoes and ask you to guess. How does c G compare with c? Laplace supposed (erroneously) that c G is much larger than c. These days, particle theorists subscribe to something known as the naturalness dogma, stating that fundamental

Greatness and audacity


constants of the same character, such as two fundamental speeds, should have roughly the same order of magnitude.4 So the default view would be that c G and c should be about the same. We now understand that the speed of propagation is a universal constant, that c G = c exactly, for the simple reason that the graviton and the photon both propagate in spacetime. The speed of propagation is a property of spacetime5 rather than of gravity or electromagnetism.6 Conceivably, some bright young guy in another civilization far far away could have proposed the existence of gravity waves propagating at the speed of light long before a complete understanding of curved spacetime was established.

Faraday’s field and our mother’s milk

For us, who took in Faraday’s ideas so to speak with our mother’s milk,7 it is hard to appreciate their greatness and audacity. —A . E I N S T E I N8 I will tell you shortly what Einstein regarded as great and audacious,9 but first I can’t resist telling you about10 Michael Faraday (1791–1867), one of the greatest experimentalists of all time. While Faraday’s genius manifested itself in the laboratory, he also introduced into theoretical physics the important and fruitful concept of a “field of force,” or “field” for short. Unlike most physicists until his time, Faraday did not come from a comfortable background. Born into almost Dickensian poverty, Faraday started as a bookseller’s errand boy, later promoted to apprentice. While rebinding a set of the Encyclopedia Britannica, he became spellbound by an article on electricity he chanced on. In Victorian London, educational lectures were often given to the public, typically

Michael Faraday (drawing by Peggy Royster after an original portrait). The field of force is represented by arrows indicating the direction a charged particle would move if placed at the location of the arrow. From Fearful Symmetry: The Search for Beauty in Modern Physics by A. Zee. Copyright ©1986 by A. Zee. Princeton University Press.

Greatness and audacity


for a charge of one shilling a lecture, a fee the young man was hard put to come up with. Fortunately, the famed Sir Humphry Davy started to give free lectures at the newly founded Royal Institution. They were highly popular. The educated public was keenly interested in science, and electricity was, well, electrifying the public. (This fine tradition of free lectures has persisted to this day in many countries, and most physics centers I know of can boast of one or two strange wild-eyed characters in regular attendance at seminars and colloquia.) Faraday, who attended the lectures religiously, eventually approached Davy. As luck would have it, Davy was at that very moment in need of a laboratory assistant. Furthermore, he embarked on a tour of European science centers a few months later and offered to take Faraday along. So Faraday did end up with an education to be envied. The Dickensian scenario was complete, however; Lady Davy was a horrid snob who insisted that Faraday eat with the servants and generally made life unpleasant. He was often reduced to performing the tasks of a valet. But it was an exciting trip, scientifically and otherwise; the Napoleonic Wars were in full swing, and, as “enemy scientists,” Davy and Faraday had to travel on safe-conduct through the lines.11 Davy’s young assistant quickly established himself, making discoveries one after another and outshining his mentor. (Davy is now a forgotten figure in physics.) Jealousy is a powerful human emotion, and unpleasantness soon developed between the two men. Among other things, Sir Davy tried to block Faraday’s membership in the Royal Society, but in vain. At the height of his career, Faraday was showered with honors. The humble apprentice was to refuse a knighthood as well as the presidency of both the Royal Institution and the Royal Society. Even Davy admitted that of all his discoveries, Faraday was the best.


Chapter 6

May the field of force be with you

But what is this field of force postulated by Faraday that Einstein considered to be so great and audacious, and now known to every child12 who has seen films on interstellar warfare? As I mentioned, in our everyday experiences, we tend to think of a force being exerted only when contact is made between material bodies. Newton’s notion of action at a distance had deeply troubled many thinkers, and now, in the 19th century, electromagnetism demonstrated this even more dramatically. That magnets could act on one another while separated by empty space is most alluring to children, and to physicists as well. Like many of his predecessors and contemporaries, Faraday grappled with this philosophical problem. He visualized what was going on by sprinkling iron filings on a piece of paper next to a wire. When a current was turned on, the iron filings would obediently form a pattern. Another pattern was formed when the filings were brought close to a magnet. Eventually, Faraday proposed that a magnet or an electric current produced what became known as a magnetic field, which exerted a force on the iron filings. Similarly, an electric charge produces around it an electric field of force. When another charge is introduced into this electric field, the field acts on this charge, exerting on it a force in accordance with Coulomb’s law. The field as a separate entity

Important point: the electric field is a separate entity. The electric field produced by an electric charge exists, regardless of whether another charge is introduced to feel the effect of the field. In effect, Faraday introduced an intermediary: two charges do not act directly on each other, but they each

Greatness and audacity


produce an electric field that, in turn, acts on the other charge. A pragmatic physicist might be inclined to dismiss all this as just talk that did not advance our knowledge one whit. Faraday’s notion does not explain Coulomb’s law; rather, it appears to be merely another way of describing Coulomb’s law. Faraday supposed the strength of the electric field produced by a charge decreases as one moves farther away from the charge, in such a way as to reproduce Coulomb’s law. But this view misses the point. As it turns out, the real content of Faraday’s picture lies in the fact that the electromagnetic field not only can be thought of as a separate entity, it is a separate physical entity. Physicists were to learn, for example, that it makes perfect physical sense to talk of the energy density in an electromagnetic field. Even more amazingly, the electromagnetic field could take off on its own and travels through spacetime. Faraday knew no mathematics, while Maxwell vowed not to read any

The notion of a field bore fruit in the hands of James Clerk Maxwell, as was mentioned earlier. Because of his upfrom-rags background, Faraday had a self-admitted blind spot—mathematics—and he was unable to transcribe his intuitive notions into precise mathematical descriptions. Just the opposite, Maxwell, scion of a distinguished family, received the best education that his era could provide, and was thereby able to achieve the grand mathematical synthesis of electromagnetism. But before he was to begin his investigations, Maxwell made a resolution: “to read no mathematics on the subject [of electricity] till I had first read through Faraday’s Experimental Researches on Electricity.” Some young


Chapter 6

contemporary theoretical physicists enamored of mathematics should take heed! Many in my world dazzle themselves (but not others) with fancy mathematics before they master the underlying physics.* Alas, a common affliction, often fatal.13 Debarred from following the French philosophers

Indeed, Maxwell was to consider Faraday’s deficiency an advantage, writing: Thus Faraday, with his penetrating intellect, his devotion to science, and his opportunities for experiments, was debarred from following the course of thought which had led to the achievements of the French philosophers, and was obliged to explain the phenomena to himself by means of a symbolism which he could understand, instead of adopting what had hitherto been the only tongue of the learned. By “symbolism,” Maxwell was referring to the notion of field (actually called “lines of force” by Faraday.) By “philosophers,” Maxwell simply meant “learned men,” following the usage of his time. Earlier, Maxwell had said, “the treatises of [the French philosophers] Poisson and Ampère [on electricity] are of so technical a form, that to derive any assistance from them the student must have been thoroughly trained in mathematics, and it is very doubtful if such a training can be begun with advantage in mature years.” Indeed, the pace at which sophisticated mathematics14 has been introduced into string theory and related areas in recent years is such that many physicists “in mature years” share heartily the sentiments expressed by Maxwell. * A note to students: Maxwell did not say not to read any mathematics. He merely

told us the appropriate order of doing things. By mathematics, Maxwell meant what we would call partial differential equations.

Greatness and audacity


The electromagnetic field leaves home and takes off on its own

By the time Maxwell burst onto the scene, a century or so of arduous experimental work had already been distilled into various laws, named after various famous physicists. Maxwell summarized these into mathematical statements, known ever since as Maxwell’s equations. For instance, one equation states how a magnetic field varying in time produces an electric field varying in space. This expresses mathematically Faraday’s law of induction: by moving a magnet around a wire, Faraday produced an electric field that pushed charges forward in the wire, thus generating a current. To Maxwell’s surprise, the equations he wrote down were not mutually consistent. Remarkably, Maxwell discovered that by adding a term to one of the equations, he could bring all of them into harmony. Armed finally with the correct equations, Maxwell made a truly amazing discovery: the existence of electromagnetic waves. Roughly speaking, if we have in a region of space an electric field changing with time, then a magnetic field is produced in the neighboring space. Its very production means that this magnetic field is also changing with time, and it generates an electric field. Thus, like a ripple on a pond spreading from a dropped pebble, an electromagnetic field propagates out as a wave, undulating between electric and magnetic energy. Let there be light! But wait, what is light?

From his equations, Maxwell was able to calculate precisely the speed of this brand new electromagnetic wave. By his time, the speed of light had been measured quite accurately, both by terrestrial experiments and by astronomical observations. The value obtained theoretically for the


Chapter 6

speed of his electromagnetic wave coincides closely with the measured speed of light! And thus Maxwell proclaimed that the mysterious phenomenon of light is just a form of electromagnetic wave. In one stroke, optics as a field of physics was subsumed under the study of electromagnetism. The laws of optics, wrested from Nature by physicists starting with Newton and Huygens, could be derived entirely from Maxwell’s equations. As I already mentioned, human vision had been hitherto limited to a narrow window in the electromagnetic spectrum, but henceforth, all forms of electromagnetic waves were ours to exploit.15 A universe of quantum fields

Maxwell’s discovery demonstrated conclusively the physical reality of the field and its claim to a separate existence. Indeed, the space around us is literally humming with packets of electromagnetic field hurrying hither and yon. The notion of field has grown from a glint in Faraday’s eyes to be all encompassing. In recent decades, physicists have come to the view that all physical reality is to be described in terms of fields. Electrons, quarks, all the fundamental constituents of matter, are but excitations of quantum fields.16 Interesting how this almost incredible view of the physical universe originated in the vague philosophical unease Newton felt with action at a distance! From electromagnetic wave to gravity wave

A deep strand ... was his total love of the idea of a field ... which made him know that there had to be a field theory of gravitation, long before the clues to that theory were securely in his hand. —F R E E M A N D Y S O N S P E A K I N G O F A L B E R T E I N S T E I N

Greatness and audacity


The lesson of electromagnetism is that as soon as we doubt the notion of action at a distance, we are almost committed to electromagnetic waves. Well, the same general considerations that demand the existence of electromagnetic waves also demand the existence of gravity waves. The gravitational force is long ranged, just as the electromagnetic force is long ranged. To communicate the movement of one massive object to another, a carrier is needed for the signal, and as soon as that carrier acquires a life of its own, it can propagate. Voilà, a gravity wave! The story is simple. We feel the gravitational effects of distant galaxies. Hence, when distant galaxies collide, we will know about it. That gravity is long ranged almost17 amounts to saying that a gravity wave can propagate across vast distances. In summary, the field triumphed over action at a distance, and once you have a field, you have a wave. You could say that Einstein is to Newton as Maxwell is to Coulomb. Thus, physicists did not doubt that gravity waves existed, since their existence followed from general considerations rather than from the details of Einstein’s theory. Historically, however, there were plenty of skeptics,18,19 including Einstein, who had a transient mental aberration. But certainly, by the time I came into the physics world, I didn’t hear wind of any doubt whatsoever. In fact, my first undergraduate research project, supervised by John Wheeler (1911–2008), involved the emission of gravity waves from a vibrating rotating neutron star.20 Soon after, Kip Thorne and his collaborator suggested that the gravity wave Wheeler and I described might be detectable with a then-existing apparatus. This turned out to be wildly optimistic. Gravitational waves from binary pulsars

Any lingering doubts were dispelled by the discovery, by R. Hulse and J. Taylor in 1974, of a binary star system in


Chapter 6

which one of the stars happens to be a pulsar. Binary star systems, in which one star orbits another, are fairly common in the universe. As the stars go round and round each other, we expect them to emit gravity waves and thus lose energy. As the stars lose energy, the time it takes the stars to complete one orbit changes. All this was well understood by the early 1970s. But the good luck came with the pulsar, which, with its regular pulse, provides a highly precise clock by which we on earth could determine the orbital period. The ratio of the observed rate at which the orbital period is changing and the predicted rate based on the emission of gravity waves turned out to be 0.997 ± 0.002. This close agreement was good enough for most physicists, but of course it would still be nice to have direct observation of gravity waves. As we now know, the wait lasted 42 years.

7 Einstein, the exterminator of relativity Truth is not relative

Einstein’s theory of relativity contains two parts, special relativity and general relativity, the former completed in 1905, the latter in 1915. After 1905, Einstein was obliged to make gravity compatible with special relativity. He had to struggle for 10 long years before he figured out how: the result was general relativity, more properly called Einstein gravity. Let us first focus on the theory of special relativity. I must now give vent to my pet peeve. Physics contains a number of unfortunate names, some due to historical confusion long since cleared up. Probably the worst name ever is relativity, as it has spawned a swarm of nonsensical statements, such as “physicists have proved that truth is relative” and “there is no absolute truth; Einstein told us so,” uttered with smug authority by numerous ignorant fools. In fact, physicists, as exemplified by Einstein, say the opposite. I like to call Einstein the exterminator of “the relativity of truth.” Just to set the record straight, Einstein did not use the term “theory of relativity” in his famous paper. The German physicist Alfred Bucherer, while criticizing Einstein’s theory, was the first to use, in 1906, the name1 “Einsteinian relativity theory.”


Chapter 7

The speed of light as seen by two observers: c = c

In 1905, Einstein insisted that the laws of physics must not depend on observers in uniform motion relative to each other. Consider two observers: a passenger sitting on a train smoothly rolling through a station at 10 meters per second without stopping, and a stationmaster standing on the ground. Suppose the passenger tosses a ball forward at 5 meters per second. To the stationmaster, the ball is evidently moving forward at 10 + 5 = 15 meters per second. That velocities add in this obvious everyday way has been known since time immemorial and is called Galilean relativity by physicists. Certainly Galileo understood it. Incidentally, Galileo talked about sailing ships, not trains, of course. In Einstein’s days, train travel was just becoming commonplace in Europe, and so it was natural for him to use trains in his discussions.2 Later, Einstein’s trains were upgraded to spaceships. In our day, perhaps one experience most readers of this book have had is walking on a moving sidewalk in a modern airport or a large subway station.3 If the sidewalk is moving ahead at 5 meters per second, and you are walking on it at 10 meters per second, then clearly relative to the terminal building, you are moving along at 10 + 5 = 15 meters per second. All this seemed beyond doubt until the end of the 19th century. Physicists were justifiably proud of their understanding of light being a particular form of electromagnetic wave. But now suppose that the passenger, instead of tossing a ball forward, shoots a beam of light forward. As always, denote the speed of light, as seen by the passenger, by c. All those photons in the laser beam are surging forward with speed c. Then the preceding discussion tells us that the speed of light, measured by the stationmaster, ought to be c + 10 meters per second.

Einstein, the exterminator of relativity


But wait! Recall that Maxwell was able to calculate the speed of light using his equations. For instance, one of these might give the strength of the magnetic field generated by an electric field, varying at such and such a rate. But a physicist performing an experiment on the train to study the magnetic field generated by an electric field varying in time should arrive at precisely the same result as a physicist performing it on the ground, since otherwise, the two physicists would perceive two different structures of physical reality. These two experimentalists can now appeal to their respective theoretical colleagues to perform Maxwell’s calculation of the speed of light. If the two theorists are both competent, they should arrive at the same answer. Thus, if Maxwell’s equations are correct, the speed of light, as measured by the passenger and by the stationmaster, should be exactly the same! In other words, c = c. There is only one speed of light, independent of observers.4

An intrinsic property of Nature

This strange behavior of light indicates that the addition of velocities cannot be simply Galilean. Maxwell’s reasoning forces us to a conclusion in violent discord with our everyday intuition: the observed speed of light is independent of how fast the observer is moving. Suppose we see a photon whizzing by and decide to give chase. We get into our starship and gun the engine until our speedometer registers 0.99 c; we are almost, but not quite, moving at the speed of light. But when we look out the window, to our astonishment we still see the photon whizzing by at the speed of light. The key point is that the speed of light is an intrinsic property of Nature, determined by the way an electric field varying in time generates a magnetic field and vice versa.


Chapter 7

In contrast, the speed of the tossed ball in our example depended on the muscular prowess and inclination of the tosser. The nature of time

To see why this caused such a crisis in the history of physics, we have to appreciate that the Galilean addition of velocities is based solidly on our fundamental understanding of the nature of time. To say that the train is traveling at 10 meters per second, we mean that when 1 second has elapsed for the stationmaster, the train has moved forward by 10 meters. To say that the ball is tossed forward at 5 meters per second, we mean that when 1 second has elapsed for the passenger, the ball has moved forward by 5 meters as measured by the passenger. Newton, and everybody else, made the unspoken but eminently reasonable assumption that when 1 second has elapsed for the passenger, precisely 1 second has also elapsed for the stationmaster. Time thus conceived is referred to as absolute Newtonian time. Given absolute Newtonian time, the stationmaster would then conclude that during the passage of 1 second, since the train has moved forward by 10 meters, the tossed ball has hurtled forward through space by 10 + 5 = 15 meters and hence is traveling at 15 meters per second. But somehow this seemingly incontrovertible logic does not work for the photon. A huge paradox! If you think hard about it, you would conclude, just like Einstein, that the only way out is to say that the passage of time is different for the passenger and for the stationmaster. More precisely, we have to reject the “eminently reasonable assumption that when 1 second has elapsed for the passenger, precisely 1 second has also elapsed for the stationmaster.” Common sense fails!

Einstein, the exterminator of relativity


For the stationmaster, the passenger is also passing through space. In other words, the stationmaster, while he feels that he is staying still, sees the passenger moving. If the train is sufficiently smooth, the passenger could also say that she is staying still but that the stationmaster is moving. Indeed, surely many readers have had this disorienting experience sitting in a vehicle moving sufficiently smoothly. By this reasoning, we conclude that the passage of time experienced by the passenger is intrinsically linked to the passage of space experienced by her. Similarly for the stationmaster. For each observer, the passage of time and the passage of space are inextricably tied. Exactly how they are linked was worked* out by Einstein in his theory of special relativity in 1905. In summary, Einstein banished space and time as separate concepts in physics. Henceforth, a new word, “spacetime,” is required to describe the world at the fundamental level. Varying not in space, but in spacetime

We will now see that the banishing of space and time as separate concepts in physics immediately resolves Newton’s vexation with action at a distance. Newton’s statement that the gravitational force exerted by a mass falls off as the square of the distance from the mass tells us how the gravitational field varies in space. Einstein now says that it is not quite kosher to say this; rather, it should be generalized to a statement about how the gravitational field varies in spacetime. In other words, knowing how the gravitational field varies in space, we immediately know how the gravitational field varies in time. In other words, we know immediately how much time it takes a gravitational disturbance to get from there to here. * Remarkably, deriving this link requires only simple high school algebra.


Chapter 7

Gravitational effects do not propagate instantaneously: no more action at a distance. That weird concept, which should bother anybody with a “competent faculty of thinking,” is now banished from physics. I have not (and could not have in the scope of this book) showed you the mathematical details of Einstein’s special relativity, but I hope that this heuristic discussion gave you a sense or flavor of how it works. In short, the insistence that the speed of light does not depend on the observer, as Maxwell told us, leads to the bizarre notion that the passage of time and the passage of space are inextricably linked. That space and time are replaced by spacetime immediately tells us how a field, be it electromagnetic or gravitational, varies in time once we know how it varies in space. Incidentally, it follows that a gravity wave propagates with precisely the same speed5 as an electromagnetic wave propagates, namely, c.6 And thus we know that the gravity wave detected in 2016 originated 1.3 billion years ago.

8 Einstein’s idea: Spacetime becomes curved A mysterious force emanating from the Bering Strait

Imagine flying from Los Angeles to Taipei. Flipping idly through the back of an in-flight magazine (or more likely the flight map on the video these days), you might notice that the plane follows a curved path arcing toward the Bering Strait. Is the Bering Strait exerting a mysterious attractive force on the plane? See figure 1. On your next trip you try another airline. This pilot follows exactly the same curved path. Don’t these pilots have any sense of originality? Why don’t they sometimes, just for the heck of it, swing south and fly over Hawaii, say? They seem to prefer flying over1 grim and unsuspecting Inuit hunters rather than cheerful Polynesian surfers. Not only is the mysterious force attractive, it is universal, independent of the make of the airplane. Should you seek enlightenment from the guy sitting next to you? Dear reader, surely you are chuckling. You know perfectly well that the Mercator projection distorts the surface of the earth, and pilots follow scrupulously the shortest possible path between Los Angeles and Taipei. The answer to the universality of the mystery force is to be sought, not in the physics, but in the economics department. We will come back to this story, but for now I digress.


Chapter 8


Figure 1. Is the Bering Strait exerting a mysterious attractive force on airplanes flying from Los Angeles to Taipei?

A number divided by itself equals 1

Earlier, I had described Newton’s law of universal gravity, stating that the force F of gravitational attraction between a mass M and a mass m is equal to a constant G (known as Newton’s gravitational constant), times the product of the two masses (namely, Mm), divided by the square of the distance R separating them: F = G Mm/R 2 . In school we also learned Newton’s law of motion that the acceleration a of a body with mass m is equal to the force F exerted on the body divided by m: a = F /m. Yes, it really is true, but tell that to a medieval peasant pushing a cart along a muddy road. He, and his educated contemporaries, would have regarded the claim that force produces acceleration as utterly loony. To all of them, and even to most of the proverbial guys and gals on our streets, Aristotle sounds much more plausible, claiming that force produces velocity. No force, no velocity. The educated among us now understand that everyday life, alas, is dominated by friction, pain, and suffering. Aristotle appears to be right, and Newton wrong. But in

Einstein’s idea


fact Newton is right, and the venerable Greek, now banished from reputable physics departments everywhere, is wrong. The fundamental laws of physics do not know about friction, pain, and suffering. So, the bottom line is: the acceleration of the moon due to the gravitational pull of the earth does not depend on the mass m of the moon at all. The force F is proportional to m, the acceleration a is given by the force F divided by m; ergo, the acceleration a does not depend on m. This profound but elementary bit of math, that something divided by itself gives 1 (m/m = 1), indicates that all falling objects on the surface of the earth rush to the ground at the same rate. Again, we all learned in school that Galileo dropped cannonballs off the Leaning Tower of Pisa2 to see whether they would all hit the ground at the same time. Only a small fraction of school children now grown up, no doubt including my dear reader, remember why he did this. The rest of our fellow citizens would guess that Galileo was either loco or high. Are inertial mass and gravitational mass really the same mass?

To Newton, mass corresponds to the amount of stuff.3 He quite naturally assumed that the mass m appearing in his law of gravity and the mass m appearing in his law of motion are one and the same. But a hair-splitting lawyer, or a habitual reader of mysteries, would surely have detected a hidden assumption here. Are the blonde4 seen kissing the butler and the blonde caught leaving the house on the night of the murder really the same blonde? Are those two masses really the same mass? To distinguish between the masses that appear in Newton’s law of gravity and in Newton’s law of motion, physicists called them the gravitational mass and the inertial


Chapter 8

mass, respectively. The former measures a couch potato’s obligation to listen to gravity, the latter his reluctance to get up and move. Conceptually, they are quite distinct and could very well not be equal. Unlike the faculty in some other university departments, we in the physics department do not accept proofs by authority, not even a single-named giant in a likely apocryphal story. And thus the Hungarian Baron Loránd Eötvös de Vásárosnamény (1848–1919), instead of doing whatever barons did in the 19th century, devoted much of his life performing ever more precise experiments establishing the equality of the gravitational mass and the inertial mass. In our days, a series of experiments, known collectively as Eötvös experiments, have established the equality of the gravitational mass and the inertial mass to a fantastic degree of accuracy. In particular, an ingenious effort, led by my former colleague Eric Adelberger at the University of Washington, is fondly referred to as the Eöt-Wash experiment.5 Nerd humor in full force here! Universality explained

That all objects fall at the same rate is known as the universality of gravity. We now flash back to you sitting on the plane chuckling at the thought of your colleagues deducing that there must a mysterious force exerted by the Bering strait on airplanes. But is it so laughably obvious? Consider the leading theoretical physicists before Einstein came along. They knew that all things fall at the same rate, be it an apple or a stone or a cannonball. To Einstein, that an apple and a stone would fall in exactly the same way in a gravitational field is no more amazing than different airlines, regardless of national or political affiliation, choosing exactly the same path getting from Los Angeles to Taipei. An apple or a stone

Einstein’s idea


traverses the same path in spacetime, just as a commercial flight follows the same path on the curved earth regardless of the airline.6 In hindsight, we might see an “obvious” connection, but hindsight7 is of course way too easy. For 300 years, the universality of gravity8 has been whispering “curved spacetime” to us. Finally, Einstein heard it. We did not go looking for curved spacetime; curved spacetime came looking for us! To Einstein, the equation m=m surely ranks as one of the two greatest equations in physics! The other is, of course, c =c No gravity, merely the curvature of spacetime

Just as there is no mysterious force emanating from the Bering Strait, one could say that there is no gravity, merely the curvature of spacetime. The gravity we observe is due to the curvature of spacetime. More accurately, gravity is equivalent to the curvature of spacetime: gravity and the curvature of spacetime are really the same thing. To summarize and emphasize the point, Einstein says that spacetime is curved and that objects take the path of least distance in getting from one point to another in spacetime. Environment dictates motion. The curvature of spacetime tells the apple, the stone, and the cannonball to follow the same path from the top of the tower to the ground. The curvature of the earth tells the pilots to follow the same path from Los Angeles to Taipei. This amazing revelation about the role of spacetime offers an elegantly simple explanation of the universality of gravity.


Chapter 8

Gravity curves spacetime. That’s it. Spacetime is curved and gravity’s job is done. It’s now up to every particle in the universe to follow the best path in this curved environment. This explains why gravity acts indiscriminately on every particle in exactly the same way. Next time you take a nasty fall, whether on the ski slope or in the bathtub, just think, every particle in your body is merely trying to get the best deal for itself. Best deal? To be explained in chapter 9. Curved spacetime

“Space tells matter how to move and matter tells space how to curve.” This memorable summary9 of Einstein gravity, due to John Wheeler, my first mentor10 in theoretical physics (as was mentioned earlier), has been widely publicized.* More accurately, for “space” we should say “spacetime.” If I were an intelligent layperson reading popular physics books, I would have been exceedingly frustrated by the term “curved spacetime.” These days, even the mass media bandy the term “curved spacetime” about with some abandon. But what exactly does it mean to say that “spacetime is curved?” I have addressed the appendix to readers like me. For those readers who do not wish to tangle with some math, no matter how slight, we can do fine proceeding by analogy. When we think about curved surfaces, such as the surface of a balloon, we see it as living in an ambient 3-dimensional flat space, the plain old Euclidean space we were born into and will die in. In math speak, the curved 2-dimensional surface is said to be embedded in a higher dimensional flat space. But as indicated in the appendix, we can perfectly * Frankly,

I do not find this formulation so exceptional. Already in Newtonian gravity, the gravitational field tells matter how to move, and matter tells the gravitational field how to behave. And in electromagnetism, the electromagnetic field tells charges how to move, and charges tell the electromagnetic field what to do.

Einstein’s idea


well conceive of, and describe, a curved space or spacetime without having to embed it in a higher dimensional space or spacetime. The metric of spacetime

Let us go back to the water wave described in chapter 4. Recall that we specify the surface of a pond by the height of the water measured from the bottom. At time t, and at the location specified by (x, y), call the height g (t, x, y), a function that depends on time t and space coordinates x and y. Without any breeze whatsoever, the surface of the pond is flat and thus g (t, x, y) = 1. But in general, g (t, x, y) varies in time and space in some complicated way according to an equation written down in the 19th century. However, when the waves are gentle and relaxed, that is, in the linear regime,* we can write g (t, x, y) = 1 + h(t, x, y) and treat h as small compared to 1. Then the equation we have to deal with simplifies. As was already mentioned in part I, the situation with Einstein gravity is closely analogous to the story of water waves. Einstein’s field equation governing the curvature of spacetime is essentially impossible to solve in general, but it simplifies enormously for gravity waves in the linear regime, so that most physics undergrads should be able to solve the corresponding equation. However, several technical, rather than conceptual, complications manage to befuddle many physics undergrads. But in a popular book, rather than a textbook,11 we can readily breeze by these complications. First, the simplest complication: in Einstein gravity, the analog of the quantity g becomes a function (or more strictly speaking, field) g (t, x, y, z) of time and three spatial * This bit of jargon was introduced in chapter 4.


Chapter 8

coordinates, namely Decartes’s x, y, z of the 3-dimensional space we live in. Second, to describe the curvature of spacetime, we need ten* such functions instead of one. But unless you want to get an advanced degree in physics, you need not be concerned. Third, in the case of water waves, when the surface of the pond is flat, g = 1. Similarly, when spacetime is flat (that is, in the absence of gravity waves), these ten fields g (t, x, y, z) are constant and equal to a simple number. The slight complication is that, of the ten, three are equal to 1, one is equal to −1, and the rest equal to 0. (Aren’t physics and math fun?) These ten fields g (t, x, y, z), known as the metric of spacetime, determine the distance between two neighboring points in spacetime. Given the metric, we can deduce† the curvature of spacetime.‡ I can give you a vague sense of how this works using an everyday example. Given an airline table of distances, you could deduce that the world is curved without ever going outside. If I tell you the three distances between Paris, Berlin, and Barcelona, you could draw a triangle on a flat piece of paper with the three cities at the vertices. But now if I also give you the distances between Rome and each of these three cities, you would find that you could not extend the triangle to a planar quadrangle. So the distances between four points suffice to prove that the world is not flat. But the metric tells you the distances between an infinite number of points. The reason is that once we know the distance between neighboring points, we can add up these tiny distances to find the distance between any two points. * The number ten will be explained in the appendix.

† That is, the mathematicians Gauss and Riemann figured out in the 19th century how to calculate the curvature given the metric. ‡ As I’ve already said, the reader who wants more can find more in the appendix to this book.

Einstein’s idea




Figure 2. Is Greenland bigger than China?

On a world map in Mercator projection,* Greenland looks bigger than China, but you know that Denmark, to which Greenland belongs, does not rank in the top ten countries by area (figure 2). From this fact alone you could deduce that the world is curved. Once the metric tells you about distance, it also tells you about area. * After

Gerardus Mercator, namely, Jerry the Merchant. In mapping the round sphere to a flat piece of paper, Mercator preserves the angles between straight lines but not the distance between points. For those who are lost, knowing the direction to your destination is more important than knowing how far you are from your destination.

9 How to detect something as ethereal as ripples in spacetime

Laser Interferometer Gravitational-Wave Observatory

At this point, different strands of our story come together. The detection of gravity waves that I opened this book with was announced by LIGO, an enormous collaborative effort led by physicists at the Massachusetts Institute of Technology and the California Institute of Technology and involving almost a thousand scientists from numerous institutions and countries. The name is a not-quite-exact acronym for the Laser Interferometer Gravitational-Wave Observatory. It has gone on for more than 40 years1 since conception and cost more than a billion dollars.2 The reason that such a gargantuan effort was required to detect gravity waves is, of course, once again, the extreme, almost ludicrous, weakness of gravity I talked about in part I. In addition, any credible sources of gravity waves are separated from us by the vast distances astronomers are so proud of. Far away, galaxies could crash into each other and black holes could suck whole civilizations up, and we would hardly notice the disturbances. It would be like detecting the water wave generated by a passing speedboat a thousand miles away.

Ripples in spacetime


Figure 1. Water waves interfering. Reprinted from http://www.physics-animations.com.

Wave interference

As the letter “I” in the name “LIGO” indicates, the detection scheme uses interference between two laser beams, as was first suggested by two Soviet physicists, M. E. Gertsenshtein and V. I. Pustovoit, back in 1962. Interference is easy to understand and can be readily observed in everyday situations, for example, when two water waves pass each other (figure 1).3 Consider superposing two waves moving in the same direction. Let the two waves have exactly the same wavelength. (The wavelength of a wave is defined as the distance from one crest to the next, or equivalently, from one trough to the next.) If the two waves are in phase (that is, if the crests and troughs of the two waves are lined up), then the result from adding the two waves will be a wave with larger amplitude: the two crests add to form a higher crest, while the two troughs add to form a deeper trough. (For example, if the two waves being superposed have the same amplitude, then the amplitude of the resulting wave would be doubled.) This is known as constructive interference.


Chapter 9

If the two waves are out of phase by exactly half a wavelength (that is, if the crests of one wave are lined up with the troughs of the other wave), then the crests and troughs tend to cancel each other. The result from adding the two waves with slightly different amplitudes would be a wave with smaller amplitude. (If the two waves have exactly the same amplitude as well as exactly the same wavelength, then they cancel each other completely, so that no wave is left at all.) This is known as destructive interference. Constructive and destructive interferences represent the two extreme cases. More generally, the two waves are not exactly in phase, nor are they out of phase by exactly half a wavelength. This case clearly leads to an interesting pattern: as the two waves move along, sometimes they reinforce each other, and sometimes they negate each other. In general, the two interfering waves will have different wavelengths and move in different directions. In fact, the situation at LIGO is just about the simplest imaginable: two electromagnetic waves with the same wavelength interfere while moving in the same direction. The LIGO detectors

The two LIGO detectors, one in Livingston, Louisiana, and one in Hanford, Washington, are identical (see figure 2). Each detector consists of a pair of 4-kilometer-long arms arranged in an L shape and enclosed in a vacuum tube. The basic design is indicated schematically in figure 2. A heavy mass is suspended at each end of an arm, four masses for each detector. A mirror is mounted on each mass, and a laser light is bounced back and forth off them to monitor the distances between the two masses on each arm of the detector.4 The light waves in the two arms are allowed to interfere, with the resulting wave sent to a photodetector. The setup

Ripples in spacetime


4 km

Test mass

Test mass 4 km

Laser source

Beam splitter

Test mass

Test mass

photodetector Figure 2. A highly schematic depiction of the Advanced LIGO detector. Certainly not to scale!

is exquisitely tuned, so that the waves interfere destructively when the two arms have exactly5 the same length, and the photodetector sees nothing. When a gravity wave from an astrophysical source passes by, one of the two arms in the L is stretched, while the other is squeezed. A half cycle later, the situation is reversed. Thus, the two arms are alternately being stretched and squeezed. Destructive interference is then no longer complete, and some laser light reaches the photodetector, signaling the passing of a gravity wave. The reason for having two detectors, as nearly identical as possible and as far apart as possible, is of course to discern local disturbances, such as a passing truck or a storm. As the reader could imagine, there is an enormous amount of noise, but this can be discerned and discarded by comparing data from the two detectors.


Chapter 9

Schematically, that is how detection of a gravity wave would work in principle. But in practice, the difficulties are daunting. The fabulous feebleness of gravity we spoke of in chapter 2 weighs on us. When people considered plausible astrophysical sources of gravity waves and put in reasonable numbers, they found that the length difference between the two arms might be as small as a billionth the size of an atom. How can you possibly measure that kind of distance change? You may gasp. The clever experimenters have come up with a scheme in which the laser light is bounced back and forth many times, thus amplifying the difference in distance the two laser beams would have to travel. You can imagine how precisely the masses have to be suspended, how well the mirrors have to be attached, how good the vacuum has to be, so on and so forth. Boys and girls, LIGO is a modern technological wonder that took decades to build (see figure 2). What to look for

One difficulty in detecting gravity waves is that the nature of the sources and their abundance in the universe are not precisely known. In contrast, Hertz could control the source of the electromagnetic waves he detected. Take, for example, two black holes merging into a single black hole. It is one thing to use Einstein’s theory to study the properties of a black hole that is just sitting around, but quite another to estimate how many black holes of a given mass have formed in the universe and to ask about the likelihood that such a black hole would have another black hole nearby. The reader can see that the latter type of questions are largely historical, in the sense that the answers depend quite a bit on happenstance in the evolution of the universe. We have only a rough idea of how often black holes merge and how far from us that would typically happen.

Ripples in spacetime


As it happened, the event detected by LIGO involves two black holes merging. Applying the known laws of physics, we can work out the different stages the merger has to go through. Initially, the two black holes orbit each other, emitting gravity waves.6 As they lose energy, they spiral in toward each other, and ultimately merge into a single black hole. The resulting black hole vibrates and eventually settles down in a process called “ringdown,” like that of a bell after being struck relaxing back to quiescence. Long before detection, physicists had calculated the precise shape of the wave emitted during each of these four stages using Einstein’s theory. After all, theorists had decades on their hands! The emission of gravity waves during orbiting and inspiral is in the linear regime, and it could be calculated analytically (that is, with pencil and paper, in layperson’s terms), but merger and ringdown are necessarily highly nonlinear and complicated; their modeling requires massive numerical work on giant computers (figure 3).7 It is by matching the observed and the calculated waveforms that the various parameters of the black hole binary, such as the masses of the two black holes and their distance from each other, can be determined. (And of course, it is the detailed match between observation and calculation that allows us to say that the gravity wave came from the merger of two black holes.) In the 2016 LIGO event, the two black holes were 29 and 36 times more massive than the sun, respectively.8,9 In writing this, I looked up what I said back in 1989, when I published my popular book on Einstein gravity, Toy.10 Here is a relevant passage: At the moment, researchers from the California Institute of Technology and the Massachusetts Institute of Technology have jointly asked the U.S. government to fund an ultrasensitive detector that should be able to pick up


Chapter 9 Livingston, Louisiana (L1)

Strain (10–21)

Hanford, Washington (H1) 1.0 0.5 0.0 –0.5 –1.0 1.0 0.5 0.0 –0.5 –1.0

L1 observed H1 observed

H1 observed (shifted, inverted)

Numerical relativity



Time (s)

Numerical relativity







Time (s)

Figure 3. Top left: The observed signal at Hanford, Washington. Top right: The observed signal at Livingston, Louisiana. The two observed signals match. Bottom left and right: The expected signals calculated using Einstein’s theory. Redrawn from B. P. Abbott, et al. “LIGO Scientific Collaboration and Virgo Collaboration” Phys Rev Lett 116, 061102. Published 11 February 2016. This article is available under the terms of the Creative Commons Attribution 3.0 License. https://creativecommons.org/licenses/by/3.0/us/legalcode.

gravity waves if the current estimate of how many gravity waves are coming in is correct. The experimenters say that if the National Science Foundation approves the project, the detector can be operating by 1991. To be sure that they have actually detected a gravity wave rather than just some local disturbance, the experimenters are asking for two detectors, to be located in California and in Maine, so that any signal picked up by one detector can be checked with the other detector. Eventually, with detectors located in different parts of the world, experimenters will be able to pinpoint the incoming direction of any gravity wave detected.” California and Maine, not Louisiana and Washington state! Did you notice that? California and Maine are relatively easy to get to from the two lead institutions, Caltech and MIT, and are almost maximally separated in the

Ripples in spacetime


continental United States. But American political and other practical considerations must have intervened. Did you also notice that optimism ran wild in those days? The detectors could be operating by 1991! But one statement from 1989 holds true. Shortly after LIGO’s detection of gravity waves in 2016, the government of India approved the building of a gravity wave detector in India. We do not doubt that we will, in due time, see detectors sprouting up around the globe, capable of pinpointing the direction of any incoming gravity wave. In fact, the 27 years between 1989 and 2016 were filled with internal struggles, with project leaders ousted or shunted aside, and competitions and bitter accusations launched against (or by) LIGO. I do not have first-hand knowledge of any of this and could only refer the reader to published accounts.11–13 In my opinion, it would be surprising indeed if any of these were absent in a project of this magnitude and duration. Indeed, over the years, the LIGO project was on several occasions at risk of being axed due to its enormous cost. Rainer Weiss of MIT, one of the leaders who endured and persisted in pushing the project through, described the development of LIGO as the perils of Pauline.14 Like many physicists, I stand in awe of LIGO. Recall the extreme feebleness of gravity compared to electromagnetism. The electromagnetic wave was detected a mere 21 (= 1886−1865) years after Maxwell’s prediction. For detection, Hertz simply used his eyeballs to see the spark caused by the passing wave. I mentioned in the prologue that, after the LIGO announcement, a reporter asked why Einstein was so far ahead of experimental confirmation. In fact, it was not so much that Einstein was ahead, but rather that the relevant experiment had to wait for the development of ultra-precise lasers, of the massive computers needed to analyze the data,


Chapter 9

and so on. Incidentally, Einstein had also laid the theoretical foundation behind the laser. Another measure of what a great intellect he possessed! The pioneer of gravity wave detection

I must pay my respects to Joseph Weber (1919–2000), the pioneer of gravity wave detection. Starting in the 1950s, when there was still considerable skepticism about the existence of gravity waves, until his death, when his detection sensitivity was almost universally scorned, Weber devoted himself to the detection of these waves. His detector consisted of a large cylindrical bar of metal, and it was hoped that a passing gravity wave would distort the bar and set up a detectable resonance. (This single sentence clearly is not intended to convey the technical sophistication that took almost half a century to develop and improve.) In hindsight, we know that Weber’s detector was simply not sensitive enough. Nevertheless, he repeatedly claimed that he had seen gravity waves. These claims were met with a storm of challenges and ended up being discredited.15 Nevertheless, the community is of the opinion that Weber deserves to be recognized as a pioneer whose efforts pushed forward the dawn of gravity wave astronomy. The reader should not get the impression, of course, that since Weber’s detector, only LIGO has been built. Quite a few detectors were built,16 but none reached the sensitivity of LIGO.

Part III

10 Getting the best possible deal To say what everyone else has already said, but better

In science, one tries to say what no one else has ever said before. In poetry, one tries to say what everyone else has already said, but better. This explains, in essence, why good poetry is as rare as good science. It would appear that science and poetry are in extreme contrast to each other. However, some theoretical physicists, like poets, do devote their creative energies to saying what has already been said, but in a different way. Their work is often dismissed by more pragmatic physicists for essentially the same reason that poetry sometimes is dismissed. A body of physics is reformulated, but the new formulation does not advance our knowledge one whit. In the vast majority of cases, in poetry as in theoretical physics, the rude dismissal is perfectly justified. The new version is more convoluted and turgid than the old. But once in a while, a poem, compact in structure and eloquent in cadence, manages to illuminate a theme more lucidly than ever before. In physics, too, formulations more in tune with the inner logic of Nature emerge from time to time. Perhaps the best example is the so-called action formulation, developed in the 18th century as an alternative to Newton’s differential formulation of physics.


Chapter 10

In Newton’s view of motion, one focuses on the moving particle at every instant in time. A force acting on the particle causes the particle’s velocity to change according to Newton’s law, F = ma. Knowing the particle’s acceleration a allows us to determine the particle’s velocity at the next instant, and then, the particle’s position at an instant after that. By repeating this procedure, one determines the position and velocity of the particle in the future. This, in short, is the standard formulation with which every beginning student of physics has to grapple. The formulation is called “differential,” since one focuses on differences in physical quantities from one instant to the next. The equations describing these changes are known as “equations of motion.” With the action formulation, in contrast, one takes an overall view of the path followed by the particle and asks for the criterion the particle “used” in choosing that particular path rather than some other path. As we have already seen in chapter 8, and as we will see in chapter 11, this notion will come to the fore when we talk about curved spacetime. The drowning beauty and the scrawny lifeguard

So, the action formulation. But first, a story about Richard Feynman,1 likely apocryphal but possibly true. The movie opens on a gorgeous Southern California beach. We zoom in on a lifeguard, noticeably scrawnier than the other lifeguards. However, we soon discover that he is considerably smarter. Egads, it is Dick Feynman, in the days before Baywatch! Perched on his high chair, he has been watching an attractive swimmer with great interest, plotting how he might win the young woman’s affection, all the while solving a field theory problem in his head. Suddenly, he notices that she is splashing about frantically. She is going under! Must be a cramp! An action hero

Getting the best possible deal




sand water



G L Figure 1. The best possible path for Feynman to follow to get to the drowning girl is along the solid lines from F to G. From Einstein Gravity in a Nutshell by A. Zee. Copyright ©2013 by Princeton University Press.

is as an action hero does: Feynman jumps down from his lookout and goes into action. Euclid2 long ago proclaimed that the shortest path between two points is a straight line. Ergo, if you are in a hurry to get from one point to another, you would want to go in a straight line. So, the other lifeguards are already proceeding in a straight line (starting from point F, the lifeguard station, in figure 1, going along the dotted line) toward the girl (at point G). That would be the path of least distance. But no, Feynman has already calculated the path that would allow him to reach the girl in the least amount of time. Time counts more than space here: least time trumps least distance. Our hero, as any other human for that matter, can run much faster, even on a soft sandy beach, than he can


Chapter 10

Figure 2. A light ray goes from the swimmer’s toes T to the observer’s eye E. Light “chooses” the path that enables it to get to its destination in the least amount of time. Since light moves faster in air than in water, the path TAE is chosen rather than the straight line path TBE. The observer’s brain, judging the direction from which the light ray comes, decides that it came from the point T . Then to the observer, the toe T appears to be at T . Therefore, the swimmer’s legs look shorter than normal. Surely you have noticed, traveling by car on a hot day, that the highway beneath a distant car often appears to be wet. But by the time you get to that spot, the road surface is in fact bone dry. This common mirage is explained in figure 3. That light is in a hurry also accounts for the observation that the air around a hot object appears to shimmer. From Fearful Symmetry: The Search for Beauty in Modern Physics by A. Zee. Copyright ©1986 by A. Zee. Princeton University Press.

swim. So the rescuer should spend more time running before plunging into the sea. A simple high school level calculation shows Feynman the best path to take (see the solid line in figure 1). Our hero beats the other guys and gets to the eternally grateful young woman (or so he hopes) first! But you don’t have to calculate to see that there is an optimal path. Clearly, only a cretin would follow the third path (the dashed line) shown in figure 1. Light in a hurry

We all know that light travels in a straight line, but we also notice that when light enters water from air, it bends. You

Getting the best possible deal



1 2


Figure 3. Summer mirage: A light ray leaving the hood H and headed downward encounters a layer of hot air near the road surface and bends upward. It ends up following Path 2 to the observer’s eye. The observer’s brain, judging the direction from which the light ray comes, concludes that it came from H . Another light ray goes directly from H to the eye, following Path 1. This is repeated for light rays leaving every point on the car, causing a reflection of the car to be seen. The brain — what a marvelous organ — deduces that the road must be wet. By the way, some readers may see that this example shows that light only cares about the local, not the global, minimum time of transit. From Fearful Symmetry: The Search for Beauty in Modern Physics by A. Zee. Copyright ©1986 by A. Zee. Princeton University Press.

can easily observe this by sticking a spoon in a glass of water. Indeed, that explains why people standing in swimming pools appear to have comically short legs. See figure 2.

Fermat’s least time principle for light

As our parable showed, the bending of light as it enters water from air can be explained if light moves more slowly in water than in air and if light is always in a hurry to get to where it is going. Light would not be so stupid as not to follow Feynman’s path, like the other lifeguards. The great mathematician Pierre Fermat (1601 or 1607/ 08?–1665)3 , he of the “last theorem,” proposed, in the year of his death, precisely this least time principle for light. That light bends is of course not just for laughs around swimming pools, but crucial for a pleasant life. To read these very words, you have (or rather your saintly mother has) cleverly positioned in your eyes a blob of watery substance


Chapter 10

(known to the cognoscenti as a lens), which you squeeze just so using tiny muscles, in order to bend light to your advantage and bring the ambient light bouncing off these words on the printed page into focus. Your mother, as the product of eons of evolution, was oh so clever giving you eyes. As we speak, you are using precisely this phenomenon of light bending to save the light entering your eyes some time (a phenomenon known as refraction) to gain yourself some knowledge about physics and the universe, an activity that evolution applauds: reading this book could conceivably boost your reproductive advantage. Material particles

After the success of the least time principle for light, physicists naturally wanted to find a similar principle for material particles. Something is minimized, but what? Matter behaves quite differently from light. For one thing, material particles do not travel at a constant speed. If a particle starts out faster, it gets to its destination faster. So a least time principle certainly does not apply. It took physicists quite a while to arrive at the correct principle, now known as the action principle. To explain this, I will invoke yet another great name, Humpty Dumpty. When Dumpty falls, he starts out at a leisurely pace, and then goes faster and faster. Not even all the King’s men and horses could make him start out fast and then slow down as he approaches the ground. A record of where Dumpty is and how fast he is falling at any instant in time is known to theoretical physicists as a history. An infinite number of histories could be contemplated (such as Napoleon defeating Wellington),4 but somehow only one history is actually realized. From everyday observation, Dumpty never starts falling fast and then slows down as if in fear of his imminent crack-up.

Getting the best possible deal


Choice of history: The action principle

What principle dictates Dumpty’s choice of history? Indeed, this is the question at the heart of physics. How does anything choose its history? Fermat answered it for light. At any instant during Dumpty’s fall, he has both kinetic and potential energy. Allow me to remind you that, in Newtonian mechanics, the kinetic energy is simply the energy associated with the movement of the particle, while the potential energy is a kind of “stored” energy that is available for conversion into kinetic energy. For example, an object near the surface of the earth has potential energy because of the earth’s gravitational pull. The higher the object is from the ground, the more potential energy it possesses. The total energy, given by the sum of kinetic energy and potential energy, is conserved; that is, it does not change. As the object falls, its potential energy decreases, while its kinetic energy increases, keeping the sum of the two constant. In other words, potential energy is converted into kinetic energy. When we go downhill skiing, we pay the lift operator to provide us with lots of potential energy, which we then convert into kinetic energy. As mentioned above, physicists had to struggle to figure out an analog of the least time principle for material particles. It turns out that the correct principle is formulated in terms of a fundamental quantity known as the action. At any instant, subtract the potential energy from the kinetic energy and call the resulting quantity the Lagrangian.5 The action is then the result of adding up the Lagrangians from the start time to the end time. (In our example, these two times would be, respectively, the time Dumpty leaves the security of the wall and the time when he spills his yolk on the ground.) Readers with a nodding acquaintance of calculus would know that “adding up” is called* “integrating.” The resulting * As was already mentioned in chapter 2.


Chapter 10

 sum is called an “integral,” denoted by the symbol , which you could see is a distorted S representing the word “sum.” The action is equal to the integral of the Lagrangian over time.* The action principle states that a material particle (as distinct from light) “chooses” the path that either maximizes or minimizes the action.6 A technical aside that most readers can simply ignore: Fermat tells us that light minimizes travel time. It turns out that in some circumstances, material particles minimize the action, as we might have guessed, but in other circumstances, they maximize the action. Physicists have coined the word “extremize” to cover both “minimize” and “maximize.” That the action principle is an extremal principle, rather than a simple minimal principle like Fermat’s, remained a mystery until the advent of quantum physics.7 Catch me if you can

The computation of the action is similar to that done by an accountant determining the total profit of a business for any given production strategy. She subtracts the total cost of production from the gross income on a weekly basis and then sums this quantity over the 52 weeks in the fiscal year. The businessperson naturally tries to maximize the total profit by following the most advantageous history. Just like the businessperson maximizing profit, Dumpty chooses the history that would minimize his action. Since the action is equal to the kinetic energy minus the potential energy summed over the duration of the fall, and since the potential energy increases with the distance from the ground, it clearly pays to spend more time high above the  S = dt L . Traditionally, the letter S is used for the action, and L for the Lagrangian.

* In math symbols,

Getting the best possible deal


ground, so that a larger potential energy could be subtracted off. In everyday life, a falling object, especially if it is fragile and valuable, appears to hesitate for a moment or so (almost as if it is saying “Catch me if you can!”) before gathering speed and crashing to the floor. That’s Galileo’s law of acceleration in action, of course. From the action point of view, we can understand what went on as follows. The object, by staying at high altitude for “as long as possible,” maximizes its potential energy and thus lowers the action. But then it has to rush at the end to get to the floor in the allotted time, and hence pays the price of a larger kinetic energy. Dumpty, therefore, starts slowly and then accelerates. With the help of elementary mathematics, one can show that the best strategy for Dumpty is to accelerate at a constant rate.8 The reader may feel that, in this case, the action formulation actually is more convoluted than the differential formulation, and indeed it is. In the latter formulation, Dumpty’s acceleration is determined immediately by Newton’s law. However, as knowledge of physics progressed beyond Newtonian mechanics, the superiority* of the action formulation became more apparent, as will be indicated below. Brevity is the soul of wit

For a long time, the action formulation was regarded as nothing more than an elegant alternative.9 Meanwhile, physics continued to be formulated largely in terms of differential equations of motion.10 However, theoretical physicists working on fundamental issues have gradually embraced the action formulation and jilted the differential formulation.11 * These days, fundamental physics is largely formulated using the action principle.


Chapter 10

All physical theories established since Newton may be formulated in terms of an action. The fundamental interactions we know about, the strong, weak, electromagnetic, and gravitational, can all be described by the action principle.* The action formulation is elegantly concise. For instance, Maxwell’s eight electromagnetic equations are replaced by a single action, specifying a single number for each possible history describing how the electromagnetic field changes. In Einstein’s theory, ten equations describing how the graviton field changes are summarized in a single action. The point is, while the equations of motion may be complicated and numerous, the action is given by a single expression. Believe me, it is much much easier to find the action, one single expression, than the ten equations of motion, as Einstein was to find out through much pain and suffering. See chapter 12. Our analogy may be helpful here. The best deal (corresponding to the action) may be easy to state, but the strategy (the equations of motion) needed to nab the best deal might be complicated to describe. A series of ever better actions

Some books describe the history of physics as a series of revolutions. I don’t like the word “revolution,” as it suggests the overthrow of the previous regime. Einstein did not show that Newton was wrong. Newtonian physics is perfectly correct when applied to objects moving slowly compared to the almost fantastic speed of light. What actually happens is that the action describing Newtonian physics has to be modified and extended. It is replaced by an Einsteinian action, which is, however, * Why this should be so represents a profound mystery. We can certainly conceive of equations of motion that do not follow from extremizing an action.

Getting the best possible deal


required to reduce to the Newtonian action when describing slowly moving objects. I prefer to think of the history of physics as a series of ever better, ever sexier actions. Often physicists simply add to an existing action. For example, in the 19th century, Maxwell’s action for electromagnetism had to be added to the Newtonian action. It is the incompatibility between the two terms in the action that led to Einstein’s special relativity, in which the Newtonian action was modified, as was just mentioned.

11 Symmetry: Physics must not depend on the physicist Everybody must agree on the action

A central theme of fundamental physics has been the overarching importance of symmetry. Indeed, I am so enamored of the concept that I devoted an entire book to symmetry,1 to which I refer the reader for details. Einstein’s special relativity offers a canonical example of a symmetry in physics. In chapter 7, I wrote that Einstein insisted that the laws of physics must not depend on observers in uniform motion relative to each other. This insistence has since been generalized and formulated as a principle: while physical reality can appear different to different observers, the structure of physical reality must be the same. I am necessarily being a bit vague here. The action principle, however, allows us to render the phrase “structure of physical reality” a bit more precisely. Different observers must agree on the action. Otherwise, different observers would be extremizing different actions and getting different deals. Symmetry implies transformation from one observer’s frame of reference to another’s. Thus, for example, in special relativity, we transform from the passenger’s conception of physics to the stationmaster’s. For example, what looks



like an electric field to the stationmaster is perceived by the passenger as a combination of an electric field and a magnetic field. Covariance versus invariance

Elementary physics is typically formulated in terms of equations, such as Newton’s equation of motion or Maxwell’s equation of electrodynamics. Under a symmetry transformation, both sides of these equations would change. To be specific, consider special relativity. A bit of useful jargon: the transformation of physical quantities from one frame of reference to another in special relativity is known as a Lorentz transformation, in honor of Hendrik Lorentz (1853–1928). For example, the equation determining the electric field generated by a bunch of charges sitting there (in other words, in the absence of any electric current) has the form (variation of electric field in a space) = (charge distribution) Under a Lorentz transformation, the quantities on the two sides of the equal sign both change, but in such a way that they remain equal. Physically, suppose the stationmaster sees some electric charges sitting on the platform, generating an electric field. The passenger on the train going through the station would see the charges moving, that is, an electric current generating a magnetic field as well as an electric field. In physicist’s jargon, the equation is said to be covariant (“changing together”), rather than invariant (“not changing”). The two sides of the equation change in the same way, rather than remain unchanged. As a result, while the physical quantities involved change, the structural relationship between them does not.


Chapter 11

As a rough analogy, one can think of a marriage in which the two partners “grow” with the years. In those rare cases in which the husband and wife both grow in the same direction and at the same rate, the relationship between them would remain the same, even though neither of them does. Unfortunately, psychologists tell us that most human relationships are not covariant in time (and certainly not invariant). In contrast to the equations of motion, the action for electromagnetism is left invariant by a Lorentz transformation. The action remains unchanged. Indeed, to say that physics possesses a certain symmetry is to say that the action is invariant under the transformation associated with that symmetry. As a result, a history seen by different observers is labeled by the same number, so there can be no dispute about which history is favored by the action principle. The action, in short, embodies the structure of physical reality.* * The power and elegance of the action formulation of physics is often admired by deep thinkers in other subjects. The eminent economist Paul Samuelson, for example, expressed his great admiration for Fermat’s least time principle in his Nobel lecture of 1970, as quoted on p. 357 of Steve Weinberg, Gravitation and Cosmology.

12 Yes, I want the best deal, but, what is the deal? Choosing a path as a metaphor for life

Now that I’ve told you quite a bit about the action principle, I can tell you how theoretical physics at the fundamental level works. What I will give you is a bit of a caricature, but it captures the spirit and is fairly close to the truth, the way a New Yorker cartoon depicts the truth. Some would see in the action principle a metaphor for life. You want to live life maximizing something, perhaps the total happiness integrated over time. You could either party now, dude, or you could study the action principle and party later in life. Of course, physics is so much simpler than real life, for which the quantity corresponding to the Lagrangian consists of a multitude of terms, each with zillions of parameters that vary from individual to individual. For example, for some geeks, studying physics has got to be way more fun than partying. There is also the minor detail that the allotted time between birth and death is not known in advance. You have to decide on what is to be maximized. Is it contribution to the well-being of others? Is it contribution to human knowledge? Is it happiness minus suffering? If so, what is the relative weight between happiness and suffering?


Chapter 12

Once you decide what you want maximized, we could fashion a life for you. Needless to say, these notions of human existence are impossible to quantify and to plan for. But you get the idea. If you tell a theoretical physicist the action governing some aspects of the physical world, then he or she can figure out how things would move by extremizing the action. So, write down an action, and, roughly speaking, the rest follows. Sometimes an action is obtained after struggling through a century of experimental work, such as the action for electromagnetism.1 At other times, an action is just postulated and is constructed to incorporate various general principles, such as the action for string theory. In more picturesque language, here is how fundamental physics, with slight exaggeration,* works. Okay, you tell me that everybody is trying to get the best possible deal. But I can’t possibly figure out what the best deal is unless you tell me what the deal is! So, tell me what the deal is, then we can talk about how to get the best possible deal. Now that you know about the action principle, we can tackle the problem of how curved spacetime tells matter to move and how matter curves spacetime. We merely have to specify the action involved. How curved spacetime tells matter to move

To determine how a particle should move in curved spacetime, you have to tell me what the action, or deal, is for the particle. What would a particle getting from one point (called “here now”) to another (called “there later”) in curved spacetime try to extremize? The histories are just curves connecting one point to another in curved spacetime. * To forestall critics, let me just say that, as a professional, I know how it actually works.

What is the deal?


What theoretical physicists do is to make the most reasonable guess for the action and check whether it works. Try your hand at this game. Take a guess! Imagine you are the particle confronted with an infinite number of curves connecting the two points. What is an intrinsic geometric quantity that distinguishes one curve from another? If you said that that the only possibility is the length of the path between here and there, then bravo or brava! You have insight, intuition, and whatever else it takes to be a theoretical physicist. For those readers who are interested, in the appendix I explain how to calculate the length of a path between two points in spacetime. Abusing ordinary language slightly, physicists think of this as the distance traversed by the particle. Since we are talking about spacetime, not space, the distance between two points involves lumping together the ordinarily separate notions of time duration and of spatial separation. The two notions are subsumed into the word “distance.” But we have yet to put in something about the particle. In fact, at this level of abstraction, the particle is a point moving through spacetime, and its only attribute is its mass. The correct action is in fact given by the mass of the particle multiplied2 by the distance it traverses. More massive 3 particles  generate more of an action. In fact, the action is S = m dτ , with dτ indicating the infinitesimal distance* between two neighboring points in spacetime. You add up all these infinitesimal distances to get the total distance between the starting and the finishing point—hence the integral.

* You may be able to see that this is getting close to Fermat’s least time principle;

the integrated distance between two spacetime points generalizes the elapsed time between two points in space.


Chapter 12

Einstein’s theory: Spacetime also wants to get the best deal

Good! You and I together have guessed roughly half of Einstein’s theory of gravity, namely, curved spacetime tells matter how to move. But that is only one half of the pas de deux. Not only does curved spacetime have to tell matter how to move, matter also has to tell spacetime how to curve. Yes, matter is getting a deal, but spacetime also wants to get the best deal. So, what is the deal for spacetime? In other words, how does matter or energy curve spacetime? Again, take a guess! You would expect in the presence of more matter, spacetime becomes more curved. You have to decide what spacetime is trying to extremize. A natural guess would be the curvature of spacetime. You got it! For those readers curious to see what the action for spacetime, known as the Einstein-Hilbert action, looks like, here it is:  √ S = d 4 x g R/G Here R denotes the curvature4 of spacetime, with the letter R chosen to honor Riemann; G is Newton’s gravitation constant. The integral is over 4-dimensional spacetime with the √ factor g constructed out of the metric, as explained in the appendix. Extremizing the action S, we obtain5 Einstein’s much celebrated field equations for gravity. They tell us how spacetime should curve around a black hole and how the universe should expand. Just to give you a flavor of how the game is played, suppose you want to do cosmology. Add to the Einstein-Hilbert action the action for a point particle S =  m dτ repeated a zillion times, each particle representing a galaxy. (On the vast scale of the universe, even an entire galaxy may be idealized as a point particle in the leading

What is the deal?


Figure 1: The action for the dance between the gravitation field and a massive particle.

approximation.) Vary to obtain the equations of motion. Solve. There you have it: a possible homework exercise for an advanced undergraduate course on Einstein gravity.6 Or perhaps you only want  one particle, say, the sun or a black hole. Then add S = m dτ only once. Solve the resulting equations of motion (figure 1). That was not so hard, was it, laid out with the benefit of hindsight! Curving space as well as time

In this modern formulation of gravity, an elegant way of summarizing Newton’s work is to say that he curved time but not space. Since Einstein had already unified time with space in his 1905 work on special relativity, he necessarily had to curve space also. Time and space are just too intimately linked for physicists to be able to curve one without curving the other (figure 2). So here is my summary of gravity as presently understood. Newton: “I curved time.” Einstein: “I curved space as well as time.” Einstein narrowly missed a career disaster

It has long puzzled me that Einstein did not use the action principle in his decade-long struggle to find the theory

Figure 2: Newton curved time; Einstein curved spacetime. Albert Einstein at Princeton Luncheon, Princeton University, NJ, 1953. Copyright 1981, Ruth Orkin.

What is the deal?


of gravity. Instead, he followed various clues on what the equations of motion for gravity must look like. For example, one clue would be that in the appropriate regime, these equations of motion must reduce to Newton’s equation for the gravitational field. Another clue would be that these equations of motion must be compatible with special relativity. As explained above, the action is one single mathematical expression, while there are ten equations of motion (depending on how you count). Furthermore, Einstein missed the subtle, and somewhat hidden, connections7 among the equations of motion. In contrast, once you write down the action, you would simply vary it to get the equations of motion, and these subtle connections would pop out automatically. This misstep by Einstein puzzled me all the more since the action principle was by then well known to physicists.8 Recall that Lagrange lived most of his life in the 18th century. Of course, as noted in chapter 8, staircase wit is easy and cheap, but still. In 1915, as Einstein was getting close to the Holy Grail after an arduous decade of work (but of course without realizing it at the time), the eminent mathematician David Hilbert (1862–1943) grasped from Einstein’s published work that all he had to do was write down the action for curved spacetime and vary it. As I explain in the appendix, in the late 19th century, mathematicians had already figured out how to determine the curvature of curved space or spacetime. Einstein did not know this, certainly not when he started his quest in 1905, but Hilbert of course did, being a mathematician. Here is the chronology of a crucial 21-day period in theoretical physics. On November 4, 1915, in a paper presented to the Royal Prussian Academy of Sciences, Einstein obtained a set of field equations. Three weeks


Chapter 12

later, on November 25, 1915, Einstein presented to the same academy his field equations, but without using the action principle. But in the meantime, Einstein was scooped! On November 20, David Hilbert presented to the Göttingen Academy the gravitational field equations he derived by varying an action. This action, as mentioned above, is now called the Einstein-Hilbert action. Quite rightly in the opinion of all physicists, Einstein is credited with this action, even though strictly speaking, he found the equations of motion that emerge from the action rather than the action itself. The theoretical physics community is not a court of law: it regards Hilbert, although he did find the action first, as playing second fiddle to Einstein.

Of incomparable beauty, but at risk of being nostrified!

I’ve hardly come to know the wretchedness of humanity better than in connection with this theory. —A . E I N S T E I N But at that time, Einstein didn’t know that history would be kind to him in this one respect. He was justifiably worried and, perhaps less justifiably, angry. In fact, he was sufficiently incensed as to dash off a letter on November 26 to a friend. In the letter, the great man also bitterly denounced his estranged wife for her influence on their children, but before launching into a diatribe about his personal life, he first accused Hilbert of stealing his theory. Einstein wrote, “the theory is of incomparable beauty. But only one colleague has really understood it, and he is trying, rather skillfully, to ‘nostrify’ it. That’s Max Abraham’s coinage. In my personal experience, I’ve hardly come to know the wretchedness of humanity better than in connection with this theory.”

What is the deal?


Well, dear reader, nostrification is not only still practiced in theoretical physics, but ever more skillfully.9 To make the whole episode all the more puzzling, Einstein and his friend Marcel Grossmann had published a paper in 1914 about a variational principle for gravity.

13 The action for Einstein gravity Physics is where the action is

Physics results from everybody in the universe striving for the best possible deal. It is a basic principle of the universe, obeyed by the universe itself in its expansion. The entire physical world is described by one single action. As physicists conquer a new area of physics, such as electromagnetism, they add to the action of the world an extra piece describing that area. At any stage in the development of physics, the action is a ragtag sum of disparate terms. Here is the term describing electromagnetism, there the one describing gravity, and so on. The ambition of fundamental physics is to unify these terms into an organic whole. While a mechanic tinkers with his engine and an architect with her design, a fundamental physicist tinkers with the action of the world. The physicist replaces a term here, modifies another there. Our search for physical understanding boils down to finding that one expression. When physicists dream of writing down the entire theory of the physical universe on a cocktail napkin, they mean to write down the action of the universe. It would take a lot more room to write down all the equations of motion, entire blackboards filled with symbols, as depicted by cartoonists.

The action for Einstein gravity


Fundamental physicists dream of writing down the design of the universe on a piece of napkin. The action formulation allows an extraordinarily compact description. From Fearful Symmetry: The Search for Beauty in Modern Physics by A. Zee. Copyright ©1986 by A. Zee. Princeton University Press.

At present, theoretical physicists believe the action looks something like what has been scrawled on the napkin in the figure. To understand what each symbol means in detail, you would have to spend years in a reputable graduate school. However, you may notice all the plus signs right away: this action consists of many pieces simply added together. For instance, the first term—R represents gravity, while the second term, F 2 , represents the other three interactions. This indicates that physicists have not yet reached a completely unified description of Nature. Physicists are struggling to find an even more compact action in which the six separate terms contained in this action will be tied together. When physicists talk about the quest for a unified theory, what they mean is that they long for an action that contains as few separate terms as possible.

14 It must be

An extremely tight theory

The fabric of modern theories of physics is tightly woven: deep, underlying symmetries mandate the design and structure of these theories. Physicists revere Einstein’s theory because it is so tight. Einstein’s theory is required to respect the constraint, almost self-evident in hindsight, that the action must not depend on the coordinates we choose to describe spacetime. (This requirement may be called “general coordinate invariance,” more commonly known as general covariance when referring to the resulting equations of motion.) To explain what this means, I appeal again to an analogy mentioned in chapter 8. In the Mercator projection, Greenland looks bigger than China, but in some other projection, it does not. But the area of Greenland and of China could not possibly depend on whether we use Mercator or some other projection. Area is an example of what mathematicians call a “geometric invariant,” meaning something which does not depend on the coordinates used.1 Similarly, curvature2 is a geometric invariant. In short, Einstein gravity is what physicists call a geometrical theory. The action is to be constructed of geometrical invariants.3 The Einstein-Hilbert action displayed in chapter 12 is definitely geometrical. But now, given our discussion about

It must be


Einstein and Beethoven. From Fearful Symmetry: The Search for Beauty in Modern Physics by A. Zee. Copyright ©1986 by A. Zee. Princeton University Press.

area being a geometrical invariant, a lightbulb clicks on, and you realize that there exists another term that we could add to the action, namely, something like the area of spacetime. And here everyday language fails us. Area is a concept applied to 2-dimensional spaces, and here we are talking about 4-dimensional spacetime. Well, physicists call the relevant quantity the “volume of spacetime,” for lack of a better term.Thus, we could add to the Einstein-Hilbert action the √ term d 4 x g . As explained in the appendix, this is equal to the volume of spacetime multiplied by an unknown constant, called the “cosmological constant” and denoted by the capital Greek letter . More later.

The rise of gravity: The paradigm for fundamental physics

Symmetry dictates design. Once the symmetry underlying gravity was discerned, physics was literally forced to Einstein’s theory. Einstein’s theory of gravity carries with it a sense of the inevitable.


Chapter 14

The notion that a particular theory is the only one possible was new to physics. For instance, Newton’s pronouncement that the gravitational attraction decreases as the square of the distance between two bodies appears quite arbitrary from a purely logical point of view. Why doesn’t the force decrease as the distance, or as the cube of the distance? Newton would have regarded this question as unanswerable. He presents his law simply as a statement whose consequences accord with the real world. In contrast, once Einstein understood the symmetry underlying gravity, the theory of gravity was fixed. The inverse square law pops out. When I first encountered Einstein’s theory of gravity, I marveled at how cleverly it is put together. With deeper understanding, I came to understand that it is essentially inevitable. It has been aptly remarked4 that Einstein’s theory of gravity has the full force of a Beethoven opus. The last movement of Beethoven’s Opus 135 carries the motto “Muss es sein? Es muss sein!” (Must it be? It must be.) Art in its perfection must be a necessity.

Part IV

15 From frozen star to black hole

A roaring desire to escape

If you throw a stone upward, it will eventually fall back down.1 If you throw harder, giving it a higher initial velocity, it will climb higher before falling down. Eventually, if the initial velocity is higher than an aptly named escape velocity, the stone will escape the earth altogether. All simple enough. Indeed, it’s an exercise in Newtonian mechanics often given to beginning students of physics. The gravitational pull by a planet of mass M and radius2 R on a stone (indeed, on any small object) of mass m on its surface, namely, G Mm/R 2 , is proportional to m, but the roaring desire to escape, the stone’s momentum, speaking loosely, is also proportional to m. Once again, that curious equality of inertial mass and gravitational mass comes into play: the mass m cancels out in balancing the desire to escape against gravity’s pull. The escape velocity of an object, interestingly, does not depend on its mass. More precisely, the object cannot escape if its initial kinetic energy, given by 12 mv 2 (where v denotes its velocity), is less than its gravitational potential energy G Mm/R at the surface of the planet. One line of high school algebra shows that this “no escape” criterion works out to be v 2 < 2G M/R 2 .


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John Michell in 1783 and Pierre-Simon de Laplace3 in 1796 independently wondered about astronomical objects massive enough so that even light could not escape from them. In other words, are there objects whose escape velocity exceeds the speed of light c? Almost 200 years later, John Wheeler gave such objects the inspired name* “black hole.” The great Newton also theorized about light: he pictured light as consisting of streams of minute particles that he called corpuscles. The escape velocity of these corpuscles, as noted above, does not depend on the unknown corpuscle mass m: the criterion for a black hole does not require us to know the mass of a hypothetical particle. Happy are we (or rather, Michell and Laplace)! Following Newton then, we simply replace v in the “no escape” criterion by the speed of light c to obtain (after multiplying by R and dividing by c 2 ) 2G M >R c2 If this inequality is satisfied, then the object is a black hole.4 For the gravitational pull to be excessively strong, either G M is unusually large, or R is unusually small. For an object to qualify as a black hole, it either has to be massive for its size, or small for its mass. (Describes obesity, no?) Remarkably, even though the physics behind the MichellLaplace argument is not correct in detail (as we now know, we should not treat light as a Newtonian “corpuscle” with a tiny mass),5 this criterion, including the factor of 2, turns out to hold even in Einstein’s theory.

* Soviet

physicists who pioneered in studying these objects called them “frozen stars.” We are all relieved that this name did not catch on. The United States is kind of a Hollywood for naming novel concepts in physics; exhibit A: quark.

From frozen star to black hole


M log10 gram

 ) universe Milky Way

40 sun earth 20

human –30





R log10 meter




–20 atom

Figure 1. Mass M is plotted along the vertical axis and characteristic size R along the horizontal axis. Note that this is a so-called log-log plot, in which both mass and characteristic size are plotted in powers of 10; otherwise, it would hardly be possible to accommodate the universe and a proton in the same figure. Adopted from GNut, p. 14. From Einstein Gravity in a Nutshell by A. Zee. Copyright © 2013 by Princeton University Press.

An obesity index for the universe

As the obesity epidemic sweeps over the developed countries, one government after another has issued some kind of obesity index, basically dividing body weight by size. Nature has her own obesity index for any object, from electron to galaxy. Plot a point for each of your favorite massive objects in the universe. (For instance, a human is taken to have height of order 1 meter and mass of order 100 kilograms.) Consider


Chapter 15

the straight line representing the equality 2G M/c 2 = R. Anything above this line, that is, inside the shaded area, would have 2G M/c 2 larger than R and so would be a black hole, and anything below this line, not. In another words, for a given R, your M had better not be too large if you don’t want to be labeled obese (figure 1).

When the oppression of gravity is too much to bear

A more modern heuristic argument incorporates Einstein’s E = mc 2 . At a distance of R from an object of mass M, a particle of mass m feels a gravitational potential energy of G Mm/R. (To fix our mental picture, think of M as much larger than m.) As the particle gets closer and closer to the massive object, that is, as R gets smaller and smaller, the gravitational potential energy gets larger and larger. At which point does the particle feel that the oppression of gravity is too much to bear? Well, according to Einstein, were the particle to be entirely converted to energy, that energy would amount to E = mc 2 . Thus, when the gravitational potential energy gets to be comparable to this characteristic energy, the particle would not be able to stand it any more. It is as if an oppressive boss is dumping on a cowering employee’s head a whole load of negative vibe that exceeds the employee’s entire inner reserve of energy. Then something’s got to give. This critical state of affairs is reached when G Mm/R  mc 2 . (The symbol means roughly equal.) Think of mc 2 as the particle’s inner reserve. Again, m cancels out (the celebrated equality between inertial mass and gravitational mass.) We recover more or less the same MichellLaplace criterion: G M/c 2  R. One advantage of this argument is that it sows the seed for Hawking radiation, as we shall see shortly.

From frozen star to black hole


The horizon of a black hole

As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas. —K A R L S C H W A R Z S C H I L D , W R I T I N G T O ALBERT EINSTEIN

A precise formulation of the black hole had to wait* for Einstein’s theory of gravity of 1915. In Einstein gravity, a massive object curves the spacetime around it. If the object is way too massive for its size, spacetime around it is curved so excessively that it essentially folds over itself, loosely speaking. Light is trapped inside. Material particles are trapped a fortiori, since they cannot move faster than light. The Michell-Laplace criterion 2G M/c 2 > R emerges from Einstein theory as follows. You sit down to solve Einstein’s equations around a massive object, in empty space. In other words, you determine what spacetime looks like. Far away from the massive object, the effect of gravity dies away, and spacetime is pretty flat, the way we like it. But you know, and have known for some centuries, that the gravitational field decreases like the square of the distance from the massive object, according to Newton. The gravitational field is very small, but not quite zero. However, where the gravitational field is small, Einstein’s theory and Newton’s theory must agree! Indeed, since you are solving Einstein’s equations in empty space outside the massive object, the requirement that Einstein must agree with Newton far away is the only way the equations know about the mass M of the object. You now examine your solution describing the curved spacetime outside the massive object. Far away, the * I am impressed that from Michell’s speculation to Einstein’s precise description took a mere 132 years.


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spacetime is hardly curved. But as you come closer and closer to the massive object, spacetime curves more and more, so that at a distance 2G M/c 2 from the object, spacetime becomes so curved that it can trap light. Since you solved Einstein’s equations in empty space outside the massive object, this distance 2G M/c 2 is relevant only if it is outside the massive object, that is, only if 2G M/c 2 is larger than the radius R of the object. The Michell-Laplace criterion 2G M/c 2 > R for a black hole pops out. To further clarify this discussion, let us understand why the earth is not a black hole. We think of the earth as incredibly massive in the context of everyday life, but M for the earth is pretty small in this context. The quantity 2G M/c 2 is incredibly tiny, way smaller than the radius of the earth. The earth does not satisfy the Michell-Laplace criterion. For an object of mass M, the quantity 2G M/c 2 defines a distance known as the horizon. If your distance from the center of the object is less than the horizon 2G M/c 2 , then you are doomed. By the way, the term “horizon” is a felicitous choice; when a ship departing from port “sinks” over the horizon, it disappears from view. If you pass through the horizon of a black hole, you disappear from the visible universe. We have now viewed the Michell-Laplace criterion from several points of view, but the basic sense remains the same. For an object to be a black hole, it has to be too massive for its size, or in other words, too small for its mass. Interestingly, the solution describing the curved spacetime outside a massive object was first obtained, not by Einstein, but by Karl Schwarzschild (1873–1916) within months of publication of Einstein’s theory. Schwarzschild’s achievement is truly remarkable, since he did it while under heavy artillery fire serving in the German army on the Russian front in World War I. (He died a year later.) I always

From frozen star to black hole


Figure 2. In popular media, the funnel is often used to represent a black hole.

tell my students that for sure they should be able to solve for the Schwarzschild black hole, as it is now known, in the peace and quiet of their rooms. Funnel: Caution

You’ve probably seen a picture of a black hole depicted as a kind of funnel, or alternatively as a rubber sheet depressed by a heavy round mass. Far away from the funnel, or the depression in the rubber sheet, the surface is supposed to be flat. This image and its variants have appeared in countless magazines, newspapers, popular books, and even on the cover of a textbook. Here it is. See figure 2. In many science museums, visitors are invited to toss a small ball onto the surface of an actual funnel-shaped construction. If you toss the ball with sufficient speed in an angular direction, it will orbit around the center of the funnel, slowly spiraling into the dark “bottomless” pit in the center. And of course, if you toss the ball in the radial direction, it will fall right in, “sucked in by the irresistible force” of the black hole, often thought of as a “source of evil” in the


Chapter 15

visitor’s mind. This museum display entertains the visitors and educates them to some extent, but it is misleading6 at best. In my experience teaching Einstein gravity, it has for sure seriously confused some students. My son the biologist informed me that his colleagues, all with high falutin’ degrees, wondered about the existence of two “gravities.” The actual force “sucking” the ball into the funnel is of course just “plain old gravity,” supplied externally by the earth. Meanwhile, the curved surface of the funnel somehow represents the profound Einsteinian vision of “true gravity.” I sure hope that you are not confused. Of course, Einstein’s vision of curved spacetime, a flavor of which I try to give you in the appendix, is considerably more profound than what a funnel could convey. For one thing, how is time curved in a funnel? It goes without saying that it is silly to insist that a toy model, which may have helped some people understand Einstein gravity, be accurate in all respects.

16 The quantum world and Hawking radiation A crash course in quantum physics

Unless you are a Papuan headhunter, you have probably heard that we actually live in a quantum world, in which everything is constantly jiggling—hence the Heisenberg uncertainty principle: you can never know exactly where anything is. The quantum world is like a daycare center: kids are zip-zapping all over the place. In contrast to classical physics, quantum physics does not allow you to locate a particle and measure its momentum to arbitrary accuracy, no matter how much you refine your instruments. More precisely, Heisenberg tells us that the uncertainty in a particle’s position multiplied by the uncertainty in its momentum is equal to a fundamental constant, known as Planck’s constant.1 Less uncertainty in one leads to more uncertainty in the other. If you know an electron’s momentum accurately (less uncertainty in momentum), you won’t know where it is (more uncertainty in position). And vice versa: if you try to locate an electron, you end up not knowing how fast it is moving. Position and momentum are known as a complementary pair in quantum physics. Time and energy form another complementary pair. What this means is that if you narrow the time interval during which you observe a system, you won’t know its precise energy. And vice versa: if you know


Chapter 16

the energy of a happening precisely, you won’t know when it is happening. Here is the sound bite: this constant uncertainty leads to Hawking radiation from a black hole. The long version now follows. Two great advances

Let us distill the two great advances of 20th-century physics into two gold-plated equations, one for each advance, with an easy-to-remember “advertising slogan” to go with them. First, let us deal with quantum physics, and then turn to special relativity a short while later. The gold-plated equation of quantum mechanics Uncertainty principle: E ∼ h¯ /t Advertising slogan: “Accounting errors can be tolerated for a short time!” An accounting error of * E can be tolerated only for the short time h¯ /E . The larger the accounting error, the sooner it will be detected and set right. In contrast, a tiny accounting error might last for a long time. In this respect, the quantum world actually accords with the garden variety everyday world: a sure-fire embezzling scheme that might not be detected for a long time is to skim off a penny at a time.2 Students of quantum physics learn to deal with these fluctuating uncertainties. But what can these fluctuations in * Physicists use the Greek letter delta  (as in the Mississippi delta and in Delta

Airlines for example) to denote uncertainty. The uncertainty in energy E is given by a constant h¯ , known as Planck’s constant, divided by the uncertainty in time t. Planck’s constant provides a measure for quantum uncertainty.

Quantum world and Hawking radiation


energy over a short duration do? Actually, nothing much. Imagine having the students in a quantum mechanics exam calculate the behavior of two electrons in a box. They could calculate till they are blue in the face, but there will still be two electrons in the box, not one more, not one less. The other great advance is special relativity, with that infamous celebrity equation about energy and mass. The gold-plated equation of special relativity Energy and matter are interchangeable: E = mc 2 Advertising slogan: “Accounting errors can be turned into stuff!” Energy can be converted into mass and hence particles according to Einstein’s famous equation E = mc 2 . Our proverbial embezzler could turn an accounting error into a Lamborghini. But only in his dreams if the world is nonrelativistic. Two separate strange worlds, but not strange enough

As noted above, in a quantum world without relativity (a world governed by what is known, in the jargon, as nonrelativistic quantum physics), nothing much happens to the quantum fluctuations. The accounting errors get noticed after time t and are rectified. In a relativistic world without the quantum (governed by what is known as relativistic classical physics), also nothing much happens. Yes, an energy fluctuation could be converted into particles, but there is no energy fluctuation in the first place. We have talked about two fascinating worlds, both strangely remote from our everyday world (which is governed by


Chapter 16


Rocketship near lightspeed, no need for quantum mechanics

The marriage of quantum mechanics & special relativity


Classical physics

Slow moving electron scattering off a proton, no need for special relativity



The square of physics. The upper right corner shows the confluence of quantum mechanics and special relativity.

nonrelativistic classical physics). Indeed, each is bizarre in its own right,* and as such, has been dramatically described in popular physics book. To recap, at the beginning of the past century, physicists uncovered two bizarre worlds, the relativistic classical world and the nonrelativistic quantum world. Each is wonderfully strange in its own way, but not strange enough. The fun really began when physicists tried to combine the two. When Doctor Heisenberg met Professor Einstein

With both quantum mechanics and special relativity, something new could happen! Now, accounting errors abound, and they can be turned into stuff. When physicists combined quantum mechanics and special relativity, around the middle of the past century, an exciting new subject, known as quantum field theory, emerged. * While

the relativistic classical world is quite well understood in spite of such mind-bending happenings as time dilation, the nonrelativistic quantum world still represents a fog of mystery to physicists after almost a century.

Quantum world and Hawking radiation


With it came profound and novel concepts, one of which was nothingness. The importance of nothingness

In quantum field theory, a state of nothingness is known as the vacuum.3 But in quantum field theory, nothingness does not merely contain nothing; on the contrary, in some sense it contains everything. The vacuum is a roiling sea of quantum fluctuations, boiling with particles and their corresponding antiparticles, coming into existence from nothing and annihilating back into nothing after a short while. How short is determined by the energy of the particle-antiparticle pair in accordance with the uncertainty principle. More precisely, when an energy fluctuation in the vacuum E exceeds 2mc 2 , with m the electron’s mass, then it can produce an electron and an anti-electron (known as a positron). With quantum mechanics and special relativity combined, particles can magically appear! But this magic lasts for only a short time* t, before the carriage (aka the Lamborghini) turns into a pumpkin, so to speak. Poof, the electron and the positron vanish into thin air! Physicists say that the electron and the positron annihilate each other. Indeed, there is nothing special about the electron in this discussion. You see that is why physicists think of nothingness as a roiling sea of pairs of particles and antiparticles of every imaginable description, popping in and out of existence. The more massive the particle, the more ephemeral its existence will be. But now we could take this argument one step farther. Instead of starting with nothingness, let us set two electrons crashing into each other with a huge amount of energy, call * Of order 1/(2mc 2 ), as some readers might realize.


Chapter 16

it E, way more than 2mc 2 . Again, in the vicinity of the two colliding electrons, a quantum fluctuation can produce an electron and a positron. But now we don’t need an accounting error: plenty of dough is already in the energy account to turn into stuff. It is all legit. In contrast to our earlier story, there is no longer any restriction on the time duration of the pair: the energy needed can simply be taken out of E. The vacuum produces an electron-positron pair costing at least 2mc 2 , taking the energy needed out of the two colliding electrons, which end up with some energy less than E − 2mc 2 . Thus, with two energetic electrons, we could end up with three electrons and a positron. The process is known as pair production and is observed routinely in the lab. The marriage of quantum mechanics and special relativity led to quantum field theory

Indeed, as long as there is enough energy, nothing in the discussion says that the pair produced has to consist of an electron and a positron. It could be a monster particle some theorist dreamed up last night and its antiparticle. Two electrons colliding with enough energy could well produce some hitherto unknown particles. This explains, in a nutshell, why physicists are constantly clamoring for resources to build ever more energetic accelerators to collide particles* with, thus producing more particles. The hope is of course that among these produced particles, there might be some that nobody has ever seen before, thus resulting in a free trip to Stockholm. The marriage of quantum mechanics and special relativity gives birth to a marvelously beautiful subject—music

* In our story, I talk about colliding electrons. For technical reasons, it is easier to collide two protons, such as at the much celebrated Large Hadion Collider (LHC).

Quantum world and Hawking radiation


please!—known as quantum field theory.4 It exhibits qualitatively new physics found neither in quantum mechanics nor in special relativity. A case of the child being vastly more scintillating than the two parents! Hawking radiation

You see that when theoretical physicists combine quantum mechanics and special relativity, qualitatively new physics appears. You might also have noticed that gravity did not come into our discussion of quantum physics thus far. The crucial question was asked by Hawking. In all that talk about a quantum fluctuation in the vacuum, what if that fluctuation is in the vicinity of a black hole? What do we mean by vicinity? Recall our earlier discussion of the horizon of a black hole. Picture a quantum fluctuation near the horizon, producing a particle and its antiparticle. Due to the uncertainty principle, we can’t be sure whether both are inside the horizon, both are outside the horizon, or one is outside but the other is inside the horizon. Of these four logical possibilities, the last two are specially interesting. (Notice that I said “four” and “two” in the preceding sentence.) To be specific, suppose the antiparticle is inside and falls to its doom, while the particle outside the horizon escapes. (Sounds like the ending of some adventure movie, doesn’t it?) An observer far away from the black hole sees the particle coming from the black hole and concludes that the black hole is radiating particles. Equally well, we could have the particle being inside the horizon and falling to its doom while the antiparticle escapes. The observer far away would actually see the black hole radiating equal* streams of particles and antiparticles, known as Hawking radiation. * Indeed, the black hole couldn’t care less which one we call, for historical reasons, particle, with the other known as the antiparticle.


Chapter 16

A quick summary of how Hawking radiation arises. If we are nowhere near a high energy collider, the particleantiparticle pairs produced by quantum fluctuations could last only for a short time. There are no colliding particles around for us to take energy from. By the uncertainty principle, the whole process can last only briefly. But if we are near the horizon of a black hole, we could sometimes flush the particle, or the antiparticle, out of sight and out of mind. To continue our analogy, an accounting error could last and be turned into stuff if a slush fund is hidden in some dark corner of the bank where no inspector has ever ventured. Or, if an inspector does venture there, she is trapped and cannot escape to tell the tale. You might wonder about the conservation of energy in the universe as a whole. Indeed, energy is conserved in the Hawking process. The black hole loses mass equal to the sum total of the mass and energy carried away in Hawking radiation, in accordance with Einstein’s relation E = mc 2 between mass and energy.

17 Gravitons and the nature of gravity The marriage of quantum mechanics and general relativity will lead to quantum gravity (we hope)

Thus far, our discussion of gravity has been based entirely on classical physics. Even Hawking radiation is, strictly speaking, based on a classical understanding of gravity. Some readers may be confused by this important point, since, in our telling of the Hawking story, we kept talking about quantum fluctuations producing particles and antiparticles. But note that these are quantum fluctuations in the field responsible for the particles and antiparticles (for example, the electron field), not quantum fluctuations in the gravitational field. Gravity’s job, so to speak, is “merely” to curve spacetime, and classical gravity is perfectly up to the task. The subject relevant to Hawking radiation is known as quantum field theory in curved spacetime. In contrast, in a true quantum theory of gravity, spacetime would be not only curved but also fluctuating like crazy. The gravitational field—namely, curved spacetime in Einstein’s theory—would itself be quantized. Thus, to “complete” our understanding of physics as we now know it, we are obliged to marry quantum mechanics and general relativity. The result that physicists have longed for would be a theory of quantum gravity, in which curved spacetime is constantly fluctuating. There—you now


Chapter 17

have a hint why a complete theory of quantum gravity is so devilishly elusive: we simply can’t make sense of wildly fluctuating time and space, whatever that means.1 Enter the graviton

Let us now go back to gravity waves, but first, we review the more familiar case of electromagnetic waves. In classical physics, a light wave is simply a wave of electromagnetic energy. In quantum physics, however, energy comes in packaged units. When we examine a light wave more closely, we see that the wave actually consists of a huge number of tiny packets of electromagnetic energy called photons (as was already mentioned in chapter 3.) The photon2 is the fundamental particle of light. The situation reminds me of those nature films with aerial shots of migrating herds of wildebeests. From a distance, we see a dark brown tide surging forward. As the lens zooms in, we see the tide differentiating into individual wildebeests thundering along. Similarly, as we zoom in and examine Nature more closely, we see what classical physicists took to be a wave of light differentiating into individual photons cruising along.3 In the same way, at the quantum level, a gravity wave consists of packets of gravitational energy called, appropriately enough, gravitons.* A swarm of gravitons

Classical physicists speak of massive objects responding to the gravitational fields generated by one another. To a * Physicists have only recently detected gravity waves, so they have certainly not seen a graviton. Indeed, to the extent that the future is foreseeable, experimentalists see no prospect of ever detecting individual gravitons. Nevertheless, as much as they believe in quantum physics, theorists believe in the graviton.

Gravitons and gravity


quantum physicist, the gravitational field consists of a swarm of gravitons. A massive object generating a gravitational field is actually emitting and absorbing these teensy-teensy bits of gravitational energy. Thus, in quantum physics, two massive objects interact gravitationally by exchanging gravitons. Similarly, two electric charges interact by exchanging photons. You could say that we are literally swimming in a swarm of gravitons generated by the earth. Ceaseless begetting leads to no end of trouble

I promised, way back in chapter 3, to tell you about a huge difference between gravity and electromagnetism, a difference that causes theoretical physicists no end of trouble. The seed for this difference is already sown by Einstein’s theory of special relativity, which states that mass and energy are the same. Consider a massive object, such as a star. The mass generates a gravitational field around it, according to Newton and Faraday. But a field contains energy. That’s fine by Newton and Faraday. But no, Einstein said that energy is the same as mass. Therefore, if mass could generate a gravitational field, then so can energy. The energy in the gravitational field in turn generates a gravitational field. A gravitational field begets another gravitational field. The process continues with no end in sight: a process described mathematically as an infinite series. It is this ceaseless begetting that could cause spacetime to literally curl up on itself,* forming a black hole, for example. Contrast the gravitational field with the electric field. An electric charge generates a electric field. The electric field * Not

so different from the water wave at the beach curling up on itself, as mentioned in chapter 4.


Chapter 17

carries energy, but not charge. It does not generate another electric field. The process ends. An electric field does not beget another electric field.* As mentioned earlier, electromagnetism is said to be linear in physics jargon, and hence, in some sense, is considered “trivial.” In contrast, gravity is highly nonlinear and a terror to deal with. For example, using traditional mathematics (by this I mean “analytic methods” in the jargon, that is, using pencil and paper), we would have no hope of computing the gravitational wave generated in the final throes of two black holes merging. Computers with enormous computing power were used to produce the theoretical curves4 to compare the detected signal with, as was actually needed for LIGO and was mentioned in chapter 9. Note two important points. First, the ceaseless begetting already arises in classical Einstein gravity, even before we try to quantize gravity. Second, this difficulty with nonlinearity is technical, not conceptual. It reflects our inability to calculate using analytic methods. Our quantum crank does not appear to work for gravity

Here I pause briefly to clarify a potential point of confusion for many readers, who might have read that physicists often speak of quantizing this or that theory. Indeed, the word “quantize” used as an active verb means to change a classical theory into a quantum theory. Thus, when we quantize Newton’s classical mechanics, we obtain quantum mechanics, and when we quantize Maxwell’s classical electrodynamics, we obtain quantum electrodynamics, and so on and so forth. By now, quantization consists of a procedure taught * This

statement is true in classical physics. In the quantum world, an electric field can generate another electric field, but under normal circumstances, the effect is weak.

Gravitons and gravity


to students: it is a crank that physicists turn to change any classical theory into a quantum theory. But there is no guarantee that the resulting quantum theory will make sense, or more technically, will behave “nicely.” When we put Einstein gravity under the quantum crank and turn it, we produce a wild man of a theory. More precisely, in processes involving gravity, quantum fluctuations grow with energy, so that when we reach the so-called Planck energy, about 1019 GeV, the fluctuations get to be so big that they go totally out of control.5 That the trusted quantum crank does not work for gravity has been of course the Mother of all headaches for theoretical physics for the past eight decades or so. Readers with a long memory will recall that I introduced, way back in chapter 2, the humongous Planck number 1019 as a measure of how feeble gravity is. Yes, the Planck energy* is simply related to the Planck number, and again reflects how “out of place” gravity is compared to the other three interactions. By the way, after the detection of gravitational waves, some in the popular press thought that the discovery would help us understand quantum gravity. But this is a bit of a misunderstanding. The gravitational wave that came to us from 1.3 billion light years away is totally a classical wave. LIGO certainly did not detect individual gravitons. The ceaseless begetting we just talked about is at least partly responsible for this giant headache, but there are other theories6 with ceaseless begetting that we have mastered. A more serious problem might be our inadequate understanding of spacetime. It may be helpful to recall the history leading from the discovery of electromagnetic waves in 1886 to an * As another way to appreciate how huge the Planck energy is, note that the Large

Hadron Collider (LHC), the world’s most powerful accelerator, can reach an energy of about 104 GeV.


Chapter 17

understanding of quantum electrodynamics. Theoretical understanding cannot be dated precisely: it is not as if theoretical physicists did not understand quantum electrodynamics one day and woke up the next morning with a complete understanding. But for the sake of the discussion, let us pick 1950, 64 years after the detection of electromagnetic waves. By this naive “reasoning,” we might expect quantum gravidynamics7 in 2080. The analogy is clearly too miserable to be trusted at all, since quantum mechanics, finally formulated in its present form in 1926, was not even a dream in 1886. The standard view about the struggle to master quantum gravity is that we have the correct crank; we are just not turning it correctly. I would suggest an alternative possibility: a new structure will have to appear in physics before we can master quantum gravity. Some might say that the structure has already arrived in the guise of string theory, but it may be far more exciting for theoretical physics to enter into a truly revolutionary framework comparable in depth to quantum mechanics. Perhaps quantum mechanics will have to be modified or extended. Pushing our silly “analogy” further than it can bear, we might expect this around 2056, 40 (= 1926 − 1886) years after the detection of gravitational waves. The quantum dance of two massive objects

Consider two massive objects, say, you and the earth. The gravitons emitted by one massive object are absorbed by the other, and vice versa, as was noted earlier. This is how quantum physicists picture the gravitational boogie-woogie between two massive objects: as they move and shake it all about, they exchange gravitons. By the way, if you have heard of Feynman diagrams and wondered what

Gravitons and gravity


A Feynman diagram describing the exchange of a graviton (the wavy line) between two particles (the solid lines with arrows on them). You may think of this as a process occurring in spacetime, with time along the vertical axis, and space along the horizontal axis.

they were, an example would be a diagram depicting the process just described in English.8 The process repeats itself rapidly. This constant exchange of gravitons between the two objects produces the observed gravitational force. (Similarly, the constant exchange of photons between two charged particles produces the observed electromagnetic force.) I have likened this constant exchange of gravitons to the marriage brokers of old traveling between two parties, telling each the other’s intentions.9 Since the early days of physics, the notion of force has been among the most basic and the most mysterious. It was thus with considerable satisfaction that physicists finally understood the origin of force as being due to the quantum exchange of mediator particles, such as the graviton and the photon.

A moral imperative but not a practical necessity

Some readers may be justifiably confused at this point. “You told us earlier that, through the decades, physicists have failed to construct a well-behaved theory of quantum gravity, but now you say that the everyday phenomenon of gravity can be understood as due to the exchange of gravitons. What is going on?” In everyday gravity, for example, that almost but not quite fatal attraction between you and the earth, the gravitons emitted by the earth play nice: each graviton gets to


Chapter 17

you without messing around with the other gravitons. Similarly, the gravitons emitted by you get to the earth directly. In more technical language, the gravitons being exchanged between you and the earth go directly from one massive body to the other, without pausing to interact with other gravitons. The gravitons are said to propagate freely. It is when the gravitons party with each other that all hell breaks loose, so to speak, and quantum gravity as we know it goes totally haywire. In this context, we are saved by the absurd weakness of gravity that we talked about in chapter 2. As was explained, the interaction of the gravitational field with matter is extremely weak. The effect of the gravitons propagating between you and the earth interacting with each other produces only a tiny correction to Newton’s law of gravity. Thus, as you would suspect, Newton’s classical gravity suffices for almost all practical purposes, from building skyscrapers to putting up satellites. The desperate search for quantum gravity is not a practical necessity, but a “moral imperative.” Indeed, while some seekers after quantum gravity are ready to kill themselves over this massive failure of will, or at least to gnash their teeth and look grim, other physicists are not in the least bothered by the failure to quantize gravity.10

18 Mysterious messages from the dark side Big news: the universe has a dark side, completely surprising physicists. First, dark matter. Then, dark energy. This is the sound bite; the long version now follows. Dark matter

Don’t shoot for the stars; we already know what’s there. Shoot for the space in between because that’s where the real mystery lies. —V E R A R U B I N E X H O R T I N G Y O U N G P H Y S I C I S T S Imagine sitting at a playground watching your child happily riding on a merry-go-round, holding on tight, as instructed. Your attention wanders. Suddenly, you notice that the merry-go-round is spinning around much faster. You instinctively rush over, fearful that your child is going to fly off.1 This was more or less what the astronomers2 observed starting in the 1920s. A galaxy typically rotates, meaning that the zillions of stars that make up the galaxy move in unison, revolving around the center of the galaxy. Astronomers could measure the speed of the stars, thanks to the Doppler effect for light. The sound version of the Doppler effect is commonly noticed in everyday life: the pitch of the siren on an


Chapter 18

approaching ambulance and on a receding ambulance sound different. Similarly, the light emitted by an approaching star is blueshifted (its frequency is raised), while the light emitted by a receding star is redshifted (its frequency is lowered). The amount of the shift is proportional to the speed of the star. When astronomers examine the Doppler data on stellar motion in rotating galaxies, they react with much of the same horror experienced by the parents in my playground analogy. The stars are moving way too fast for their own good! Actually, Fritz Zwicky3 first suggested (and coined the term) “dark matter” in 1933 by observing the motion of galaxies in a cluster of galaxies, rather than the motion of stars in an individual galaxy. But the underlying principle is the same. The individual galaxies are moving much too fast, so that, unless a large amount of unseen matter is holding them back by means of gravitational attraction, they would fly away from the cluster. Observational techniques improved by the 1960s, so that Vera Rubin4 (1928–2016) and Kent Ford were able to measure the collective5 motions of stars in different regions of rotating galaxies and thus firmly established that galaxies were themselves suffused by this unseen dark matter. As explained in chapter 5, we require contact with the forces in everyday life. The children on the merry-go-round are told to hold on tight to the handrail. The stars do not have anything to hold onto, of course; instead they are kept from flying off into deep silent space by the gravitational attraction exerted on them by the zillions of other stars in the galaxy. It is a collective enterprise: the stars form a conglomerate known as a galaxy by virtue of their mutual gravitational attraction for one another. Notice that although the gravitational force decreases, according to Newton, like the square of the distance and so is minuscule on the galactic

Messages from the dark side


scale, the pull of the zillions of other stars in the galaxy really adds up and keeps the stars bound to the galaxy. This was the expectation. The data show that, yes, the gravitational pull of the other stars on any given star does add up, but the total is not quite enough. The galaxy should have fallen apart with all the stars flying off into deep space, each chasing after its own destiny rather than remaining part of the greater good. I have simplified the story slightly, but only slightly. Astronomers actually had data on how the speed of the stars as they revolve around the galactic center depends on their distances from the center, and this also disagreed with theoretical expectations.6 An important point: note that the dark matter story does not have anything to do with Einstein gravity as such. Newtonian gravity is completely adequate to account for motion on the galactic scale. Thus was born the notion of dark matter.7 Galaxies, including our very own Milky Way, must be suffused by a mysterious type of matter with quite a bit of mass. This unknown matter is called dark because it neither emits nor absorbs light. Clearly, it can’t emit light (otherwise, we would have seen it), and it can’t absorb light, since we can see the stars on the other side of the galaxy (after accounting for various observed interstellar clouds of dust particles). Notice that the universality of gravity, in contrast to electromagnetism, that we spoke of in chapter 8, is crucial here. Whatever dark matter is, while it is free not to have anything to do with light, it must listen to gravity, because gravity is just curved spacetime. The orthodox view is that dark matter consists of hitherto unknown elementary particles that do not interact with light. These are in fact very easy to introduce into the standard theory of particle physics; simply do not couple these particles to the electromagnetic field; that is, let them be


Chapter 18

electrically neutral. Thus began a tremendous effort to detect such particles in earth-bound laboratories. Lately, a bit of skepticism of this view has crept in, merely because, after years of intense search, nothing has been sighted. Even so, I still much prefer this view of dark matter to the one highly speculative view on the market. Back in 1983, the Israeli physicist Mordehai Milgrom proposed modifying Newton’s laws to account for the rotation of galaxies.8 You might think that after this many centuries, Newtonian physics has been thoroughly tested and verified. Yes, but the acceleration experienced by stars in rotating galaxies is much smaller than any that has been measured on earth and in the solar system. I remarked in chapter 14 that Einstein’s theory is extremely tight: it cannot be easily modified without messing up the various celebrated tests (such as the bending of light) that the theory has passed with flying colors, not to mention our daily use of GPS, which has to take into account corrections due to Einstein gravity. In contrast, Newtonian laws are quite loose. You feel like modifying Newtonian physics? Go ahead, but make sure that the effects of your modification are so tiny so that they show up only on galactic scales. I personally find such ad hoc modification of Newton’s laws contrived and distasteful. In becoming theoretical physicists, students are told to keep an open mind and not to dismiss unorthodox suggestions (provided that they are consistent with known facts, of course) out of hand. But still, one’s mind should not be so open that it leaks, possibly leaving an empty mind. Here I might mention another advantage of the action principle over the equation of motion approach. It is considerably more difficult to modify the action for Newtonian physics than to modify the equations of motion for it.

Messages from the dark side


Dark energy

When Einstein triumphantly completed his theory of gravity, he missed predicting that the universe would expand, as was discovered later by Vesto Slipher, Milton Humason, Edwin Hubble, and others. With the Einstein-Hilbert action given in chapter 12, if we fill the universe with known particles (that is, atoms and molecules, electrons, protons, photons, and what not) it will expand. We simply take the metric describing an expanding universe given in the appendix, plug it into the equations resulting from the action, and solve for the behavior of the function a(t) measuring the size of the universe. In fact, now, more than a century after Einstein gravity was proposed, an advanced undergrad would be capable of doing this calculation. He or she would find that a(t) increases, but at an ever decreasing rate.9 In other words, the universe expands but decelerates in its expansion. This can be understood heuristically: the known particles, in their uncoordinated motion, exert a pressure outward, leading to expansion, but gravitational attraction between the particles tends to pull everybody back, and hence slows down the expansion. The big surprise was that observation of distant supernova in the 1990s indicated that the expansion of the universe was actually speeding up rather than slowing down. Contrary to the impression given by some popular media, this effect can be readily accommodated in Einstein gravity. Recall that I explained in chapter 13 that in Einstein gravity, the action has to be composed of geometric invariants, and that besides the curvature, the volume of spacetime is also clearly an invariant. We are free to add to the Einstein-Hilbert action the so-called cosmological constant term, consisting of the volume of spacetime multiplied by a constant . Incidentally, Einstein was quite aware of the possibility of


Chapter 18

including this term (and I would be exceedingly surprised if Hilbert, being a mathematician, did not know about it). Including the cosmological term leads to an additional term in the equation governing the expansion of the universe. Again, our bright undergrad could readily show that with the appropriate choice of , he or she could make the universe expand at an ever-increasing rate. This proverbial undergrad10 would also notice that the cosmological constant term, as its name suggests, has an effect only on cosmological distance scales. Thus, it would not affect any of our exceedingly successful calculations involving gravity from the solar system scale all the way up to the galactic scale. For completeness, I should mention that other explanations for the accelerating expansion of the universe have been floated.11 But since the cosmological constant is ready made and available, I believe that most theoretical physicists prefer, for the sake of simplicity, to use the cosmological constant rather than to have to invent some other far-fromcompelling constructs. The cosmological constant  has a rather convoluted history12 in theoretical physics. As I mentioned, its existence was known to be possible since the time of Einstein. But since its only effect was on the expansion of the universe, for many decades  was postulated to be mathematically zero. Unfortunately, while many theoretical physicists tried, nobody managed to come up with a convincing reason why that should be so. Now that observational data has indicated that it is extremely small13 but not zero, the mystery has only deepened. Here and in chapter 14, I extolled the virtue of Einstein gravity as being an impressively tight theory. But in the present context, one could also say that Einstein gravity is too loose: it allows for two fundamental constants, Newton’s constant G and the cosmological constant . Perhaps the situation echoes a pseudo-philosophical utterance of

Messages from the dark side


Niels Bohr, that the opposite of a great truth is also a great truth. Annoyingly, the cosmological constant  only reveals itself on cosmological scales. Leaving these deep issues aside for the moment, I can dispose of a triviality that has confused the lay public a bit for no good reason. The equation of motion for the gravitational field in Einstein’s theory has the schematic form (variation of the gravitational field in spacetime) = (distribution of energy in spacetime) When we include in the action a term equal to the volume of spacetime multiplied by a constant  and extremize the action to obtain the equation of motion, then of course an extra term will pop up in the equation of motion. This term is usually included in the distribution of energy on the right hand side of the equation. Indeed, that is the origin of the term “dark energy,” a form of energy that can’t be seen except in the expansion of the universe. But as any high school student could tell you, the equation a = b + c can perfectly well also be written as a − c = b. Thus, some people with nothing better to do prefer to move the dark energy term from the right side of Einstein’s equation to the left side, and regard it as some kind of force other than gravity. A few even go so far as to call it antigravity, a term that is unenlightening at best and misleading at worst. Is it a new form of energy? Is it a new force? Somehow, this debate, which raged for a while in the popular media (or blogosphere, or whatever you call it) barely stirred a ripple in the theoretical physics community. Dear reader, you can understand why. Whether you put a term on the right or on the left of an equation does not change the physics one iota. The situation reminds me of creative corporate accounting as humorously portrayed: depending on whether you put


Chapter 18

a tax write-off on the left or right side of the ledger, you could either make a huge profit or sustain a bad loss. Here then is another advantage of the action formulation of physics versus the equation of motion formulation. There is no left or right side to the action; it is just a sum of a bunch of terms, the fewer the better, according to theoretical physicists hellbent on unification. You tell the universe what the deal is (that is, what the action is), and the universe will do what it takes to find the best possible deal. The concordance model

Throughout history, our conception of the cosmos has changed a great deal. Currently, the consensus is known as the CDM model, also referred to as the concordance model. You already know what  stands for, and CDM stands for cold dark matter, cold meaning that the postulated dark matter particles are moving around much slower than the speed of light. Current measurements indicate that, of the total14 energy and mass of the universe, dark energy contributes 68%, dark matter 27%, and ordinary matter (which you and I are made of) only 5%, as was already mentioned way back in chapter 1. It was really quite a shock: until recent times, this enormous dark side of the universe was largely unsuspected, although hints of it existed.15 The long history of our growing understanding of the universe has been a humbling process, a steady erosion of anthropocentrism and geocentrism. The ancient Chinese thought that their Middle Kingdom occupied the center of the world. The Greek Anaxagoras was ridiculed for suggesting that the sun may be as large as the Peloponnesus. Eventually, Copernicus instigated a revolution by suggesting that the earth is not at the center of the world. Yet the belief that the sun was at the center of the galaxy persisted until 1915,

Messages from the dark side


when Harlow Shapley determined that we are out near the edge. For years afterward, astronomers believed that ours was the only galaxy, thinking that what we now recognize as other galaxies were merely clouds of luminous gas in our galaxy. But just as we finally came to recognize ourselves as passengers on a smallish planet circling an insignificant star lost somewhere near the edge of an ordinary-looking galaxy drifting inside a relatively sparse cluster of galaxies in some region of the universe resembling any other region, we learn that the matter out of which you and I and stars and galaxies are made may not even be the main component of the universe. How humble do we have to be?

19 A new window to the cosmos To know the universe better

The excitement over the detection of gravity waves stems from their promise to open up another window to the world out there. For eons, our knowledge of the cosmos has come to us in the form of light. And then Maxwell and Hertz discovered light is only one form of electromagnetic waves. With the development of detectors for the other forms of electromagnetic waves, microwave astronomy, radio astronomy, infrared astronomy, ultraviolet astronomy, X-ray astronomy, and gamma-ray astronomy were born one after another. After all, astronomical bodies have no reason to radiate electromagnetic waves only in those frequencies detectable by certain creatures on a particular speck of a planet. The universe is humming across the entire electromagnetic spectrum. It is as if we had been peering at the cosmos through a narrow window and all of a sudden, the curtain was pulled back to reveal that the window was in fact quite wide. Still, wide as the electromagnetic window is, we are now in the wonderful situation that another window is suddenly open to us. The year 2016 heralded the dawning of a fabulous new epoch in our age-old exploration of the cosmos. I am reminded of those sound and light shows at tourist attractions.

New window to the cosmos


The universe is putting on a sound and light show also— more accurately, a gravitational wave and electromagnetic wave show. But until 2016, it had been like a silent movie. Suddenly, the sound was switched on. To develop gravity wave astronomy would be akin to our collectively growing a second set of eyes. New types of signals will be received.1 An exciting prospect is that gravitational wave astronomy might give us information about the dark side otherwise seemingly destined to be forever hidden from us. The coming of gravity wave astronomy will reveal the universe as we have never seen it before. Detectors more sensitive than LIGO are planned. Indeed, they have been planned for a long time, since people were at one point growing increasingly pessimistic that LIGO would be able to see anything. In particular, the European Space Agency has proposed the Laser Interferometer Space Antenna (LISA) and the Evolved Laser Interferometer Space Antenna (eLISA). Three spacecraft, one each at the tip of an equilateral triangle whose sides measure millions of kilometers in length, will fly in a near-earth orbit around the sun. The distances between the three spacecraft are to be accurately measured by laser interferometry in order to detect passing gravitational waves. Some fascinating proposals have been aired. One attractive possibility is to launch two satellites, each carrying an atomic clock linked by laser light.2 The idea is that since gravity affects the flow of time in Einstein’s theory, a passing gravitational wave would cause the highly accurate clocks to tick at slightly different rates. When and if eLISA flies, according to design specifications, hundreds of events will be expected on the very first day it becomes operational. As an enthusiast exclaimed, “physics doesn’t get much better than this!” Yes, better living through gravitational waves!


Chapter 19

A child asks a panel of experts on gravity

A child asks: Why do we all3 fall down? A panel of experts replies. Aristotle: Well, the earth is the natural home for rocks and men. Rocks fall because they want to go home. As rocks fall, they go faster and faster, much as recalcitrant rental horses will break into a gallop as they approach the stable, dragging the terrified tourists with them. When you jump out of a jungle gym, you are expressing your inner desire to go home. Newton: That Aristotle fellow is full of baloney. I have interviewed plenty of rocks, and they never said anything about going home. Rocks and apples fall because they and the earth and every other object in the universe exert an attractive force on one another. By the way, as you jump out of a jungle gym, you are actually also pulling the earth up. Einstein: Newton is so right, but there is more to the story. The force Newton talks about results from the curvature of space and time, which, by the way, are just two aspects of spacetime. The earth warps spacetime around the jungle gym, so that when you jump, you are actually looking for the best deal in town, seeking to extremize your action. The quantum theorist of gravity: Einstein somehow finds the quantum world distasteful, even though he was one of the founders of quantum physics. If he weren’t so stubborn, he might have realized that his curved spacetime is due to gazillions of gravitons sashaying around. When you jump out of the jungle gym, gravitons zing back and forth like crazy between you and the earth. Leaving Aristotle aside—I really don’t think what he said is right—the other three are all truth sayers.4

Appendix What does curved spacetime mean? I know full well that many otherwise intelligent persons find math frightening, but math is an indispensable language for describing abstract concepts like curved and spacetime. As I said in the preface, this book is meant to be slightly above a popular physics book and somewhat below a physics textbook. The level of math needed here is comparable to that of an introductory calculus course. Since you are holding this book in your hands, I can safely bet that you are vastly more sophisticated than the proverbial guy and gal in the street. What I can promise you is that, if you have enough patience to get through this appendix, you will understand what curved spacetime is about. However, you can also enjoy reading this book without slugging through the appendix, if that is not your thing. I will go extremely, perhaps excruciatingly, slowly. One step at a time. First, flat space. Then curved space. Next, flat spacetime. Finally, curved spacetime. Walk before you fly and all that. The cast consists of five great men: René Descartes, Pythagoras (he of the single name), Bernhard Riemann (1826–1866), Hermann Minkowski, and, of course, Albert Einstein.

Flat space The story is that Descartes, whom we already met in chapter 4 in connection with water waves, was lying in bed when he realized that he could locate a buzzing fly with three numbers. Thus were Cartesian coordinates1 born. Instead of the 3-dimensional space the Cartesian fly was buzzing around in, let us keep it simple and think about 2-dimensional space. Consider a point specified by the coordinates (x, y). See figure A.1. A nearby point is then specified by (x + dx, y + dy). In math speak, dx (known as the differential of x) simply means a very small2 change in x. For our purposes here, dx should be thought of as one symbol, not d multiplied by x. (For example, it may be that x = 1.78 centimeters, and dx = 0.001 centimeter.) Similarly, dy means a very small change in y. In other words, x + dx is a number very close to x, and y + dy is a number very close to y. What is the distance between the two neighboring points? Pythagoras knows the answer.3 The distance ds is given by ds 2 = dx 2 + dy 2



( x+ dx , y +dy ) ds dy (x ,y ) dx Figure A.1. Two nearby points have Cartesian coordinates (x, y) and (x + dx, y + dy) respectively. Pythagoras tells us how to determine the distance ds between the two points. In the text, dx and dy are described as very small, infinitesimal in fact. They are blown up here for clarity. Since dx and dy are both very small, evidently ds is also very small. This formula for ds characterizes the flat 2-dimensional space known as the plane. All fine and dandy. But how do we describe a curved space? How about writing ds 2 = dx 2 + ( f dy)2 , with f some number not equal to 1? Nope, this is still not curved. We have effectively denoted the distance in the y-direction by f dy instead of dy. This merely amounts to something like measuring distance in the x-direction using some metal bar provided by some French revolutionary and distance in the y-direction using some English king’s foot. We need to be more clever, what the French call “malin” (which translates to “tricky” but not exactly). Onward to curved space!

From flat space to curved surface Let us stick to 2-dimensional space, namely, surfaces. The curved surface most familiar from everyday life is the sphere. See figure A.2. Set the radius of the sphere to 1. Otherwise, we would have the radius littering our formulas. In other words, we measure length and distance in terms of the radius of the sphere. It is also convenient to think of this mathematical sphere as the globe we live on, so that I can use ready-made words, like “latitude,” “equator,” and “north pole.” Denote latitude and longitude by the Greek letters θ and ϕ, respectively. Picture a point on the sphere, and call it Paris, just for ease of reference. Denote the latitude and longitude of Paris by θ P and ϕ P , respectively.4 Consider a place with the same longitude as Paris but a slightly different latitude, namely, θ P + dθ . The distance between this place and Paris is then given by5 dθ . This is because the lines of longitude define “great circles” of radius 1. This place and Paris both lie on a circle of radius 1. In contrast, lines of fixed latitude do not define great circles, except for the equator. In other words, consider a place with the same latitude as Paris but a slightly different longitude, namely, ϕ P + dϕ. The distance between this place and Paris is definitely not given by dϕ. What I just said is that the distance ds between a point with coordinates (θ, ϕ) and a neighboring point with coordinates (θ, ϕ + dϕ) is not equal to simply dϕ, but

Curved spacetime


North pole

Figure A.2. The distance between two nearby points with the same latitude but with longitudes differing slightly by dϕ is given by f (θ )dϕ. The function f (θ ) is equal to 1 at the equator, decreases steadily as we move north, and vanishes at the north pole. rather it is equal to f (θ )dϕ. The distance depends on the latitude θ as indicated by the function f (θ ). This function is equal to 1 at the equator, but is considerably less than 1 at the latitude of Paris. See figure A.2. As we go north, this function keeps on decreasing, until it vanishes at the north pole. (Why? Think about this for a moment. It is because longitude ceases to be defined at the north pole.) Thus, on a sphere, the distance ds between two neighboring points, one with coordinates (θ, ϕ) and the other with coordinates (θ + dθ, ϕ + dϕ) is given by ds 2 = dθ 2 + ( f (θ ) dϕ)2 The key point is that f (θ ) is not merely a number, but a function6 of θ , that is, a number that varies depending on θ . You can think of this as a generalization of the Pythagorean formula ds 2 = dθ 2 + dϕ 2 . Aha, we’ve got it! From this example, we learned that to go from the flat plane, for which ds 2 = dx 2 + dy 2 , to a curved surface, we should write7 ds 2 = dx 2 + ( f (x) dy)2 , which of course can also be written as ds 2 = dx 2 + f (x)2 dy 2 , not ds 2 = dx 2 + f 2 dy 2 with f a constant. Enter Bernhard Riemann. He said, “now that we have inserted a function in front of dy 2 , why not insert a function in front of dx 2 also? In fact, why not include dx dy and insert a function in front of him also? These three functions could all depend on



both x and y!” So, here is Bernie’s proposal: ds 2 = a(x, y)dx 2 + b(x, y)dx dy + c(x, y)dy 2 You specify the three8 functions a, b, c, and each one of your choices characterizes a curved surface known as a Riemann surface. From curved surface to curved space And thus Riemann started a branch of mathematics known as Riemannian geometry. To summarize, we’ve done flat and curved space. But all of this is in two dimensions, you say. How about higher dimensional spaces? Easy! Just add another coordinate z. Flat 3-dimensional space is then described by ds 2 = dx 2 + dy 2 + dz2 , generalizing Pythagoras. How about a curved 3-dimensional space? Not that hard either. Instead of three functions a, b, c, we now have six functions, each a function of x, y, z. (We need three more functions, because we now have not only dz2 , but also dx dz and dy dz.) If you picture x as describing east and west, y as describing north and south, then z describes up and down. That wasn’t so hard, was it? Good, we now move on to spacetime. From flat space to flat spacetime Enter Minkowski, who proposed that we regard time, denoted by t in physics, as the fourth coordinate, after x, y, and z. (This was after Einstein established special relativity; by the way, Einstein said that this proposal never occurred to him.) How would you do it? What would be ds 2 in spacetime? Dear reader, think for a moment before reading on. Since we went from ds 2 = dx 2 + dy 2 to ds 2 = dx 2 + dy 2 + dz2 , our first guess might be ds 2 = dx 2 + dy 2 + dz2 + dt 2 . But this is wrong for two important reasons. First, how would time differ from space? You and I (and they, too) all know that we can go east and west, north and south, and up and down as we please, but we cannot go back to when we were young. We must somehow distinguish time from space in our equation! The solution proposed by physicists would make nonphysicists laugh. Instead of adding dt 2 , how about subtracting dt 2 instead? It seems so naive and childish, but it turns out to be right. Nature actually works that way. Amazing! So try ds 2 = dx 2 + dy 2 + dz2 − dt 2 . This is still not quite right. I alluded to two reasons just now. The second reason is that, as any school child could tell you, you cannot subtract one second squared from one centimeter squared. Makes no sense. We have to convert an interval of time dt into a length segment by multiplying it by the speed of light c, namely, c dt.*

* Well, c is so many zillions of centimeter per second, so, with dt say 0.001 second, the product c dt would come out as some number of centimeters. No mystery here.

Curved spacetime


Now we are cooking with gas: write down ds 2 = dx 2 + dy 2 + dz2 − (c dt)2 Note the minus sign and the appearance of c. This describes what is known as Minkowskian flat spacetime. I have made it all look easy by not mentioning various delicacies. Let me emphasize one important point, however. This would not have made any sense if the speed of light c were not a fundamental constant in the universe.

You are ready to curve spacetime! Now that Minkowski has gone from space to spacetime, Einstein is ready to curve spacetime. Dear reader, since you know how to go from flat space to curved curve, you may be able to go from flat spacetime to curved spacetime. Try it! Introduce a function of x, y, z, insert it in front of c dt, and write ds 2 = dx 2 + dy 2 + dz2 − ( f (x, y, z) c dt)2 Yes, it is that easy. With the appropriate9 f , Einstein was able to obtain Newtonian gravity as a special case, predict that gravity affects the flow of time, and calculate the precession of the orbits of Mercury. Easy, no? Now that you get the idea, you can write down all sorts of curved spacetimes, for example, an expanding universe. Instead of sticking a function of space in front of dt 2 as we just did, we could stick a function of time in front of dx 2 + dy 2 + dz2 : ds 2 = (a(t))2 (dx 2 + dy 2 + dz2 ) − c 2 dt 2 Notice that while this spacetime is curved, the space contained in it is flat. At time t, the square of the distance between a point with coordinates (x, y, z) and a neighboring point with coordinates (x + dx, y + dy, z + dz) is given by (a(t))2 (dx 2 + dy 2 + dz2 ), namely, what Pythagoras said it is, multiplied by a factor a(t). Thus, if a(t) increases with time, this spacetime describes an expanding universe. Cosmological observations indicate that the universe we live in is fairly well described by this curved spacetime if a(t) is an exponentially growing function of time. These two curved spacetimes are among the most important in Einstein gravity. See how easy10 it is to learn Einstein gravity!

Curved spacetimes in general Not that hard, is it? It does sound a bit too easy. Indeed, you might wonder why we could get away with such minimal modifications of flat Minkowski spacetime. The general 3-dimensional curved space already requires six functions to describe. In contrast, in each of these two curved spacetimes described here, only one function was needed. That is because these two curved spacetimes are highly symmetric.



You are right to wonder. These two spacetimes have particularly simple forms. In general, 4-dimensional curved spacetime requires ten functions to describe it. Dear reader, can you figure out why ten before reading on? Yes, indeed. In addition to four functions multiplying dx 2 , dy 2 , dz2 , and dt 2 , there is a function multiplying each of the six combinations dx dy, dx dz, dx dt, dy dz, dy dt, and dz dt. Hence ten functions altogether. Each of these functions could depend on x, y, z, t. Elsewhere, I have spoken quite a bit11 about Nature’s kindness to theoretical physicists. In my career in theoretical physics, I have often been struck by how Nature keeps it simple at the fundamental level, so that physicists would be able to figure Her out. We just came across one of numerous examples: the expanding universe we live in can be described by one single function a(t) depending on one single variable t. A more compact notation Theoretical physicists are an impressively lazy lot, and so they easily tire of writing ten functions together with ten quantities, such as dz2 and dy dt. With the help of their mathematician friends, they came up with a marvelous invention known as index notation. Instead of writing x, y, z, they write x 1 , x 2 , x 3 . The letter x, which denoted one of the three spatial coordinates, is now drafted to do triple duty, representing all three spatial coordinates. In other words, x 1 = x, x 2 = y, x 3 = z. (You see that the index notation frees us from the trivial constraint that the English alphabet contains only 26 letters. If you like, you could talk about 27-dimensional space by simply writing x 1 , x 2 , · · · ,x 26 , x 27 .) Even better, at this point, you can include the time coordinate t also: simply call it12 x 0 . That’s right, the letter x is now doing quadruple duty: with the nifty index notation, it can represent time as well as space. So, instead of t, x, y, z, we now write x 0 , x 1 , x 2 , x 3 , with x 0 = t, x 1 = x, x 2 = y, x 3 = z. Collectively, these four coordinates, x 0 , x 1 , x 2 , x 3 , can be denoted more compactly by x μ , where the index* μ takes on the values 0, 1, 2, 3. I said that theoretical physicists (and mathematicians) got tired of writing quantities such as dz2 and dy dt. Now they can write simply dx μ dx ν . As μ and ν separately take on values 0, 1, 2, 3, the expression dx μ dx ν ranges over all ten of these quantities. For example, dx 3 dx 3 = dz2 , and dx 2 dx 0 = dy dt. Using this notation (merely notation: neither physics nor math, nothing profound at all, just bookkeeping), we can then write the most general curved spacetime concisely as ds 2 = g μν (x)dx μ dx ν with the indices μ and ν ranging over 0, 1, 2, 3. It is also implied† that the terms are to be summed over. In other words, g μν (x)dx μ dx ν is shorthand for13 g 00 (x)(dx 0 )2 + g 11 (x)(dx 1 )2 + · · · + 2g 01 (x)dx 0 dx 1 + 2g 02 (x)dx 0 dx 2 + · · · + 2g 23 (x)dx 2 dx 3 . Note that, instead of stupidly inventing names for each of the

* The Greek letters μ and ν (to be used below) correspond to the Latin letters m

and n, respectively, and are traditionally used by physicists in this context. † This notation is known as the Einstein repeated index summation, said by some to be one of Einstein’s greatest hits.

Curved spacetime


ten functions that appear in front of dt 2 , dt dx, dt dy, · · · , dy dz, dz2 , we simply denote them collectively by g μν (x). The ten functions g μν (x) are known collectively as the spacetime metric: as the jargon suggests, they measure spacetime. Let me forestall a potential confusion here. The notation g μν (x) is shorthand for g μν (x 0 , x 1 , x 2 , x 3 ). The letter x is used to denote x 0 , x 1 , x 2 , x 3 collectively. In other words, each of the ten functions g μν (x) is a function of t, x, y, z, that is, a function of spacetime. (Indeed, it would be absurd to say that they are functions of the coordinate x only. What the heck is so special about the x direction?) You may be impatient to hear about gravity waves. We are almost there. First, to make sure that we understand Einstein’s notation, let us see how flat Minkowski spacetime is a special case of the general curved spacetime described here. Flat Minkowski spacetime corresponds to a particularly simple form of g μν (x). The ten functions actually are not functions, just numbers, and all but four are equal to 0. These four are g 00 = −1, g 11 = +1, g 22 = +1, g 33 = +1. In other words, ds 2 = −(dx 0 )2 + (dx 1 )2 + (dx 2 )2 + (dx 3 )2 , which I trust you recognize as flat Minkowski spacetime written using indices rather than using x, y, z, t. Since the earth’s gravitational field is so weak, most of the time we are hanging out in a spacetime that is very close to flat Minkowski spacetime. Thus, flat Minkowski spacetime is by far the most important spacetime to know and love. Not surprisingly, then, theoretical physicists traditionally assign the Minkowski metric, that is, the metric I just described, a special symbol, namely, ημν , using the Greek letter η (pronounced “eta”). Nothing profound here: we merely define ημν by specifying that the only nonzero components of η are η00 = −1, η11 = +1, η22 = +1, η33 = +1. In other words, we can write the flat spacetime we wrote as ds 2 = −(dx 0 )2 + (dx 1 )2 + (dx 2 )2 + (dx 3 )2 more compactly as ds 2 = ημν dx μ dx ν I emphasize that all of this is just a compact notation to keep track of the large number of quantities needed to describe spacetime. Learning a notation is a bit like learning a language, speaking loosely. In the present context, you need it to know what physicists are talking about.

How to describe a gravity wave Finally we are ready for a serious discussion of gravity waves! We simply modify flat spacetime by a tiny bit. We invite ourselves to consider a curved spacetime described by   ds 2 = ημν + h μν (x) dx μ dx ν In other words, we have merely added to the bunch of 1s and 0s in ημν some functions h μν (x), which we are going to regard as small compared to 1. The metric of this (slightly) curved spacetime is given by g μν (x) = ημν + h μν (x). Does this ring a bell? Yes? Indeed, that’s why I spent some time in chapter 4 talking about water waves on the surface of a placid lake. Without any wind, the surface of the lake is flat, and the



depth of the water is given by g (t, x, y) = 1. When a breeze whips up some waves, g (t, x, y) = 1 + h(t, x, y). The surface undulates in space and time. If the amplitude of the wave is small, then we treat h(t, x, y) as small compared to 1. As I mentioned earlier, in this case the nasty equations for fluid dynamics simplify to an equation that undergrads can solve. Surely it has not escaped your notice that the form of the metric of spacetime g μν (x) = ημν + h μν (x) is structurally14 the same as g (t, x, y) = 1 + h(t, x, y). Einstein gave us a set of equations* for determining g μν . When we substitute g μν = ημν + h μν into these equations, they simplify enormously, leaving us with equations for determining h μν that are only marginally more complicated than the equations for electromagnetic waves. Needless to say, this is merely a simplified first description. In real life, the spacetime around two black holes merging could hardly be taken to be flat Minkowski spacetime. But once the gravity wave leaves this region, then the description given should be more or less adequate, except for the fact that the universe has expanded some during the 1 billion years or so that the waves took to reach us. From metric to curvature Since the metric determines the distance between any two points, once the metric of a curved space (or spacetime) is given, we can deduce all that we need to know, such as the curvature of that space. Here is an operational procedure a civilization of mites15 living on a curved surface would follow to determine how curved their world is. (Remember, they cannot go outside their surface to take a look, any more that we can go outside our universe to see whether it is curved.) Given a point P, find all the points that are located a small distance r away from P. This defines a circle of radius r around that point P. Moving around the circle and adding up the distances between points on the circle infinitesimally separated from each other gives the circumference of the circle. Divide the circumference by the radius r . If this ratio is equal to 2π  6.28 . . . in the limit r becomes very small, then the surface is flat. If not, then the surface is curved. In fact, Riemann saved us from having to do all this; given a metric, he found a formula for calculating what is now called the Riemann curvature tensor. So nowadays, any bright undergraduate would be able16 to calculate the curvature of the spacetimes described by the metrics specified earlier in this appendix. Out of the Riemann curvature tensor, a quantity called the scalar curvature and denoted by R can be obtained. Einstein’s action for gravity is simply the scalar curvature R of spacetime. See chapter 13. Another important geometrical quantity is the volume of an infinitesimal region of space or spacetime. As expected, this is determined by the metric and is written by √ physicists and mathematicians as g , with g a mathematical expression constructed from the metric. This quantity also appears in Einstein’s action as given in chapter 13.

* Since you know the action for Einstein gravity from chapter 13, you could in principle vary that action to obtain these equations.

Postscript While this book was going through production, it was announced that the Nobel Prize in Physics for 2017 had been awarded to Rainer Weiss, Barry C. Barish, and Kip S. Thorne for leading LIGO to its historic discovery.

Photos of the three Nobel prize winners side-by-side. © Molly Riley/AFP/Getty Images. From Getty Images / Photographer: Molly Riley / Collection: AFP.

On August 17, 2017, a little less than two years after the first detection of gravity wave, another burst of gravity wave was detected from the merger of two neutron stars. Since neutron stars, in contrast to black holes, do emit electromagnetic waves, the event, cataloged as GW170817, was also seen by various observatories tuned to different regions of the electromagnetic spectrum. The era of “multi-messenger astrophysics” has dawned. It has long been known theoretically that elements heavier than* iron (Fe:26) were synthesized in neutron star mergers. Some of these elements, such as silver (Ag:47), platinum (Pt:78), gold (Au:79), and uranium (U:92) have played, and continue to play, important roles in human affairs.

* The number after the scientific symbol indicates the number of protons in the corresponding atomic nucleus.

Notes Preface 1. Actually, it weighs less than the classic text by Misner, Thorne, and Wheeler: MTW weighs 5.6 pounds, significantly more than GNut’s paltry 4.6 pounds. 2. Except in passing. 3. Facsimiles of Einstein’s manuscript are available in The Road to Relativity, by H. Gutfreund and J. Renn, Princeton University Press, 2015. Prologue 1. 2. 3. 4.

To set the time scale, dinosaurs roamed about 0.24 billion years ago. See chapter 17. Hence the detection event, the first of its kind, is being cataloged as GW150914. This formula did not appear in Einstein’s original papers on special relativity. Einstein discovered it a few months later, and published it in a 2-page paper, writing L v2 K0 − K1 = 2 V 2

What! It doesn’t look like E = mc 2 to you? Einstein is telling us that, when an object moving at velocity v radiates, its kinetic energy K changes by (in modern 2 notation) δ K = δcE2 v2 . (In his paper, L denotes the energy emitted in radiation and V the speed of light.) He then goes on to say, a couple of paragraphs later, “It is not excluded that it will prove possible to test this theory using bodies whose energy content is variable to a high degree (e.g., radium salts).” Einstein wrote to a friend excitedly: “One more consequence of the paper on electrodynamics has also occurred to me. .... The argument is amusing and seductive; but for all I know the Lord might be laughing over it and leading me around by the nose.” As we all know, the Lord did not lead Einstein around by the nose. Many years later, in 1946, Einstein gave an elegant derivation, which, surprisingly, is omitted from most textbooks (I like Einstein’s 1946 derivation much better than his original 1905 derivation) and so is in danger of being forgotten. See page 232 of GNut. The same derivation was given on page 125 of A. Einstein’s Out of My Later Years, Philosophical Library, 1956. 5. A remarkably modern paper, it derives gravity waves in clear logical steps almost exactly as how a modern textbook would present the subject. 6. This notation was first introduced by Weber and Kohlrausch in 1856, long before Albert was born. By the way, “celeritas,” being Latin, is not related to “celery,” which comes from the Greek word for “parsley.” Meanwhile, “Kohl” means “cabbage” in German.


Notes to chapter 2

7. When the rumors of the impending discovery of gravity waves started flying around cyberspace, I emailed my correspondents to name me a theoretical physicist who does not believe in gravity waves. Nobody could come up with a name. Still, it is crucial that physics be based on observational evidence. 8. An extreme example may be the speculation of Democritus (“chosen by the people,” c. 460–c. 370 BC) about atoms. It took over two millennia for it to be verified. In our own times, it is anybody’s guess whether string theory will ever be experimentally verified, and how long it will take. 9. Including my own GNut. 10. Einstein committed a serious error in his 1916 paper, which led the English physicist Arthur Eddington to jest that gravity waves propagate with the speed of thought. Einstein’s 1918 paper, in contrast, contains more or less the essence of the treatment given in modern textbooks. 11. M. Bartusiak, Einstein’s Unfinished Symphony. 12. A. Zee, An Old Man’s Toy (hereafter cited as Toy). 13. It is of course not always American: witness “kiwi” beating out “Chinese gooseberry.” Chapter 1. A friendly contest between the four interactions 1. See GNut, chapter VIII.2. 2. We still devote one day a week to electromagnetism: Thursday is Thor’s day. Chapter 2. Gravity is absurdly weak 1. We do know when Newton died; the discrepancy is due to the difference between the Julian and Gregorian calendars. 2. Gravity is responsible for a number of ailments, in particular gout. Molecules of uric acid in the bloodstream are driven downward by gravity and congregate in the lower extremities, typically around the big toes. When the concentration of uric acid reaches a critical value, it can suddenly crystallize, causing excruciating pain. 3. Witness the popularity of the idea in science fiction, notably Jules Verne’s Journey to the Center of the Earth (1864). 4. N. Kollerstrom, “The Hollow World of Edmond Halley,” J. Hist. Astronomy 23 (1992) p. 185. 5. For a popular account, see Toy. 6. The details were worked out by Sir James Jeans (1877–1946). Here is what I just said in more technical language. In stellar physics, the Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter. 7. In honor of Max Planck, who first introduced this number into physics. 8. The curious reader can find this worked out in Zee, Unity of Forces in the Universe (hereafter cited as Unity), volume 2.

Notes to chapter 4


Chapter 3. Detection of electromagnetic waves 1. His fundamental contributions range from physics to physiology. During his visit to the United States, Helmholtz was treated like royalty, but on the ship returning to Europe he fell, hit his head, and died soon after. See B. Brown, Planck. 2. Note that this is also the year Maxwell died and Einstein was born. It was also the year the American tycoon and philanthropist John Hertz was born. See a later endnote about him. 3. One of the few examples of an apparatus named after a city rather than a person. 4. With frequency around 100 million hertz (MHz). 5. Look at photos of Hertz’s apparatus, as shown in the text. It would make an easy project for ambitious high school students. 6. Given the role played by electromagnetic waves in our society, I am often astonished that they were discovered a mere 130 years ago. 7. Sadly, the Nazis saw fit to remove Hertz’s portrait from the Hamburg Rathaus, even though his father and paternal grandparents had converted from Judaism to Christianity in the early 19th century. His mother was the daughter of a Lutheran pastor. 8. I was saddened somewhat, but perhaps I shouldn’t have been, that when I searched online for Hertz, a rental car company soundly beat out the person who brought us our electromagnetic age. It is a comment on what is valued in our society. The mogul John Hertz (1879–1961), founder of the rental car company, was actually a remarkable character. Born Sndor Herz in what is now Slovakia, he was five when his family emigrated to Chicago. As a young man, he boxed under the name “Dan Donnelly,” and after winning several championships, eventually fought under his own name. He literally fought his way up in the world. 9. Now discussed in almost any introductory quantum mechanics textbook. 10. In his youth, Planck was much vexed by his inability to obtain a desirable job. Every time such a position opened up, it would be offered to Hertz, with Planck coming in as second choice. See B. Brown, Planck. 11. It still exists in certain areas of physics, but no longer in the so-called Big Science, with the letter “b,” as in a billion dollars. Chapter 4. From water waves to gravity waves 1. You can see his skull, once housing his big brain, in the Museum of Man in Paris. 2. What this means is the following. Suppose the pond is 13 feet deep. Let us define 1 phathom as 13 feet. Then in terms of phathoms, g is equal to exactly 1 phathom. Historical units such as fathoms, hands, and stones are defined precisely in this spirit. 3. And was written down by Claude-Louis Navier (1785–1836) and George Stokes (1819–1903). 4. The Clay prize; see Wikipedia, for example. 5. For example, the exponential e h is approximately h to leading order.


Notes to chapter 6

Chapter 5. Spooky action at a distance 1. I am not sure when gravity was first explicitly recognized as a force. To the ancients, gravity, ubiquitous and ever present, must have been subsumed into a general consciousness of existence. 2. There is perhaps a lesson here for the young theoretical physicists reading this book. Newton was content to postulate the inverse square law and then explore its consequences. He left its dynamical origin to others, like Descartes, whose theory of vortices sweeping the planets along was swept into the dustbin of history. I might call the Descartes approach the “all or nothing approach,” which some theoretical physicists still indulge in. At any stage in the development of physics, certain questions are not appropriate; for instance somebody could always demand of Newton, “Hey Isaac, so why inverse square?” Chapter 6. Greatness and audacity: Enter the field 1. He has been immortalized in the term “Laplacian,” which physics students mutter all the time. 2. The notation used here is obviously not the one the Marquis used. 3. Physics textbooks tend to introduce Newton’s idea about gravity, work out the moon moving around the earth in a circular orbit, and leave it at that. But if you consider that by then, people had already observed the moon for several millennia, you would realize that a great deal was known about the motion of the moon. There were quite a few discrepancies left unexplained by Newton, which no doubt caused him and his contemporaries and successors some major headaches. (We now know that some of these are due to the pull of the other planets and the sun and to tidal effects.) Well, Laplace thought that if gravity were due to some tiny particles zipping back and forth at the speed c G , the time delay could solve some outstanding puzzles about the moon’s orbit (I am impressed that Laplace’s picture is eerily similar to the modern quantum field theoretic view of the graviton zipping back and forth). 4. Some sort of democratic impulse. 5. This was already mandated by special relativity. 6. But before this understanding, it would seem strange, perhaps even bizarre, that gravity waves and electromagnetic waves would propagate at precisely the same speed c. 7. Those of you who were bottle fed may be excused. 8. Einstein, Out of My Later Years. 9. I must say that the latest and the brashest ideas on the cutting edge of theoretical physics today often seem neither great nor audacious. 10. This passage about Faraday is adapted from my book Fearful. 11. I was amazed when I read this. In more recent times, enemy scientists typically have been captured and interned. 12. My son Max, five as of the writing of this sentence, often asked me to exert the force on him. I would stretch out my hand like the Emperor, and he would grasp his neck and pretend to choke like Luke.

Notes to chapter 7


13. A friendly word of advice to those readers of my textbooks who complained on the jungle river that they are not mathematical enough. 14. The American school of theoretical physics by tradition has stressed physical intuition, at the expense of what is sometimes referred to as “fancy shmancy mathematics.” I will refrain from exploring the historical and sociological origins of this emphasis, which has both strengths and weaknesses. Generally speaking, European physicists receive a much more vigorous training in contemporary mathematics than their American counterparts. The French philosophers, now referred to as the French physicists, still are regarded by many Americans as overly mathematical. Of course, what is considered fancy by one generation is often thought basic by the next. The mathematics used by Poisson et al. now looks like child’s play and is familiar to any undergraduate student of physics. 15. Physicists have often used the birth of telecommunications to illustrate the importance of funding basic research. They can easily imagine the Royal Navy official charged with allocating funds to improve communications deciding it would be folly to support these strange types fooling around with wires and frogs’ legs in their gloomy laboratories. Obviously, he might have reasoned, the money would be better spent on breeding a speedier strain of carrier pigeon. 16. QFT Nut. 17. It is actually somewhat more subtle than that, hence my use of the word “almost” twice in this section. The key point, as physics undergrads learn, is that while the static gravitational and electric potential fall off with 1/r , in the propagating wave these potentials fall off like e i kr /r . 18. An excellent account, with detailed explanations for the skepticism, has been given by Daniel Kennefick, Traveling at the Speed of Thought. 19. It was during this period, in which general relativity had made relatively little progress, that Richard Feynman participated in a conference on the subject. After hearing some lectures devoted to formalisms, a bored and disgusted Feynman wrote a famous letter to his wife telling her to never allow him to attend a conference on this subject again. Some physicists at the conference tried to convince the others that gravity waves do not exist. In a ludicrously unfair judgment, Feynman referred to the other participants as worms trying to crawl out of a bottle and classified them into six different kinds. I have on occasion classified my Amazon critics using a similar scheme. 20. See J. A. Wheeler, “Superdense Stars,” Annual Review of Astronomy and Astrophysics, vol. 4, 1966, p. 423. See also later work by K. Thorne and A. Campolattaro, Astrophysical Journal, 1967, vol. 149, p. 591. Chapter 7. Einstein, the exterminator of relativity 1. 2. 3. 4.

In German, Einsteinsche Relativittstheorie. P. Galison, Einstein’s Clocks, Poincaré’s Maps. Such as Châtelet Les Halles in Paris. Here we reached this conclusion using Maxwell’s equations. Historically, this was also established empirically by the famous Michelson-Morley experiment. 5. Given that a gravity wave propagates at the speed of light, some wit has suggested that a levity wave should propagate at the speed of dark.


Notes to chapter 8

6. Oliver Heaviside in 1893, and independently Henri Poincaré in 1905, anticipated the existence of gravity waves, arguing by analogy with electromagnetic waves. Poincaré understood that Lorentz invariance is a property of spacetime, not solely of electromagnetism, and thus even stated that gravity waves propagated with the speed of light. But only Einstein had the actual relativistic theory of gravity, and so only he was able to determine the properties of gravity waves. Chapter 8. Einstein’s idea: Spacetime becomes curved 1. 2. 3. 4. 5. 6. 7. 8.




I am abusing geography slightly. This celebrated experiment was also performed by Simon Stevins of Bruges. Newtonian physics cannot entertain the existence of massless particles. As to what type of blonde, see the classification and scholarly study Blonde Like Me by Natalia Ilyin. See https://www.npl.washington.edu/eotwash/node/1. If the gravitational mass were not equal to the inertial mass, this would correspond to, in our analogy, different airplanes seeing a different curvature of the earth. Staircase wit, l’esprit d’escalier, Treppenwitz, firing the cannon after the cavalry has already charged by you. Einstein fastened on the universality of gravity as the one essential fact. A priori, it was certainly not clear, out of the known facts about gravity, which one we should fasten onto. When I first learned about gravity, I wondered about the inverse square law, why it was the square of the distance and not the cube, say. No doubt many students have had the same thought. That it is inverse square is now understood in quantum field theory as due to the masslessness of the photon and of the graviton. In fact, one could have started with the masslessness of the graviton and, knowing how it couples to mass in Newtonian gravity, recovered Einstein gravity. But that’s another story for another evening. See GNut, chapter IX.5 See Box 1 in the article about Wheeler by C. Misner, K. Thorne, and W. Zurek: http://authors.library.caltech.edu/15184/1/Misner2009p1638PhysToday.pdf. Note that according to this article, Wheeler was not the first to come up with the term “black hole.” Incidentally, reference 14 in this article contains a description of my work mentioned in an earlier endnote. One reason I went to Princeton was because I had read about John Wheeler. I learned physics from him starting on day one, until the end of my junior year, when Murph Goldberger told me that I had better abandon gravity and study something more interesting called quantum field theory instead. And so I devoted the entire summer learning quantum field theory. Then I spent my senior year working with Arthur Wightman on his particular approach, known as axiomatic field theory, complete with theorems, proofs, and all that kind of stuff. I remember Goldberger saying to Sam Treiman in my presence, “I saved the kid from Wheeler’s clutches only to see him fall into a worse trap.” When it came time for graduate studies, I went to Wheeler for advice, and he was gracious enough to pick up the phone and got me into the appropriate school. Readers who wish to feast on these niceties could read GNut. It is explained in detail there how g blossoms into ten different functions.

Notes to chapter 9


Chapter 9. How to detect something as ethereal as ripples in spacetime 1. I recommend getting the history of LIGO from one of its founders, Rainer Weiss. See http://news.mit.edu/2016/rainer-weiss-ligo-origins-0211. 2. With this kind of time and cost involved, the reader can readily surmise that considerable infighting has occurred, with one scientist after another outsted from the project. For a short book like this, I have to assume that the reader is not terribly interested in gossipy details. More important, the question of which institution deserves the most recognition might occur. For this, I refer the reader to the actual press release: http://ligo.org/detections/GW150914/pressrelease/english.pdf. I quote two passages here: The discovery ... was made by the LIGO Scientific Collaboration (which includes the GEO Collaboration and the Australian Consortium for Interferometric Gravitational Astronomy) and the Virgo Collaboration using data from the two LIGO detectors. The discovery was made possible by the enhanced capabilities of Advanced LIGO, a major upgrade. ... The US National Science Foundation leads in financial support for Advanced LIGO. Funding organizations in Germany (Max Planck Society), the U.K. (Science and Technology Facilities Council, STFC) and Australia (Australian Research Council) also have made significant commitments to the project. Several of the key technologies that made Advanced LIGO so much more sensitive have been developed and tested by the German-UK GEO collaboration. Significant computer resources have been contributed by the AEI Hannover Atlas Cluster, the LIGO Laboratory, Syracuse University, and the University of Wisconsin Milwaukee. Several universities designed, built, and tested key components for Advanced LIGO: The Australian National University, the University of Adelaide, the University of Florida, Stanford University, Columbia University in the City of New York, and Louisiana State University. 3. In physics, wave interference has played—and continues to play—a crucial role. The phenomenon, characteristic of waves, was used in a crucial experiment by Thomas Young (1773–1829) to establish that light was a wave. By the way, the breadth of Young’s interests was such that he was referred to as “the last man who knew everything.” 4. Interestingly, each detector is conceptually similar to the famous MichelsonMorley interferometer that established special relativity. In that case, the experiment was to see whether the speed of light was different in the two arms. 5. The word “exactly” is of course a mathematical abstraction and is used here to simplify the discussion. Two kilometers-long arms could hardly be built to have “exactly” the same length, but the slight difference in lengths can be adjusted for. 6. QFT Nut, chapter N.1. 7. In fact, much of this modeling can also be done analytically using the perturbation theory first developed by S. Chandrasekhar. 8. The masses of the two black holes involved surprised astrophysicists somewhat. The black holes are considerably more massive than the stellar-mass black holes that should result when massive stars die, but are many orders of magnitude less


9. 10. 11. 12. 13. 14.



Notes to chapter 10

massive than the million- to billion-solar-mass giant black hole that is expected to sit at the center of each galaxy. The merger radiated away 3 solar masses worth of energy in gravity waves, resulting in a black hole with 29 + 36 − 3 = 62 times the mass of the sun. In a later edition, the title was changed to Einstein’s Universe by a new publisher. M. Bartusiak, Einstein’s Unfinished Symphony. Kennefick, Traveling at the Speed of Thought. See also H. Collins, Gravity’s Ghost. Note especially the reference to the “Italians,” a codeword that may or may not refer to people born in Italy. The Perils of Pauline is an American melodrama film serial shown in 1914 in weekly installments, featuring a damsel named Pauline in constant distress and always saved at the last minute. Richard Garwin, one of Weber’s most vocal critics, simply built a replica of Weber’s detector and showed that he could not pick up any signal. At a physics conference, Garwin and Weber almost came to blows. For example, in recent times, the TAMA 300 in Japan, the GEO 600 in Germany, and Virgo in Italy. In fact, members of the Virgo team worked on LIGO and were listed on the discovery paper.

Chapter 10. Getting the best possible deal 1. R. P. Feynman, QED, with a new introduction by A. Zee. 2. Babies have no need for Euclid; as soon as they can crawl, they move toward the obscure objects of their desire along a straight line. 3. The bitter academic controversy over Fermat’s birth year stems from his father marrying twice and naming two sons from two different wives both Pierre. K. Barner, NTM, 2001, vol. 9, no. 4, p. 209. 4. Historians have fun exploring counterfactual histories. See Cowley, What If? 5. Two small stories about two towering figures connected with the action principle: Lagrange and Feynman. Starting when he was 18, Joseph Louis, the Comte de Lagrange (1736–1813), (who, by the way, was born Giuseppe Lodovico Lagrangia before the term “Italian” existed), worked on the problem of the tautochrone, which nowadays we would describe as the problem of finding the extremum of functionals. A year or so later, he sent a letter to Leonhard Euler (1707–1783), the leading mathematician of the time, to say that he had solved the isoperimetrical problem: for curves of a given perimeter, find the one that would maximize the area enclosed. Euler had been struggling with the same problem, but he generously gave the teenager full credit. Later, he recommended that Lagrange should succeed him as the director of mathematics at the Prussian Academy of Sciences. Richard Feynman (1918–1988) recalled that when he first learned of the action principle, he was blown away. Indeed, the action principle underlies some of Feynman’s deepest contributions to theoretical physics. In particular, his formulation of quantum mechanics depends very much on the action. 6. The reader should not confuse extremization of the action with the everyday observation that matter likes to minimize energy, which is just the principle of

Notes to chapter 10


“water always flows downhill” and “a couch potato will stay on the couch.” Throw a child’s marble into a bowl. Come back later, and you would be astonished if it is not resting at the bottom of the bowl. The marble has minimized its total energy by setting its kinetic energy to zero and lowering its potential energy as much as possible. (Occasionally, a bright student might wonder if this minimization of energy contradicts the conservation of energy. In fact, while the latter is absolute and sacred to physicists, the former is merely apparent because we choose to ignore other forms of energy. By rattling in the bowl, the marble has generated sound and heat, both of which escaped into the environment.) 7. I consider this to be one of the great triumphs of quantum physics: the explanation of why the action is extremized, rather than minimized or maximized. 8. I must emphasize that the action principle of mechanics says no more, and no less, than Newton’s laws of motion. The action formulation, although more compact and aesthetically more appealing, is physically entirely equivalent to Newton’s formulation. The outlook, however, is quite different in the two formulations. In the action formulation, one takes a structural view, comparing different ways by which the particle could have gotten from here to there. To the 17th- and 18th- century mind, the least time and least action principles provided comforting evidence of Divine guidance. A voice told each particle in the universe to follow the most advantageous path and history. Not surprisingly, the least action principle has inspired a considerable amount of quasi-philosophical, quasi-theological writing, a body of writing, which, while intriguing, proves to be sterile ultimately. Nowadays, physicists generally adopt the conservative, pragmatic position that the least action principle is simply a more compact way to formulate physics, and that the quasi-theological interpretation suggested by it is neither admissible nor relevant. Next time you are invited to a dinner party at the home of a philosophy professor, say the word “teleological” in the middle of the main course. After these guys have stopped clawing at each other, utter, with nonchalant total self-assurance, “the ontological is distinct from the epistemological, while the tautological is antithetical to the logical,” and watch the fun start again. That statement is of course what is known in polite circles as “utter nonsense” and in less polite circles as total BS, but it gives you an idea of how some academics talk. The philosophy-R-us version, which I could give you for no charge, is that things are teleological if they have a purpose, or at least act as if they have a purpose. That’s a big no-no in modern science. You see that Fermat’s least time principle (incidentally, if it ever comes to a priority dispute, Fermat would have to cede to Heron of Alexandria, circa AD 65) has a strongly teleological flavor— that light, and particularly, daylight, somehow knows how to save time—a flavor totally distasteful to the post rational palate. In contrast, at the time of Fermat, there was lots of quasi-theological talk about Divine Providence and Harmonious Nature, so no one questioned that light would be guided to follow the most prudent path. 9. The title of this section is my reminder to the author to keep this book brief.


Notes to chapter 12

10. In the differential formulation, we specify the initial position and velocity of a particle and ask where it will be at a later time T and how fast it will be moving then. In the action formulation, we specify the initial position of a particle and its final position at some time T . Note that the initial velocity is not specified as in the differential formulation; rather, the initial velocity is to be determined by the action principle. The particle has to “find” the initial velocity needed to get it to the specified final position at time T , sort of like the protagonist in the Western 3:10 to Yuma. 11. Newton’s equation of motion is described as “local” in time: it tells us what is going to happen in the next instant. In contrast, the action principle is “global”: one integrates over various possible trajectories and chooses the best one. While the two formulations are mathematically entirely equivalent, the action principle offers numerous advantages over the equation of motion approach. For example, the action leads directly to an understanding of quantum mechanics via the so-called Dirac-Feynman path integral formulation. Indeed, the discussion here gives a premonition of the emergence of probability in the quantum world. Which path will the particle choose? Betting odds, anybody? See, for example, R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals; also, QFT Nut, chapter I.2. Chapter 11. Symmetry: Physics must not depend on the physicist 1. Fearful. Chapter 12. Yes, I want the best deal, but what is the deal? 1. For the curious reader, here is the action governing the electromagnetic field: S =  4 2 d x F . That’s it. Simple, eh? By extremizing this action, we obtain Maxwell’s equations for the electromagnetic field. I can deconstruct the action for you. Actions are traditionally denoted by  capital S; the integral sign is known to students of calculus; the symbol d 4 x indicates that the integration is over spacetime, with the 4 saying that spacetime is 4-dimentional in the sense of Minkowski. The hard part is F 2 : F is actually a tensor denoting the electromagnetic field. For this, you will have to get a textbook on electromagnetism at the appropriate level and self-study—believe me, it’s not that hard (so I say, having learned it decades ago)—or pay tuition at an institution of learning to have it explained to you. 2. By the way, this follows from high school level dimensional analysis. 3. The astute reader might worry about massless particles, such as the photon. See GNut for details. 4. Specifically, R the scalar curvature. There are other measures of curvature, known as the Riemann curvature tensor and the Ricci tensor, but the requirement that the action be invariant picks out the scalar curvature. 5. GNut, page 390. 6. For details, see GNut, chapter IV.2. . . .

Notes to chapter 15


7. The so-called Bianchi identities. 8. The material here is adapted from my book GNut. See p. 396. 9. I am reminded of a New Yorker cartoon showing a hapless employee standing before the boss’s big desk, with the boss saying “Yes, it was your idea, but I am the one who recognized that it was a good idea.” Chapter 14. It must be 1. I can now explain the error in Einstein’s 1916 paper, which I mentioned in the prologue. Some physical properties of gravity waves deduced by Einstein were not invariant. In other words, they depended on the coordinates used to describe them and so could not be physical. 2. Strictly speaking, the scalar curvature mentioned in an earlier endnote in chapter 12 is an invariant. 3. Some readers may wonder why other geometrical invariants besides  and R are not included in the action. Surely, if R is invariant, then R 2 , for example, will also be invariant. The answer is that in modern formulation of quantum field theory, possible terms are ordered according to how important they are expected to be over long distances in spacetime. These other terms you might worry about are all (expected to be) negligible compared to  and R. See GNut, chapter X.3. 4. By Abraham Pais, the leading biographer of Einstein. Chapter 15. From frozen star to black hole 1. Hence the warning against firing guns in the air in celebration in certain countries. 2. Back in chapter 2, I stated that the gravitational attraction between two objects, of mass M and m, is equal to G Mm/R 2 with R the distance between the two objects. When applied to the earth and the moon, since the sizes of the earth and of the moon are both tiny compared to the distance between them, it is clear what R means. But when Newton applied his law to the gravitation attraction between the earth and the apple, what should he have taken for R? Should R have been the height of the apple tree? In fact, as was explained in chapter 2 Newton spent years proving that R should be the distance between the apple and the center of the earth. Since the height of the apple tree is completely negligible compared to the radius of the earth, R is equal to the radius of the earth. Similarly here, R should be taken to be the radius of the planet. 3. Interestingly, Laplace removed this speculation from later editions of his book. 4. For a modern treatment of black hole, see GNut, Part VII. 5. Furthermore, this often-cited argument actually does not establish the existence of a black hole defined as an object from which nothing can escape. The escape velocity refers to the initial speed with which we attempt to fling something into outer space. In a Newtonian world, we could certainly escape from any massive planet in a rocket with a powerful enough engine. 6. See p. 432 of GNut for a more technical reason, pointed out by D. Marolf, for objecting to this analogy.


Notes to chapter 18

Chapter 16. The quantum world and Hawking radiation 1. We already met the Planck number in chapter 2. For a fascinating biography of Max Planck, see B. Brown, Planck. 2. It has been done; the key is of course to repeat this theft for millions of accounts. 3. The use of that particular word is consonant with its use in everyday parlance. But considering that air contains zillions of molecules to begin with, the vacua produced by even the best commercially available pump will still have quite a few molecules in them. A quantum field theorist simply conceptualizes a quantum state with nothing in it. 4. For the interested reader who already knows some quantum mechanics and special relativity, many textbooks stand ready to teach you quantum field theory. In particular, see QFT Nut. Chapter 17. Gravitons and the nature of gravity 1. Which naturally does not inhibit people from writing about it. Quite to the contrary. 2. In a vague sense, but only in a vague sense, you could say that Newton’s corpuscles never went away. 3. This picture is somewhat oversimplified, but is, however, adequate for our purposes here. 4. This type of problem has given birth to a new area of physics known as “numerical relativity.” 5. For more, see Toy, p. 203 and subsequent pages. For a technical, yet more or less accessible account, see QFTNut, chapter III.2, and GNut, chapter X.8. 6. Notably, non-abelian gauge theories. You can think of the begetting in these theories as being more restrained than in Einstein gravity. 7. Can’t resist a truly sophomoric nod to quantum gravydynamics, especially as I am writing this shortly after Thanksgiving, when I had a discussion with a French friend on the role of gravy in French cuisine, as distinct from sauce. 8. See QFTNut, chapter I.5 and I.7. 9. See Fearful p. 164. 10. Notably, Freeman Dyson of the Institute for Advanced Study in Princeton. For further discussion of the struggle to quantize gravity, see chapter X.8 of GNut. Chapter 18. Mysterious messages from the dark side 1. I actually saw this at a playground in Paris. A couple of big kids, perhaps aged nine or ten, came over and spun the merry-go-round hard. All the little kids aged five or less went flying off and started crying like crazy. You can imagine the parents dropping their cell phones and rushing over. 2. An early suggestion was by Jacobus Kapteyn, later confirmed by Jan Oort. 3. A cantankerous character, Zwicky also invented the term “spherical bastards” to describe his colleagues who were bastards no matter how he looked at them.

Notes to appendix


4. As I was working on the final draft of this chapter, the sad news came that Vera Rubin had died at the age of 88. See http://www.latimes.com/local/obituaries/lame-vera-rubin-20161226-story.html. 5. Note that it is not necessary to resolve the motion of individual stars, of which there are zillions. 6. Astronomers have also discovered some extremely diffuse galaxies containing almost no stars, which may be composed entirely of dark matter. 7. For more, see chapters 10 and 11 of Toy. 8. This proposal goes by the acronym MOND for modified Newtonian dynamics. 9. See GNut, p. 495, and chapter VIII.2. 10. This person is hardly mythical, because, as I said in the preface to this book and in the preface to GNut, more than once I have taught Einstein gravity as an advanced undergraduate course. 11. I am aware that a vast literature exists out there, but given the size and nature of this book, I must refrain from further comment. 12. See, for example, QFT Nut or GNut. 13. On some scale that theoretical physicists are very fond of. See QFTNut, p. 449, and GNut, p. 746 14. Photons and neutrinos contribute negligibly. 15. For a cartoon depiction of the situation in the late 1980s, see p. 185 of Toy. Chapter 19. A new window to the cosmos 1. See Bartusiak, Einstein’s Unfinished Symphony. 2. Shimon Kolkowitz, Igor Pikovski, Nicholas Langellier, Mikhail D. Lukin, Ronald L. Walsworth, Jun Ye, “Gravitational Wave Detection with Optical Lattice Atomic Clocks,” arXiv:1606.01859. See this article for references to other proposals as well. 3. Ashes, ashes, we all fall down! 4. I was tempted to invite Darwin to join the panel also. Charles Darwin: There used to be apples that fell down and others that flew up into outer space. Those that flew up did not get to reproduce. So apples evolved to fall down. I am not a geologist and so I don’t know about rocks. Appendix: What does curved spacetime mean? 1. When I was in high school, I got the erroneous impression that the notion of coordinates originated with Descartes. In fact, by the time of Ptolemy, astronomers in the West certainly had defined latitudes and longitudes. In China, Chang Heng, roughly a contemporary of Ptolemy, was said to have derived, by watching a woman weaving, a system of coordinates to map heaven and earth. The Chinese words for “latitudes” and “longitudes,” “jing” and “wei,” are just the terms for warp and weft in weaving. 2. For those readers who know calculus, “very small” means so small that it actually approaches zero. 3. Actually also known in several other ancient civilizations, including those of Babylonia, China, and Egypt.


Notes to appendix

4. Of course, the French had insisted that ϕ P should be set to 0, but unfortunately for them, the Brits were more powerful when these things were determined. 5. I am not worrying about the additional technicality that dθ might be negative, while distance is usually understood to be positive. This problem is taken care of, because all the terms appear as squares in the generalized Pythagorean formula given later in this appendix. 6. For the mathematically sophisticated reader, f (θ ) = cos θ , with θ defined to be 0 at the equator and π/2 at the north pole. 7. If you are at all into math, you will have fun figuring out the properties of the spaces described by various metrics. For example, consider ds 2 = (dx 2 + dy 2 )/y 2 with y > 0. The space it describes is called the Poincaré half plane and has some weird properties. See GNut, p. 67. 8. Note that dy dx is the same as dx dy and should not be counted separately. 9. That’s the hard part, but still not that hard. It is easily mastered by undergrads. I should know, since I have taught it to undergrads. 10. Seriously. I kid you not: way way easier than learning quantum mechanics. The math involved only goes a bit beyond what is discussed here. 11. See Fearful, QFT Nut, and GNut. 12. Historically, the time coordinate t was written as x 4 , but later it was realized that it was more sensibly written as x 0 . 13. There are ten terms altogether but I have not bothered to write them all out; the ones I did not write out are indicated by dots. 14. I say “structurally,” because there are clearly some difference in the details. For one thing, g μν (x) consists of ten functions, instead of the one function g (t, x, y). For another, x is now a compact notation denoting (t, x, y, z), but that is just because we live in 3-dimensional space, while the surface of the lake is 2-dimensional. 15. See GNut, p. 6 and p. 77. 16. I mention this to encourage you. If you feel that you are comparable to a bright undergrad at a large American state university, then for sure you can learn how to derive the Riemann curvature tensor. That is an experimentally established fact.

Bibliography Marcia Bartusiak, Einstein’s Unfinished Symphony: Listening to the Sounds of SpaceTime, Joseph Henry Press, 2000. Brandon Brown, Planck: Driven by Vision, Broken by War, Oxford University Press, 2015. Harry Collins, Gravity’s Shadow: The Search for Gravitational Waves, University of Chicago Press, 2004 Harry Collins, Gravity’s Ghost: Scientific Discovery in the Twenty-first Century, University of Chicago Press, 2011. Robert Cowley, The Collected What If? Eminent Historians Imagining What Might Have Been, Putnam, 2001. Albert Einstein, Out of My Later Years, 1993. Richard P. Feynman and Albert R. Hibbs, Quantum Mechanics and Path Integrals, Dover, 2012. Richard P. Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press, 2014. Peter Galison, Einstein’s Clocks, Poincaré’s Maps: Empires of Time, W. W. Norton, 2004. H. Gutfreund and J. Renn, The Road to Relativity, Princeton University Press, 2015. Daniel Kennefick, Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves, Princeton University Press, 2016. Tony Rothman, Everything’s Relative: And Other Fables from Science and Technology, Wiley, 2003. Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, 1972. Books by This Author Fearful Symmetry: The Search for Beauty in Modern Physics, Princeton University Press, 2016. An Old Man’s Toy: Gravity at Work and Play in Einstein’s Universe, Macmillan, 1990; later published as Einstein’s Universe: Gravity at Work and Play, Oxford University Press, 2001 (referred to as Toy or Toy/Universe). Unity of Forces in the Universe, World Scientific, 1982 (referred to as Unity). Quantum Field Theory in a Nutshell, Princeton University Press, 2010 (referred to as QFT Nut). Einstein Gravity in a Nutshell, Princeton University Press, 2013 (referred to as GNut).

Index italic pages refer to figures and tables. Abraham, Max, 98 acceleration, 10n, 23, 58–59, 85, 120, 127, 134, 136 accelerators, 120, 127n action at a distance, 56; Coulomb’s law and, 15, 44–45, 49; Einstein and, 39; electromagnetic waves and, 49; gravity and, 37–39; magnets and, 44; Newton and, 38–39, 44, 48–49, 55; quantum entanglement and, 38n; spooky, 37–39 action formulation; advantage of, 138; differential formulation and, 85–86, 162n10; elegant precision of, 85–86, 90n, 101; Newton’s formulation and, 77–78, 161n8; structural view of, 161n8 action principle: choice of history and, 83–84; Dirac-Feynman path integral formulation and, 162n11; Einstein and, 94, 97–103, 150n; Einstein-Hilbert action and, 94, 98, 102–3, 135; electromagnetism and, 86, 92; energy and, 83; equation of motion approach and, 134; Feynman and, 160n5, 162n11; fundamental physics and, 85n; gravity and, 86; Hilbert and, 97–98; initial velocity and, 162n10; Lagrange and, 160n5; light and, 83–84, 86; as metaphor for life, 91–92; Newton and, 85–87, 161n8, 162n11; particles and, 78, 83–84, 161n8, 162n11; quantum mechanics and, 160n5, 161n7; spacetime and, 92–95; special relativity and, 97–98; strong interaction and, 86; structure of physical reality and, 88; symmetry and, 88–89; theology and, 161n8; weak interaction and, 86

Adelberger, Eric, 60 advertising slogans, 116–17 AEI Hannover Atlas Cluster, 159n2 algebra, 55n, 107 Amazon, 157n19 American school, 157n14 Ampère, André-Marie, 46 Anaxagoras, 138 Andromeda, 13 antennae, 26, 141 antigravity, 137 antiparticles, 119–23 Aristotle, 1, 58–59, 142 atomic clock, 141 atoms: electrons and, 9–12, 17–18, 22–23, 30, 48, 109, 115, 117–20, 123, 135; fission and, 14; fusion and, 12, 14, 51; matter and, 3, 9–11, 14, 17, 22–23, 70, 109, 135, 141, 151n, 154n8, 165n2; neutrons and, 9, 12, 14; nucleus of, 9, 14, 151n; protons and, 9–14, 16–18, 22–24, 109, 118, 120n, 135, 151n; quarks and, 9, 22, 48, 108n Australian Consortium for Interferometric Gravitational Astronomy, 159n2 Australian National University, 159n2 Australian Research Council, 159n2 Babylonia, 165n3 Barish, Barry C., 151 Bartusiak, Marcia, 165n1 Beethoven, Ludwig van, 103, 104 Bentley, Richard, 38–39 Bering Strait, 57, 58, 60–61 Berlin Prize, 26 Bianchi identities, 163n7 Big Science, 155n11 binary stars, 50



black holes: detection of, 66–74; Einstein and, 108, 111–14; electromagnetic waves and, 151; energy and, 1, 71, 107, 110, 115–17, 119, 121–23, 160n9; escape velocity and, 107–8, 163n5; event horizon of, 111–13; fields and, 124–25, 130; funnel image of, 113–14; galaxy centers and, 159n8; gravity waves and, 1–2, 160n9; Hawking radiation and, 110, 115–17, 119, 121–23; mass and, 70–71, 107–13, 122, 159n8, 160n9; merging of, 1–2, 70–71, 126, 160n9; Michell-Laplace criterion and, 108, 110–12; motion and, 95; neutron stars and, 151; Newton and, 163n5; Schwarzschild and, 111–13; spacetime and, 1–2, 66–74, 94–95, 111–14, 125, 150; speed of light and, 108; Wheeler and, 108, 158n9 blogosphere, 137 Blonde Like Me (Ilyin), 158n4 blue shift, 132 Bohr, Niels, 137 book weight, ix, 153n1 Brown, B., 164n1 Bucherer, Alfred, 51 c=fλ, 27 calculus, 19, 83, 143, 162n1, 165n2 California Institute of Technology (Caltech), 66, 71–72 capacitors, 26 Cartesian coordinates, 143, 144 centrifugal forces, 10n Chandrasekhar, S., 159n7 Chang Heng, 165n1 China, 165n1, 165n3 classical physics, 4, 30, 115, 117–18, 123–24, 126n Clay prize, 155n4 clouds, 11, 22, 133, 139, 154n6 Coleman, Sidney, 21 Columbia University, 159n2 common sense, 54 computers, 71, 73, 126, 159n2

concordance model, 138–39 conservation of energy, 122, 161n6 constructive interference, 67–68 Copernicus, 138 copper, 26 cosmological constant, 103, 135–37 Coulomb’s law, 15, 44–45, 49 covariance, 89–90, 102 current, 44, 47, 89 curved surfaces, 62, 114, 144–46, 150 dark energy, 9, 131, 138; cosmological constant and, 135–37; Einstein gravity and, 135–37; electrons and, 135; expansion of universe and, 135–37; photons and, 135; protons and, 135 dark matter, 9; composition of, 133–34; concordance model and, 138–39; Einstein gravity and, 133–34, 165n10; galaxies and, 131–34, 165n6; mass and, 133, 138; Newton and, 133; spacetime and, 133, 135, 137; speed of light and, 138; Zwicky and, 132 Davy, Humphrey, 43 delta, 116 Democritus, 154n8 density, 21–23, 45 Descartes, René, 31, 64, 143, 155n1, 156n2, 165n1 destructive interference, 68–69 detectors, x; gravitons and, 2, 124n, 126–27; gravity waves and, 2, 27, 66–74, 126–27, 140–41, 151, 159n1, 159n2, 159n4, 160n15, 160n16; LIGO, 66–74, 126–27, 141, 151, 159n1, 159n2, 160n16; LISA, 141; Michelson-Morley interferometer and, 159n4; more sensitive, 141; wave interference and, 67–69, 159n3; Weber and, 74, 153n6, 160n15 differential equations, 46n, 77–78, 85, 143 162n10 Dirac-Feynman path integral formulation, 162n11 Doppler effect, 131–32 Dyson, Freeman, 49, 164n10

Index E=mc2 , 2, 40, 110, 117, 119–20, 122, 153n4 Eddington, Arthur, x, 154n10 Egypt, 165n3 Einstein, Albert, 1; accuses Hilbert of theft, 98; action at a distance and, 39; action principle and, 94, 97–103, 150n; aging of, 3n; black holes and, 108, 111–14; close career disaster of, 97–98; cosmological constant and, 103, 135–37; dark energy and, 135–37; dark matter and, 133–34; Dyson and, 49; E=mc2 and, 2, 40, 110, 117, 119–20, 122, 153n4; Eddington and, 154n10; error in paper of, 154n10, 163n1; expansion of universe and, 135; Faraday and, 41, 44; field equations of, 33, 63, 94, 98; general relativity and, 2, 51, 123–24, 157n19; gravitational fields and, 60, 137; gravitons and, 86, 97, 123, 125–27; gravity waves and, 3–6, 153n5, 154n10, 163n1; Grossman and, 99; Heisenberg and, 118; Maxwell and, 155n2; Michell-Laplace argument and, 108, 111–12; Newton and, 5, 44, 49, 55, 59, 86–87, 95–97, 104, 108, 111, 125, 133–34, 136, 142, 147, 158n8; Out of My Later Years and, 153n4, 156n8; paper of 1916 and, 3, 6, 154n10, 163n1; photoelectric effect and, 29–30; quantum mechanics and, 29, 142; Schwarzschild and, 111–13; spacetime and, 2, 4, 33, 55–57, 61–63, 70–74, 94–97, 102–3, 111, 114, 123, 135, 141–43, 146–50, 157n6; special relativity and, 2–3, 40, 51, 55–56, 87–89, 95, 97, 116–21, 125, 146, 153n4, 156n5, 159n4, 164n4; speed of light and, 52–54, 56; symmetry and, 88; theoretical physicists before, 60–61; theory of gravity by, 2, 33, 49–51, 62–63, 70–71, 94–95, 97, 102–4, 111, 114, 123, 126–27, 133–36, 141–42, 147, 150n, 157n6, 158n8, 164n6, 165n10; tightness of theory by, 102–4, 136–37


Einstein gravity, 5; action principle and, 100–1, 150n; black holes and, 111, 114; dark energy and, 135–36; dark matter and, 133–34, 165n10; equations of motion and, 95; general relativity as, 51; as geometrical theory, 102; gravitons and, 126–27, 164n6; gravity waves and, 33; quantum mechanics and, ix-x; spacetime and, 33, 62, 147, 158n8; three classic tests of, x Einstein Gravity in a Nutshell (Zee), ix-x, 109, 153n1, 153n4, 154n1, 154n9, 158n8, 158n11, 162n3, 162n5, 162n6, 163n3, 163n4, 163n6, 163n8, 164n5, 164n10, 165n9, 165n10, 165n12, 165n13, 166n7, 166n11, 166n15 Einstein-Hilbert action, 94, 98, 102–3, 135 Einstein repeated index summation, 148n Einstein’s Unfinished Symphony (Bartusiak), 165n1 Einstein’s Universe: Gravity at Work and Play (Zee), ix, 71, 154n5, 154n12, 164n5, 165n7, 165n15 electricity: capacitors and, 26; charge and, 12–18, 26, 29, 43–45, 47, 89, 125–26, 134; Coulomb’s law and, 15, 44–45, 49; current and, 44, 47, 89; F=e2 /R2 and, 17, 23; Faraday and, 41–48, 125, 156n10; fields and, 30, 44–47, 53, 88–89, 125–26; photon exchange and, 125; potential fall off, 157n17; symmetry and, 88–89 electromagnetic waves, 10, 13, 141; action at a distance and, 49; black holes and, 151; c=fλ, 27; detection of, 25–40, 47, 68, 70, 73, 127–28, 155n6; equations for, 27, 150; fields and, 40–41; gravitons and, 30; Heaviside and, 157n6; Hertz and, 26–27, 29; interference of, 68; light as form of, 52, 124, 140; Maxwell and, 26, 47–48, 140; Poincaré and, 157n6; quantum mechanics and, 29–30,



155n9; spectrum of, 28, 48, 140, 151; speed of, 26–27, 56, 156n6 electromagnetism: action principle and, 86, 92; attraction and, 1, 12–18, 22, 38–39, 57–58, 104, 129, 132, 135, 142, 163n2; charge and, 10–18, 26, 29, 42–47, 62, 89, 125–26, 129, 157n15; dark matter and, 131–39; detecting waves of, 25–30; electricity and, 41, 43, 45–46; energy and, 30, 45, 47, 125–26; Faraday and, 41–48, 125, 156n10; fields and, 30, 40–51, 53, 56, 62, 86, 89, 133, 162n1; Hawking radiation and, 110, 116, 121–23; Hertz and, 4–5, 7–8, 10, 25–29, 70, 73, 140, 155nn2; light and, 11 (see also light); Maxwell and, 25–29, 40, 45–49, 53, 56, 73, 86–87, 89, 126, 140, 155n2, 157n4, 162n1; Maxwell’s equations and, 26–27, 46–48, 53, 86, 89, 157n4, 162n1; photoelectric effect and, 29–30; photons and, 125 (see also photons); quantum field theory and, 48, 118–20, 123, 156n3, 158n8, 158n10, 163n3, 164n3, 164n4; repulsion and, 12–18; strength compared to gravity, 15–18, 22–23 electrons: attraction to protons, 22; dark energy and, 135; mass of, 17n, 119–20; momentum of, 115; as nucleons, 9; obesity index and, 109; Planck’s constant and, 30, 115; quantum cloud and, 11; quantum field theory and, 48; quantum mechanics and, 117–20, 123; quarks and, 9, 22, 48, 108n; repulsion between, 18; star formation and, 12, 23; string theory and, 9n Encyclopedia Britannica, 41 energy: action principle and, 83; atomic structure and, 11; binary pulsars and, 50; black holes and, 1, 71, 107, 110, 115–17, 119, 121–23, 160n9; conservation of, 122, 161n6; dark, 9, 131, 135–38; E=mc2 and, 2, 40, 110, 117, 119–20, 122, 153n4; electromagnetism and, 30, 45, 47,

125–26; fission, 14; fusion, 12, 14, 51; gravitons and, 124–25; Hawking radiation and, 110, 115–17, 119, 121–23; kinetic, 83–85, 107, 153n4, 161n6; momentum and, 9, 107, 115; nuclear, 12, 14, 23; particle-antiparticle pairs and, 119–20; Planck, 127; potential, 83–85, 107, 110, 161n6; quantum field theory and, 120–21; spacetime and, 2, 94; special relativity and, 2–3, 117, 119, 125; strong interaction and, 10; time and, 115–16 Eötvos de Vásárosnamény, Loránd, 60 Eöt-Wash experiment, 60 escape velocity, 107–8, 163n5 Euclidean geometry, 62, 79, 160n2 Euler, Leonhard, 160n5 European Space Agency, 141 Evolved Laser Interferometer Space Antenna (eLISA), 141 expansion of universe, 22, 94, 100, 135–37, 147–48, 150 exponentials, 155n5 F=e2 /R2 , 17, 23 F=G Mm/R2 , 15, 19–20, 27n, 58 F=ma, 78 Faraday, Michael, x; background of, 41, 43; Coulomb’s law and, 45; Davy and, 27n, 41, 43, 90n, 157n19; Einstein and, 41, 44; electromagnetic fields and, 41–48, 125, 156n10; honors of, 43; magnets and, 44; mathematics and, 45–46; Maxwell and, 45–47; Newton and, 125 Fearful Symmetry: The Search for Beauty in Modern Physics (Zee), 13, 42, 80–81, 101, 103, 156n10, 162n1, 164n9, 166n11 Fermat, Pierre, 81–84, 90n, 93n, 160n3, 161n8 Feynman, Richard, 78–81, 128–29, 157n19, 160n1, 160n5, 162n11 field equations: Einstein’s, 33, 63, 94, 98; Hilbert and, 98

Index fields: concept of, 33n; covariance and, 89–90; electric, 30, 44–47, 53, 88–89, 125–26; electromagnetism and, 30, 44–48, 53, 56, 62, 86, 89, 133, 162n1; Faraday and, 41–48, 125, 156n10; gravitational, 30, 55, 60, 62n, 97–98, 111, 123–25, 130, 137, 149; gravitons and, 86–87; gravity waves and, 41, 48–50; invariance and, 89–90; as lines of force, 46; magnetism and, 44–45, 47, 53, 89; Maxwell’s equations and, 26–27, 46–48, 53, 86, 89, 157n4, 162n1; particles and, 40, 42; physical reality and, 48; as separate entities, 44–45; spacetime and, 41, 45, 55 fission, 14 flat space, 62, 143–49 fluid dynamics, 10n, 32, 150 forces: acceleration and, 10n, 23, 58–59, 85, 120, 127, 134, 136; action at a distance and, 38–39, 44, 48–49, 55–56; attraction, 1, 12–18, 22, 38–39, 57–58, 104, 129, 132, 135, 142, 163n2; centrifugal, 10n; Coulomb’s law and, 15, 44–45, 49; electromagnetism and, 10 (see also electromagnetism); F=e2 /R2 and, 17, 23; F=G Mm/R2 and, 15, 19–20, 27n, 58; F=ma and, 78; fields and, 33, 46 (see also fields); gravity and, 10 (see also gravity); Newton and, 15, 19–20, 22, 32, 39, 44, 55, 58, 78, 104, 132, 142; perpetual contest between, 18–19; range vs. strength and, 12–14; repulsion, 12–18; strong interaction and, 10–14, 23, 86; weak interaction and, 10, 12, 86 Ford, Kent, 132 formalisms, 157n19 French philosophers, 46–47, 157n14 frequency, 27–30, 131–32, 155n4 friction, 58–59 functions, 31, 33, 63–64, 135, 145–49, 158n11, 160n5, 166n14 fusion, 12, 14, 51 galaxies: Andromeda, 13; black holes and, 159n8; center of, 138–39, 159n8;


clusters of, 132, 139; collision of, 49, 66; concordance model and, 138–39; dark matter and, 131–34, 165n6; diffuse, 165n6; Doppler effect and, 131–32; Einstein-Hilbert action and, 94–95; formation of, 22; Milky Way, 13, 109, 133, 138; motion and, 39, 131–34; obesity index and, 109; rotation of, 131–34 Galilean relativity, 52 Galileo, 52, 59, 85 Garwin, Richard, 160n15 Gauss, Carl Friedrich, 64n general relativity, 2, 51, 123–24, 157n19. See also Einstein gravity GEO Collaboration, 159n2, 160n16 geometry, 62, 79, 93, 102–3, 135, 143–46, 150, 160n2, 163n3 Gertsenshtein, M. E., 67 Gnerlich, Ingrid, ix gold, 151 Goldberger, Murph, 158n10 gout, 154n2 gravitational constant, 15, 58, 94, 136 gravitational fields: Einstein and, 60, 137; gravitons and, 124–25, 130; Hilbert and, 98; Newtonian gravity and, 62n, 97, 111; quantum mechanics and, 30, 123–25, 130; spacetime and, 137; variance of, 55, 111; weakness of, 130, 149 “Gravitational Wave Detection with Optical Lattice Atomic Clocks” (Kolkowitz, et al), 165n2 Gravitation and Cosmology (Weinberg), 90n gravitons: detection of, 2, 124n, 126–27; Einstein and, 86, 97, 123, 125–27, 164n6; electromagnetic waves and, 30; exchange of, 128–30; Feynman diagrams and, 128–29; field changes and, 86; free propagation of, 130; gravity waves and, 30, 124; mass and, 125, 128–30, 158n8; matter and, 130; numerical relativity and, 164n4; as packets of energy, 124–25;



photons and, 30, 41, 124–25, 129, 158n8; quantum field theory and, 123, 156n3; quantum gravity and, 123–24, 127–30; quantum mechanics and, 123–30, 156n3; spacetime and, 123, 125, 127, 129, 142; special relativity and, 125; speed of, 2; swarm of, 124–25 gravity: action at a distance and, 37–39; action principle and, 86; ailments from, 154n2; antigravity and, 137; Einstein, 5 (see also Einstein gravity); expansion of universe and, 22, 94, 100, 135–37, 147–48, 150; F=G Mm/R2 and, 15, 19–20, 27n, 58; general relativity and, 2, 51, 123, 157n19; Hawking radiation and, 110, 116, 121–23; inverse square law and, 13, 15, 39, 55, 104, 111, 132, 156n2, 158n8; mass and, 13, 107 (see also mass); Michell-Laplace criterion and, 108, 110–12; Newtonian, 4–5, 15, 23, 27n, 38n, 38–40, 39, 58–59, 62n, 95, 97, 104, 107, 111, 126, 130, 133–34, 136, 147, 156n3, 158n8; quantum, 123–24, 127–30; special relativity and, 95, 97, 121, 125; speed of light and, 40–41, 56, 154n10, 156n3, 156n6, 157n6; stars and, 133; strength compared to electromagnetism, 15–18, 22–23; time and, 39; universality of, 15, 23, 41, 58–61, 133, 158n8; weakness of, 4–5, 15–18, 22–23, 66, 127, 130, 149 gravity waves, 9; atomic clock and, 141; belief in, 154n7, 157n19; black holes and, 1–2, 160n9; describing, 149–50; detection of, x, 2, 4, 25, 27, 66–74, 126–27, 140–41, 151, 159n1, 159n2, 159n4, 160n15, 160n16; direct observation of, 50; Eddington and, 154n10; Einstein and, 3–6, 33, 153n5, 154n10, 163n1; fields and, 41, 48–50; as “gravitational waves”, 5, 50, 66, 126–28, 141; gravitons and, 30, 124; GW150914 and, 153; GW170817 and, 151; Heaviside and, 157n6; impact of

studying, 140–41; LIGO Laboratory and, 66–74, 126–27, 141, 151, 159n1, 159n2, 160n16; neutron stars and, 151; Poincaré and, 157n6; postulating existence of, 3–4; quantum mechanics and, 30; skeptics of, 49; spacetime and, 63–74, 149–50; speed of light and, 40–41, 56, 154n10, 156n3, 156n6, 157n6; understanding, 31, 33; wave interference and, 67–69, 159n3 Greek, 103, 116n, 144, 148n, 149, 153n6 Gregorian calendar, 154n1 Grossman, Marcel, 99 Gutfreund, H., 153n3 GW150914, 153 GW170817, 151 Halley, Edmond, 21–22 Hawking radiation: black holes and, 110, 115–17, 119, 121–23; energy and, 110, 115–17, 119, 121–23; quantum field theory and, 119, 123; special relativity and, 121 heat, 161n6 Heaviside, Oliver, 157n6 Heisenberg, 115, 118 Hell, 21–22 Helmholtz, Ludwig Ferdinand von, 26, 155n1 Heron of Alexandria, 161n8 Hertz, John, 155n2, 155n8 Hertz, Rudolf, x; death of, 26n; electromagnetic waves and, 26–27, 29; electromagnetism and, 4–8, 10, 25–29, 70, 73, 140, 155nn2; Helmholtz and, 26; Karlsruhe and, 26; quantum mechanics and, 29; transmitter of, 26 hertz (Hz), 27–29, 155n4 Hertz (rental car company), 155n8 Hilbert, David, 94, 97–98, 102–3, 135–36 hollow earth theory, 21–22 Hooke, Robert, 20–21 Hulse, R., 50 Humpty Dumpty, 82–85 Huygens, Christiaan, 48 hydrogen, 22–23

Index Ilyin, Natalia, 158n4 index notation, 148–49 inertia, 59–60, 107, 110, 158n6 Institute for Advanced Study, Princeton, 164n10 integration, 19n, 83–84, 91, 93n, 162n1, 162n11 interferometers, 66–74, 126–27, 141, 157n4, 159n1, 159n2, 159n4, 160n16 invariants, 89–90, 102–3, 135, 157n6, 162n4, 163n1, 163n2, 163n3 inverse square law, 13, 15, 39, 55, 104, 111, 132, 156n2, 158n8 iron, 44, 151 Jeans, James, 154n6 Journey to the Center of the Earth (Verne), 154n3 Julian calendar, 154n1 Kapteyn, Jacobus, 164n2 Kennefick, Daniel, 157n18 kinetic energy, 83–85, 107, 153n4, 161n6 Kohlrausch, Rudolf, 153n6 Kolkowitz, Shimon, 165n2 Lagrange, Joseph Louis, Comte de, 160n5 Lagrangians, 83–84, 91 Langellier, Nicholas, 165n2 Laplace, Pierre-Simon de, 40, 108, 110–12, 156n3, 163n3 Large Hadron Collider (LHC), 120n, 127n Laser Interferometer Gravitational-Wave Observatory (LIGO): California Institute of Technology and, 66, 71; detectors and, 66–74, 126–27, 141, 151, 159n1, 159n2, 160n16; gravitons and, 126–27; LIGO Scientific Collaboration and, 159n2; Louisiana detector and, 2, 68, 72; MIT and, 66, 71–73; project difficulties of, 72–73; Washington detector and, 2, 68, 72; wave interference and, 67–69, 159n3 Laser Interferometer Space Antenna (LISA), 141


Latin, 148n, 153n6 law of acceleration, 85 leading approximations, 33, 95 Leaning Tower of Pisa, 59 least time principle, 81–84, 90n, 93n, 161n8 lectures, 27n, 41, 43, 90n, 157n19 Leyden jars, 26 light: action principle and, 83–84, 86; bending, 27, 80–82, 134; black holes and, 108, 111–12 (see also black holes); blue shift and, 132; dark matter and, 9 (see also dark matter); Doppler effect and, 131–32; E=mc2 and, 2, 40, 110, 117, 119–20, 122, 153n4; Fermat’s least time principle for, 81–84, 90n, 93n, 161n8; as form of electromagnetic waves, 52, 124, 140, 159n3; gravitons and, 124; information from, 140; laser, 52, 66–70, 73–74, 141; law of optics and, 48; matter and, 82; Maxwell’s equations and, 26–27, 46–48, 53, 86, 89, 157n4, 162n1; Michell-Laplace criterion and, 108, 110–12; nature of, 47–48; Newton and, 108; particles of, 82, 84, 108, 111, 124, 133; photoelectric effect and, 29–30; photons and, 18, 30, 41, 52–54, 124–25, 129, 135, 158n8, 162n3, 165n14; pulsars and, 50; refracted, 27, 82; shortest path taken by, 78–82; shows of, 140–41; speed of, 2–3, 26–27 (see also speed of light); strong interaction and, 10; teleology and, 161n8; wave interference and, 67–69, 159n3; wavelengths of, 29; Young and, 159n3 light years, 1, 127 linear regimes, 33, 63, 71 Louisiana State University, 159n2 Lukin, Mikhail D., 165n2 magnetism: fields and, 44–45, 47, 53, 89; Maxwell’s equations and, 26–27, 46–48, 53, 86, 89, 157n4, 162n1; motion and, 18; strength compared to



gravity, 18–20, 37; waves and, 25, 27, 140 Marolf, D., 163n6 mass: acceleration and, 59; black holes and, 70–71, 107–13, 122, 159n8, 160n9; center of, 29, 112, 163n2; composition of universe by, 9; dark energy and, 9; dark matter and, 9, 133, 138; E=mc2 and, 2, 40, 110, 117, 119–20, 122, 153n4; electrons and, 17n, 119–20; Eöt-Wash experiment and, 60; F=G Mm/R2 and, 15, 19–20, 27n, 58; gravitons and, 125, 128–30, 158n8; inertia and, 59–60, 107, 110, 158n6; inverse square law and, 13, 15, 39, 55, 104, 111, 156n2, 158n8; LIGO detectors and, 68–70, 73; momentum and, 9, 107, 115; Newtonian gravity and, 39, 55, 58–60, 107–8, 163n2, 163n5; ordinary matter and, 9; particles and, 93, 108, 110–11, 117, 119, 122, 133, 138, 158n3, 162n3; photons and, 158n8, 162n3; positive force and, 17; protons and, 16, 17n, 24, 109; as quantity of matter, 59; solar, 23–24, 159n8; stars and, 125 Massachusetts Institute of Technology (MIT), 66, 71–73 mathematics, ix-x; algebra, 55n, 107; American school and, 157n14; c=fλ, 27; calculus, 19, 83, 143, 162n1, 165n2; Cartesian coordinates, 143, 144; differential equations, 46n, 77, 78, 85, 143, 162n10; E=mc2 , 2, 40, 110, 117, 119–20, 122, 153n4; Einstein-Hilbert action, 94–95; equations of motion and, 78, 85–86, 89–90, 95, 97–98, 100, 102, 134, 137–38, 162n11; Euclid and, 62, 79, 160n2; exponentials, 155n5; F=e2 /R2 , 17, 23; F=G Mm/R2 , 15, 19–20, 27n, 58; F=ma, 15, 19–20, 27n, 58, 78; Faraday and, 45–46; field equations, 98; functions, 31, 33, 63–64, 135, 145–49, 158n11, 160n5, 166n14; geometry, 62, 79, 93, 102–3, 135, 143–46, 150, 160n2, 163n3;

integration, 19n, 83–84, 91, 93n, 162n1, 162n11; inverse square law, 13, 15, 39, 55, 104, 111, 156n2, 158n8; Lagrangians, 83–84, 91; leading approximation, 33, 95; Maxwell and, 45–46; Michell-Laplace criterion, 108, 110–12; notation, 16, 148–49, 153n4, 153n6, 156n2, 166n14; partial differential equations, 46n matter: action principle and, 83–84 (see also action principle); atoms and, 3, 9–11, 14, 17, 22–23, 70, 109, 135, 141, 151n, 154n8, 165n2; dark, 9, 131–34, 138, 165n6; E=mc2 and, 2, 40, 110, 117, 119–20, 122, 153n4; gravitons and, 130; gravity and, 38 (see also gravity); Jeans instability and, 154n6; light and, 82; molecules and, 9, 18–19, 37, 135, 154n2, 164n3; ordinary, 9; quantum field theory and, 48; spacetime and, 62, 92–94 Max Planck Society, 159n2 Maxwell, James Clerk, x; Coulomb and, 49; Einstein and, 155n2; electromagnetic waves and, 26, 47–48, 140; electromagnetism and, 25–29, 40, 45–49, 53, 56, 73, 86–87, 89, 126, 140, 155n2, 157n4, 162n1; Faraday and, 45–47; law of optics and, 48; light and, 48; mathematics and, 45–46 media, 62, 113, 135, 137 Mercator projection, 57, 65, 102 Michell-Laplace criterion, 108, 110–12 Michelson-Morley experiment, 157n4, 159n4 Milgrom, Mordehai, 134 Milky Way, 13, 109, 133, 138 Minkowski, Hermann, 143, 146–47, 149–50, 162n1 Misner, C., 153n1, 158n9 modified Newtonian dynamics (MOND), 165n8 molecules, 9, 18–19, 37, 135, 154n2, 164n3 momentum, 9; inertia and, 59–60, 107, 110, 158n6; position and, 115

Index moon, 15, 21, 38–40, 59, 156n3, 163n2 motion, 165n5; acceleration and, 10n, 23, 58–59, 85, 120, 127, 134, 136; action principle and, 83–84, 161n8; additive speed and, 52–53; black holes and, 95; blue shift and, 132; circular, 18, 50, 71, 113, 141, 147, 156n3; differential equations of, 85; Doppler effect and, 131–32; environment and, 61; equations of, 78, 85–86, 89–90, 95–98, 100, 102, 134, 136, 137–38, 162n11; expansion of universe and, 22, 94, 100, 135–37, 147–48, 150; galaxies and, 39, 131–32, 134; inertia and, 59–60, 107, 110, 158n6; initial position and, 162n10; local time and, 162n11; magnetism and, 18; momentum and, 9, 107, 115; nature of time and, 54–55; Newton and, 58–59, 78, 85–86, 89, 97, 133–34, 156n3, 161n8, 162n11; observation of, 52–53; orbital, 50, 71, 113, 141, 147, 156n3; position and, 3, 78, 115, 162n10; rotational, 49, 131–32, 134; spacetime and, 92–93; special relativity and, 97 (see also special relativity); speed of light and, 2–3, 26–27, 40–41, 48, 52–56, 86, 108, 138, 147, 153n4, 157n5, 157n6, 159n4; uniform, 52–53, 88; velocity and, 58, 78, 107–8, 153n4, 162n10, 163n5 multi-messenger astrophysics, 151 Museum of Man, 155n1 naturalness dogma, 40–41 Navier, Claude-Louis, 155n2 Nazis, 155n7 neutrinos, 165n14 neutrons, 9, 12, 14 neutron stars, 49, 151 Newton, Isaac: acceleration and, 85, 134; action at a distance and, 38–39, 44, 48–49, 55; action formulation and, 77, 85–86, 161n8, 162n11; action principle and, 85–87, 161n8; Aristotle and, 58–59, 142; Bentley and, 38–39; black holes and, 163n5; corpuscles of,


108, 164n2; dark matter and, 133; death of, 154n1; differential formulation of, 77–78; Einstein and, 5, 44, 49, 55, 59, 86–87, 95–97, 104, 108, 111, 125, 133–34, 136, 142, 147, 158n8; F=G Mm/R2 and, 15, 19–20, 27n, 58; F=ma and, 78; Faraday and, 125; forces and, 15, 19–20, 22, 32, 39, 44, 55, 58, 78, 104, 132, 142; gravitational constant and, 15, 58, 94, 136; gravity and, 4–5, 15, 23, 27n, 38–40, 58–59, 62n, 95, 97, 104, 107, 111, 126, 130, 133–34, 136, 147, 156n3, 158n8; Hooke and, 20–21; inverse square law and, 13, 15, 39, 55, 104, 111, 132, 156n2, 158n8; light and, 108; local time and, 162n11; location of hell and, 21–22; mass and, 59–60; massless particles and, 158n3; modified Newtonian dynamics (MOND) and, 165n8; motion and, 58–59, 78, 85–86, 89, 97, 133–34, 156n3, 161n8, 162n11; optics and, 48; Principia and, 21, 39; rotation of galaxies and, 134; spacetime and, 4, 55, 94, 96, 111, 142; time and, 39, 54–55, 95, 96; two superb theorems of, 19–21 Newtonian mechanics, 83, 85, 107, 126 New Yorker, 91, 163n9 Nobel Prize, 90n, 151 nonlinear regimes, 33 nonrelativistic classical physics, 117–18 nonrelativistic quantum physics, 117–18 nostrification, 98–99 notation, 16, 148–49, 153n4, 153n6, 156n2, 166n14 nuclear physics, 12, 14, 23 nucleus, 9, 14, 151n numerical relativity, 164n4 obesity index, 108–10 observational protocol, x Old Man’s Toy, An (Zee), ix, 71–72, 154n5, 154n12, 164n5, 165n7, 165n15 Oort, Jan, 164n2 optics, 48



orbital motion, 50, 71, 113, 141, 147, 156n3 Orkin, Ruth, 96 Out of My Later Years (Einstein), 153n4, 156n8 Pais, Abraham, 163n4 partial differential equations, 46n particles: action principle and, 78, 83–84, 161n8, 162n11; antiparticles and, 119–23; charged, 18, 42, 129; dark matter and, 131–39; Dirac-Feynman path and, 162n11; E=mc2 and, 110; escape velocity and, 108; fields and, 40, 42; initial velocity and, 162n10; Laplace and, 40; least time principle for, 82; of light, 82, 84, 108, 111, 124, 133; mass and, 93, 108, 110–11, 117, 119, 122, 133, 138, 158n3, 162n3; material, 82, 84; momentum of, 115; motion and, 78 (see also motion); naturalness dogma and, 40–41; Newton’s laws and, 78; photons and, 124 (see also photons); position and, 3, 78, 115, 162n10; spacetime and, 62, 92–95; standard theory of, 133 phase, 67–68 photoelectric effect, 29–30 photons: dark energy and, 135; electromagnetic force and, 129; gravitons and, 30, 41, 124–25, 129, 158n8; infrared, 18; light and, 18, 30, 41, 52–54, 124–25, 129, 135, 158n8, 162n3, 165n14; mass and, 158n8, 162n3; paradox of, 54; Planck’s constant and, 30; spacetime and, 41; speed of, 2–3, 26–27, 40–41, 48, 52–54, 56, 86, 108, 138, 147, 153n4, 157n5, 157n6, 159n4 Pikovski, Igor, 165n2 Planck, Max, 30, 154n7, 155n10, 158n2 Planck (Brown), 155n1, 155n10, 164n1 Planck energy, 127 Planck number, 23–24, 127, 164n1 Planck’s constant, 30, 115, 116n platinum, 151

poetry, 31, 77 Poincaré, Henri, 157n6, 166n7 Poisson, Siméon Denis, 46, 157n14 position, 3, 39, 78, 115, 162n10 positrons, 119–20 potential energy, 83–85, 107, 110, 161n6 Principia (Newton), 21, 39 protons: atomic number and, 151n; attraction to electrons, 22; dark energy and, 135; electric charge and, 11; hydrogen and, 22; Large Hadron Collider and, 120n; mass of, 16, 17n, 24, 109; as nucleons, 9; quarks and, 9, 22, 48, 108n; repulsion between, 18; special relativity and, 118; star formation and, 12, 23; strong attraction and, 12, 14, 16–18 Prussian academy, 26 Ptolemy, 165n1 pulsars, 50 Pustovoit, V. I., 67 Pythagoras, 143–47, 166n5 QED (Feynman), 160n1 quantum clouds, 11 quantum crank, 126–28 quantum electrodyamics, 126–28 quantum entanglement, 38n quantum field theory: electromagnetism and, 48, 118–23, 156n3, 158n8, 158n10, 163n3, 164n3, 164n4; electrons and, 48; geometrical invariants and, 163n3; gravitons and, 123, 156n3; Hawking radiation and, 119, 123; as marriage of quantum mechanics and special relativity, 120–21; matter and, 48; modern formulation of, 163n3; nothingness and, 119–20; physical reality and, 48; spacetime and, 123, 163n3; vacuums and, 119, 164n3 Quantum Field Theory in a Nutshell (Zee), 33n, 157n16, 159n6, 162n11, 164n4, 164n5, 164n8, 165n12, 165n13, 166n11 quantum gravidynamics, 128

Index quantum gravity, 123–24, 127–30 quantum gravydynamics, 164n7 quantum mechanics, ix-x; action at a distance and, 38n; action principle and, 160n5, 161n7; classical physics and, 4, 30, 115, 117, 118, 123–24, 126n; difficulty of, 166n10; Einstein and, 29, 142; electromagnetic waves and, 29–30, 155n9; electrons and, 117–20, 123; energy and, 120–21; Feynman and, 160n5; gravitational fields and, 30, 123–25, 130; gravitons and, 123–30, 156n3; gravity waves and, 30; Hawking radiation and, 110, 115–17, 119, 121–23; Heisenberg and, 115, 118; Hertz and, 29; nothingness and, 119–20, 158n8; path integrals and, 162n11; photoelectric effect and, 29–30; Planck and, 23–24, 30, 109, 115–16, 127, 154n7, 155n10, 164n1; special relativity and, 116–21, 164n4; uncertainty principle and, 115–16, 119, 121–22; ushering in era of, 29–30 quantum states, 164n3 quarks, 9, 22, 48, 108n radio waves, 10n, 26–29, 140 red shift, 132 Renn, J., 153n3 repulsion, 12–18 Riemann, Bernhard, 64n, 94, 143, 145 Riemann curvature tensor, 150, 162n4, 166n16 Riemannian geometry, 146 Riemann surface, 146 Road to Relativity, The (Gutfreund and Renn), 153n3 round off errors, 18 Royal Institution, 43 Royal Prussian Academy of Sciences, 98 Royal Society, 43 Rubin, Vera, 131–32, 165n4 Samuelson, Paul, 90n satellites, 141 Schwarzschild, Karl, 111–13


scientific notation, 16 Shapley, Harlow, 139 silver, 151 solar mass, 23–24, 159n8 solar system, 134, 136 song of the universe, 1–2 spacetime: action principle and, 92–95; black holes and, 1–2, 66–74, 94–95, 111–14, 125, 150; Cartesian coordinates and, 143, 144; compact notation and, 148–49; cosmological constant and, 103, 135–37; curved, 2, 33, 41, 57, 61–65, 78, 92–97, 102, 111–12, 114, 123, 133, 142–50; dark matter and, 133, 135, 137; Einstein and, 2, 4, 33, 55–57, 61–63, 70–74, 94–98, 102–3, 111, 114, 123, 135, 141–43, 146–50, 157n6, 158n8; Einstein-Hilbert action and, 94, 98, 102–3, 135; elasticity of, 4; energy and, 2, 94; expansion of universe and, 22, 94, 100, 135–37, 147–48, 150; fields and, 41, 45, 55; flat space and, 62, 143–49; funnel image of, 113–14; gravitational fields and, 137; gravitons and, 123, 125, 127, 129, 142; gravity waves and, 63–64, 66–74, 149–50; matter and, 62, 92–94; meaning of, 143–50; Mercator projection and, 57, 65, 102; metric for, 63–65, 149–50; Minkowski and, 143, 146–47, 149–50, 162n1; motion and, 92–93; Newton and, 4, 55, 94, 96, 111, 142; particles and, 62, 92–95; photons and, 41; quantum field theory and, 123, 163n3; Riemann and, 64n, 94, 143, 145–46, 150, 162n4, 166n16; ripples in, 1, 65–73; special relativity and, 2–3, 40, 51, 55–56, 87–89, 95, 97, 116–21, 125, 146, 153n4, 156n5, 159n4, 164n4; speed of light and, 41, 56, 146–47, 157n6; symmetry and, 147; universality of gravity and, 61, 133 special relativity, 156n5; action principle and, 97–98; completion of, 51; E=mc2 and, 2, 40, 110, 117, 119–20, 122,



153n4; energy and, 2–3, 117, 119, 125; gravitons and, 125; gravity and, 95, 97, 121, 125; Hawking radiation and, 121; Minkowski’s time coordinate and, 146; Newtonian action and, 87; protons and, 118; quantum mechanics and, 116–21, 164n4; spacetime and, 2–3, 40, 51, 55–56, 87–89, 95, 97, 116–21, 125, 146, 153n4, 156n5, 159n4, 164n4; speed of light and, 3, 40, 52–53, 56, 153n4, 159n4; symmetry and, 88–89 speed of light: as absolute limit, 40–41; black holes and, 108; dark matter and, 138; electromagnetic waves and, 26–27; escape velocity and, 108; gravitons and, 2; gravity waves and, 40–41, 56, 154n10, 156n3, 156n6, 157n6; as intrinsic property of nature, 53–54; measurement of, 48; Michelson-Morley interferometer and, 157n4, 159n4; nature of time and, 54–55; Newtonian physics and, 86; observation of, 52–56; as possible limit, 3; radiation and, 153; spacetime and, 41, 56, 146–47, 157n6 spherical bastards, 164n3 Stanford University, 159n2 stars, x; binary, 50, 71; black holes and, 151 (see also black holes); blue shift and, 132; electrons and, 12, 23; formation of, 12, 23, 154n6; frozen, 108n; galaxies and, 131 (see also galaxies); gravity and, 133; mass and, 125; mechanistic view of, 38; neutron, 49, 151; protons and, 12, 23–24; pulsars and, 50; Sun, 10, 12–13, 24, 39, 71, 95, 109, 138–39, 141, 160n9 Stevins, Simon, 158n2 Stokes, George, 155n2 string theory, 9n, 23, 46, 92, 128, 154n8 strong interaction, 10–14, 23, 86 Sun, 10, 12–13, 24, 39, 71, 95, 109, 138–39, 141, 160n9 superposition, 67 symbolism, 46

symmetry, 102; action principle and, 88–90; as dictating design, 103–4; Einstein and, 88; inverse square law and, 103–4; spacetime and, 147; special relativity and, 88–89 Syracuse University, 159n2 TAMA 300, 160n16 Taylor, J., 50 telecommunications, 157n15 telegraph, 29 teleology, 161n8 television, 29 theology, 161n8 theoretical physics, x, 165n13; action at a distance and, 38–39, 44, 48–49, 55–56; action principle and, 134 (see also action principle); American school of, 157n14; antigravity and, 137; cosmological constant and, 103, 135–37; Einstein’s field equations and, 97–98; fancy mathematics and, 46; Faraday’s field concept and, 41; Feynman and, 160n5; fundamentals of, 85, 91, 148; gravitons and, 125, 128; gravity waves and, 4, 154 (see also gravity waves); Hawking radiation and, 121; Hilbert’s field equations and, 98; histories and, 82; index notation and, 148; Minkowski metric and, 149; Newton and, 156n2; nostrification in, 98–99; open mind for, 134; poetry and, 77; quantum crank and, 127; quantum mechanics and, 127–28, 160n5 (see also quantum mechanics); simplicity of Nature and, 148; taking things to extremes and, 3–4; universality and, 60–61; Wheeler and, 62 Thorne, Kip, 49, 151, 153n1 time: curved, 95, 96; energy and, 115–16; Fermat’s least time principle for light and, 81–84, 90n, 93n, 161n8; Lagrangians and, 83–84, 91; Minkowski and, 146–47; nature of, 54–55; speed of light and, 54–55. See also spacetime

Index transmitters, 26 Traveling at the Speed of Thought (Kennefick), 157n18, 160n12 Treiman, Sam, 158n10 truth, 51, 91, 137, 142 U.K. Science and Technology Facilities Council (STFC), 159n2 uncertainty principle, 115–16, 119, 121–22 Unity of Forces in the Universe (Zee), 154n8 University of Adelaide, 159n2 University of Florida, 159n2 University of Washington, 60 University of Wisconsin Milwaukee, 159n2 uranium, 14, 151 vacuums, 38, 68, 70, 119–21, 164n3 velocity, 58, 78, 107–8, 153n4, 162n10, 163n5

Verne, Jules, 154n3 Virgo Collaboration, 159n2, 160n16 Walsworth, Ronald L., 165n2 water waves, 5, 31–32, 63–64, 66–67, 125n, 143, 149 wavelength, 25, 27–29, 67–68 weak interaction, 10–12, 13, 86 Weber, Joseph, 74, 153n6, 160n15 Weinberg, Steve, 90n Weiss, Rainer, 73, 151, 159n1 Wheeler, John, 49, 62, 108, 153n1, 158n9, 158n10 Wightman, Arthur, 158n10 Wikipedia, 155n4 World War I, 112 Ye, Jun, 165n2 Young, Thomas, 159n3 zinc, 26 Zwicky, Fritz, 132, 164n3