A Zee - On Gravity _ A Brief Tour of a Weighty Subject-Princeton University Press (2018)

Prévia do material em texto

on
gravity
on
gravity
a brief tour of a
weighty subject a. zee
Princeton University Press 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
http://www.press.princeton.edu
To all those who taught me about gravity
Contents
Preface ix
Timeline xiii
Prologue: The song of the universe 1
Part I
1 A friendly contest between the four interactions 9
2 Gravity is absurdly weak 15
3 Detection of electromagnetic waves 25
4 From water waves to gravity waves 31
Part II
5 Spooky action at a distance 37
6 Greatness and audacity: Enter the field 40
7 Einstein, the exterminator of relativity 51
8 Einstein’s idea: Spacetime becomes curved 57
9 How to detect something as ethereal as ripples
in spacetime 66
Part III
10 Getting the best possible deal 77
11 Symmetry: Physics must not depend on the physicist 88
12 Yes, I want the best deal, but what is the deal? 91
13 The action for Einstein gravity 100
14 It must be 102
viii Contents
Part IV
15 From frozen star to black hole 107
16 The quantum world and Hawking radiation 115
17 Gravitons and the nature of gravity 123
18 Mysterious messages from the dark side 131
19 A new window to the cosmos 140
Appendix: What does curved spacetime mean? 143
Postscript 151
Notes 153
Bibliography 167
Index 169
Preface
After writing a massive textbook on Einstein gravity, called
appropriately enough Einstein Gravity in a Nutshell and
referred to henceforth asGNut, 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 long-
time 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 longbookonEinstein gravity, I hada license towrite
a short book on Einstein gravity.
I had also published in 1989 a popular book about
Einsteingravity titledAnOldMan’sToy and later republished
as Einstein’s Universe: Gravity at Work and Play, referred to
henceforth asToy. 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 math-
ematically 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, youcould still enjoy this bookas apopularbook
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 nomen-
tion 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 pub-
lic. 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
gravitywaves, at leastwith thebenefitofhindsight. For a short
book such as this, I am obliged to pick and choose.
Preface xi
Acknowledgments
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 manu-
script 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 computerwas indispens-
able. 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 theÉcoleNormale Supérieure for their hospitality,
and Jean-Philippe Bouchaud for financing my chair through
the Foundation of the ÉcoleNormale Supérieure. Needless to
say, but as always, I appreciate the support of mywife, 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
JosephWeber 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 rathermalevolent but somewhat clever species, cannow
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 firstdemand 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. Histori-
cally, 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 JohnWheeler
(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 waveWheeler and
I described might be detectable with a then-existing appara-
tus. 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
50 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 physi-
cists, 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 rel-
ativity 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 con-
fusion long since cleared up. Probably the worst name ever
is relativity, as it has spawned a swarm of nonsensical state-
ments, 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.”
52 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 sec-
ond 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 under-
standing of light being a particular form of electromag-
netic 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 53
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 themagnetic
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 cal-
culation 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 mea-
sured 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 every-
day intuition: the observed speed of light is independent
of how fast the observer is moving. Suppose we see a pho-
ton 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 aston-
ishment 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.
54 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 at10 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 pas-
sage 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 passen-
ger, precisely 1 second has also elapsed for the stationmas-
ter.” Common sense fails!
Einstein, the exterminator of relativity 55
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, know-
ing how the gravitational field varies in space, we imme-
diately 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.
56 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 prop-
agates, 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
Amysterious 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
flightmap on the video these days), youmight 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 fol-
lows 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.
58 Chapter 8
LAX
TPE
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 dis-
tance R separating them: F = GMm/R2.
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 bym: 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 every-
day life, alas, is dominated by friction, pain, and suffering.
Aristotle appears to be right, and Newton wrong. But in
Einstein’s idea 59
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
massm of the moon at all. The force F is proportional tom,
the acceleration a is given by the force F divided bym; ergo,
the acceleration a does not depend onm.
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 massm appearingin his law
of gravity and the massm appearing in his law of motion are
one and the same.
But a hair-splitting lawyer, or a habitual reader of myster-
ies, 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 twomasses really the samemass?
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
60 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 au-
thority, not even a single-named giant in a likely apoc-
ryphal 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 de-
gree 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 experi-
ment.5 Nerd humor in full force here!
Universality explained
That all objects fall at the same rate is known as the univer-
sality 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 the-
oretical 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 na-
tional or political affiliation, choosing exactly the same path
getting from Los Angeles to Taipei. An apple or a stone
Einstein’s idea 61
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” connec-
tion, but hindsight7 is of course way too easy.
For 300 years, the universality of gravity8 has been whis-
pering “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 equiv-
alent 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.
62 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 JohnWheeler, 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 gravi-
tational 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 63
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, com-
plicationsmanage to befuddlemany 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.
64 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 con-
cerned.
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 ofgravity waves), these ten fields g (t, x, y, z)
are constant and equal to a simple number. The slight com-
plication is that, of the ten, three are equal to 1, one is equal
to−1, and the rest equal to 0. (Aren’t physics andmath 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 dis-
tance 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 65
GREENLAND
CHINA
Figure 2. Is Greenland bigger than China?
On aworldmap inMercator 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 themetric 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 Techno-
logy and the California Institute of Technology and involv-
ing 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 sep-
arated 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 67
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 situa-
tions, 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 wave-
length. (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.
http://www.physics-animations.com
68 Chapter 9
If the two waves are out of phase by exactly half a wave-
length (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 69
Test
mass
Test
mass
Laser source
Beam
splitter
photodetector
Test
mass Test
mass
4 
km
4 km
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.
70 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 smallas 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 dis-
tance the two laser beams would have to travel. You can
imagine how precisely themasses 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 happen-
stance 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 71
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, emit-
ting 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 pre-
cise 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 wave-
forms that the various parameters of the black hole binary,
such as the masses of the two black holes and their dis-
tance 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 Tech-
nology have jointly asked the U.S. government to fund
an ultrasensitive detector that should be able to pick up
72 Chapter 9
Hanford, Washington (H1)
Time (s) Time (s)
Livingston, Louisiana (L1)
St
ra
in
 (1
0–2
1 )
1.0
0.5
–0.5
1.0
–1.0
0.0
1.0
0.5
0.30 0.35 0.40 0.45 0.30 0.35 0.40 0.45
–0.5
1.0
–1.0
0.0
L1 observed
H1 observed (shifted, inverted)H1 observed
Numerical relativity Numerical relativity
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 andMIT, and are almost maximally separated in the
https://creativecommons.org/licenses/by/3.0/us/legalcode
Ripples in spacetime 73
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 detec-
tors 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 sev-
eral 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 electromag-
netism. 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,
74 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 con-
sisted 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 claimedthat 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 illumi-
nate 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 cen-
tury as an alternative to Newton’s differential formulation of
physics.
78 Chapter 10
In Newton’s view of motion, one focuses on the moving
particle at every instant in time. A force acting on the par-
ticle causes the particle’s velocity to change according to
Newton’s law, F = ma. Knowing the particle’s accelera-
tion 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 begin-
ning 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 equa-
tions 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 franti-
cally. She is going under! Must be a cramp! An action hero
Getting the best possible deal 79
sand
water
L
M
F
C
G
x
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 pro-
ceeding 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
80 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 spendmore 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 eter-
nally 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 81
1H
H1
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 swim-
ming 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 movesmore 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 readthese
very words, you have (or rather your saintly mother has)
cleverly positioned in your eyes a blob of watery substance
82 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, physi-
cists 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 parti-
cle 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 contem-
plated (such as Napoleon defeating Wellington),4 but some-
how 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 83
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 any-
thing 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 en-
ergy 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 pro-
vide 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 parti-
cles. 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.
84 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 dis-
tinct 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 mini-
mize 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 prin-
ciple, 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 pro-
duction 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
* In math symbols, S = ∫ dtL . Traditionally, the letter S is used for the action,
and L for the Lagrangian.
Getting the best possible deal 85
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 accel-
eration 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 formula-
tion actually is more convoluted than the differential formu-
lation, 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 differ-
ential 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.
86 Chapter 10
All physical theories established since Newtonmay be for-
mulated in terms of an action. The fundamental interactions
we know about, the strong, weak, electromagnetic, and grav-
itational, 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 describinghow 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 (corre-
sponding 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 cor-
rect 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 87
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 over-
arching 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 rel-
ativity 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 gener-
alized 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 prin-
ciple, 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
Symmetry 89
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 equa-
tions, such as Newton’s equation of motion or Maxwell’s
equation of electrodynamics. Under a symmetry transfor-
mation, 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 chargesmoving, 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 chang-
ing”). The two sides of the equation change in the same way,
rather than remain unchanged. As a result, while the phys-
ical quantities involved change, the structural relationship
between them does not.
90 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. Unfor-
tunately, 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 elec-
tromagnetism 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 in-
variant under the transformation associated with that sym-
metry. 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 indivi-
dual. 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?
92 Chapter 12
Once you decide what you want maximized, we could fash-
ion 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 space-
time, 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? 93
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 par-
ticle. 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 par-
ticle multiplied2 by the distance it traverses. More massive
particles generate more of an action.3 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.
94 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, space-
time becomes more curved. You have to decide what space-
time 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 space-
time, known as the Einstein-Hilbert action, looks like, here
it is:
S =
∫
d4x
√
g R/G
Here R denotes the curvature4 of spacetime, with the letter
R chosen to honor Riemann; G is Newton’s gravitation con-
stant. 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 space-
time 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? 95
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 result-
ing 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 inti-
mately 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 ac-
tion 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? 97
of gravity. Instead, he followed various clues on what the
equations of motion for gravity must look like. For ex-
ample, one clue would be that in the appropriate regime,
these equations of motion must reduce to Newton’s equa-
tion 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 mathemat-
ical 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 equa-
tions 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 livedmost 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 fromEinstein’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
98 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. EINSTEIN
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? 99
Well, dear reader, nostrification is not only still practiced
in theoretical physics, but ever more skillfully.9
Tomake the whole episode all themore 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 develop-
ment 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 find-
ing 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 equa-
tions of motion, entire blackboards filled with symbols, as
depicted by cartoonists.
The action for Einstein gravity 101
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 sec-
ond 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 struc-
ture 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 invari-
ance,” 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, Green-
land 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 geomet-
rical 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 103
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 space-
time. 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 rel-
evant quantity the “volume of spacetime,” for lack of a better
term. Thus, we could add to the Einstein-Hilbert action the
term
∫
d4x√g�. As explained in the appendix, this is equal
to the volume of spacetime multiplied by an unknown con-
stant, 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 underly-
ing gravity was discerned, physics was literally forced to
Einstein’s theory. Einstein’s theory of gravity carries with it
a sense of the inevitable.
104 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 grav-
ity 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,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 out-
ward, 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.
2 Prologue
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 incomprehen-
sible 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, kilome-
ters 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 thepulsewithamillisecond 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
In1915, as theseparticulargravitonsapproachedearth—after
1.3 billion years, only a hundredmore years to go!—an earth-
lingnamedAlbertEinstein (1879–1955)finally completedhis
theoryof gravity, alsoknownas 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 rel-
ativity: E = mc2, 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 3
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 thewater towave 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 atomscrowd theirneighborsdownthe line,
who are in turn compressed. Thus information gets trans-
mitted 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 thewavemoves.Youcould think
of rigidityasameasureof theeagernessof theatoms tobounce
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
4 Prologue
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 onceEinsteindeclared that spacetime is elas-
tic, not absolutely rigid, gravitywaves 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 ques-
tion, but it would be more accurate to ask why experiments
are so far behind the theory in this case.* In physics, some-
times 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 nec-
essary 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 5
fantastically weak. To understandwhy, we need to appreciate
that, in spite of our everyday experiences, gravity is fantasti-
callyweak.This factwill be explained in thenext twochapters.
You say gravitational wave, I say gravity wave
Youmight 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 “grav-
ity waves” long before Einstein came onto the scene. The
excesswater in the crest of awave is pulleddownby 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.
Iwas curiouswhich termpopularphysics bookswoulduse.
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
interactionsgiving 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 massm on its surface,
namely, GMm/R2, 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 1
2mv2 (where v denotes its veloc-
ity), is less than its gravitational potential energy GMm/R
at the surface of the planet. One line of high school algebra
shows that this “no escape” criterion works out to be v2 R
If this inequality is satisfied, then the object is a black
hole.4 For the gravitational pull to be excessively strong,
either GM is unusually large, or R is unusually small. For an
object to qualify as a black hole, it either has to bemassive for
its size, or small for its mass. (Describes obesity, no?)
Remarkably, even though the physics behind the Michell-
Laplace argument is not correct in detail (as we now know,
we should not treat light as a Newtonian “corpuscle” with a
tinymass),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 109
–20
20
40
universe
Milky Way
sun
earth
human
atom
Planck
–30 –20 –10 10 20
proton
log10 meter)R
log10 gram)M
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 coun-
tries, 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
110 Chapter 15
the straight line representing the equality 2GM/c2 = R.
Anything above this line, that is, inside the shaded area,
would have 2GM/c2 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 = mc2. At a distance of R from an object of mass M, a
particle of mass m feels a gravitational potential energy of
GMm/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 = mc2. Thus, when the gravita-
tional potential energy gets to be comparable to this charac-
teristic 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 some-
thing’s got to give. This critical state of affairs is reached
when GMm/R � mc2. (The symbol means roughly equal.)
Think ofmc2 as the particle’s inner reserve. Again,m cancels
out (the celebrated equality between inertial mass and grav-
itational mass.) We recover more or less the same Michell-
Laplace criterion: GM/c2 � 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 111
The horizon of a black hole
As you see, the war treated me kindly enough, in spite of
the heavy gunfire, to allowme to get away from it all and
take this walk in the land of your ideas.
—KARL SCHWARZSCHILD, WRITING TO
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 speak-
ing. Light is trapped inside. Material particles are trapped a
fortiori, since they cannot move faster than light.
The Michell-Laplace criterion 2GM/c2 > R emerges
from Einstein theory as follows.
You sit down to solve Einstein’s equations around a mas-
sive 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 themassive object, the requirement that
Einsteinmust agree withNewton 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.
112 Chapter 15
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 2GM/c2 from the object,
spacetime becomes so curved that it can trap light.
Since you solved Einstein’s equations in empty space out-
side the massive object, this distance 2GM/c2 is relevant
only if it is outside themassive object, that is, only if 2GM/c2
is larger than the radius R of the object.
The Michell-Laplace criterion 2GM/c2 > R for a black
holepops out.
To further clarify this discussion, let us understand why
the earth is not a black hole. We think of the earth as incred-
ibly massive in the context of everyday life, but M for the
earth is pretty small in this context. The quantity 2GM/c2 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 2GM/c2 defines
a distance known as the horizon. If your distance from the
center of the object is less than the horizon 2GM/c2, then
you are doomed. By the way, the term “horizon” is a felic-
itous 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 space-
time 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 inWorldWar I. (He died a year later.) I always
From frozen star to black hole 113
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 count-
less 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 direc-
tion, 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
114 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 ex-
ternally 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 clas-
sical 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 tomore un-
certainty 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
116 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 quantummechanics
Uncertainty principle: �E ∼ h̄/�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 h̄/�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 h̄, 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 117
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 = mc2
Advertising slogan: “Accounting errors can be
turned into stuff!”
Energy can be converted into mass and hence parti-
cles according to Einstein’s famous equation E = mc2. Our
proverbial embezzler could turn an accounting error into a
Lamborghini. But only in his dreams if the world is non-
relativistic.
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 noth-
ing much happens. Yes, an energy fluctuation could be con-
verted into particles, but there is no energy fluctuation in the
first place.
We have talked about two fascinating worlds, both stra-
ngely remote from our everyday world(which is governed by
118 Chapter 16
FAST
SLOW
BIG SMALL
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, some-
thing new could happen!
Now, accounting errors abound, and they can be turned
into stuff.
When physicists combined quantum mechanics and spe-
cial relativity, around the middle of the past century, an ex-
citing 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 119
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
notmerely 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 corres-
ponding 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 vac-
uum �E exceeds 2mc2, 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 annihi-
late each other.
Indeed, there is nothing special about the electron in this
discussion. You see that is why physicists think of nothing-
ness as a roiling sea of pairs of particles and antiparticles of
every imaginable description, popping in and out of exis-
tence. 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/(2mc2), as some readers might realize.
120 Chapter 16
it E , way more than 2mc2. 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 account-
ing 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 re-
striction 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 2mc2, taking the en-
ergy needed out of the two colliding electrons, which end
up with some energy less than E − 2mc2. Thus, with two
energetic electrons, we could end upwith three electrons and
a positron. The process is known as pair production and is
observed routinely in the lab.
The marriage of quantummechanics 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 elec-
trons 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 ac-
celerators 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 rela-
tivity 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 121
please!—known as quantum field theory.4 It exhibits qualita-
tively 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 fluctua-
tion is in the vicinity of a black hole?
What do we mean by vicinity? Recall our earlier discus-
sion of the horizon of a black hole. Picture a quantum fluc-
tuation near the horizon, producing a particle and its an-
tiparticle. 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 pre-
ceding 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 parti-
cles 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.
122 Chapter 16
A quick summary of how Hawking radiation arises. If
we are nowhere near a high energy collider, the particle-
antiparticle 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 prin-
ciple, 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 = mc2
between mass and energy.
17
Gravitons andthe
nature of gravity
The marriage of quantummechanics 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 speak-
ing, 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 talk-
ing 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, space-
time 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 mechan-
ics 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
124 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 pack-
aged 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 al-
ready mentioned in chapter 3.) The photon2 is the funda-
mental 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 con-
sists 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, experimental-
ists 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 125
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 gravi-
tons. 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 differ-
ence that causes theoretical physicists no end of trouble. The
seed for this difference is already sown by Einstein’s the-
ory of special relativity, which states thatmass and energy are
the same.
Consider a massive object, such as a star. The mass gener-
ates 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 beget-
ting 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.
126 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 lin-
ear 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
holesmerging. 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 calcu-
late 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 classi-
cal theory into a quantum theory. Thus, when we quantize
Newton’s classical mechanics, we obtain quantum mechan-
ics, and when we quantizeMaxwell’s classical electrodynam-
ics, 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 127
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 fluctua-
tions 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
from1.3 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 understand-
ing 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.
128 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 quan-
tum 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 129
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. (Simi-
larly, 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.
Amoral imperative but not a practical necessity
Some readers may be justifiably confused at this point.
“You told us earlier that, through the decades, physi-
cists have failed to construct a well-behaved theory of quan-
tum gravity, but now you say that the everyday phenomenon
of gravity can be understood as due to the exchange of gravi-
tons. What is going on?”
In everyday gravity, for example, that almost but not
quite fatal attraction between you and the earth, the gravi-
tons emitted by the earth play nice: each graviton gets to
130 Chapter 17
you without messing around with the other gravitons. Sim-
ilarly, 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 grav-
ity 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.
—VERA RUBIN EXHORTING YOUNG PHYSICISTS
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 in-
stinctively 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 uni-
son, 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
132 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 emit-
ted 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 mea-
sure 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 con-
glomerate known as a galaxy by virtue of their mutual gravi-
tational 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 133
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 elec-
tromagnetism, 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 par-
ticles to the electromagnetic field; that is, let them be
134 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 modify-
ing 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 extreme-
ly tight: it cannot be easily modified without messing up the
various celebrated tests (such as the bending of light) that the
theory has passedwith flying colors, not tomention 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 sug-
gestions (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 135
Dark energy
When Einstein triumphantly completed his theory of grav-
ity, 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 appen-
dix, 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
136 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 uni-
verse. 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 con-
stant term, as its name suggests, has an effect only on cosmo-
logical 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 explana-
tions 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-from-
compelling 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 gravityis
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 137
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 gravita-
tional 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 vol-
ume 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 equa-
tion 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 equa-
tion 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 rip-
ple 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 account-
ing as humorously portrayed: depending onwhether you put
138 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 darkmatter, coldmeaning that the postulated
dark matter particles are moving around much slower than
the speed of light. Current measurements indicate that, of
the total14 energy andmass of the universe, dark energy con-
tributes 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 enor-
mous 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 suggest-
ing that the sun may be as large as the Peloponnesus. Even-
tually, 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 139
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 pas-
sengers 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 galax-
ies 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 discov-
ered light is only one form of electromagnetic waves. With
the development of detectors for the other forms of electro-
magnetic waves, microwave astronomy, radio astronomy,
infrared astronomy, ultraviolet astronomy, X-ray astron-
omy, and gamma-ray astronomy were born one after an-
other. After all, astronomical bodies have no reason to
radiate electromagnetic waves only in those frequencies de-
tectable by certain creatures on a particular speck of a planet.
The universe is humming across the entire electromag-
netic 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 re-
minded of those sound and light shows at tourist attractions.
New window to the cosmos 141
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 as-
tronomy 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 LIGOwould
be able to see anything. In particular, the European Space
Agency has proposed the Laser Interferometer Space An-
tenna (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 kilo-
meters in length, will fly in a near-earth orbit around the
sun. The distances between the three spacecraft are to be ac-
curately measured by laser interferometry in order to detect
passing gravitational waves.
Some fascinating proposals have been aired. One attrac-
tive 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 specifica-
tions, 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!
142 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 homefor 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 = dx2 + dy2
144 Appendix
(x+dx , y+dy )
(x ,y )
dx
ds
dy
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 = dx2 + ( 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 θP and ϕP , respectively.4 Consider a place with the same
longitude as Paris but a slightly different latitude, namely, θP + 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, ϕP + 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 145
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 = dθ2 + ( 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 = dθ2 + dϕ2.
Aha, we’ve got it! From this example, we learned that to go from the flat plane, for
which ds 2 = dx2 + dy2, to a curved surface, we should write7 ds 2 = dx2 + ( f (x) dy)2,
which of course can also be written as ds 2 = dx2 + f (x)2dy2, not ds 2 = dx2 + f 2dy2
with f a constant.
Enter Bernhard Riemann. He said, “now that we have inserted a function in front
of dy2, why not insert a function in front of dx2 also? In fact, why not include dx dy
and insert a function in front of him also? These three functions could all depend on
146 Appendix
both x and y!” So, here is Bernie’s proposal:
ds 2 = a(x, y)dx2 + b(x, y)dx dy +c(x, y)dy2
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 = dx2 + dy2 + 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 = dx2 + dy2 to ds 2 = dx2 + dy2 + dz2, our first guess
might be ds 2 = dx2 + dy2 + dz2 + dt2.
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 dt2, how about subtracting dt2 instead? It seems so naive and childish, but it
turns out to be right. Nature actually works that way. Amazing!
So try ds 2 = dx2 + dy2 + dz2 − dt2.
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 147
Now we are cooking with gas: write down
ds 2 = dx2 + dy2 + 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 = dx2 + dy2 + 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
dt2 as we just did, we could stick a function of time in front of dx2 + dy2 + dz2:
ds 2 = (a(t))2(dx2 + dy2 + dz2)− c2dt2
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(dx2 + dy2 + 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.
148 Appendix
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 dx2, dy2, dz2, and dt2,
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 x1, x2, x3. 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, x1 = x, x2 = y, x3 = 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
x1, x2, · · · ,x26, x27.)
Even better, at this point, you can include the time coordinate t also: simply call
it12 x0. 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
x0, x1, x2, x3, with x0 = t, x1 = x, x2 = y, x3 = z.
Collectively, these four coordinates, x0, x1, x2, x3, can be denoted more compactly
by xμ, where the index* μ takes on the values 0, 1, 2, 3.
I said that theoretical physicists (and mathematicians) got tired of writing quanti-
ties such as dz2 and dy dt. Now they can write simply dxμdxν . As μ and ν separately
take on values 0, 1, 2, 3, the expression dxμdxν ranges over all ten of these quantities.
For example, dx3dx3 = dz2, and dx2dx0 = dy dt.
Using this notation (merely notation: neither physics nor math, nothing profound
at all, just bookkeeping), we can then write themost general curved spacetime concisely
as ds 2 = gμν (x)dxμdxν 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, gμν (x)dxμdxν is short-
hand for13 g00(x)(dx0)2 + g11(x)(dx1)2 + · · · + 2g01(x)dx0dx1 + 2g02(x)dx0dx2 +
· · · + 2g23(x)dx2dx3. 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
andn, 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 149
ten functions that appear in front of dt2, dt dx, dt dy, · · · , dy dz, dz2, we simply
denote them collectively by gμν (x). The ten functions gμν (x) are known collectively
as the spacetime metric: as the jargon suggests, they measure spacetime.
Let me forestall a potential confusion here. The notation gμν (x) is shorthand for
gμν (x0, x1, x2, x3). The letter x is used to denote x0, x1, x2, x3 collectively. In other
words, each of the ten functions gμν (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 gμν (x).
The ten functions actually are not functions, just numbers, and all but four are
equal to 0. These four are g00 = −1, g11 = +1, g22 = +1, g33 = +1. In other
words, ds 2 = −(dx0)2 + (dx1)2 + (dx2)2 + (dx3)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 hang-
ing 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 theMinkowski 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 = −(dx0)2 + (dx1)2 +
(dx2)2 + (dx3)2 more compactly as
ds 2 = ημνdxμdxν
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 + hμν (x)
)
dxμdxν
In other words, we have merely added to the bunch of 1s and 0s in ημν some functions
hμν (x), which we are going to regard as small compared to 1. The metric of this
(slightly) curved spacetime is given by gμν (x) = ημν + hμν (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
150 Appendix
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
gμν (x) = ημν + hμν (x) is structurally14 the same as g (t, x, y) = 1+ h(t, x, y).
Einstein gave us a set of equations* for determining gμν . When we substitute
gμν = ημν + hμν into these equations, they simplify enormously, leaving us with
equations for determining hμν 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 grav-
ity 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. To set the time scale, dinosaurs roamed about 0.24 billionMatter 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 mole-
cules 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 par-
ticles. 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.”
10 Chapter 1
summary of the four forces, known as gravity, electromag-
netism, 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 inter-
actions. 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 quan-
tum 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 electromag-
netism. What we see in everyday life is by and large due to
some residual effect of the electromagnetic force. Since com-
mon everyday objects are all electrically neutral, consisting
of equal numbers of protons and electrons, the electromag-
netic 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 electromag-
netism shakes us is when thunder and lightning fill the sky.
While we modern dudes have totally enslaved electromag-
netism, 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 sum-
marizing centuries of painstaking investigations. For exam-
ple, 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
12 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 electromag-
netic 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 nu-
clear attraction between protons a tiny bit stronger, two pro-
tons 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 pro-
ton 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” sug-
gests. 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 be-
tween two protons abruptly falls to zero as soon as theymove
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/R2, 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
14 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 strongyears ago.
2. See chapter 17.
3. Hence the detection event, the first of its kind, is being cataloged as GW150914.
4. 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
K0 − K1 = L
V2
v2
2
What! It doesn’t look like E = mc2 to you? Einstein is telling us that, when an
object moving at velocity v radiates, its kinetic energy K changes by (in modern
notation) δK = δE
c2
v2
2 . (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, sur-
prisingly, 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.
154 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 physi-
cist 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 4 155
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 eh is approximately h to leading order.
156 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 developmentof 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 cG , 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 7 157
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 eikr /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 Astro-
physics, 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. In German, Einsteinsche Relativittstheorie.
2. P. Galison, Einstein’s Clocks, Poincaré’s Maps.
3. Such as Châtelet Les Halles in Paris.
4. 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.
158 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. I am abusing geography slightly.
2. This celebrated experiment was also performed by Simon Stevins of Bruges.
3. Newtonian physics cannot entertain the existence of massless particles.
4. As to what type of blonde, see the classification and scholarly study Blonde Like
Me by Natalia Ilyin.
5. See https://www.npl.washington.edu/eotwash/node/1.
6. 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.
7. Staircase wit, l’esprit d’escalier, Treppenwitz, firing the cannon after the cavalry
has already charged by you.
8. 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
9. 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.
10. 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 ofmy 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 entiresummer 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.
11. Readers who wish to feast on these niceties could read GNut. It is explained in
detail there how g blossoms into ten different functions.
https://www.npl.washington.edu/eotwash/node/1
http://authors.library.caltech.edu/15184/1/Misner2009p1638PhysToday.pdf
Notes to chapter 9 159
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/press-
release/english.pdf. I quote two passages here:
The discovery ... was made by the LIGO Scientific Collaboration (which in-
cludes 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 Michelson-
Morley interferometer that established special relativity. In that case, the experi-
ment 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
http://news.mit.edu/2016/rainer-weiss-ligo-origins-0211
http://ligo.org/detections/GW150914/press-release/english.pdf
http://ligo.org/detections/GW150914/press-release/english.pdf
160 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.
9. 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.
10. In a later edition, the title was changed to Einstein’s Universe by a new publisher.
11. M. Bartusiak, Einstein’s Unfinished Symphony.
12. Kennefick, Traveling at the Speed of Thought.
13. 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.
14. 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.
15. Richard Garwin, one ofWeber’s most vocal critics, simply built a replica ofWeber’s
detector and showed that he could not pick up any signal. At a physics conference,
Garwin and Weber almost came to blows.
16. 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 10 161
“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 beone 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.
162 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 =∫
d4xF 2. 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 d4x
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 15 163
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 R2, 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 GMm/R2 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
ofa 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.
164 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 quantummechanics 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 165
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/la-
me-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.
http://www.latimes.com/local/obituaries/lame-vera-rubin-20161226-story.html
http://www.latimes.com/local/obituaries/lame-vera-rubin-20161226-story.html
166 Notes to appendix
4. Of course, the French had insisted that ϕP 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 = (dx2 + dy2)/y2
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 x4, but later it was realized that
it was more sensibly written as x0.
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, gμν (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 Space-
Time, 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
170 Index
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 171
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,
172 Index
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=GMm/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 173
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=GMm/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;
174 Index
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=GMm/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 alsomass);
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 175
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
176 Index
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=GMm/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=GMm/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 177
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=GMm/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
178 Index
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 alsomotion);
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 179
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,
180 Index
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 181
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
	Dedication
	Contents
	Preface
	Timeline
	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?
	Postscript
	Notes
	Bibliography
	Indexattraction 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 neu-
trons right next to it, but each proton feels the electric repul-
sion 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 electromag-
netic 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 grav-
itational 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 dis-
tance R separating them. Or, in a more concise language,
F = GMm/R2.
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 prod-
uct of the two charges (namely, q1q2) divided by the square
of the distance R separating them. Or, in a more concise
language, F = q1q2/R2.
A striking mystery: the fall-off of the force with increasing
distance—the 1/R2 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.
16 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 ren-
dered 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 17
them is equal to Fgravitation = Gmp
2/R2, with mp the mass
of the proton. The electric repulsion between them, on the
other hand, is equal to Felectric = e2/R2, where e denotes the
fundamental unit of charge carried by the proton.
Thus, the ratio of the two forces Felectric/Fgravitation =
e2/(Gmp
2). Note that the factors of R2 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 pro-
tons, 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 compen-
sates 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.
18 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 elec-
trons, 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 electro-
magnetism 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 19
molecules cannot escape. The electric force easily over-
whelms the pull of the entire earth.
The escape route is through the top of the glass. Absorb-
ing 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 objectionLet’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 = GMm/R2
to the magnet and the earth, we should mentally cut up the
earth into a multitude of infinitesimal pieces, each some dis-
tance R from the magnet and each pulling on the magnet,
and add up the individual forces.
20 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 for-
mula F = GMm/R2, 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 21
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 shoul-
ders 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 arro-
gant 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 address-
ing 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 physi-
cist colleagues would agree that this represents the simplest
extension of an existing theory. A rudimentary understand-
ing of volcanoes (plus a close reading of the Bible) pro-
vided strong observational evidence, confirming the theory
for sure.
22 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, publishedNewton’s Principia at his expense, to pro-
pose 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 hydro-
gen 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 23
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 hy-
drogen 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 col-
lapse and destined to become a star. In modern understand-
ing, 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 interac-
tion becomes effective. A star is born!
What could work against gravity? In other words, what
must gravity overcome to form structure in the primor-
dial 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. Grav-
ity’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
Felectric/Fgravitation = e2/(Gmp
2) ∼ 1036. Here the electric
force is not in the game, and e2 does not enter. So, we
should measure the feebleness of gravity by the number
1/(Gmp
2) ∼ 1038 = (1019)2. The huge number 1019, which
we might call the Planck7 number, indicates the intrinsic
24 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 emer-
gence 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 humongousnumber 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 electromag-
netic 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, whichmay 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 electro-
magnetic 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,
26 Chapter 3
he deduced the existence of electromagnetic waves. Perhaps
toMaxwell’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 no-
ticed, 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 re-
ceiver 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 ad-
justable gap between the two ends.5 A spark from his trans-
mitter 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 re-
ceiver 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 27
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 electro-
magnetic waves had two polarizations.
Compared to Hertz, people working on gravity wave de-
tection 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 respec-
tive governments to build more detectors. See later.
A window to the outside world
The unit for frequency, the hertz (written asHz), 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 = GMm/R2, 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.
http://www.americanradiohistory.com
1000 m
100 m
10 m
1 m
10 cm
1 cm
1 mm
1000 µm
100 µm
10 µm
1 µm
1000 nm
100 nm
10 nm
0.1 Å
0.1 nm
1Å
1 nm
WavelengthFrequency (Hz)
Gamma-rays
X-rays
Ultraviolet
Visible
Infrared
Thermal IR
Microwaves
Radio, TV
Longwaves
1019
1018
1017
1016
1015
1014
1013
1012
1011
1010
109
108
107
106
Near IR
Far IR
Radar
AM
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 29
I don’t have to belabor the impact of electromagnetic
waves on human civilization. As we all know, the discov-
ery 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 Per-
haps 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.” Andwhen
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
30 Chapter 3
We now understand that an electromagnetic wave of fre-
quency f is actually composed of a stampeding herd of pho-
tons, 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 higherthe frequency, the more vigorous the kick be-
comes. In contrast, in classical physics, the amplitude of the
wave corresponds to the size of the electric field, which deter-
mines 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 photoelec-
tric effect to cease abruptly when the frequency drops be-
low 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 pro-
logue. Indeed, there is an easy parallel between electromag-
netism and gravity: the photon is to the graviton as the elec-
tromagnetic 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 col-
lectively 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 gov-
erning 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 spec-
ified 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 gov-
erning 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.
32 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 them-
selves, 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 nowwrite 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 33
encounter a term involving h multiplied by itself (namely
h2, the square of h) you can chuck that term out the win-
dow. 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 lin-
ear 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 foot-
ball, 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 act-
ing 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 experi-
ence.* 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.
38 Chapter 5
Gravity is the glaring exception. When the earth pulls
Newton’s apple down, no hand comes out of the earth grab-
bing 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 world-
view, 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 distanceis 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 dis-
tance through a vacuum without the mediation of any-
thing 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 39
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 con-
stant 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 =
mc2 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 cG 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 cG 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 cG compare with c?
Laplace supposed (erroneously) that cG is much larger
than c . These days, particle theorists subscribe to something
known as the naturalness dogma, stating that fundamental
Greatness and audacity 41
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 cG and c should be about
the same. We now understand that the speed of propagation
is a universal constant, that cG = 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. EINSTEIN8
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 43
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 tradi-
tion of free lectures has persisted to this day in many coun-
tries, 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, mak-
ing discoveries one after another and outshining his mentor.
(Davy is now a forgotten figure in physics.) Jealousy is a pow-
erful 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.
44 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 apiece 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 45
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 elec-
tric field produced by a charge decreases as one moves far-
ther 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 electro-
magnetic 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 amaz-
ingly, 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 up-
from-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
46 Chapter 6
contemporary theoretical physicists enamored of mathe-
matics should take heed! Many in my world dazzle them-
selves (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 devo-
tion 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 symbolismwhich 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 mathemat-
ics14 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 47
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 pro-
duces an electric field varying in space. This expresses math-
ematically Faraday’s law of induction: by moving a mag-
net around a wire, Faraday produced an electric field that
pushed charges forward in the wire, thus generating a cur-
rent. To Maxwell’s surprise, the equations he wrote down
were not mutually consistent. Remarkably, Maxwell discov-
ered 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 pre-
cisely 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 astrono-
mical observations. The value obtained theoretically for the
48 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 physi-
cal reality of the field and its claim to a separate existence.
Indeed, the space around us is literally humming with pack-
ets 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. Elec-
trons, 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.
—FREEMAN DYSON SPEAKING OF ALBERT EINSTEIN
Greatness and audacity 49
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 consider-
ations that