A Brief History of Time – Stephen Hawking

A Brief History of Time – Stephen Hawking
Chapter 1 – Our Picture of the Universe
Chapter 2 – Space and Time
Chapter 3 – The Expanding Universe
Chapter 4 – The Uncertainty Principle
Chapter 5 – Elementary Particles and the Forces of Nature
Chapter 6 – Black Holes
Chapter 7 – Black Holes Ain’t So Black
Chapter 8 – The Origin and Fate of the Universe
Chapter 9 – The Arrow of Time
Chapter 10 – Wormholes and Time Travel
Chapter 11 – The Unification of Physics
Chapter 12 – Conclusion
Glossary
Acknowledgments & About The Author
FOREWARD
I didn’t write a foreword to the original edition of A Brief History of Time.
That was done by Carl Sagan. Instead, I wrote a short piece titled
“Acknowledgments” in which I was advised to thank everyone. Some of
the foundations that had given me support weren’t too pleased to have been
mentioned, however, because it led to a great increase in applications.
I don’t think anyone, my publishers, my agent, or myself, expected the book
to do anything like as well as it did. It was in the London Sunday Times
best-seller list for 237 weeks, longer than any other book (apparently, the
Bible and Shakespeare aren’t counted). It has been translated into
something like forty languages and has sold about one copy for every 750
men, women, and children in the world. As Nathan Myhrvold of Microsoft
(a former post-doc of mine) remarked: I have sold more books on physics
than Madonna has on sex. The success of A Brief History indicates that
there is widespread interest in the big questions like: Where did we come
from? And why is the universe the way it is?
I have taken the opportunity to update the book and include new theoretical
and observational results obtained since the book was first published (on
April Fools’ Day, 1988). I have included a new chapter on wormholes and
time travel. Einstein’s General Theory of Relativity seems to offer the
possibility that we could create and maintain wormholes, little tubes that
connect different regions of space-time. If so, we might be able to use them
for rapid travel around the galaxy or travel back in time. Of course, we have
not seen anyone from the file:///C|/WINDOWS/Desktop/blahh/Stephen
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A Brief History of Time – Stephen Hawking
future (or have we?) but I discuss a possible explanation for this.
I also describe the progress that has been made recently in finding
“dualities” or correspondences between apparently different theories of
physics. These correspondences are a strong indication that there is a
complete unified theory of physics, but they also suggest that it may not be
possible to express this theory in a single fundamental formulation. Instead,
we may have to use different reflections of the underlying theory in
different situations. It might be like our being unable to represent the
surface of the earth on a single map and having to use different maps in
different regions. This would be a revolution in our view of the unification
of the laws of science but it would not change the most important point: that
the universe is governed by a set of rational laws that we can discover and
understand.
On the observational side, by far the most important development has been
the measurement of fluctuations in the cosmic microwave background
radiation by COBE (the Cosmic Background Explorer satellite) and other
collaborations. These fluctuations are the finger-prints of creation, tiny
initial irregularities in the otherwise smooth and uniform early universe that
later grew into galaxies, stars, and all the structures we see around us. Their
form agrees with the predictions of the proposal that the universe has no
boundaries or edges in the imaginary time direction; but further
observations will be necessary to distinguish this proposal from other
possible explanations for the fluctuations in the background. However,
within a few years we should know whether we can believe that we live in a
universe that is completely self-contained and without beginning or end.
Stephen Hawking
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CHAPTER 1
OUR PICTURE OF THE UNIVERSE
A well-known scientist (some say it was Bertrand Russell) once gave a
public lecture on astronomy. He described how the earth orbits around the
sun and how the sun, in turn, orbits around the center of a vast collection of
stars called our galaxy. At the end of the lecture, a little old lady at the back
of the room got up and said: “What you have told us is rubbish. The world
is really a flat plate supported on the back of a giant tortoise.” The scientist
gave a superior smile before replying, “What is the tortoise standing on.”
“You’re very clever, young man, very clever,” said the old lady. “But it’s
turtles all the way down!”
Most people would find the picture of our universe as an infinite tower of
tortoises rather ridiculous, but why do we think we know better? What do
we know about the universe, and how do we know it? Where did the
universe come from, and where is it going? Did the universe have a
beginning, and if so, what happened before then? What is the nature of
time? Will it ever come to an end? Can we go back in time? Recent
breakthroughs in physics, made possible in part by fantastic new
technologies, suggest answers to some of these longstanding questions.
Someday these answers may seem as obvious to us as the earth orbiting the
sun – or perhaps as ridiculous as a tower of tortoises. Only time (whatever
that may be) will tell. As long ago as 340 BC the Greek philosopher
Aristotle, in his book On the Heavens, was able to put forward two good
arguments for believing that the earth was a round sphere rather than a Hat
plate. First, he realized that eclipses of the moon were caused by the earth
coming between the sun and the moon. The earth’s shadow on the moon
was always round, which would be true only if the earth was spherical. If
the earth had been a flat disk, the shadow would have been elongated and
elliptical, unless the eclipse always occurred at a time when the sun was
directly under the center of the disk. Second, the Greeks knew from their
travels that the North Star appeared lower in the sky when viewed in the
south than it did in more northerly regions. (Since the North Star lies over
the North Pole, it appears to be directly above an observer at the North
Pole, but to someone looking from the equator, it appears to lie just at the
horizon. From the difference in the apparent position of the North Star in
Egypt and Greece, Aristotle even quoted an estimate that the distance
around the earth was 400,000 stadia. It is not known exactly what length a
stadium was, but it may have been about 200
yards, which would make Aristotle’s estimate about twice the currently
accepted figure. The Greeks even had a third argument that the earth must
be round, for why else does one first see the sails of a ship coming over the
horizon, and only later see the hull?
Aristotle thought the earth was stationary and that the sun, the moon, the
planets, and the stars moved in circular orbits about the earth. He believed
this because he felt, for mystical reasons, that the earth was the center of the
universe, and that circular motion was the most perfect. This idea was
elaborated by Ptolemy in the second century AD into a complete
cosmological model. The earth stood at the center, surrounded by eight
spheres that carried the moon, the sun, the stars, and the five planets known
at the time, Mercury, Venus, Mars, Jupiter, and Saturn.
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Figure 1:1
The planets themselves moved on smaller circles attached to their
respective spheres in order to account for their rather complicated observed
paths in the sky. The outermost sphere carried the so-called fixed stars,
which always stay in the same positions relative to each other but which
rotate together across the sky. What lay beyond the last sphere was never
made very clear, but it certainly was not part of mankind’s observable
universe.
Ptolemy’s model provided a reasonably accurate system for predicting the
positions of heavenly bodies in the sky. But in order to predict these
positions correctly, Ptolemy had to make an assumption that the moon
followed a path that sometimes brought it twice as close to the earth as at
other times. And that meant that the moon ought sometimes to appear twice
as big as at other times! Ptolemy recognized this flaw, but nevertheless his
model was generally, although not universally, accepted. It was adopted by
the Christian church as the picture of the universe that was in accordance
with Scripture, for it had the great advantage that it left lots of room outside
the sphere of fixed stars for heaven and hell.
A simpler model, however, was proposed in 1514 by a Polish priest,
Nicholas Copernicus. (At first, perhaps for fear of being branded a heretic
by his church, Copernicus circulated his model anonymously.) His idea was
that the sun was stationary at the center and that the earth and the planets
moved in circular orbits around the sun. Nearly a century passed before this
idea was taken seriously. Then two astronomers – the German, Johannes
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Kepler, and the Italian, Galileo Galilei – started publicly to support the
Copernican theory, despite the fact that the orbits it predicted did not quite
match the ones observed. The death blow to the Aristotelian/Ptolemaic
theory came in 1609. In that year, Galileo started observing the night sky
with a telescope, which had just been invented. When he looked at the
planet Jupiter, Galileo found that it was accompanied by several small
satellites or moons that orbited around it. This implied that everything did
not have to orbit directly around the earth, as Aristotle and Ptolemy had
thought. (It was, of course, still possible to believe that the earth was
stationary at the center of the universe and that the moons of Jupiter moved
on extremely complicated paths around the earth, giving the appearance that
they orbited Jupiter. However, Copernicus’s theory was much simpler.) At
the same time, Johannes Kepler had modified Copernicus’s theory,
suggesting that the planets moved not in circles but in ellipses (an ellipse is
an elongated circle). The predictions now finally matched the observations.
As far as Kepler was concerned, elliptical orbits were merely an ad hoc
hypothesis, and a rather repugnant one at that, because ellipses were clearly
less perfect than circles. Having discovered almost by accident that
elliptical orbits fit the observations well, he could not reconcile them with
his idea that the planets were made to orbit the sun by magnetic forces. An
explanation was provided only much later, in 1687, when Sir Isaac Newton
published his Philosophiae Naturalis Principia Mathematica, probably the
most important single work ever published in the physical sciences. In it
Newton not only put forward a theory of how bodies move in space and
time, but he also developed the complicated mathematics needed to analyze
those motions. In addition, Newton postulated a law of universal gravitation
according to which each body in the universe was attracted toward every
other body by a force that was stronger the more massive the bodies and the
closer they were to each other. It was this same force that caused objects to
fall to the ground. (The story that Newton was inspired by an apple hitting
his head is almost certainly apocryphal. All Newton himself ever said was
that the idea of gravity came to him as he sat “in a contemplative mood”
and “was occasioned by the fall of an apple.”) Newton went on to show
that, according to his law, gravity causes the moon to move in an elliptical
orbit around the earth and causes the earth and the planets to follow
elliptical paths around the sun. The Copernican model got rid of Ptolemy’s
celestial spheres, and with them, the idea that the universe had a natural
boundary. Since “fixed stars” did not appear to change their positions apart
from a rotation across the sky caused by the earth spinning on its axis, it
became natural to suppose that the fixed stars were objects like our sun but
very much farther away.
Newton realized that, according to his theory of gravity, the stars should
attract each other, so it seemed they could not remain essentially
motionless. Would they not all fall together at some point? In a letter in
1691 to Richard Bentley, another leading thinker of his day, Newton argued
that this would indeed happen if there were only a finite number of stars
distributed over a finite region of space. But he reasoned that if, on the
other hand, there were an infinite number of stars, distributed more or less
uniformly over infinite space, this would not happen, because there would
not be any central point for them to fall to. This argument is an instance of
the pitfalls that you can encounter in talking about infinity. In an infinite
universe, every point can be regarded as the center, because every point has
an infinite number of stars on each side of it. The correct approach, it was
realized only much later, is to consider the finite situation, in which the stars
all fall in on each other, and then to ask how things change if one adds more
stars roughly uniformly distributed outside this region. According to
Newton’s law, the extra stars would make no difference at all to the original
ones on average, so the stars would fall in just as fast. We can add as many
stars as we like, but they will still always collapse in on themselves. We
now know it is impossible to have an infinite static model of the universe in
which gravity is always attractive.
It is an interesting reflection on the general climate of thought before the
twentieth century that no one had suggested that the universe was
expanding or contracting. It was generally accepted that either the universe
had existed forever in an unchanging state, or that it had been created at a
finite time in the past more or less as we observe it today. In part this may
have been due to people’s tendency to believe in eternal truths, as well as
the comfort they found in the thought that even though they may grow old
and die, the universe is eternal and unchanging.
Even those who realized that Newton’s theory of gravity showed that the
universe could not be static did not think to suggest that it might be
expanding. Instead, they attempted to modify the theory by making the
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gravitational force repulsive at very large distances. This did not
significantly affect their predictions of the motions of the planets, but it
allowed an infinite distribution of stars to remain in equilibrium – with the
attractive forces between nearby stars balanced by the repulsive forces from
those that were farther away. However, we now believe such an equilibrium
would be unstable: if the stars in some region got only slightly nearer each
other, the attractive forces between them would become stronger and
dominate over the repulsive forces so that the stars would continue to fall
toward each other. On the other hand, if the stars got a bit farther away from
each other, the repulsive forces would dominate and drive them farther
apart. Another objection to an infinite static universe is normally ascribed to
the German philosopher Heinrich Olbers, who wrote about this theory in
1823. In fact, various contemporaries of Newton had raised the problem,
and the Olbers article was not even the first to contain plausible arguments
against it. It was, however, the first to be widely noted. The difficulty is that
in an infinite static universe nearly every line of sight would end on the
surface of a star. Thus one would expect that the whole sky would be as
bright as the sun, even at night. Olbers’ counter-argument was that the light
from distant stars would be dimmed by absorption by intervening matter.
However, if that happened the intervening matter would eventually heat up
until it glowed as brightly as the stars. The only way of avoiding the
conclusion that the whole of the night sky should be as bright as the surface
of the sun would be to assume that the stars had not been shining forever
but had turned on at some finite time in the past. In that case the absorbing
matter might not have heated up yet or the light from distant stars might not
yet have reached us. And that brings us to the question of what could have
caused the stars to have turned on in the first place.
The beginning of the universe had, of course, been discussed long before
this. According to a number of early cosmologies and the
Jewish/Christian/Muslim tradition, the universe started at a finite, and not
very distant, time in the past. One argument for such a beginning was the
feeling that it was necessary to have “First Cause”
to explain the existence of the universe. (Within the universe, you always
explained one event as being caused by some earlier event, but the
existence of the universe itself could be explained in this way only if it had
some beginning.) Another argument was put forward by St. Augustine in
his book The City of God. He pointed out that civilization is progressing
and we remember who performed this deed or developed that technique.
Thus man, and so also perhaps the universe, could not have been around all
that long. St. Augustine accepted a date of about 5000 BC for the Creation
of the universe according to the book of Genesis. (It is interesting that this
is not so far from the end of the last Ice Age, about 10,000 BC, which is
when archaeologists tell us that civilization really began.)
Aristotle, and most of the other Greek philosophers, on the other hand, did
not like the idea of a creation because it smacked too much of divine
intervention. They believed, therefore, that the human race and the world
around it had existed, and would exist, forever. The ancients had already
considered the argument about progress described above, and answered it
by saying that there had been periodic floods or other disasters that
repeatedly set the human race right back to the beginning of civilization.
The questions of whether the universe had a beginning in time and whether
it is limited in space were later extensively examined by the philosopher
Immanuel Kant in his monumental (and very obscure) work Critique of
Pure Reason, published in 1781. He called these questions antinomies (that
is, contradictions) of pure reason because he felt that there were equally
compelling arguments for believing the thesis, that the universe had a
beginning, and the antithesis, that it had existed forever. His argument for
the thesis was that if the universe did not have a beginning, there would be
an infinite period of time before any event, which he considered absurd.
The argument for the antithesis was that if the universe had a beginning,
there would be an infinite period of time before it, so why should the
universe begin at any one particular time? In fact, his cases for both the
thesis and the antithesis are really the same argument. They are both based
on his unspoken assumption that time continues back forever, whether or
not the universe had existed forever. As we shall see, the concept of time
has no meaning before the beginning of the universe. This was first pointed
out by St. Augustine. When asked:
“What did God do before he created the universe?” Augustine didn’t reply:
“He was preparing Hell for people who asked such questions.” Instead, he
said that time was a property of the universe that God created, and that time
did not exist before the beginning of the universe.
When most people believed in an essentially static and unchanging
universe, the question of whether or not it had a beginning was really one of
metaphysics or theology. One could account for what was observed equally
well on the theory that the universe had existed forever or on the theory that
it was set in motion at some finite
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time in such a manner as to look as though it had existed forever. But in
1929, Edwin Hubble made the landmark observation that wherever you
look, distant galaxies are moving rapidly away from us. In other words, the
universe is expanding. This means that at earlier times objects would have
been closer together. In fact, it seemed that there was a time, about ten or
twenty thousand million years ago, when they were all at exactly the same
place and when, therefore, the density of the universe was infinite. This
discovery finally brought the question of the beginning of the universe into
the realm of science.
Hubble’s observations suggested that there was a time, called the big bang,
when the universe was infinitesimally small and infinitely dense. Under
such conditions all the laws of science, and therefore all ability to predict
the future, would break down. If there were events earlier than this time,
then they could not affect what happens at the present time. Their existence
can be ignored because it would have no observational consequences. One
may say that time had a beginning at the big bang, in the sense that earlier
times simply would not be defined. It should be emphasized that this
beginning in time is very different from those that had been considered
previously. In an unchanging universe a beginning in time is something that
has to be imposed by some being outside the universe; there is no physical
necessity for a beginning. One can imagine that God created the universe at
literally any time in the past. On the other hand, if the universe is
expanding, there may be physical reasons why there had to be a beginning.
One could still imagine that God created the universe at the instant of the
big bang, or even afterwards in just such a way as to make it look as though
there had been a big bang, but it would be meaningless to suppose that it
was created before the big bang. An expanding universe does not preclude a
creator, but it does place limits on when he might have carried out his job!
In order to talk about the nature of the universe and to discuss questions
such as whether it has a beginning or an end, you have to be clear about
what a scientific theory is. I shall take the simpleminded view that a theory
is just a model of the universe, or a restricted part of it, and a set of rules
that relate quantities in the model to observations that we make. It exists
only in our minds and does not have any other reality (whatever that might
mean). A theory is a good theory if it satisfies two requirements. It must
accurately describe a large class of observations on the basis of a model that
contains only a few arbitrary elements, and it must make definite
predictions about the results of future observations. For example, Aristotle
believed Empedocles’s theory that everything was made out of four
elements, earth, air, fire, and water. This was simple enough, but did not
make any definite predictions. On the other hand, Newton’s theory of
gravity was based on an even simpler model, in which bodies attracted each
other with a force that was proportional to a quantity called their mass and
inversely proportional to the square of the distance between them. Yet it
predicts the motions of the sun, the moon, and the planets to a high degree
of accuracy.
Any physical theory is always provisional, in the sense that it is only a
hypothesis: you can never prove it. No matter how many times the results
of experiments agree with some theory, you can never be sure that the next
time the result will not contradict the theory. On the other hand, you can
disprove a theory by finding even a single observation that disagrees with
the predictions of the theory. As philosopher of science Karl Popper has
emphasized, a good theory is characterized by the fact that it makes a
number of predictions that could in principle be disproved or falsified by
observation. Each time new experiments are observed to agree with the
predictions the theory survives, and our confidence in it is increased; but if
ever a new observation is found to disagree, we have to abandon or modify
the theory.
At least that is what is supposed to happen, but you can always question the
competence of the person who carried out the observation.
In practice, what often happens is that a new theory is devised that is really
an extension of the previous theory. For example, very accurate
observations of the planet Mercury revealed a small difference between its
motion and the predictions of Newton’s theory of gravity. Einstein’s general
theory of relativity predicted a slightly different motion from Newton’s
theory. The fact that Einstein’s predictions matched what was seen, while
Newton’s did not, was one of the crucial confirmations of the new theory.
However, we still use Newton’s theory for all practical purposes because
the difference between its predictions and those of general relativity is very
small in the situations that we normally deal with. (Newton’s theory also
has the great advantage that it is much simpler to work with than
Einstein’s!)
The eventual goal of science is to provide a single theory that describes the
whole universe. However, the file:///C|/WINDOWS/Desktop/blahh/Stephen
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approach most scientists actually follow is to separate the problem into two
parts. First, there are the laws that tell us how the universe changes with
time. (If we know what the universe is like at any one time, these physical
laws tell us how it will look at any later time.) Second, there is the question
of the initial state of the universe. Some people feel that science should be
concerned with only the first part; they regard the question of the initial
situation as a matter for metaphysics or religion. They would say that God,
being omnipotent, could have started the universe off any way he wanted.
That may be so, but in that case he also could have made it develop in a
completely arbitrary way. Yet it appears that he chose to make it evolve in a
very regular way according to certain laws. It therefore seems equally
reasonable to suppose that there are also laws governing the initial state.
It turns out to be very difficult to devise a theory to describe the universe all
in one go. Instead, we break the problem up into bits and invent a number
of partial theories. Each of these partial theories describes and predicts a
certain limited class of observations, neglecting the effects of other
quantities, or representing them by simple sets of numbers. It may be that
this approach is completely wrong. If everything in the universe depends on
everything else in a fundamental way, it might be impossible to get close to
a full solution by investigating parts of the problem in isolation.
Nevertheless, it is certainly the way that we have made progress in the past.
The classic example again is the Newtonian theory of gravity, which tells us
that the gravitational force between two bodies depends only on one
number associated with each body, its mass, but is otherwise independent of
what the bodies are made of. Thus one does not need to have a theory of the
structure and constitution of the sun and the planets in order to calculate
their orbits. Today scientists describe the universe in terms of two basic
partial theories – the general theory of relativity and quantum mechanics.
They are the great intellectual achievements of the first half of this century.
The general theory of relativity describes the force of gravity and the largescale structure of the universe, that is, the structure on scales from only a
few miles to as large as a million million million million (1 with twentyfour zeros after it) miles, the size of the observable universe. Quantum
mechanics, on the other hand, deals with phenomena on extremely small
scales, such as a millionth of a millionth of an inch. Unfortunately,
however, these two theories are known to be inconsistent with each other –
they cannot both be correct. One of the major endeavors in physics today,
and the major theme of this book, is the search for a new theory that will
incorporate them both – a quantum theory of gravity. We do not yet have
such a theory, and we may still be a long way from having one, but we do
already know many of the properties that it must have. And we shall see, in
later chapters, that we already know a fair amount about the predications a
quantum theory of gravity must make.
Now, if you believe that the universe is not arbitrary, but is governed by
definite laws, you ultimately have to combine the partial theories into a
complete unified theory that will describe everything in the universe. But
there is a fundamental paradox in the search for such a complete unified
theory. The ideas about scientific theories outlined above assume we are
rational beings who are free to observe the universe as we want and to draw
logical deductions from what we see.
In such a scheme it is reasonable to suppose that we might progress ever
closer toward the laws that govern our universe. Yet if there really is a
complete unified theory, it would also presumably determine our actions.
And so the theory itself would determine the outcome of our search for it!
And why should it determine that we come to the right conclusions from the
evidence? Might it not equally well determine that we draw the wrong
conclusion.? Or no conclusion at all?
The only answer that I can give to this problem is based on Darwin’s
principle of natural selection. The idea is that in any population of selfreproducing organisms, there will be variations in the genetic material and
upbringing that different individuals have. These differences will mean that
some individuals are better able than others to draw the right conclusions
about the world around them and to act accordingly. These individuals will
be more likely to survive and reproduce and so their pattern of behavior and
thought will come to dominate. It has certainly been true in the past that
what we call intelligence and scientific discovery have conveyed a survival
advantage. It is not so clear that this is still the case: our scientific
discoveries may well destroy us all, and even if they don’t, a complete
unified theory may not make much difference to our chances of survival.
However, provided the universe has evolved in a regular way, we might
expect that the reasoning abilities that natural selection has given us would
be valid also in our search for a complete unified theory, and so would not
lead us to the wrong conclusions.
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Because the partial theories that we already have are sufficient to make
accurate predictions in all but the most extreme situations, the search for the
ultimate theory of the universe seems difficult to justify on practical
grounds. (It is worth noting, though, that similar arguments could have been
used against both relativity and quantum mechanics, and these theories have
given us both nuclear energy and the microelectronics revolution!) The
discovery of a complete unified theory, therefore, may not aid the survival
of our species. It may not even affect our lifestyle. But ever since the dawn
of civilization, people have not been content to see events as unconnected
and inexplicable. They have craved an understanding of the underlying
order in the world. Today we still yearn to know why we are here and where
we came from. Humanity’s deepest desire for knowledge is justification
enough for our continuing quest. And our goal is nothing less than a
complete description of the universe we live in.
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CHAPTER 2
SPACE AND TIME
Our present ideas about the motion of bodies date back to Galileo and
Newton. Before them people believed Aristotle, who said that the natural
state of a body was to be at rest and that it moved only if driven by a force
or impulse. It followed that a heavy body should fall faster than a light one,
because it would have a greater pull toward the earth.
The Aristotelian tradition also held that one could work out all the laws that
govern the universe by pure thought: it was not necessary to check by
observation. So no one until Galileo bothered to see whether bodies of
different weight did in fact fall at different speeds. It is said that Galileo
demonstrated that Aristotle’s belief was false by dropping weights from the
leaning tower of Pisa. The story is almost certainly untrue, but Galileo did
do something equivalent: he rolled balls of different weights down a smooth
slope. The situation is similar to that of heavy bodies falling vertically, but
it is easier to observe because the Speeds are smaller. Galileo’s
measurements indicated that each body increased its speed at the same rate,
no matter what its weight. For example, if you let go of a ball on a slope
that drops by one meter for every ten meters you go along, the ball will be
traveling down the slope at a speed of about one meter per second after one
second, two meters per second after two seconds, and so on, however heavy
the ball. Of course a lead weight would fall faster than a feather, but that is
only because a feather is slowed down by air resistance. If one drops two
bodies that don’t have much air resistance, such as two different lead
weights, they fall at the same rate. On the moon, where there is no air to
slow things down, the astronaut David R. Scott performed the feather and
lead weight experiment and found that indeed they did hit the ground at the
same time. Galileo’s measurements were used by Newton as the basis of his
laws of motion. In Galileo’s experiments, as a body rolled down the slope it
was always acted on by the same force (its weight), and the effect was to
make it constantly speed up. This showed that the real effect of a force is
always to change the speed of a body, rather than just to set it moving, as
was previously thought. It also meant that whenever a body is not acted on
by any force, it will keep on moving in a straight line at the same speed.
This idea was first stated explicitly in Newton’s Principia Mathematica,
published in 1687, and is known as Newton’s first law. What happens to a
body when a force does act on it is given by Newton’s second law. This
states that the body will accelerate, or change its speed, at a rate that is
proportional to the force. (For example, the acceleration is twice as great if
the force is twice as great.) The acceleration is also smaller the greater the
mass (or quantity of matter) of the body. (The same force acting on a body
of twice the mass will produce half the acceleration.) A familiar example is
provided by a car: the more powerful the engine, the greater the
acceleration, but the heavier the car, the smaller the acceleration for the
same engine. In addition to his laws of motion, Newton discovered a law to
describe the force of gravity, which states that every body attracts every
other body with a force that is proportional to the mass of each body. Thus
the force between two bodies would be twice as strong if one of the bodies
(say, body A) had its mass doubled. This is what you might expect because
one could think of the new body A as being made of two bodies with the
original mass. Each would attract body B with the original force. Thus the
total force between A and B would be twice the original force. And if, say,
one of the bodies had twice the mass, and the other had three times the
mass, then the force would be six times as strong. One can now see why all
bodies fall at the same rate: a body of twice the weight will have twice the
force of gravity pulling it down, but it will also have twice the mass.
According to Newton’s second law, these two effects will exactly cancel
each other, so the acceleration will be the same in all cases. Newton’s law
of gravity also tells us that the farther apart the bodies, the smaller the force.
Newton’s law of gravity says that the gravitational attraction of a star is
exactly one quarter that of a similar star at half the distance. This law
predicts the orbits of the earth, the moon, and the planets with great
accuracy. If the law were that the gravitational attraction of a star went
down faster or increased more rapidly with distance, the orbits of the
planets would not be elliptical, they would either spiral in to the sun or
escape from the sun. The big difference between the ideas of Aristotle and
those of Galileo and Newton is that Aristotle believed in a preferred state of
rest, which any body would take up if it were not driven by some force Or
impulse. In particular, he thought that the earth was at rest. But it follows
from Newton’s laws that there is no unique
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standard of rest. One could equally well say that body A was at rest and
body B was moving at constant speed with respect to body A, or that body B
was at rest and body A was moving. For example, if one sets aside for a
moment the rotation of the earth and its orbit round the sun, one could say
that the earth was at rest and that a train on it was traveling north at ninety
miles per hour or that the train was at rest and the earth was moving south
at ninety miles per hour. If one carried out experiments with moving bodies
on the train, all Newton’s laws would still hold. For instance, playing PingPong on the train, one would find that the ball obeyed Newton’s laws just
like a ball on a table by the track. So there is no way to tell whether it is the
train or the earth that is moving. The lack of an absolute standard of rest
meant that one could not determine whether two events that took place at
different times occurred in the same position in space. For example,
suppose our Ping-Pong ball on the train bounces straight up and down,
hitting the table twice on the same spot one second apart. To someone on
the track, the two bounces would seem to take place about forty meters
apart, because the train would have traveled that far down the track between
the bounces. The nonexistence of absolute rest therefore meant that one
could not give an event an absolute position in space, as Aristotle had
believed. The positions of events and the distances between them would be
different for a person on the train and one on the track, and there would be
no reason to prefer one person’s position to the other’s.
Newton was very worried by this lack of absolute position, or absolute
space, as it was called, because it did not accord with his idea of an absolute
God. In fact, he refused to accept lack of absolute space, even though it was
implied by his laws. He was severely criticized for this irrational belief by
many people, most notably by Bishop Berkeley, a philosopher who believed
that all material objects and space and time are an illusion. When the
famous Dr. Johnson was told of Berkeley’s opinion, he cried, “I refute it
thus!” and stubbed his toe on a large stone.
Both Aristotle and Newton believed in absolute time. That is, they believed
that one could unambiguously measure the interval of time between two
events, and that this time would be the same whoever measured it, provided
they used a good clock. Time was completely separate from and
independent of space. This is what most people would take to be the
commonsense view. However, we have had to change our ideas about space
and time. Although our apparently commonsense notions work well when
dealing with things like apples, or planets that travel comparatively slowly,
they don’t work at all for things moving at or near the speed of light. The
fact that light travels at a finite, but very high, speed was first discovered in
1676 by the Danish astronomer Ole Christensen Roemer. He observed that
the times at which the moons of Jupiter appeared to pass behind Jupiter
were not evenly spaced, as one would expect if the moons went round
Jupiter at a constant rate. As the earth and Jupiter orbit around the sun, the
distance between them varies. Roemer noticed that eclipses of Jupiter’s
moons appeared later the farther we were from Jupiter. He argued that this
was because the light from the moons took longer to reach us when we
were farther away. His measurements of the variations in the distance of the
earth from Jupiter were, however, not very accurate, and so his value for the
speed of light was 140,000 miles per second, compared to the modern value
of 186,000 miles per second. Nevertheless, Roemer’s achievement, in not
only proving that light travels at a finite speed, but also in measuring that
speed, was remarkable – coming as it did eleven years before Newton’s
publication of Principia Mathematica. A proper theory of the propagation
of light didn’t come until 1865, when the British physicist James Clerk
Maxwell succeeded in unifying the partial theories that up to then had been
used to describe the forces of electricity and magnetism. Maxwell’s
equations predicted that there could be wavelike disturbances in the
combined electromagnetic field, and that these would travel at a fixed
speed, like ripples on a pond. If the wavelength of these waves (the distance
between one wave crest and the next) is a meter or more, they are what we
now call radio waves. Shorter wavelengths are known as microwaves (a
few centimeters) or infrared (more than a ten-thousandth of a centimeter).
Visible light has a wavelength of between only forty and eighty millionths
of a centimeter. Even shorter wavelengths are known as ultraviolet, X rays,
and gamma rays. Maxwell’s theory predicted that radio or light waves
should travel at a certain fixed speed. But Newton’s theory had got rid of
the idea of absolute rest, so if light was supposed to travel at a fixed speed,
one would have to say what that fixed speed was to be measured relative to.
It was therefore suggested that there was a substance called the "ether” that
was present everywhere, even in
"empty” space. Light waves should travel through the ether as sound waves
travel through air, and their speed should therefore be relative to the ether.
Different observers, moving relative to the ether, would see light
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coming toward them at different speeds, but light’s speed relative to the
ether would remain fixed. In particular, as the earth was moving through the
ether on its orbit round the sun, the speed of light measured in the direction
of the earth’s motion through the ether (when we were moving toward the
source of the light) should be higher than the speed of light at right angles
to that motion (when we are not moving toward the source). In 1887Albert
Michelson (who later became the first American to receive the Nobel Prize
for physics) and Edward Morley carried out a very careful experiment at the
Case School of Applied Science in Cleveland. They compared the speed of
light in the direction of the earth’s motion with that at right angles to the
earth’s motion. To their great surprise, they found they were exactly the
same!
Between 1887 and 1905 there were several attempts, most notably by the
Dutch physicist Hendrik Lorentz, to explain the result of the MichelsonMorley experiment in terms of objects contracting and clocks slowing down
when they moved through the ether. However, in a famous paper in 1905, a
hitherto unknown clerk in the Swiss patent office, Albert Einstein, pointed
out that the whole idea of an ether was unnecessary, providing one was
willing to abandon the idea of absolute time. A similar point was made a
few weeks later by a leading French mathematician, Henri Poincare.
Einstein’s arguments were closer to physics than those of Poincare, who
regarded this problem as mathematical. Einstein is usually given the credit
for the new theory, but Poincare is remembered by having his name
attached to an important part of it. The fundamental postulate of the theory
of relativity, as it was called, was that the laws of science should be the
same for all freely moving observers, no matter what their speed. This was
true for Newton’s laws of motion, but now the idea was extended to include
Maxwell’s theory and the speed of light: all observers should measure the
same speed of light, no matter how fast they are moving. This simple idea
has some remarkable consequences. Perhaps the best known are the
equivalence of mass and energy, summed up in Einstein’s famous equation
E=mc2 (where E is energy, m is mass, and c is the speed of light), and the
law that nothing may travel faster than the speed of light. Because of the
equivalence of energy and mass, the energy which an object has due to its
motion will add to its mass. In other words, it will make it harder to
increase its speed. This effect is only really significant for objects moving at
speeds close to the speed of light. For example, at 10
percent of the speed of light an object’s mass is only 0.5 percent more than
normal, while at 90 percent of the speed of light it would be more than
twice its normal mass. As an object approaches the speed of light, its mass
rises ever more quickly, so it takes more and more energy to speed it up
further. It can in fact never reach the speed of light, because by then its
mass would have become infinite, and by the equivalence of mass and
energy, it would have taken an infinite amount of energy to get it there. For
this reason, any normal object is forever confined by relativity to move at
speeds slower than the speed of light. Only light, or other waves that have
no intrinsic mass, can move at the speed of light.
An equally remarkable consequence of relativity is the way it has
revolutionized our ideas of space and time. In Newton’s theory, if a pulse of
light is sent from one place to another, different observers would agree on
the time that the journey took (since time is absolute), but will not always
agree on how far the light traveled (since space is not absolute). Since the
speed of the light is just the distance it has traveled divided by the time it
has taken, different observers would measure different speeds for the light.
In relativity, on the other hand, all observers must agree on how fast light
travels. They still, however, do not agree on the distance the light has
traveled, so they must therefore now also disagree over the time it has
taken. (The time taken is the distance the light has traveled – which the
observers do not agree on – divided by the light’s speed – which they do
agree on.) In other words, the theory of relativity put an end to the idea of
absolute time! It appeared that each observer must have his own measure of
time, as recorded by a clock carried with him, and that identical clocks
carried by different observers would not necessarily agree.
Each observer could use radar to say where and when an event took place
by sending out a pulse of light or radio waves. Part of the pulse is reflected
back at the event and the observer measures the time at which he receives
the echo. The time of the event is then said to be the time halfway between
when the pulse was sent and the time when the reflection was received
back: the distance of the event is half the time taken for this round trip,
multiplied by the speed of light. (An event, in this sense, is something that
takes place at a single point in space, at a specified point in time.) This idea
is shown here, which is an example of a space-time diagram…
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Figure 2:1
Using this procedure, observers who are moving relative to each other will
assign different times and positions to the same event. No particular
observer’s measurements are any more correct than any other observer’s,
but all the measurements are related. Any observer can work out precisely
what time and position any other observer will assign to an event, provided
he knows the other observer’s relative velocity. Nowadays we use just this
method to measure distances precisely, because we can measure time more
accurately than length. In effect, the meter is defined to be the distance
traveled by light in 0.000000003335640952 second, as measured by a
cesium clock. (The reason for that particular number is that it corresponds
to the historical definition of the meter – in terms of two marks on a
particular platinum bar kept in Paris.) Equally, we can use a more
convenient, new unit of length called a light-second. This is simply defined
as the distance that light travels in one second. In the theory of relativity, we
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terms of time and the speed of light, so it follows automatically that every
observer will measure light to have the same speed (by definition, 1 meter
per 0.000000003335640952 second). There is no need to introduce the idea
of an ether, whose presence anyway cannot be detected, as the MichelsonMorley experiment showed. The theory of relativity does, however, force us
to change fundamentally our ideas of space and time. We must accept that
time is not completely separate from and independent of space, but is
combined with it to form an object called space-time.
It is a matter of common experience that one can describe the position of a
point in space by three numbers, or coordinates. For instance, one can say
that a point in a room is seven feet from one wall, three feet from another,
and five feet above the floor. Or one could specify that a point was at a
certain latitude and longitude and a certain height above sea level. One is
free to use any three suitable coordinates, although they have only a limited
range of validity. One would not specify the position of the moon in terms
of miles north and miles west of Piccadilly Circus and feet above sea level.
Instead, one might describe it in terms of distance from the sun, distance
from the plane of the orbits of the planets, and the angle between the line
joining the moon to the sun and the line joining the sun to a nearby star
such as Alpha Centauri. Even these coordinates would not be of much use
in describing the position of the sun in our galaxy or the position of our
galaxy in the local group of galaxies. In fact, one may describe the whole
universe in terms of a collection of overlapping patches. In each patch, one
can use a different set of three coordinates to specify the position of a point.
An event is something that happens at a particular point in space and at a
particular time. So one can specify it by four numbers or coordinates.
Again, the choice of coordinates is arbitrary; one can use any three welldefined spatial coordinates and any measure of time. In relativity, there is
no real distinction between the space and time coordinates, just as there is
no real difference between any two space coordinates. One could choose a
new set of coordinates in which, say, the first space coordinate was a
combination of the old first and second space coordinates. For instance,
instead of measuring the position of a point on the earth in miles north of
Piccadilly and miles west of Piccadilly, one could use miles northeast of
Piccadilly, and miles north-west of Piccadilly. Similarly, in relativity, one
could use a new time coordinate that was the old time (in seconds) plus the
distance (in light-seconds) north of Piccadilly.
It is often helpful to think of the four coordinates of an event as specifying
its position in a four-dimensional space called space-time. It is impossible
to imagine a four-dimensional space. I personally find it hard enough to
visualize three-dimensional space! However, it is easy to draw diagrams of
two-dimensional spaces, such as the surface of the earth. (The surface of the
earth is two-dimensional because the position of a point can be specified by
two coordinates, latitude and longitude.) I shall generally use diagrams in
which time increases upward and one of the spatial dimensions is shown
horizontally. The other two spatial dimensions are ignored or, sometimes,
one of them is indicated by perspective. (These are called space-time
diagrams, like Figure 2:1.)
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Figure 2:2
For example, in Figure 2:2 time is measured upward in years and the
distance along the line from the sun to Alpha Centauri is measured
horizontally in miles. The paths of the sun and of Alpha Centauri through
space-time are shown as the vertical lines on the left and right of the
diagram. A ray of light from the sun follows the diagonal line, and takes
four years to get from the sun to Alpha Centauri. As we have seen,
Maxwell’s equations predicted that the speed of light should be the same
whatever the speed of the source, and this has been confirmed by accurate
measurements. It follows from this that if a pulse of light is emitted at a
particular time at a particular point in space, then as time goes on it will
spread out as a sphere of light whose size and position are independent of
the speed of the source. After one millionth of a second the light will have
spread out to form a sphere with a radius of 300 meters; after two millionths
of a second, the radius will be 600 meters; and so on. It will be like the
ripples that spread out on the surface of a pond when a stone is thrown in.
The ripples spread out as a circle that gets bigger as time goes on. If one
stacks snapshots of the ripples at different times one above the other, the
expanding circle of ripples will mark out a cone whose tip is at the place
and time at which the stone hit the water Figure 2:3.
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Figure 2:3
Similarly, the light spreading out from an event forms a (three-dimensional)
cone in (the four-dimensional) space-time. This cone is called the future
light cone of the event. In the same way we can draw another cone, called
the past light cone, which is the set of events from which a pulse of light is
able to reach the given event Figure 2:4.
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Figure 2:4
Given an event P, one can divide the other events in the universe into three
classes. Those events that can be reached from the event P by a particle or
wave traveling at or below the speed of light are said to be in the future of
P. They will lie within or on the expanding sphere of light emitted from the
event P. Thus they will lie within or on the future light cone of P in the
space-time diagram. Only events in the future of P can be affected by what
happens at P because nothing can travel faster than light.
Similarly, the past of P can be defined as the set of all events from which it
is possible to reach the event P
traveling at or below the speed of light. It is thus the set of events that can
affect what happens at P. The events that do not lie in the future or past of P
are said to lie in the elsewhere of P Figure 2:5.
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Figure 2:5
What happens at such events can neither affect nor be affected by what
happens at P. For example, if the sun were to cease to shine at this very
moment, it would not affect things on earth at the present time because they
would be in the elsewhere of the event when the sun went out Figure 2:6.
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Figure 2:6
We would know about it only after eight minutes, the time it takes light to
reach us from the sun. Only then would events on earth lie in the future
light cone of the event at which the sun went out. Similarly, we do not know
what is happening at the moment farther away in the universe: the light that
we see from distant galaxies left them millions of years ago, and in the case
of the most distant object that we have seen, the light left some eight
thousand million years ago. Thus, when we look at the universe, we are
seeing it as it was in the past. If one neglects gravitational effects, as
Einstein and Poincare did in 1905, one has what is called the special theory
of relativity. For every event in space-time we may construct a light cone
(the set of all possible paths of light in space-time emitted at that event),
and since the speed of light is the same at every event and in every
direction, all the light cones will be identical and will all point in the same
direction. The theory also tells us that nothing can travel faster than light.
This means that the path of any object through space and time must be
represented by a line that lies within the light cone at each event on it (Fig.
2.7). The special theory of relativity was very successful in explaining that
the speed of light appears the same to all observers (as shown by the
Michelson-Morley experiment) and in describing what happens when
things move at speeds close to the speed of light. However, it was
inconsistent with the Newtonian theory of gravity, which said that objects
attracted each other with a force that depended on the distance between
them. This meant that if one moved one of the objects, the force on the
other one would change instantaneously. Or in other gravitational effects
should travel with infinite velocity, instead of at or below the speed of light,
as the special theory of relativity required. Einstein made a number of
unsuccessful attempts between 1908 and 1914 to find a theory of gravity
that was consistent with special relativity. Finally, in 1915, he proposed
what we now call the general theory of relativity. Einstein made the
revolutionary suggestion that gravity is not a force like other forces, but is a
consequence of the fact that space-time is not flat, as had been previously
assumed: it is curved, or “warped,” by the distribution
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of mass and energy in it. Bodies like the earth are not made to move on
curved orbits by a force called gravity; instead, they follow the nearest thing
to a straight path in a curved space, which is called a geodesic. A geodesic
is the shortest (or longest) path between two nearby points. For example,
the surface of the earth is a two-dimensional curved space. A geodesic on
the earth is called a great circle, and is the shortest route between two points
(Fig. 2.8). As the geodesic is the shortest path between any two airports,
this is the route an airline navigator will tell the pilot to fly along. In general
relativity, bodies always follow straight lines in four-dimensional spacetime, but they nevertheless appear to us to move along curved paths in our
three-dimensional space. (This is rather like watching an airplane flying
over hilly ground. Although it follows a straight line in three-dimensional
space, its shadow follows a curved path on the two-dimensional ground.)
The mass of the sun curves space-time in such a way that although the earth
follows a straight path in four-dimensional space-time, it appears to us to
move along a circular orbit in three-dimensional space. Fact, the orbits of
the planets predicted by general relativity are almost exactly the same as
those predicted by the Newtonian theory of gravity. However, in the case of
Mercury, which, being the nearest planet to the sun, feels the strongest
gravitational effects, and has a rather elongated orbit, general relativity
predicts that the long axis of the ellipse should rotate about the sun at a rate
of about one degree in ten thousand years. Small though this effect is, it had
been noticed before 1915 and served as one of the first confirmations of
Einstein’s theory. In recent years the even smaller deviations of the orbits of
the other planets from the Newtonian predictions have been measured by
radar and found to agree with the predictions of general relativity. Light
rays too must follow geodesics in space-time. Again, the fact that space is
curved means that light no longer appears to travel in straight lines in space.
So general relativity predicts that light should be bent by gravitational
fields. For example, the theory predicts that the light cones of points near
the sun would be slightly bent inward, on account of the mass of the sun.
This means that light from a distant star that happened to pass near the sun
would be deflected through a small angle, causing the star to appear in a
different position to an observer on the earth (Fig. 2.9). Of course, if the
light from the star always passed close to the sun, we would not be able to
tell whether the light was being deflected or if instead the star was really
where we see it. However, as the earth orbits around the sun, different stars
appear to pass behind the sun and have their light deflected. They therefore
change their apparent position relative to other stars. It is normally very
difficult to see this effect, because the light from the sun makes it
impossible to observe stars that appear near to the sun the sky. However, it
is possible to do so during an eclipse of the sun, when the sun’s light is
blocked out by the moon. Einstein’s prediction of light deflection could not
be tested immediately in 1915, because the First World War was in
progress, and it was not until 1919 that a British expedition, observing an
eclipse from West Africa, showed that light was indeed deflected by the
sun, just as predicted by the theory. This proof of a German theory by
British scientists was hailed as a great act of reconciliation between the two
countries after the war. It is ionic, therefore, that later examination of the
photographs taken on that expedition showed the errors were as great as the
effect they were trying to measure. Their measurement had been sheer luck,
or a case of knowing the result they wanted to get, not an uncommon
occurrence in science. The light deflection has, however, been accurately
confirmed by a number of later observations.
Another prediction of general relativity is that time should appear to slower
near a massive body like the earth. This is because there is a relation
between the energy of light and its frequency (that is, the number of waves
of light per second): the greater the energy, the higher frequency. As light
travels upward in the earth’s gravitational field, it loses energy, and so its
frequency goes down. (This means that the length of time between one
wave crest and the next goes up.) To someone high up, it would appear that
everything down below was making longer to happen. This prediction was
tested in 1962, using a pair of very accurate clocks mounted at the top and
bottom of a water tower. The clock at the bottom, which was nearer the
earth, was found to run slower, in exact agreement with general relativity.
The difference in the speed of clocks at different heights above the earth is
now of considerable practical importance, with the advent of very accurate
navigation systems based on signals from satellites. If one ignored the
predictions of general relativity, the position that one calculated would be
wrong by several miles!
Newton’s laws of motion put an end to the idea of absolute position in
space. The theory of relativity gets rid of absolute time. Consider a pair of
twins. Suppose that one twin goes to live on the top of a mountain while the
other stays at sea level. The first twin would age faster than the second.
Thus, if they met again, one would be older than the other. In this case, the
difference in ages would be very small, but it would be much larger if one
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of the twins went for a long trip in a spaceship at nearly the speed of light.
When he returned, he would be much younger than the one who stayed on
earth. This is known as the twins paradox, but it is a paradox only if one has
the idea of absolute time at the back of one’s mind. In the theory of
relativity there is no unique absolute time, but instead each individual has
his own personal measure of time that depends on where he is and how he
is moving.
Before 1915, space and time were thought of as a fixed arena in which
events took place, but which was not affected by what happened in it. This
was true even of the special theory of relativity. Bodies moved, forces
attracted and repelled, but time and space simply continued, unaffected. It
was natural to think that space and time went on forever.
The situation, however, is quite different in the general theory of relativity.
Space and time are now dynamic quantities: when a body moves, or a force
acts, it affects the curvature of space and time – and in turn the structure of
space-time affects the way in which bodies move and forces act. Space and
time not only affect but also are affected by everything that happens in the
universe. Just as one cannot talk about events in the universe without the
notions of space and time, so in general relativity it became meaningless to
talk about space and time outside the limits of the universe.
In the following decades this new understanding of space and time was to
revolutionize our view of the universe. The old idea of an essentially
unchanging universe that could have existed, and could continue to exist,
forever was replaced by the notion of a dynamic, expanding universe that
seemed to have begun a finite time ago, and that might end at a finite time
in the future. That revolution forms the subject of the next chapter. And
years later, it was also to be the starting point for my work in theoretical
physics. Roger Penrose and I showed that Einstein’s general theory of
relativity implied that the universe must have a beginning and, possibly, an
end.
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CHAPTER 3
THE EXPANDING UNIVERSE
If one looks at the sky on a clear, moonless night, the brightest objects one
sees are likely to be the planets Venus, Mars, Jupiter, and Saturn. There will
also be a very large number of stars, which are just like our own sun but
much farther from us. Some of these fixed stars do, in fact, appear to
change very slightly their positions relative to each other as earth orbits
around the sun: they are not really fixed at all! This is because they are
comparatively near to us. As the earth goes round the sun, we see them
from different positions against the background of more distant stars. This
is fortunate, because it enables us to measure directly the distance of these
stars from us: the nearer they are, the more they appear to move. The
nearest star, called Proxima Centauri, is found to be about four light-years
away (the light from it takes about four years to reach earth), or about
twenty-three million million miles. Most of the other stars that are visible to
the naked eye lie within a few hundred light-years of us. Our sun, for
comparison, is a mere light-minutes away! The visible stars appear spread
all over the night sky, but are particularly concentrated in one band, which
we call the Milky Way. As long ago as 1750, some astronomers were
suggesting that the appearance of the Milky Way could be explained if most
of the visible stars lie in a single disklike configuration, one example of
what we now call a spiral galaxy. Only a few decades later, the astronomer
Sir William Herschel confirmed this idea by painstakingly cataloging the
positions and distances of vast numbers of stars. Even so, the idea gained
complete acceptance only early this century.
Our modern picture of the universe dates back to only 1924, when the
American astronomer Edwin Hubble demonstrated that ours was not the
only galaxy. There were in fact many others, with vast tracts of empty space
between them. In order to prove this, he needed to determine the distances
to these other galaxies, which are so far away that, unlike nearby stars, they
really do appear fixed. Hubble was forced, therefore, to use indirect
methods to measure the distances. Now, the apparent brightness of a star
depends on two factors: how much light it radiates (its luminosity), and how
far it is from us. For nearby stars, we can measure their apparent brightness
and their distance, and so we can work out their luminosity. Conversely, if
we knew the luminosity of stars in other galaxies, we could work out their
distance by measuring their apparent brightness. Hubble noted that certain
types of stars always have the same luminosity when they are near enough
for us to measure; therefore, he argued, if we found such stars in another
galaxy, we could assume that they had the same luminosity – and so
calculate the distance to that galaxy. If we could do this for a number of
stars in the same galaxy, and our calculations always gave the same
distance, we could be fairly confident of our estimate. In this way, Edwin
Hubble worked out the distances to nine different galaxies. We now know
that our galaxy is only one of some hundred thousand million that can be
seen using modern telescopes, each galaxy itself containing some hundred
thousand million stars. Figure 3:1 shows a picture of one spiral galaxy that
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Figure 3:1
We live in a galaxy that is about one hundred thousand light-years across
and is slowly rotating; the stars in its spiral arms orbit around its center
about once every several hundred million years. Our sun is just an ordinary,
average-sized, yellow star, near the inner edge of one of the spiral arms. We
have certainly come a long way since Aristotle and Ptolemy, when thought
that the earth was the center of the universe!
Stars are so far away that they appear to us to be just pinpoints of light. We
cannot see their size or shape. So how can we tell different types of stars
apart? For the vast majority of stars, there is only one characteristic feature
that we can observe – the color of their light. Newton discovered that if
light from the sun passes through a triangular-shaped piece of glass, called a
prism, it breaks up into its component colors (its spectrum) as in a rainbow.
By focusing a telescope on an individual star or galaxy, one can similarly
observe the spectrum of the light from that star or galaxy. Different stars
have different spectra, but the relative brightness of the different colors is
always exactly what one would expect to find in the light emitted by an
object that is glowing red hot. (In fact, the light emitted by any opaque
object that is glowing red hot has a characteristic spectrum that depends
only on its temperature – a thermal spectrum. This means that we can tell a
star’s temperature from the spectrum of its light.) Moreover, we find that
certain very specific colors are missing from stars’
spectra, and these missing colors may vary from star to star. Since we know
that each chemical element absorbs a characteristic set of very specific
colors, by matching these to those that are missing from a star’s spectrum,
we can determine exactly which elements are present in the star’s
atmosphere. In the 1920s, when astronomers began to look at the spectra of
stars in other galaxies, they found something most peculiar: there were the
same characteristic sets of missing colors as for stars in our own galaxy, but
they were all shifted by the same relative amount toward the red end of the
spectrum. To understand the implications of this, we must first understand
the Doppler effect. As we have seen, visible light consists of fluctuations, or
waves, in the electromagnetic field. The wavelength (or distance from one
wave crest to the next) of light is extremely small, ranging from four to
seven ten-millionths of a meter. The different wavelengths of light are what
the human eye sees as different colors, with the longest wavelengths
appearing at the red end of the spectrum and the shortest wavelengths at the
blue end. Now imagine a source of light at a constant distance from us, such
as a star, emitting waves of light at a constant wavelength. Obviously the
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the waves we receive will be the same as the wavelength at which they are
emitted (the gravitational field of the galaxy will not be large enough to
have a significant effect). Suppose now that the source starts moving toward
us. When the source emits the next wave crest it will be nearer to us, so the
distance between wave crests will be smaller than when the star was
stationary. This means that the wavelength of the waves we receive is
shorter than when the star was stationary. Correspondingly, if the source is
moving away from us, the wavelength of the waves we receive will be
longer. In the case of light, therefore, means that stars moving away from us
will have their spectra shifted toward the red end of the spectrum (redshifted) and those moving toward us will have their spectra blue-shifted.
This relationship between wavelength and speed, which is called the
Doppler effect, is an everyday experience. Listen to a car passing on the
road: as the car is approaching, its engine sounds at a higher pitch
(corresponding to a shorter wavelength and higher frequency of sound
waves), and when it passes and goes away, it sounds at a lower pitch. The
behavior of light or radio waves is similar. Indeed, the police make use of
the Doppler effect to measure the speed of cars by measuring the
wavelength of pulses of radio waves reflected off them.
ln the years following his proof of the existence of other galaxies, Rubble
spent his time cataloging their distances and observing their spectra. At that
time most people expected the galaxies to be moving around quite
randomly, and so expected to find as many blue-shifted spectra as redshifted ones. It was quite a surprise, therefore, to find that most galaxies
appeared red-shifted: nearly all were moving away from us! More
surprising still was the finding that Hubble published in 19 29: even the size
of a galaxy’s red shift is not random, but is directly proportional to the
galaxy’s distance from us. Or, in other words, the farther a galaxy is, the
faster it is moving away! And that meant that the universe could not be
static, as everyone previously had thought, is in fact expanding; the distance
between the different galaxies is changing all the time. The discovery that
the universe is expanding was one of the great intellectual revolutions of the
twentieth century. With hindsight, it is easy wonder why no one had thought
of it before. Newton, and others should have realized that a static universe
would soon start to contract under the influence of gravity. But suppose
instead that the universe is expanding. If it was expanding fairly slowly, the
force of gravity would cause it eventually to stop expanding and then to
start contracting. However, if it was expanding at more than a certain
critical rate, gravity would never be strong enough to stop it, and the
universe would continue to expand forever. This is a bit like what happens
when one fires a rocket upward from the surface of the earth. If it has a
fairly low speed, gravity will eventually stop the rocket and it will start
falling back. On the other hand, if the rocket has more than a certain critical
speed (about seven miles per second), gravity will not be strong enough to
pull it back, so it will keep going away from the earth forever. This
behavior of the universe could have been predicted from Newton’s theory
of gravity at any time in the nineteenth, the eighteenth, or even the late
seventeenth century. Yet so strong was the belief in a static universe that it
persisted into the early twentieth century. Even Einstein, when he
formulated the general theory of relativity in 1915, was so sure that the
universe had to be static that he modified his theory to make this possible,
introducing a so-called cosmological constant into his equations. Einstein
introduced a new “antigravity” force, which, unlike other forces, did not
come from any particular source but was built into the very fabric of spacetime. He claimed that space-time had an inbuilt tendency to expand, and
this could be made to balance exactly the attraction of all the matter in the
universe, so that a static universe would result. Only one man, it seems, was
willing to take general relativity at face value, and while Einstein and other
physicists were looking for ways of avoiding general relativity’s prediction
of a nonstatic universe, the Russian physicist and mathematician Alexander
Friedmann instead set about explaining it.
Friedmann made two very simple assumptions about the universe: that the
universe looks identical in whichever direction we look, and that this would
also be true if we were observing the universe from anywhere else. From
these two ideas alone, Friedmann showed that we should not expect the
universe to be static. In fact, in 1922, several years before Edwin Hubble’s
discovery, Friedmann predicted exactly what Hubble found!
The assumption that the universe looks the same in every direction is
clearly not true in reality. For example, as we have seen, the other stars in
our galaxy form a distinct band of light across the night sky, called the
Milky Way. But if we look at distant galaxies, there seems to be more or
less the same number of them. So the universe does seem to be roughly the
same in every direction, provided one views it on a large scale compared to
the distance between galaxies, and ignores the differences on small scales.
For a long time, this was sufficient justification for Friedmann’s assumption
– as a rough approximation to the real universe. But more recently a lucky
accident uncovered the fact that Friedmann’s assumption is in fact a
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description of our universe.
In 1965 two American physicists at the Bell Telephone Laboratories in New
Jersey, Arno Penzias and Robert Wilson, were testing a very sensitive
microwave detector. (Microwaves are just like light waves, but with a
wavelength of around a centimeter.) Penzias and Wilson were worried when
they found that their detector was picking up more noise than it ought to.
The noise did not appear to be coming from any particular direction. First
they discovered bird droppings in their detector and checked for other
possible malfunctions, but soon ruled these out. They knew that any noise
from within the atmosphere would be stronger when the detector was not
pointing straight up than when it was, because light rays travel through
much more atmosphere when received from near the horizon than when
received from directly overhead. The extra noise was the same whichever
direction the detector was pointed, so it must come from outside the
atmosphere. It was also the same day and night and throughout the year,
even though the earth was rotating on its axis and orbiting around the sun.
This showed that the radiation must come from beyond the Solar System,
and even from beyond the galaxy, as otherwise it would vary as the
movement of earth pointed the detector in different directions. In fact, we
know that the radiation must have traveled to us across most of the
observable universe, and since it appears to be the same in different
directions, the universe must also be the same in every direction, if only on
a large scale. We now know that whichever direction we look, this noise
never varies by more than a tiny fraction: so Penzias and Wilson had
unwittingly stumbled across a remarkably accurate confirmation of
Friedmann’s first assumption. However, because the universe is not exactly
the same in every direction, but only on average on a large scale, the
microwaves cannot be exactly the same in every direction either. There
have to be slight variations between different directions. These were first
detected in 1992 by the Cosmic Background Explorer satellite, or COBE, at
a level of about one part in a hundred thousand. Small though these
variations are, they are very important, as will be explained in Chapter 8. At
roughly the same time as Penzias and Wilson were investigating noise in
their detector, two American physicists at nearby Princeton University, Bob
Dicke and Jim Peebles, were also taking an interest in microwaves. They
were working on a suggestion, made by George Gamow (once a student of
Alexander Friedmann), that the early universe should have been very hot
and dense, glowing white hot. Dicke and Peebles argued that we should still
be able to see the glow of the early universe, because light from very distant
parts of it would only just be reaching us now. However, the expansion of
the universe meant that this light should be so greatly red-shifted that it
would appear to us now as microwave radiation. Dicke and Peebles were
preparing to look for this radiation when Penzias and Wilson heard about
their work and realized that they had already found it. For this, Penzias and
Wilson were awarded the Nobel Prize in 1978 (which seems a bit hard on
Dicke and Peebles, not to mention Gamow!).
Now at first sight, all this evidence that the universe looks the same
whichever direction we look in might seem to suggest there is something
special about our place in the universe. In particular, it might seem that if
we observe all other galaxies to be moving away from us, then we must be
at the center of the universe. There is, however, an alternate explanation: the
universe might look the same in every direction as seen from any other
galaxy too. This, as we have seen, was Friedmann’s second assumption. We
have no scientific evidence for, or against, this assumption. We believe it
only on grounds of modesty: it would be most remarkable if the universe
looked the same in every direction around us, but not around other points in
the universe! In Friedmann’s model, all the galaxies are moving directly
away from each other. The situation is rather like a balloon with a number
of spots painted on it being steadily blown up. As the balloon expands, the
distance between any two spots increases, but there is no spot that can be
said to be the center of the expansion. Moreover, the farther apart the spots
are, the faster they will be moving apart. Similarly, in Friedmann’s model
the speed at which any two galaxies are moving apart is proportional to the
distance between them. So it predicted that the red shift of a galaxy should
be directly proportional to its distance from us, exactly as Hubble found.
Despite the success of his model and his prediction of Hubble’s
observations, Friedmann’s work remained largely unknown in the West
until similar models were discovered in 1935 by the American physicist
Howard Robertson and the British mathematician Arthur Walker, in
response to Hubble’s discovery of the uniform expansion of the universe.
Although Friedmann found only one, there are in fact three different kinds
of models that obey Friedmann’s two fundamental assumptions. In the first
kind (which Friedmann found) the universe is expanding sufficiently slowly
that the gravitational attraction between the different galaxies causes the
expansion to slow down and eventually to stop. The galaxies then start to
move toward each other and the universe contracts.
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Figure 3:2
Figure 3:2 shows how the distance between two neighboring galaxies
changes as time increases. It starts at zero, increases to a maximum, and
then decreases to zero again. In the second kind of solution, the universe is
expanding so rapidly that the gravitational attraction can never stop it,
though it does slow it down a bit. Figure 3:3
Figure 3:3 Shows the Separation between neighboring galaxies in this
model. It starts at zero and eventually the galaxies are moving apart at a
steady speed. Finally, there is a third kind of solution, in which the universe
is expanding only just fast enough to avoid recollapse.
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Figure 3:4
In this case the separation, shown in Figure 3:4, also starts at zero and
increases forever. However, the speed at which the galaxies are moving
apart gets smaller and smaller, although it never quite reaches zero. A
remarkable feature of the first kind of Friedmann model is that in it the
universe is not infinite in space, but neither does space have any boundary.
Gravity is so strong that space is bent round onto itself, making it rather like
the surface of the earth. If one keeps traveling in a certain direction on the
surface of the earth, one never comes up against an impassable barrier or
falls over the edge, but eventually comes back to where one started.
In the first kind of Friedmann model, space is just like this, but with three
dimensions instead of two for the earth’s surface. The fourth dimension,
time, is also finite in extent, but it is like a line with two ends or boundaries,
a beginning and an end. We shall see later that when one combines general
relativity with the uncertainty principle of quantum mechanics, it is possible
for both space and time to be finite without any edges or boundaries.
The idea that one could go right round the universe and end up where one
started makes good science fiction, but it doesn’t have much practical
significance, because it can be shown that the universe would recollapse to
zero size before one could get round. You would need to travel faster than
light in order to end up where you started before the universe came to an
end – and that is not allowed!
In the first kind of Friedmann model, which expands and recollapses, space
is bent in on itself, like the surface of the earth. It is therefore finite in
extent. In the second kind of model, which expands forever, space is bent
the other way, like the surface of a saddle. So in this case space is infinite.
Finally, in the third kind of Friedmann model, with just the critical rate of
expansion, space is flat (and therefore is also infinite). But which
Friedmann model describes our universe? Will the universe eventually stop
expanding and start contracting, or will it expand forever? To answer this
question we need to know the present rate of expansion of the universe and
its present average density. If the density is less than a certain critical value,
determined by the rate of expansion, the gravitational attraction will be too
weak to halt the expansion. If the density is greater than the critical value,
gravity will stop the expansion at some time in the future and cause the
universe to recollapse.
We can determine the present rate of expansion by measuring the velocities
at which other galaxies are moving away from us, using the Doppler effect.
This can be done very accurately. However, the distances to the galaxies are
not very well known because we can only measure them indirectly. So all
we know is that the universe is expanding by between 5 percent and 10
percent every thousand million years. However, our uncertainty about the
present average density of the universe is even greater. If we add up the
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the stars that we can see in our galaxy and other galaxies, the total is less
than one hundredth of the amount required to halt the expansion of the
universe, even for the lowest estimate of the rate of expansion. Our galaxy
and other galaxies, however, must contain a large amount of “dark matter”
that we cannot see directly, but which we know must be there because of
the influence of its gravitational attraction on the orbits of stars in the
galaxies. Moreover, most galaxies are found in clusters, and we can
similarly infer the presence of yet more dark matter in between the galaxies
in these clusters by its effect on the motion of the galaxies. When we add up
all this dark matter, we still get only about one tenth of the amount required
to halt the expansion. However, we cannot exclude the possibility that there
might be some other form of matter, distributed almost uniformly
throughout the universe, that we have not yet detected and that might still
raise the average density of the universe up to the critical value needed to
halt the expansion. The present evidence therefore suggests that the
universe will probably expand forever, but all we can really be sure of is
that even if the universe is going to recollapse, it won’t do so for at least
another ten thousand million years, since it has already been expanding for
at least that long. This should not unduly worry us: by that time, unless we
have colonized beyond the Solar System, mankind will long since have died
out, extinguished along with our sun!
All of the Friedmann solutions have the feature that at some time in the past
(between ten and twenty thousand million years ago) the distance between
neighboring galaxies must have been zero. At that time, which we call the
big bang, the density of the universe and the curvature of space-time would
have been infinite. Because mathematics cannot really handle infinite
numbers, this means that the general theory of relativity (on which
Friedmann’s solutions are based) predicts that there is a point in the
universe where the theory itself breaks down. Such a point is an example of
what mathematicians call a singularity. In fact, all our theories of science
are formulated on the assumption that space-time is smooth and nearly fiat,
so they break down at the big bang singularity, where the curvature of
space-time is infinite. This means that even if there were events before the
big bang, one could not use them to determine what would happen
afterward, because predictability would break down at the big bang.
Correspondingly, if, as is the case, we know only what has happened since
the big bang, we could not determine what happened beforehand. As far as
we are concerned, events before the big bang can have no consequences, so
they should not form part of a scientific model of the universe. We should
therefore cut them out of the model and say that time had a beginning at the
big bang.
Many people do not like the idea that time has a beginning, probably
because it smacks of divine intervention. (The Catholic Church, on the
other hand, seized on the big bang model and in 1951officially pronounced
it to be in accordance with the Bible.) There were therefore a number of
attempts to avoid the conclusion that there had been a big bang. The
proposal that gained widest support was called the steady state theory. It
was suggested in 1948 by two refugees from Nazi-occupied Austria,
Hermann Bondi and Thomas Gold, together with a Briton, Fred Hoyle, who
had worked with them on the development of radar during the war. The idea
was that as the galaxies moved away from each other, new galaxies were
continually forming in the gaps in between, from new matter that was being
continually created. The universe would therefore look roughly the same at
all times as well as at all points of space. The steady state theory required a
modification of general relativity to allow for the continual creation of
matter, but the rate that was involved was so low (about one particle per
cubic kilometer per year) that it was not in conflict with experiment. The
theory was a good scientific theory, in the sense described in Chapter 1: it
was simple and it made definite predictions that could be tested by
observation. One of these predictions was that the number of galaxies or
similar objects in any given volume of space should be the same wherever
and whenever we look in the universe. In the late 1950s and early 1960s a
survey of sources of radio waves from outer space was carried out at
Cambridge by a group of astronomers led by Martin Ryle (who had also
worked with Bondi, Gold, and Hoyle on radar during the war). The
Cambridge group showed that most of these radio sources must lie outside
our galaxy (indeed many of them could be identified with other galaxies)
and also that there were many more weak sources than strong ones. They
interpreted the weak sources as being the more distant ones, and the
stronger ones as being nearer. Then there appeared to be less common
sources per unit volume of space for the nearby sources than for the distant
ones. This could mean that we are at the center of a great region in the
universe in which the sources are fewer than elsewhere. Alternatively, it
could mean that the sources were more numerous in the past, at the time
that the radio waves left on their journey to us, than they are now. Either
explanation contradicted the predictions of the steady state theory.
Moreover, the discovery of the microwave radiation by Penzias and Wilson
in 1965 also indicated that the universe must have been much denser in the
past. The steady state theory therefore had to be abandoned.
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Another attempt to avoid the conclusion that there must have been a big
bang, and therefore a beginning of time, was made by two Russian
scientists, Evgenii Lifshitz and Isaac Khalatnikov, in 1963. They suggested
that the big bang might be a peculiarity of Friedmann’s models alone, which
after all were only approximations to the real universe. Perhaps, of all the
models that were roughly like the real universe, only Friedmann’s would
contain a big bang singularity. In Friedmann’s models, the galaxies are all
moving directly away from each other – so it is not surprising that at some
time in the past they were all at the same place. In the real universe,
however, the galaxies are not just moving directly away from each other –
they also have small sideways velocities. So in reality they need never have
been all at exactly the same place, only very close together. Perhaps then
the current expanding universe resulted not from a big bang singularity, but
from an earlier contracting phase; as the universe had collapsed the particles
in it might not have all collided, but had flown past and then away from
each other, producing the present expansion of the the universe that were
roughly like Friedmann’s models but took account of the irregularities and
random velocities of galaxies in the real universe. They showed that such
models could start with a big bang, even though the galaxies were no longer
always moving directly away from each other, but they claimed that this
was still only possible in certain exceptional models in which the galaxies
were all moving in just the right way. They argued that since there seemed
to be infinitely more Friedmann-like models without a big bang singularity
than there were with one, we should conclude that there had not in reality
been a big bang. They later realized, however, that there was a much more
general class of Friedmann-like models that did have singularities, and in
which the galaxies did not have to be moving any special way. They
therefore withdrew their claim in 1970. The work of Lifshitz and
Khalatnikov was valuable because it showed that the universe could have
had a singularity, a big bang, if the general theory of relativity was correct.
However, it did not resolve the crucial question: Does general relativity
predict that our universe should have had a big bang, a beginning of time?
The answer to this carne out of a completely different approach introduced
by a British mathematician and physicist, Roger Penrose, in 1965. Using
the way light cones behave in general relativity, together with the fact that
gravity is always attractive, he showed that a star collapsing under its own
gravity is trapped in a region whose surface eventually shrinks to zero size.
And, since the surface of the region shrinks to zero, so too must its volume.
All the matter in the star will be compressed into a region of zero volume,
so the density of matter and the curvature of space-time become infinite. In
other words, one has a singularity contained within a region of space-time
known as a black hole.
At first sight, Penrose’s result applied only to stars; it didn’t have anything
to say about the question of whether the entire universe had a big bang
singularity in its past. However, at the time that Penrose produced his
theorem, I was a research student desperately looking for a problem with
which to complete my Ph.D. thesis. Two years before, I had been diagnosed
as suffering from ALS, commonly known as Lou Gehrig’s disease, or motor
neuron disease, and given to understand that I had only one or two more
years to live. In these circumstances there had not seemed much point in
working on my Ph.D.–I did not expect to survive that long. Yet two years
had gone by and I was not that much worse. In fact, things were going
rather well for me and I had gotten engaged to a very nice girl, Jane Wilde.
But in order to get married, I needed a job, and in order to get a job, I
needed a Ph.D.
In 1965 I read about Penrose’s theorem that any body undergoing
gravitational collapse must eventually form a singularity. I soon realized
that if one reversed the direction of time in Penrose’s theorem, so that the
collapse became an expansion, the conditions of his theorem would still
hold, provided the universe were roughly like a Friedmann model on large
scales at the present time. Penrose’s theorem had shown that any collapsing
star must end in a singularity; the time-reversed argument showed that any
Friedmann-like expanding universe must have begun with a singularity. For
technical reasons, Penrose’s theorem required that the universe be infinite in
space. So I could in fact, use it to prove that there should be a singularity
only if the universe was expanding fast enough to avoid collapsing again
(since only those Friedmann models were infinite in space). During the next
few years I developed new mathematical techniques to remove this and
other technical conditions from the theorems that proved that singularities
must occur. The final result was a joint paper by Penrose and myself in
1970, which at last proved that there must have been a big bang singularity
provided only that general relativity is correct and the universe contains as
much matter as we observe. There was a lot of opposition to our work,
partly from the Russians because of their Marxist belief in scientific
determinism, and partly from people who felt that the whole idea of
singularities was repugnant and spoiled the beauty of Einstein’s theory.
However, one cannot really argue with a mathematical theorem. So in the
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became generally accepted and nowadays nearly everyone assumes that the
universe started with a big bang singularity. It is perhaps ironic that, having
changed my mind, I am now trying to convince other physicists that there
was in fact no singularity at the beginning of the universe – as we shall see
later, it can disappear once quantum effects are taken into account.
We have seen in this chapter how, in less than half a century, man’s view of
the universe formed over millennia has been transformed. Hubble’s
discovery that the universe was expanding, and the realization of the
insignificance of our own planet in the vastness of the universe, were just
the starting point. As experimental and theoretical evidence mounted, it
became more and more clear that the universe must have had a beginning in
time, until in 1970 this was finally proved by Penrose and myself, on the
basis of Einstein’s general theory of relativity. That proof showed that
general relativity is only an incomplete theory: it cannot tell us how the
universe started off, because it predicts that all physical theories, including
itself, break down at the beginning of the universe. However, general
relativity claims to be only a partial theory, so what the singularity theorems
really show is that there must have been a time in the very early universe
when the universe was so small that one could no longer ignore the smallscale effects of the other great partial theory of the twentieth century,
quantum mechanics. At the start of the 1970s, then, we were forced to turn
our search for an understanding of the universe from our theory of the
extraordinarily vast to our theory of the extraordinarily tiny. That theory,
quantum mechanics, will be described next, before we turn to the efforts to
combine the two partial theories into a single quantum theory of gravity.
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CHAPTER 4
THE UNCERTAINTY PRINCIPLE
The success of scientific theories, particularly Newton’s theory of gravity,
led the French scientist the Marquis de Laplace at the beginning of the
nineteenth century to argue that the universe was completely deterministic.
Laplace suggested that there should be a set of scientific laws that would
allow us to predict everything that would happen in the universe, if only we
knew the complete state of the universe at one time. For example, if we
knew the positions and speeds of the sun and the planets at one time, then we
could use Newton’s laws to calculate the state of the Solar System at any
other time. Determinism seems fairly obvious in this case, but Laplace went
further to assume that there were similar laws governing everything else,
including human behavior.
The doctrine of scientific determinism was strongly resisted by many people,
who felt that it infringed God’s freedom to intervene in the world, but it
remained the standard assumption of science until the early years of this
century. One of the first indications that this belief would have to be
abandoned came when calculations by the British scientists Lord Rayleigh
and Sir James Jeans suggested that a hot object, or body, such as a star, must
radiate energy at an infinite rate. According to the laws we believed at the
time, a hot body ought to give off electromagnetic waves (such as radio
waves, visible light, or X rays) equally at all frequencies. For example, a hot
body should radiate the same amount of energy in waves with frequencies
between one and two million million waves a second as in waves with
frequencies between two and three million million waves a second. Now
since the number of waves a second is unlimited, this would mean that the
total energy radiated would be infinite.
In order to avoid this obviously ridiculous result, the German scientist Max
Planck suggested in 1900 that light, X rays, and other waves could not be
emitted at an arbitrary rate, but only in certain packets that he called quanta.
Moreover, each quantum had a certain amount of energy that was greater the
higher the frequency of the waves, so at a high enough frequency the
emission of a single quantum would require more energy than was available.
Thus the radiation at high frequencies would be reduced, and so the rate at
which the body lost energy would be finite. The quantum hypothesis
explained the observed rate of emission of radiation from hot bodies very
well, but its implications for determinism were not realized until 1926, when
another German scientist, Werner Heisenberg, formulated his famous
uncertainty principle. In order to predict the future position and velocity of a
particle, one has to be able to measure its present position and velocity
accurately. The obvious way to do this is to shine light on the particle. Some
of the waves of light will be scattered by the particle and this will indicate its
position. However, one will not be able to determine the position of the
particle more accurately than the distance between the wave crests of light,
so one needs to use light of a short wavelength in order to measure the
position of the particle precisely. Now, by Planck’s quantum hypothesis, one
cannot use an arbitrarily small amount of light; one has to use at least one
quantum. This quantum will disturb the particle and change its velocity in a
way that cannot be predicted. moreover, the more accurately one measures
the position, the shorter the wavelength of the light that one needs and hence
the higher the energy of a single quantum. So the velocity of the particle will
be disturbed by a larger amount. In other words, the more accurately you try
to measure the position of the particle, the less accurately you can measure
its speed, and vice versa. Heisenberg showed that the uncertainty in the
position of the particle times the uncertainty in its velocity times the mass of
the particle can never be smaller than a certain quantity, which is known as
Planck’s constant. Moreover, this limit does not depend on the way in which
one tries to measure the position or velocity of the particle, or on the type of
particle: Heisenberg’s uncertainty principle is a fundamental, inescapable
property of the world. The uncertainty principle had profound implications
for the way in which we view the world. Even after more than seventy years
they have not been fully appreciated by many philosophers, and are still the
subject of much controversy. The uncertainty principle signaled an end to
Laplace’s dream of a theory of science, a model of the universe that would
be completely deterministic: one certainly cannot predict future events
exactly if one cannot even measure the present state of the universe
precisely! We could still imagine that there is a set of laws that determine
events completely for some supernatural being, who could observe the
present state of the universe without disturbing it. However, such models of
the universe are not of much interest to us ordinary mortals. It seems better
to employ the principle of economy known as Occam’s razor and cut out all
the features of the theory that cannot be observed. This approach led
Heisenberg, Erwin Schrodinger, and Paul Dirac in the 1920s to reformulate
mechanics into a new theory called quantum mechanics, based on the
uncertainty principle. In this theory particles no longer had separate, welldefined positions and velocities that could not be observed, Instead, they had
a quantum state, which was a combination of position and velocity. In
general, quantum mechanics does not predict a single definite result for an
observation. Instead, it predicts a number of different possible outcomes and
tells us how likely each of these is. That is to say, if one made the same
measurement on a large number of similar systems, each of which started off
in the same way, one would find that the result of the
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measurement would be A in a certain number of cases, B in a different
number, and so on. One could predict the approximate number of times that
the result would be A or B, but one could not predict the specific result of an
individual measurement. Quantum mechanics therefore introduces an
unavoidable element of unpredictability or randomness into science. Einstein
objected to this very strongly, despite the important role he had played in the
development of these ideas. Einstein was awarded the Nobel Prize for his
contribution to quantum theory. Nevertheless, Einstein never accepted that
the universe was governed by chance; his feelings were summed up in his
famous statement “God does not play dice.” Most other scientists, however,
were willing to accept quantum mechanics because it agreed perfectly with
experiment. Indeed, it has been an outstandingly successful theory and
underlies nearly all of modern science and technology. It governs the
behavior of transistors and integrated circuits, which are the essential
components of electronic devices such as televisions and computers, and is
also the basis of modern chemistry and biology. The only areas of physical
science into which quantum mechanics has not yet been properly
incorporated are gravity and the large-scale structure of the universe.
Although light is made up of waves, Planck’s quantum hypothesis tells us
that in some ways it behaves as if it were composed of particles: it can be
emitted or absorbed only in packets, or quanta. Equally, Heisenberg’s
uncertainty principle implies that particles behave in some respects like
waves: they do not have a definite position but are “smeared out” with a
certain probability distribution. The theory of quantum mechanics is based
on an entirely new type of mathematics that no longer describes the real
world in terms of particles and waves; it is only the observations of the
world that may be described in those terms. There is thus a duality between
waves and particles in quantum mechanics: for some purposes it is helpful to
think of particles as waves and for other purposes it is better to think of
waves as particles. An important consequence of this is that one can observe
what is called interference between two sets of waves or particles. That is to
say, the crests of one set of waves may coincide with the troughs of the other
set. The two sets of waves then cancel each other out rather than adding up
to a stronger wave as one might expect Figure 4:1. Figure 4:1
A familiar example of interference in the case of light is the colors that are
often seen in soap bubbles. These are caused by reflection of light from the
two sides of the thin film of water forming the bubble. White light consists
of light waves of all different wavelengths, or colors, For certain
wavelengths the crests of the waves reflected from one side of the soap film
coincide with the troughs reflected from the other side. The colors
corresponding to these wavelengths are absent from the reflected light,
which therefore appears to be colored. Interference can also occur for
particles, because of the duality introduced by quantum mechanics. A
famous example is the so-called two-slit experiment Figure 4:2.
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Figure 4:2
Consider a partition with two narrow parallel slits in it. On one side of the
partition one places a source of fight of a particular color (that is, of a
particular wavelength). Most of the light will hit the partition, but a small
amount will go through the slits. Now suppose one places a screen on the far
side of the partition from the light. Any point on the screen will receive
waves from the two slits. However, in general, the distance the light has to
travel from the source to the screen via the two slits will be different. This
will mean that the waves from the slits will not be in phase with each other
when they arrive at the screen: in some places the waves will cancel each
other out, and in others they will reinforce each other. The result is a
characteristic pattern of light and dark fringes. The remarkable thing is that
one gets exactly the same kind of fringes if one replaces the source of light
by a source of particles such as electrons with a definite speed (this means
that the corresponding waves have a definite length). It seems the more
peculiar because if one only has one slit, one does not get any fringes, just a
uniform distribution of electrons across the screen. One might therefore
think that opening another slit would just increase the number of electrons
hitting each point of the screen, but, because of interference, it actually
decreases it in some places. If electrons are sent through the slits one at a
time, one would expect each to pass through one slit or the other, and so
behave just as if the slit it passed through were the only one there – giving a
uniform distribution on the screen. In reality, however, even when the
electrons are sent one at a time, the fringes still appear. Each electron,
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passing through both slits at the same time!
The phenomenon of interference between particles has been crucial to our
understanding of the structure of atoms, the basic units of chemistry and
biology and the building blocks out of which we, and everything around us,
are made. At the beginning of this century it was thought that atoms were
rather like the planets orbiting the sun, with electrons (particles of negative
electricity) orbiting around a central nucleus, which carried positive
electricity. The attraction between the positive and negative electricity was
supposed to keep the electrons in their orbits in the same way that the
gravitational attraction between the sun and the planets keeps the planets in
their orbits. The trouble with this was that the laws of mechanics and
electricity, before quantum mechanics, predicted that the electrons would
lose energy and so spiral inward until they collided with the nucleus. This
would mean that the atom, and indeed all matter, should rapidly collapse to a
state of very high density. A partial solution to this problem was found by
the Danish scientist Niels Bohr in 1913. He suggested that maybe the
electrons were not able to orbit at just any distance from the central nucleus
but only at certain specified distances. If one also supposed that only one or
two electrons could orbit at any one of these distances, this would solve the
problem of the collapse of the atom, because the electrons could not spiral in
any farther than to fill up the orbits with e least distances and energies.
This model explained quite well the structure of the simplest atom,
hydrogen, which has only one electron orbiting around the nucleus. But it
was not clear how one ought to extend it to more complicated atoms.
Moreover, the idea of a limited set of allowed orbits seemed very arbitrary.
The new theory of quantum mechanics resolved this difficulty. It revealed
that an electron orbiting around the nucleus could be thought of as a wave,
with a wavelength that depended on its velocity. For certain orbits, the length
of the orbit would correspond to a whole number (as opposed to a fractional
number) of wavelengths of the electron. For these orbits the wave crest
would be in the same position each time round, so the waves would add up:
these orbits would correspond to Bohr’s allowed orbits. However, for orbits
whose lengths were not a whole number of wavelengths, each wave crest
would eventually be canceled out by a trough as the electrons went round;
these orbits would not be allowed.
A nice way of visualizing the wave/particle duality is the so-called sum over
histories introduced by the American scientist Richard Feynman. In this
approach the particle is not supposed to have a single history or path in
space-time, as it would in a classical, nonquantum theory. Instead it is
supposed to go from A to B by every possible path. With each path there are
associated a couple of numbers: one represents the size of a wave and the
other represents the position in the cycle (i.e., whether it is at a crest or a
trough). The probability of going from A to B is found by adding up the
waves for all the paths. In general, if one compares a set of neighboring
paths, the phases or positions in the cycle will differ greatly. This means that
the waves associated with these paths will almost exactly cancel each other
out. However, for some sets of neighboring paths the phase will not vary
much between paths. The waves for these paths will not cancel out Such
paths correspond to Bohr’s allowed orbits.
With these ideas, in concrete mathematical form, it was relatively
straightforward to calculate the allowed orbits in more complicated atoms
and even in molecules, which are made up of a number of atoms held
together by electrons in orbits that go round more than one nucleus. Since
the structure of molecules and their reactions with each other underlie all of
chemistry and biology, quantum mechanics allows us in principle to predict
nearly everything we see around us, within the limits set by the uncertainty
principle. (In practice, however, the calculations required for systems
containing more than a few electrons are so complicated that we cannot do
them.)
Einstein’s general theory of relativity seems to govern the large-scale
structure of the universe. It is what is called a classical theory; that is, it does
not take account of the uncertainty principle of quantum mechanics, as it
should for consistency with other theories. The reason that this does not lead
to any discrepancy with observation is that all the gravitational fields that we
normally experience are very weak. How-ever, the singularity theorems
discussed earlier indicate that the gravitational field should get very strong in
at least two situations, black holes and the big bang. In such strong fields the
effects of quantum mechanics should be important. Thus, in a sense,
classical general relativity, by predicting points of infinite density, predicts
its own downfall, just as classical (that is, nonquantum) mechanics predicted
its downfall by suggesting that atoms should collapse to infinite density. We
do not yet have a complete consistent theory that unifies general relativity
and quantum mechanics, but we do know a number of the features it should
have. The consequences that these would have for black holes and the big
bang will be described in later chapters. For the moment, however, we shall
turn to the recent attempts to bring together our understanding of the other
forces of nature into a single, unified quantum theory.
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CHAPTER 5
ELEMENTARY PARTICLES AND THE FORCES OF NATURE
Aristotle believed that all the matter in the universe was made up of four
basic elements – earth, air, fire, and water. These elements were acted on by
two forces: gravity, the tendency for earth and water to sink, and levity, the
tendency for air and fire to rise. This division of the contents of the universe
into matter and forces is still used today. Aristotle believed that matter was
continuous, that is, one could divide a piece of matter into smaller and
smaller bits without any limit: one never came up against a grain of matter
that could not be divided further. A few Greeks, however, such as
Democritus, held that matter was inherently grainy and that everything was
made up of large numbers of various different kinds of atoms. (The word
atom means “indivisible” in Greek.) For centuries the argument continued
without any real evidence on either side, but in 1803 the British chemist and
physicist John Dalton pointed out that the fact that chemical compounds
always combined in certain proportions could be explained by the grouping
together of atoms to form units called molecules. However, the argument
between the two schools of thought was not finally settled in favor of the
atomists until the early years of this century. One of the important pieces of
physical evidence was provided by Einstein. In a paper written in 1905, a
few weeks before the famous paper on special relativity, Einstein pointed out
that what was called Brownian motion – the irregular, random motion of
small particles of dust suspended in a liquid – could be explained as the
effect of atoms of the liquid colliding with the dust particles. By this time
there were already suspicions that these atoms were not, after all, indivisible.
Several years previously a fellow of Trinity College, Cambridge, J. J.
Thomson, had demonstrated the existence of a particle of matter, called the
electron, that had a mass less than one thousandth of that of the lightest
atom. He used a setup rather like a modern TV
picture tube: a red-hot metal filament gave off the electrons, and because
these have a negative electric charge, an electric field could be used to
accelerate them toward a phosphor-coated screen. When they hit the screen,
flashes of light were generated. Soon it was realized that these electrons
must be coming from within the atoms themselves, and in 1911 the New
Zealand physicist Ernest Rutherford finally showed that the atoms of matter
do have internal structure: they are made up of an extremely tiny, positively
charged nucleus, around which a number of electrons orbit. He deduced this
by analyzing the way in which alpha-particles, which are positively charged
particles given off by radioactive atoms, are deflected when they collide with
atoms.
At first it was thought that the nucleus of the atom was made up of electrons
and different numbers of a positively charged particle called the proton, from
the Greek word meaning “first,” because it was believed to be the
fundamental unit from which matter was made. However, in 1932 a
colleague of Rutherford’s at Cambridge, James Chadwick, discovered that
the nucleus contained another particle, called the neutron, which had almost
the same mass as a proton but no electrical charge. Chadwick received the
Nobel Prize for his discovery, and was elected Master of Gonville and Caius
College, Cambridge (the college of which I am now a fellow). He later
resigned as Master because of disagreements with the Fellows. There had
been a bitter dispute in the college ever since a group of young Fellows
returning after the war had voted many of the old Fellows out of the college
offices they had held for a long time. This was before my time; I joined the
college in 1965 at the tail end of the bitterness, when similar disagreements
forced another Nobel Prize – winning Master, Sir Nevill Mott, to resign.
Up to about thirty years ago, it was thought that protons and neutrons were
“elementary” particles, but experiments in which protons were collided with
other protons or electrons at high speeds indicated that they were in fact
made up of smaller particles. These particles were named quarks by the
Caltech physicist Murray Gell-Mann, who won the Nobel Prize in 1969 for
his work on them. The origin of the name is an enigmatic quotation from
James Joyce: “Three quarks for Muster Mark!” The word quark is supposed
to be pronounced like quart, but with a k at the end instead of a t, but is
usually pronounced to rhyme with lark.
There are a number of different varieties of quarks: there are six “flavors,”
which we call up, down, strange, charmed, bottom, and top. The first three
flavors had been known since the 1960s but the charmed quark was
discovered only in 1974, the bottom in 1977, and the top in 1995. Each
flavor comes in three “colors,” red, green, and blue. (It should be
emphasized that these terms are just labels: quarks are much smaller than the
wavelength of visible light and so do not have any color in the normal sense.
It is just that modern physicists seem to have more imaginative ways of
naming new particles and phenomena – they no longer restrict themselves to
Greek!) A proton or neutron is made up of three quarks, one of each color. A
proton contains two up quarks and one down quark; a neutron contains two
down and one up. We can create particles made up of the other quarks
(strange, charmed, bottom, and top), but these all have a much greater mass
and decay very rapidly into protons and neutrons.
We now know that neither the atoms nor the protons and neutrons within
them are indivisible. So the question is: what are the truly elementary
particles, the basic building blocks from which everything is made? Since
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is much larger than the size of an atom, we cannot hope to “look” at the parts
of an atom in the ordinary way. We need to use something with a much
smaller wave-length. As we saw in the last chapter, quantum mechanics tells
us that all particles are in fact waves, and that the higher the energy of a
particle, the smaller the wavelength of the corresponding wave. So the best
answer we can give to our question depends on how high a particle energy
we have at our disposal, because this determines on how small a length scale
we can look. These particle energies are usually measured in units called
electron volts. (In Thomson’s experiments with electrons, we saw that he
used an electric field to accelerate the electrons. The energy that an electron
gains from an electric field of one volt is what is known as an electron volt.)
In the nineteenth century, when the only particle energies that people knew
how to use were the low energies of a few electron volts generated by
chemical reactions such as burning, it was thought that atoms were the
smallest unit. In Rutherford’s experiment, the alpha-particles had energies of
millions of electron volts. More recently, we have learned how to use
electromagnetic fields to give particles energies of at first millions and then
thousands of millions of electron volts. And so we know that particles that
were thought to be “elementary” thirty years ago are, in fact, made up of
smaller particles. May these, as we go to still higher energies, in turn be
found to be made from still smaller particles? This is certainly possible, but
we do have some theoretical reasons for believing that we have, or are very
near to, a knowledge of the ultimate building blocks of nature.
Using the wave/particle duality discussed in the last chapter, every-thing in
the universe, including light and gravity, can be described in terms of
particles. These particles have a property called spin. One way of thinking of
spin is to imagine the particles as little tops spinning about an axis. However,
this can be misleading, because quantum mechanics tells us that the particles
do not have any well-defined axis. What the spin of a particle really tells us
is what the particle looks like from different directions. A particle of spin 0
is like a dot: it looks the same from every direction Figure 5:1-i. On the other
hand, a particle of spin 1 is like an arrow: it looks different from different
directions Figure 5:1-ii. Only if one turns it round a complete revolution
(360 degrees) does the particle look the same. A particle of spin 2 is like a
double-headed arrow Figure 5:1-iii: it looks the same if one turns it round
half a revolution (180 degrees). Similarly, higher spin particles look the
same if one turns them through smaller fractions of a complete revolution.
All this seems fairly straightforward, but the remark-able fact is that there
are particles that do not look the same if one turns them through just one
revolution: you have to turn them through two complete revolutions! Such
particles are said to have spin ½.
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Figure 5:1
All the known particles in the universe can be divided into two groups:
particles of spin ½, which make up the matter in the universe, and particles
of spin 0, 1, and 2, which, as we shall see, give rise to forces between the
matter particles. The matter particles obey what is called Pauli’s exclusion
principle. This was discovered in 1925 by an Austrian physicist, Wolfgang
Pauli – for which he received the Nobel Prize in 1945. He was the archetypal
theoretical physicist: it was said of him that even his presence in the same
town would make experiments go wrong! Pauli’s exclusion principle says
that two similar particles can-not exist in the same state; that is, they cannot
have both the same position and the same velocity, within the limits given by
the uncertainty principle. The exclusion principle is crucial because it
explains why matter particles do not collapse to a state of very high density
under the influence of the forces produced by the particles of spin 0, 1, and
2: if the matter particles have very nearly the same positions, they must have
different velocities, which means that they will not stay in the same position
for long. If the world had been created without the exclusion principle,
quarks would not form separate, well-defined protons and neutrons. Nor
would these, together with electrons, form separate, well-defined atoms.
They would all collapse to form a roughly uniform, dense “soup.”
A proper understanding of the electron and other spin-½ particles did not
come until 1928, when a theory was proposed by Paul Dirac, who later was
elected to the Lucasian Professorship of Mathematics at Cambridge (the
same professorship that Newton had once held and that I now hold). Dirac’s
theory was the first of its kind that was consistent with both quantum
mechanics and the special theory of relativity. It explained mathematically
why the electron had spin-½; that is, why it didn’t look the same if you
turned it through only one complete revolution, but did if you turned it
through two revolutions. It also predicted that the electron should have a
partner: an anti-electron, or positron. The discovery of the positron in 1932
confirmed Dirac’s theory and led to his being awarded the Nobel Prize for
physics in 1933. We now know that every particle has an antiparticle, with
which it can annihilate. (In the case of the force-carrying particles, the
antiparticles are the same as the particles themselves.) There could be whole
antiworlds and antipeople made out of antiparticles. However, if you meet
your antiself, don’t shake hands! You would both vanish in a great flash of
light. The question of why there seem to be so many more particles than
antiparticles around us is extremely
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important, and I shall return to it later in the chapter.
In quantum mechanics, the forces or interactions between matter particles
are all supposed to be carried by particles of integer spin – 0, 1, or 2. What
happens is that a matter particle, such as an electron or a quark, emits a
force-carrying particle. The recoil from this emission changes the velocity of
the matter particle. The force-carrying particle then collides with another
matter particle and is absorbed. This collision changes the velocity of the
second particle, just as if there had been a force between the two matter
particles. It is an important property of ‘ the force-carrying particles that they
do not obey the exclusion principle. This means that there is no limit to the
number that can be exchanged, and so they can give rise to a strong force.
However, if the force-carrying particles have a high mass, it will be difficult
to produce and exchange them over a large distance. So the forces that they
carry will have only a short range. On the other hand, if the force-carrying
particles have no mass of their own, the forces will be long range. The forcecarrying particles exchanged between matter particles are said to be virtual
particles because, unlike “real” particles, they cannot be directly detected by
a particle detector. We know they exist, however, because they do have a
measurable effect: they give rise to forces between matter particles. Particles
of spin 0, 1, or 2 do also exist in some circumstances as real particles, when
they can be directly detected. They then appear to us as what a classical
physicist would call waves, such as waves of light or gravitational waves.
They may sometimes be emitted when matter particles interact with each
other by exchanging virtual force-carrying particles. (For example, the
electric repulsive force between two electrons is due to the exchange of
virtual photons, which can never be directly detected; but if one electron
moves past another, real photons may be given off, which we detect as light
waves.)
Force-carrying particles can be grouped into four categories according to the
strength of the force that they carry and the particles with which they
interact. It should be emphasized that this division into four classes is manmade; it is convenient for the construction of partial theories, but it may not
correspond to anything deeper. Ultimately, most physicists hope to find a
unified theory that will explain all four forces as different aspects of a single
force. Indeed, many would say this is the prime goal of physics today.
Recently, successful attempts have been made to unify three of the four
categories of force – and I shall describe these in this chapter. The question
of the unification of the remaining category, gravity, we shall leave till later.
The first category is the gravitational force. This force is universal, that is,
every particle feels the force of gravity, according to its mass or energy.
Gravity is the weakest of the four forces by a long way; it is so weak that we
would not notice it at all were it not for two special properties that it has: it
can act over large distances, and it is always attractive. This means that the
very weak gravitational forces between the individual particles in two large
bodies, such as the earth and the sun, can all add up to produce a significant
force. The other three forces are either short range, or are sometimes
attractive and some-times repulsive, so they tend to cancel out. In the
quantum mechanical way of looking at the gravitational field, the force
between two matter particles is pictured as being carried by a particle of spin
2 called the graviton. This has no mass of its own, so the force that it carries
is long range. The gravitational force between the sun and the earth is
ascribed to the exchange of gravitons between the particles that make up
these two bodies. Although the exchanged particles are virtual, they certainly
do produce a measurable effect – they make the earth orbit the sun! Real
gravitons make up what classical physicists would call gravitational waves,
which are very weak – and so difficult to detect that they have not yet been
observed.
The next category is the electromagnetic force, which interacts with
electrically charged particles like electrons and quarks, but not with
uncharged particles such as gravitons. It is much stronger than the
gravitational force: the electromagnetic force between two electrons is about
a million million million million million million million (1 with forty-two
zeros after it) times bigger than the gravitational force. However, there are
two kinds of electric charge, positive and negative. The force between two
positive charges is repulsive, as is the force between two negative charges,
but the force is attractive between a positive and a negative charge. A large
body, such as the earth or the sun, contains nearly equal numbers of positive
and negative charges. Thus the attractive and repulsive forces between the
individual particles nearly cancel each other out, and there is very little net
electromagnetic force. However, on the small scales of atoms and molecules,
electromagnetic forces dominate. The electromagnetic attraction between
negatively charged electrons and positively charged protons in the nucleus
causes the electrons to orbit the nucleus of the atom, just as gravitational
attraction causes the earth to orbit the sun. The electromagnetic attraction is
pictured as being caused by the exchange of large numbers of virtual
massless particles of spin 1, called photons. Again, the photons that are
exchanged are virtual particles. However, when an electron changes from
one allowed orbit to another one nearer to the nucleus, energy is released and
a real photon is emitted – which can be observed as visible light by the
human eye, if it has the right wave-length, or by a photon detector such as
photographic film. Equally, if a real photon collides with an atom, it may
move an electron from an orbit nearer the nucleus to one farther away. This
uses up the energy of the photon, so it is absorbed.
The third category is called the weak nuclear force, which is responsible for
radioactivity and which acts on all matter particles of spin-½, but not on
particles of spin 0, 1, or 2, such as photons and gravitons. The weak nuclear
force was not well understood until 1967, when Abdus Salam at Imperial
College, London, and Steven Weinberg at Harvard both
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proposed theories that unified this interaction with the electromagnetic force,
just as Maxwell had unified electricity and magnetism about a hundred years
earlier. They suggested that in addition to the photon, there were three other
spin-1
particles, known collectively as massive vector bosons, that carried the weak
force. These were called W+ (pronounced W plus), W-(pronounced W
minus), and Zº (pronounced Z naught), and each had a mass of around 100
GeV (GeV
stands for gigaelectron-volt, or one thousand million electron volts). The
Weinberg-Salam theory exhibits a property known as spontaneous symmetry
breaking. This means that what appear to be a number of completely
different particles at low energies are in fact found to be all the same type of
particle, only in different states. At high energies all these particles behave
similarly. The effect is rather like the behavior of a roulette ball on a roulette
wheel. At high energies (when the wheel is spun quickly) the ball behaves in
essentially only one way – it rolls round and round. But as the wheel slows,
the energy of the ball decreases, and eventually the ball drops into one of the
thirty-seven slots in the wheel. In other words, at low energies there are
thirty-seven different states in which the ball can exist. If, for some reason,
we could only observe the ball at low energies, we would then think that
there were thirty-seven different types of ball!
In the Weinberg-Salam theory, at energies much greater than 100 GeV, the
three new particles and the photon would all behave in a similar manner. But
at the lower particle energies that occur in most normal situations, this
symmetry between the particles would be broken. WE, W, and Zº would
acquire large masses, making the forces they carry have a very short range.
At the time that Salam and Weinberg proposed their theory, few people
believed them, and particle accelerators were not powerful enough to reach
the energies of 100 GeV required to produce real W+, W-, or Zº particles.
However, over the next ten years or so, the other predictions of the theory at
lower energies agreed so well with experiment that, in 1979, Salam and
Weinberg were awarded the Nobel Prize for physics, together with Sheldon
Glashow, also at Harvard, who had suggested similar unified theories of the
electromagnetic and weak nuclear forces. The Nobel committee was spared
the embarrassment of having made a mistake by the discovery in 1983 at
CERN
(European Centre for Nuclear Research) of the three massive partners of the
photon, with the correct predicted masses and other properties. Carlo
Rubbia, who led the team of several hundred physicists that made the
discovery, received the Nobel Prize in 1984, along with Simon van der Meer,
the CERNengineer who developed the antimatter storage system employed.
(It is very difficult to make a mark in experimental physics these days unless
you are already at the top! ) The fourth category is the strong nuclear force,
which holds the quarks together in the proton and neutron, and holds the
protons and neutrons together in the nucleus of an atom. It is believed that
this force is carried by another spin-1 particle, called the gluon, which
interacts only with itself and with the quarks. The strong nuclear force has a
curious property called confinement: it always binds particles together into
combinations that have no color. One cannot have a single quark on its own
because it would have a color (red, green, or blue). Instead, a red quark has
to be joined to a green and a blue quark by a “string” of gluons (red + green
+ blue = white). Such a triplet constitutes a proton or a neutron. Another
possibility is a pair consisting of a quark and an antiquark (red + antired, or
green + antigreen, or blue + antiblue = white). Such combinations make up
the particles known as mesons, which are unstable because the quark and
antiquark can annihilate each other, producing electrons and other particles.
Similarly, confinement prevents one having a single gluon on its own,
because gluons also have color. Instead, one has to have a collection of
gluons whose colors add up to white. Such a collection forms an unstable
particle called a glueball.
The fact that confinement prevents one from observing an isolated quark or
gluon might seem to make the whole notion of quarks and gluons as
particles somewhat metaphysical. However, there is another property of the
strong nuclear force, called asymptotic freedom, that makes the concept of
quarks and gluons well defined. At normal energies, the strong nuclear force
is indeed strong, and it binds the quarks tightly together. However,
experiments with large particle accelerators indicate that at high energies the
strong force becomes much weaker, and the quarks and gluons behave
almost like free particles.
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Figure 5:2
Figure 5:2 shows a photograph of a collision between a high-energy proton
and antiproton. The success of the unification of the electromagnetic and
weak nuclear forces led to a number of attempts to combine these two forces
with the strong nuclear force into what is called a grand unified theory (or
GUT). This title is rather an exaggeration: the resultant theories are not all
that grand, nor are they fully unified, as they do not include gravity. Nor are
they really complete theories, because they contain a number of parameters
whose values cannot be predicted from the theory but have to be chosen to
fit in with experiment. Nevertheless, they may be a step toward a complete,
fully unified theory. The basic idea of GUTs is as follows: as was mentioned
above, the strong nuclear force gets weaker at high energies. On the other
hand, the electromagnetic and weak forces, which are not asymptotically
free, get stronger at high energies. At some very high energy, called the
grand unification energy, these three forces would all have the same strength
and so could just be different aspects of a single force. The GUTs also
predict that at this energy the different spin-½ matter particles, like quarks
and electrons, would also all be essentially the same, thus achieving another
unification. The value of the grand unification energy is not very well
known, but it would probably have to be at least a thousand million million
GeV. The present generation of particle accelerators can collide particles at
energies of about one hundred GeV, and machines are planned that would
raise this to a few thousand GeV. But a machine that was powerful enough to
accelerate particles to the grand unification energy would have to be as big
as the Solar System – and would be unlikely to be funded in the present
economic climate. Thus it is impossible to test grand unified theories directly
in the laboratory. However, just as in the case of the electromagnetic and
weak unified theory, there are low-energy consequences of the theory that
can be tested.
The most interesting of these is the prediction that protons, which make up
much of the mass of ordinary matter, can spontaneously decay into lighter
particles such as antielectrons. The reason this is possible is that at the grand
unification energy there is no essential difference between a quark and an
antielectron. The three quarks inside a proton normally do not have enough
energy to change into antielectrons, but very occasionally one of them may
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sufficient energy to make the transition because the uncertainty principle
means that the energy of the quarks inside the proton cannot be fixed
exactly. The proton would then decay. The probability of a quark gaining
sufficient energy is so low that one is likely to have to wait at least a million
million million million million years (1 followed by thirty zeros). This is
much longer than the time since the big bang, which is a mere ten thousand
million years or so (1 followed by ten zeros). Thus one might think that the
possibility of spontaneous proton decay could not be tested experimentally.
However, one can increase one’s chances of detecting a decay by observing
a large amount of matter containing a very large number of protons. (If, for
example, one observed a number of protons equal to 1 followed by thirtyone zeros for a period of one year, one would expect, according to the
simplest GUT, to observe more than one proton decay.) A number of such
experiments have been carried out, but none have yielded definite evidence
of proton or neutron decay. One experiment used eight thousand tons of
water and was performed in the Morton Salt Mine in Ohio (to avoid other
events taking place, caused by cosmic rays, that might be confused with
proton decay). Since no spontaneous proton decay had been observed during
the experiment, one can calculate that the probable life of the proton must be
greater than ten million million million million million years (1 with thirtyone zeros). This is longer than the lifetime predicted by the simplest grand
unified theory, but there are more elaborate theories in which the predicted
lifetimes are longer. Still more sensitive experiments involving even larger
quantities of matter will be needed to test them. Even though it is very
difficult to observe spontaneous proton decay, it may be that our very
existence is a consequence of the reverse process, the production of protons,
or more simply, of quarks, from an initial situation in which there were no
more quarks than antiquarks, which is the most natural way to imagine the
universe starting out. Matter on the earth is made up mainly of protons and
neutrons, which in turn are made up of quarks. There are no antiprotons or
antineutrons, made up from antiquarks, except for a few that physicists
produce in large particle accelerators. We have evidence from cosmic rays
that the same is true for all the matter in our galaxy: there are no antiprotons
or antineutrons apart from a small number that are produced as particle/
antiparticle pairs in high-energy collisions. If there were large regions of
antimatter in our galaxy, we would expect to observe large quantities of
radiation from the borders between the regions of matter and antimatter,
where many particles would be colliding with their anti-particles,
annihilating each other and giving off high-energy radiation.
We have no direct evidence as to whether the matter in other galaxies is
made up of protons and neutrons or antiprotons and anti-neutrons, but it
must be one or the other: there cannot be a mixture in a single galaxy
because in that case we would again observe a lot of radiation from
annihilations. We therefore believe that all galaxies are composed of quarks
rather than antiquarks; it seems implausible that some galaxies should be
matter and some antimatter. Why should there be so many more quarks than
antiquarks? Why are there not equal numbers of each? It is certainly
fortunate for us that the numbers are unequal because, if they had been the
same, nearly all the quarks and antiquarks would have annihilated each other
in the early universe and left a universe filled with radiation but hardly any
matter. There would then have been no galaxies, stars, or planets on which
human life could have developed. Luckily, grand unified theories may
provide an explanation of why the universe should now contain more quarks
than antiquarks, even if it started out with equal numbers of each. As we
have seen, GUTs allow quarks to change into antielectrons at high energy.
They also allow the reverse processes, antiquarks turning into electrons, and
electrons and antielectrons turning into antiquarks and quarks. There was a
time in the very early universe when it was so hot that the particle energies
would have been high enough for these transformations to take place. But
why should that lead to more quarks than antiquarks? The reason is that the
laws of physics are not quite the same for particles and antiparticles. Up to
1956 it was believed that the laws of physics obeyed each of three separate
symmetries called C, P, and T. The symmetry C means that the laws are the
same for particles and antiparticles. The symmetry P means that the laws are
the same for any situation and its mirror image (the mirror image of a
particle spinning in a right-handed direction is one spinning in a left-handed
direction). The symmetry T means that if you reverse the direction of motion
of all particles and antiparticles, the system should go back to what it was at
earlier times; in other words, the laws are the same in the forward and
backward directions of time. In 1956 two American physicists, Tsung-Dao
Lee and Chen Ning Yang, suggested that the weak force does not in fact
obey the symmetry P. In other words, the weak force would make the
universe develop in a different way from the way in which the mirror image
of the universe would develop. The same year, a colleague, Chien-Shiung
Wu, proved their prediction correct. She did this by lining up the nuclei of
radioactive atoms in a magnetic field, so that they were all spinning in the
same direction, and showed that the electrons were given off more in one
direction than another. The following year, Lee and Yang received the Nobel
Prize for their idea. It was also found that the weak force did not obey the
symmetry C. That is, it would cause a universe composed of antiparticles to
behave differently from our universe. Nevertheless, it seemed that the weak
force did obey the combined symmetry CP. That is, the universe would
develop in the same way as its mirror image if, in addition, every particle
was swapped with its antiparticle! However, in 1964 two more Americans, J.
W. Cronin and Val Fitch, discovered that even the CP
symmetry was not obeyed in the decay of certain particles called K-mesons.
Cronin and Fitch eventually received the Nobel Prize for their work in 1980.
(A lot of prizes have been awarded for showing that the universe is not as
simple as we might have thought!)
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There is a mathematical theorem that says that any theory that obeys
quantum mechanics and relativity must always obey the combined symmetry
CPT. In other words, the universe would have to behave the same if one
replaced particles by antiparticles, took the mirror image, and also reversed
the direction of time. But Cronin and Fitch showed that if one replaces
particles by antiparticles and takes the mirror image, but does not reverse the
direction of time, then the universe does not behave the same. The laws of
physics, therefore, must change if one reverses the direction of time –
they do not obey the symmetry T.
Certainly the early universe does not obey the symmetry T: as time runs
forward the universe expands – if it ran backward, the universe would be
contracting. And since there are forces that do not obey the symmetry T, it
follows that as the universe expands, these forces could cause more
antielectrons to turn into quarks than electrons into antiquarks. Then, as the
universe expanded and cooled, the antiquarks would annihilate with the
quarks, but since there would be more quarks than antiquarks, a small excess
of quarks would remain. It is these that make up the matter we see today and
out of which we ourselves are made. Thus our very existence could be
regarded as a confirmation of grand unified theories, though a qualitative
one only; the uncertainties are such that one cannot predict the numbers of
quarks that will be left after the annihilation, or even whether it would be
quarks or antiquarks that would remain. (Had it been an excess of
antiquarks, however, we would simply have named antiquarks quarks, and
quarks antiquarks.) Grand unified theories do not include the force of
gravity. This does not matter too much, because gravity is such a weak force
that its effects can usually be neglected when we are dealing with elementary
particles or atoms. However, the fact that it is both long range and always
attractive means that its effects all add up. So for a sufficiently large number
of matter particles, gravitational forces can dominate over all other forces.
This is why it is gravity that determines the evolution of the universe. Even
for objects the size of stars, the attractive force of gravity can win over all
the other forces and cause the star to collapse. My work in the 1970s focused
on the black holes that can result from such stellar collapse and the intense
gravitational fields around them. It was this that led to the first hints of how
the theories of quantum mechanics and general relativity might affect each
other – a glimpse of the shape of a quantum theory of gravity yet to come.
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CHAPTER 6
BLACK HOLES
The term black hole is of very recent origin. It was coined in 1969 by the
American scientist John Wheeler as a graphic description of an idea that
goes back at least two hundred years, to a time when there were two theories
about light: one, which Newton favored, was that it was composed of
particles; the other was that it was made of waves. We now know that really
both theories are correct. By the wave/particle duality of quantum
mechanics, light can be regarded as both a wave and a particle. Under the
theory that light is made up of waves, it was not clear how it would respond
to gravity. But if light is composed of particles, one might expect them to be
affected by gravity in the same way that cannonballs, rockets, and planets
are. At first people thought that particles of light traveled infinitely fast, so
gravity would not have been able to slow them down, but the discovery by
Roemer that light travels at a finite speed meant that gravity might have an
important effect.
On this assumption, a Cambridge don, John Michell, wrote a paper in 1783
in the Philosophical Transactions of the Royal Society of London in which
he pointed out that a star that was sufficiently massive and compact would
have such a strong gravitational field that light could not escape: any light
emitted from the surface of the star would be dragged back by the star’s
gravitational attraction before it could get very far. Michell suggested that
there might be a large number of stars like this. Although we would not be
able to see them because the light from them would not reach us, we would
still feel their gravitational attraction. Such objects are what we now call
black holes, because that is what they are: black voids in space. A similar
suggestion was made a few years later by the French scientist the Marquis de
Laplace, apparently independently of Michell. Interestingly enough, Laplace
included it in only the first and second editions of his book The System of the
World, and left it out of later editions; perhaps he decided that it was a crazy
idea. (Also, the particle theory of light went out of favor during the
nineteenth century; it seemed that everything could be explained by the
wave theory, and according to the wave theory, it was not clear that light
would be affected by gravity at all.)
In fact, it is not really consistent to treat light like cannonballs in Newton’s
theory of gravity because the speed of light is fixed. (A cannonball fired
upward from the earth will be slowed down by gravity and will eventually
stop and fall back; a photon, however, must continue upward at a constant
speed. How then can Newtonian grav-ity affect light?) A consistent theory of
how gravity affects light did not come along until Einstein proposed general
relativity in 1915. And even then it was a long time before the implications
of the theory for massive stars were understood. To understand how a black
hole might be formed, we first need an understanding of the life cycle of a
star. A star is formed when a large amount of gas (mostly hydrogen) starts to
collapse in on itself due to its gravitational attraction. As it contracts, the
atoms of the gas collide with each other more and more frequently and at
greater and greater speeds –
the gas heats up. Eventually, the gas will be so hot that when the hydrogen
atoms collide they no longer bounce off each other, but instead coalesce to
form helium. The heat released in this reaction, which is like a controlled
hydrogen bomb explosion, is what makes the star shine. This additional heat
also increases the pressure of the gas until it is sufficient to balance the
gravitational attraction, and the gas stops contracting. It is a bit like a balloon
– there is a balance between the pressure of the air inside, which is trying to
make the balloon expand, and the tension in the rubber, which is trying to
make the balloon smaller. Stars will remain stable like this for a long time,
with heat from the nuclear reactions balancing the gravitational attraction.
Eventually, however, the star will run out of its hydrogen and other nuclear
fuels. Paradoxically, the more fuel a star starts off with, the sooner it runs
out. This is because the more massive the star is, the hotter it needs to be to
balance its gravitational attraction. And the hotter it is, the faster it will use
up its fuel. Our sun has probably got enough fuel for another five thousand
million years or so, but more massive stars can use up their fuel in as little as
one hundred million years, much less than the age of the universe. When a
star runs out of fuel, it starts to cool off and so to contract. What might
happen to it then was first understood only at the end of the 1920s.
In 1928 an Indian graduate student, Subrahmanyan Chandrasekhar, set sail
for England to study at Cambridge with the British astronomer Sir Arthur
Eddington, an expert on general relativity. (According to some accounts, a
journalist told Eddington in the early 1920s that he had heard there were
only three people in the world who understood general relativity. Eddington
paused, then replied, “I am trying to think who the third person is.”) During
his voyage from India, Chandrasekhar worked out how big a star could be
and still support itself against its own gravity after it had used up all its fuel.
The idea was this: when the star becomes small, the matter particles get very
near each other, and so according to the Pauli exclusion principle, they must
have very different velocities. This makes them move away from each other
and so tends to make the star expand. A star can therefore maintain itself at a
constant radius by a balance between the attraction of gravity and the
repulsion that arises from the exclusion principle, just as earlier in its life
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gravity was balanced by the heat.
Chandrasekhar realized, however, that there is a limit to the repulsion that
the exclusion principle can provide. The theory of relativity limits the
maximum difference in the velocities of the matter particles in the star to the
speed of light. This means that when the star got sufficiently dense, the
repulsion caused by the exclusion principle would be less than the attraction
of gravity. Chandrasekhar calculated that a cold star of more than about one
and a half times the mass of the sun would not be able to support itself
against its own gravity. (This mass is now known as the Chandrasekhar
limit.) A similar discovery was made about the same time by the Russian
scientist Lev Davidovich Landau. This had serious implications for the
ultimate fate of massive stars. If a star’s mass is less than the Chandrasekhar
limit, it can eventually stop contracting and settle down to a possible final
state as a “white dwarf” with a radius of a few thousand miles and a density
of hundreds of tons per cubic inch. A white dwarf is supported by the
exclusion principle repulsion between the electrons in its matter. We observe
a large number of these white dwarf stars. One of the first to be discovered is
a star that is orbiting around Sirius, the brightest star in the night sky. Landau
pointed out that there was another possible final state for a star, also with a
limiting mass of about one or two times the mass of the sun but much
smaller even than a white dwarf. These stars would be supported by the
exclusion principle repulsion between neutrons and protons, rather than
between electrons. They were therefore called neutron stars. They would
have a radius of only ten miles or so and a density of hundreds of millions of
tons per cubic inch. At the time they were first predicted, there was no way
that neutron stars could be observed. They were not actually detected until
much later.
Stars with masses above the Chandrasekhar limit, on the other hand, have a
big problem when they come to the end of their fuel. In some cases they may
explode or manage to throw off enough matter to reduce their mass below
the limit and so avoid catastrophic gravitational collapse, but it was difficult
to believe that this always happened, no matter how big the star. How would
it know that it had to lose weight? And even if every star managed to lose
enough mass to avoid collapse, what would happen if you added more mass
to a white dwarf ‘or neutron star to take it over the limit?
Would it collapse to infinite density? Eddington was shocked by that
implication, and he refused to believe Chandrasekhar’s result. Eddington
thought it was simply not possible that a star could collapse to a point. This
was the view of most scientists: Einstein himself wrote a paper in which he
claimed that stars would not shrink to zero size. The hostility of other
scientists, particularly Eddington, his former teacher and the leading
authority on the structure of stars, persuaded Chandrasekhar to abandon this
line of work and turn instead to other problems in astronomy, such as the
motion of star clusters. However, when he was awarded the Nobel Prize in
1983, it was, at least in part, for his early work on the limiting mass of cold
stars.
Chandrasekhar had shown that the exclusion principle could not halt the
collapse of a star more massive than the Chandrasekhar limit, but the
problem of understanding what would happen to such a star, according to
general relativity, was first solved by a young American, Robert
Oppenheimer, in 1939. His result, however, suggested that there would be no
observational consequences that could be detected by the telescopes of the
day. Then World War II intervened and Oppenheimer himself became
closely involved in the atom bomb project. After the war the problem of
gravitational collapse was largely forgotten as most scientists became caught
up in what happens on the scale of the atom and its nucleus. In the 1960s,
however, interest in the large-scale problems of astronomy and cosmology
was revived by a great increase in the number and range of astronomical
observations brought about by the application of modern technology.
Oppenheimer’s work was then rediscovered and extended by a number of
people. The picture that we now have from Oppenheimer’s work is as
follows. The gravitational field of the star changes the paths of light rays in
space-time from what they would have been had the star not been present.
The light cones, which indicate the paths followed in space and time by
flashes of light emitted from their tips, are bent slightly inward near the
surface of the star. This can be seen in the bending of light from distant stars
observed during an eclipse of the sun. As the star contracts, the gravitational
field at its surface gets stronger and the light cones get bent inward more.
This makes it more difficult for light from the star to escape, and the light
appears dimmer and redder to an observer at a distance. Eventually, when
the star has shrunk to a certain critical radius, the gravitational field at the
surface becomes so strong that the light cones are bent inward so much that
light can no longer escape Figure 6:1.
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Figure 6:1
According to the theory of relativity, nothing can travel faster than light.
Thus if light cannot escape, neither can anything else; everything is dragged
back by the gravitational field. So one has a set of events, a region of spacetime, from which it is not possible to escape to reach a distant observer. This
region is what we now call a black hole. Its boundary is called the event
horizon and it coincides with the paths of light rays that just fail to escape
from the black hole. In order to understand what you would see if you were
watching a star collapse to form a black hole, one has to remember that in
the theory of relativity there is no absolute time. Each observer has his own
measure of time. The time for someone on a star will be different from that
for someone at a distance, because of the gravitational field of the star.
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Suppose an intrepid astronaut on the surface of the collapsing star,
collapsing inward with it, sent a signal every second, according to his watch,
to his spaceship orbiting about the star. At some time on his watch, say
11:00, the star would shrink below the critical radius at which the
gravitational field becomes so strong nothing can escape, and his signals
would no longer reach the spaceship. As 11:00 approached his companions
watching from the spaceship would find the intervals between successive
signals from the astronaut getting longer and longer, but this effect would be
very small before 10:59:59. They would have to wait only very slightly more
than a second between the astronaut’s 10:59:58
signal and the one that he sent when his watch read 10:59:59, but they would
have to wait forever for the 11:00 signal. The light waves emitted from the
surface of the star between 10:59:59 and 11:00, by the astronaut’s watch,
would be spread out over an infinite period of time, as seen from the
spaceship. The time interval between the arrival of successive waves at the
spaceship would get longer and longer, so the light from the star would
appear redder and redder and fainter and fainter. Eventually, the star would
be so dim that it could no longer be seen from the spaceship: all that would
be left would be a black hole in space. The star would, however, continue to
exert the same gravitational force on the spaceship, which would continue to
orbit the black hole. This scenario is not entirely realistic, however, because
of the following problem. Gravity gets weaker the farther you are from the
star, so the gravitational force on our intrepid astronaut’s feet would always
be greater than the force on his head. This difference in the forces would
stretch our astronaut out like spaghetti or tear him apart before the star had
contracted to the critical radius at which the event horizon formed! However,
we believe that there are much larger objects in the universe, like the central
regions of galaxies, that can also undergo gravitational collapse to produce
black holes; an astronaut on one of these would not be torn apart before the
black hole formed. He would not, in fact, feel anything special as he reached
the critical radius, and could pass the point of no return without noticing it
However, within just a few hours, as the region continued to collapse, the
difference in the gravitational forces on his head and his feet would become
so strong that again it would tear him apart.
The work that Roger Penrose and I did between 1965 and 1970 showed that,
according to general relativity, there must be a singularity of infinite density
and space-time curvature within a black hole. This is rather like the big bang
at the beginning of time, only it would be an end of time for the collapsing
body and the astronaut. At this singularity the laws of science and our ability
to predict the future would break down. However, any observer who
remained outside the black hole would not be affected by this failure of
predictability, because neither light nor any other signal could reach him
from the singularity. This remarkable fact led Roger Penrose to propose the
cosmic censorship hypothesis, which might be paraphrased as “God abhors a
naked singularity.” In other words, the singularities produced by
gravitational collapse occur only in places, like black holes, where they are
decently hidden from outside view by an event horizon. Strictly, this is what
is known as the weak cosmic censorship hypothesis: it protects observers
who remain outside the black hole from the consequences of the breakdown
of predictability that occurs at the singularity, but it does nothing at all for
the poor unfortunate astronaut who falls into the hole.
There are some solutions of the equations of general relativity in which it is
possible for our astronaut to see a naked singularity: he may be able to avoid
hitting the singularity and instead fall through a "wormhole” and come out in
another region of the universe. This would offer great possibilities for travel
in space and time, but unfortunately it seems that these solutions may all be
highly unstable; the least disturbance, such as the presence of an astronaut,
may change them so that the astronaut could not see the singularity until he
hit it and his time came to an end. In other words, the singularity would
always lie in his future and never in his past. The strong version of the
cosmic censorship hypothesis states that in a realistic solution, the
singularities would always lie either entirely in the future (like the
singularities of gravitational collapse) or entirely in the past (like the , big
bang). I strongly believe in cosmic censorship so I bet Kip Thorne and John
Preskill of Cal Tech that it would always hold. I lost the bet on a technicality
because examples were produced of solutions with a singularity that was
visible from a long way away. So I had to pay up, which according to the
terms of the bet meant I had to clothe their nakedness. But I can claim a
moral victory. The naked singularities were unstable: the least disturbance
would cause them either to disappear or to be hidden behind an event
horizon. So they would not occur in realistic situations.
The event horizon, the boundary of the region of space-time from which it is
not possible to escape, acts rather like a one-way membrane around the
black hole: objects, such as unwary astronauts, can fall through the event
horizon into the black hole, but nothing can ever get out of the black hole
through the event horizon. (Remember that the event horizon is the path in
space-time of light that is trying to escape from the black hole, and nothing
can travel faster than light.) One could well say of the event horizon what
the poet Dante said of the entrance to Hell: “All hope abandon, ye who enter
here.” Anything or anyone who falls through the event horizon will soon
reach the region of infinite density and the end of time.
General relativity predicts that heavy objects that are moving will cause the
emission of gravitational waves, ripples in the curvature of space that travel
at the speed of light. These are similar to light waves, which are ripples of
the electromagnetic field, but they are much harder to detect. They can be
observed by the very slight change in separation they produce between
neighboring freely moving objects. A number of detectors are being built in
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Europe, and Japan that will measure displacements of one part in a thousand
million million million (1 with twenty-one zeros after it), or less than the
nucleus of an atom over a distance of ten miles. Like light, gravitational
waves carry energy away from the objects that emit them. One would
therefore expect a system of massive objects to settle down eventually to a
stationary state, because the energy in any movement would be carried away
by the emission of gravitational waves. (It is rather like dropping a cork into
water: at first it bobs up and down a great deal, but as the ripples carry away
its energy, it eventually settles down to a stationary state.) For example, the
movement of the earth in its orbit round the sun produces gravitational
waves. The effect of the energy loss will be to change the orbit of the earth
so that gradually it gets nearer and nearer to the sun, eventually collides with
it, and settles down to a stationary state. The rate of energy loss in the case
of the earth and the sun is very low – about enough to run a small electric
heater. This means it will take about a thousand million million million
million years for the earth to run into the sun, so there’s no immediate cause
for worry! The change in the orbit of the earth is too slow to be observed,
but this same effect has been observed over the past few years occurring in
the system called PSR 1913 +
16 (PSR stands for “pulsar,” a special type of neutron star that emits regular
pulses of radio waves). This system contains two neutron stars orbiting each
other, and the energy they are losing by the emission of gravitational waves
is causing them to spiral in toward each other. This confirmation of general
relativity won J. H. Taylor and R. A. Hulse the Nobel Prize in 1993. It will
take about three hundred million . years for them to collide. Just before they
do, they will be orbiting so fast that they will emit enough gravitational
waves for detectors like LIGO to pick up. During the gravitational collapse
of a star to form a black hole, the movements would be much more rapid, so
the rate at which energy is carried away would be much higher. It would
therefore not be too long ‘ before it settled down to a stationary state. What
would this final stage look like? One might suppose that it would depend on
all the complex features of the star from which it had formed – not only its
mass and rate of rotation, but also the different densities of various parts of
the star, and the complicated movements of the gases within the star. And if
black holes were as varied as the objects that collapsed to form them, it
might be very difficult to make any predictions about black holes in general.
In 1967, however, the study of black holes was revolutionized by Werner
Israel, a Canadian scientist (who was born in Berlin, brought up in South
Africa, and took his doctoral degree in Ireland). Israel showed that,
according to general relativity, non-rotating black holes must be very simple;
they were perfectly spherical, their size depended only on their mass, and
any two such black holes with the same mass were identical. They could, in
fact, be described by a particular solution of Einstein’s equations that had
been known since 1917, found by Karl Schwarzschild shortly after the
discovery of general relativity. At first many people, including Israel
himself, argued that since black holes had to be perfectly spherical, a black
hole could only form from the collapse of a perfectly spherical object. Any
real star – which would never be perfectly spherical – could therefore only
collapse to form a naked singularity. There was, however, a different
interpretation of Israel’s result, which was advocated by Roger Penrose and
John Wheeler in particular. They argued that the rapid movements involved
in a star’s collapse would mean that the gravitational waves it gave off
would make it ever more spherical, and by the time it had settled down to a
stationary state, it would be precisely spherical. According to this view, any
non-rotating star, however complicated its shape and internal structure,
would end up after gravitational collapse as a perfectly spherical black hole,
whose size would depend only on its mass. Further calculations supported
this view, and it soon came to be adopted generally. Israel’s result dealt with
the case of black holes formed from non-rotating bodies only. In 1963, Roy
Kerr, a New Zealander, found a set of solutions of the equations of general
relativity that described rotating black holes. These
“Kerr” black holes rotate at a constant rate, their size and shape depending
only on their mass and rate of rotation. If the rotation is zero, the black hole
is perfectly round and the solution is identical to the Schwarzschild solution.
If the rotation is non-zero, the black hole bulges outward near its equator
(just as the earth or the sun bulge due to their rotation), and the faster it
rotates, the more it bulges. So, to extend Israel’s result to include rotating
bodies, it was conjectured that any rotating body that collapsed to form a
black hole would eventually settle down to a stationary state described by
the Kerr solution. In 1970 a colleague and fellow research student of mine at
Cambridge, Brandon Carter, took the first step toward proving this
conjecture. He showed that, provided a stationary rotating black hole had an
axis of symmetry, like a spinning top, its size and shape would depend only
on its mass and rate of rotation. Then, in 1971, I proved that any stationary
rotating black hole would indeed have such an axis of symmetry. Finally, in
1973, David Robinson at Kings College, London, used Carter’s and my
results to show that the conjecture had been correct: such a black hole had
indeed to be the Kerr solution. So after gravitational collapse a black hole
must settle down into a state in which it could be rotating, but not pulsating.
Moreover, its size and shape would depend only on its mass and rate of
rotation, and not on the nature of the body that had collapsed to form it. This
result became known by the maxim: “A black hole has no hair.” The “no
hair” theorem is of great practical importance, because it so greatly restricts
the possible types of black holes. One can therefore make detailed models of
objects that might contain black holes and compare the predictions of the
models with observations. It also means that a very large amount of
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the body that has collapsed must be lost when a black hole is formed,
because afterward all we can possibly measure about the body is its mass
and rate of rotation. The significance of this will be seen in the next chapter.
Black holes are one of only a fairly small number of cases in the history of
science in which a theory was developed in great detail as a mathematical
model before there was any evidence from observations that it was correct.
Indeed, this used to be the main argument of opponents of black holes: how
could one believe in objects for which the only evidence was calculations
based on the dubious theory of general relativity? In 1963, however, Maarten
Schmidt, an astronomer at the Palomar Observatory in California, measured
the red shift of a faint starlike object in the direction of the source of radio
waves called 3C273 (that is, source number 273 in the third Cambridge
catalogue of radio sources). He found it was too large to be caused by a
gravitational field: if it had been a gravitational red shift, the object would
have to be so massive and so near to us that it would disturb the orbits of
planets in the Solar System. This suggested that the red shift was instead
caused by the expansion of the universe, which, in turn, meant that the object
was a very long distance away. And to be visible at such a great distance, the
object must be very bright, must, in other words, be emitting a huge amount
of energy. The only mechanism that people could think of that would
produce such large quantities of energy seemed to be the gravitational
collapse not just of a star but of a whole central region of a galaxy. A
number of other similar “quasi-stellar objects,” or quasars, have been
discovered, all with large red shifts. But they are all too far away and
therefore too difficult to observe to provide conclusive evidence of black
holes. Further encouragement for the existence of black holes came in 1967
with the discovery by a research student at Cambridge, Jocelyn Bell-Burnell,
of objects in the sky that were emitting regular pulses of radio waves. At
first Bell and her supervisor, Antony Hewish, thought they might have made
contact with an alien civilization in the galaxy! Indeed, at the seminar at
which they announced their discovery, I remember that they called the first
four sources to be found LGM 1 – 4, LGM standing for “Little Green Men.”
In the end, however, they and everyone else came to the less romantic
conclusion that these objects, which were given the name pulsars, were in
fact rotating neutron stars that were emitting pulses of radio waves because
of a complicated interaction between their magnetic fields and surrounding
matter. This was bad news for writers of space westerns, but very hopeful for
the small number of us who believed in black holes at that time: it was the
first positive evidence that neutron stars existed. A neutron star has a radius
of about ten miles, only a few times the critical radius at which a star
becomes a black hole. If a star could collapse to such a small size, it is not
unreasonable to expect that other stars could collapse to even smaller size
and become black holes. How could we hope to detect a black hole, as by its
very definition it does not emit any light? It might seem a bit like looking for
a black cat in a coal cellar. Fortunately, there is a way. As John Michell
pointed out in his pioneering paper in 1783, a black hole still exerts a
gravitational fierce on nearby objects. Astronomers have observed many
systems in which two stars orbit around each other, attracted toward each
other by gravity. They also observe systems in which there is only one
visible star that is orbiting around some unseen companion. One cannot, of
course, immediately conclude that the companion is a black hole: it might
merely be a star that is too faint to be seen. However, some of these systems,
like the one called Cygnus X-1 Figure 6:2, are also strong sources of X-rays.
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Figure 6:2
The best explanation for this phenomenon is that matter has been blown off
the surface of the visible star. As it falls toward the unseen companion, it
develops a spiral motion (rather like water running out of a bath), and it gets
very hot, emitting X-rays Figure 6:3.
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Figure 6:3
For this mechanism to work, the unseen object has to be very small, like a
white dwarf, neutron star, or black hole. From the observed orbit of the
visible star, one can determine the lowest possible mass of the unseen object.
In the case of Cygnus X-l, this is about six times the mass of the sun, which,
according to Chandrasekhar’r result, is too great for the unseen object to be a
white dwarf. It is also too large a mass to be a neutron star. It seems,
therefore, that it must be a black hole.
There are other models to explain Cygnus X-1 that do not include a black
hole, but they are all rather far-fetched. A black hole seems to be the only
really natural explanation of the observations. Despite this, I had a bet with
Kip Thorne of the California Institute of Technology that in fact Cygnus X-1
does not contain a black hole! This was a form f insurance policy for me. I
have done a lot of work on black holes, and it would all be wasted if it
turned out that black holes do not exist. But in that case, I would have the
consolation of winning my bet, which would bring me four years of the
magazine Private Eye. In fact, although the situation with Cygnus X-1 has
not changed much since we made the bet in 1975, there is now so much
other observational evidence in favor of black holes that I have conceded the
bet. I paid the specified penalty, which was a one-year subscription to
Penthouse, to the outrage of Kip’s liberated wife. We also now have
evidence for several other black holes in systems like Cygnus X-1 in our
galaxy and in two neighboring galaxies called the Magellanic Clouds. The
number of black holes, however, is almost certainly very much higher; in the
long history of the universe, many stars must have burned all their nuclear
fuel and have had to collapse. The number of black holes may well be
greater even than the number of visible stars, which totals about a hundred
thousand million in our galaxy alone. The extra gravitational attraction of
such a large number of black holes could explain why our galaxy rotates at
the rate it does: the mass of the visible stars is insufficient to account for this.
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have some evidence that there is a much larger black hole, with a mass of
about a hundred thousand times that of the sun, at the center of our galaxy.
Stars in the galaxy that come too near this black hole will be torn apart by
the difference in the gravitational forces on their near and far sides. Their
remains and gas that is thrown off other stars, will fall toward the black hole.
As in the case of Cygnus X-l, the gas will spiral inward and will heat up,
though not as much as in that case. It will not get hot enough to emit X rays,
but it could account for the very compact source of radio waves and infrared
rays that is observed at the galactic center.
It is thought that similar but even larger black holes, with masses of about a
hundred million times the mass of the sun, occur at the centers of quasars.
For example, observations with the Hubble telescope of the galaxy known as
M87
reveal that it contains a disk of gas 130 light-years across rotating about a
central object two thousand million times the mass of the sun. This can only
be a black hole. Matter falling into such a supermassive black hole would
provide the only source of power great enough to explain the enormous
amounts of energy that these objects are emitting. As the matter spirals into
the black hole, it would make the black hole rotate in the same direction,
causing it to develop a magnetic field rather like that of the earth. Very highenergy particles would be generated near the black hole by the in-falling
matter. The magnetic field would be so strong that it could focus these
particles into jets ejected outward along the axis of rotation of the black hole,
that is, in the directions of its north and south poles. Such jets are indeed
observed in a number of galaxies and quasars. One can also consider the
possibility that there might be black holes with masses much less than that of
the sun. Such black holes could not be formed by gravitational collapse,
because their masses are below the Chandrasekhar mass limit: stars of this
low mass can support themselves against the force of gravity even when they
have exhausted their nuclear fuel. Low-mass black holes could form only if
matter was compressed to enormous densities by very large external
pressures. Such conditions could occur in a very big hydrogen bomb: the
physicist John Wheeler once calculated that if one took all the heavy water
in all the oceans of the world, one could build a hydrogen bomb that would
compress matter at the center so much that a black hole would be created.
(Of course, there would be no one left to observe it!) A more practical
possibility is that such low-mass black holes might have been formed in the
high temperatures and pressures of the very early universe. Black holes
would have been formed only if the early universe had not been perfectly
smooth and uniform, because only a small region that was denser than
average could be compressed in this way to form a black hole. But we know
that there must have been some irregularities, because otherwise the matter
in the universe would still be perfectly uniformly distributed at the present
epoch, instead of being clumped together in stars and galaxies. Whether the
irregularities required to account for stars and galaxies would have led to the
formation of a significant number of “primordial” black holes clearly
depends on the details of the conditions in the early universe. So if we could
determine how many primordial black holes there are now, we would learn a
lot about the very early stages of the universe. Primordial black holes with
masses more than a thousand million tons (the mass of a large mountain)
could be detected only by their gravitational influence on other, visible
matter or on the expansion of the universe. However, as we shall learn in the
next chapter, black holes are not really black after all: they glow like a hot
body, and the smaller they are, the more they glow. So, paradoxically,
smaller black holes might actually turn out to be easier to detect than large
ones!
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CHAPTER 7
BLACK HOLES AIN’T SO BLACK
Before 1970, my research on general relativity had concentrated mainly on
the question of whether or not there had been a big bang singularity.
However, one evening in November that year, shortly after the birth of my
daughter, Lucy, I started to think about black holes as I was getting into bed.
My disability makes this rather a slow process, so I had plenty of time. At
that date there was no precise definition of which points in space-time lay
inside a black hole and which lay outside. I had already discussed with
Roger Penrose the idea of defining a black hole as the set of events from
which it was not possible to escape to a large distance, which is now the
generally accepted definition. It means that the boundary of the black hole,
the event horizon, is formed by the light rays that just fail to escape from the
black hole, hovering forever just on the edge Figure 7:1. It is a bit like
running away from the police and just managing to keep one step ahead but
not being able to get clear away!
Figure 7:1
Suddenly I realized that the paths of these light rays could never approach
one another. If they did they must eventually run into one another. It would
be like meeting someone else running away from the police in the opposite
direction – you would both be caught! (Or, in this case, fall into a black
hole.) But if these light rays were swallowed up by the black hole, then they
could not have been on the boundary of the black hole. So the paths of light
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event horizon had always to be moving parallel to, or away from, each other.
Another way of seeing this is that the event horizon, the boundary of the
black hole, is like the edge of a shadow – the shadow of impending doom. If
you look at the shadow cast by a source at a great distance, such as the sun,
you will see that the rays of light in the edge are not approaching each other.
If the rays of light that form the event horizon, the boundary of the black
hole, can never approach each other, the area of the event horizon might stay
the same or increase with time, but it could never decrease because that
would mean that at least some of the rays of light in the boundary would
have to be approaching each other. In fact, the area would increase whenever
matter or radiation fell into the black hole Figure 7:2. Figures 7:2 & 7:3
Or if two black holes collided and merged together to form a single black
hole, the area of the event horizon of the final black hole would be greater
than or equal to the sum of the areas of the event horizons of the original
black holes Figure 7:3. This nondecreasing property of the event horizon’s
area placed an important restriction on the possible behavior of black holes. I
was so excited with my discovery that I did not get much sleep that night.
The next day I rang up Roger Penrose. He agreed with me. I think, in fact,
that he had been aware of this property of the area. However, he had been
using a slightly different definition of a black hole. He had not realized that
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hole according to the two definitions would be the same, and hence so would
their areas, provided the black hole had settled down to a state in which it
was not changing with time.
The nondecreasing behavior of a black hole’s area was very reminiscent of
the behavior of a physical quantity called entropy, which measures the
degree of disorder of a system. It is a matter of common experience that
disorder will tend to increase if things are left to themselves. (One has only
to stop making repairs around the house to see that!) One can create order
out of disorder (for example, one can paint the house), but that requires
expenditure of effort or energy and so decreases the amount of ordered
energy available.
A precise statement of this idea is known as the second law of
thermodynamics. It states that the entropy of an isolated system always
increases, and that when two systems are joined together, the entropy of the
combined system is greater than the sum of the entropies of the individual
systems. For example, consider a system of gas molecules in a box. The
molecules can be thought of as little billiard balls continually colliding with
each other and bouncing off the walls of the box. The higher the temperature
of the gas, the faster the molecules move, and so the more frequently and
harder they collide with the walls of the box and the greater the outward
pressure they exert on the walls. Suppose that initially the molecules are all
confined to the left-hand side of the box by a partition. If the partition is then
removed, the molecules will tend to spread out and occupy both halves of
the box. At some later time they could, by chance, all be in the right half or
back in the left half, but it is overwhelmingly more probable that there will
be roughly equal numbers in the two halves. Such a state is less ordered, or
more disordered, than the original state in which all the molecules were in
one half. One therefore says that the entropy of the gas has gone up.
Similarly, suppose one starts with two boxes, one containing oxygen
molecules and the other containing nitrogen molecules. If one joins the
boxes together and removes the intervening wall, the oxygen and the
nitrogen molecules will start to mix. At a later time the most probable state
would be a fairly uniform mixture of oxygen and nitrogen molecules
throughout the two boxes. This state would be less ordered, and hence have
more entropy, than the initial state of two separate boxes. The second law of
thermodynamics has a rather different status than that of other laws of
science, such as Newton’s law of gravity, for example, because it does not
hold always, just in the vast majority of cases. The probability of all the gas
molecules in our first box
found in one half of the box at a later time is many millions of millions to
one, but it can happen. However, if one has a black hole around there seems
to be a rather easier way of violating the second law: just throw some matter
with a lot of entropy such as a box of gas, down the black hole. The total
entropy of matter outside the black hole would go down. One could, of
course, still say that the total entropy, including the entropy inside the black
hole, has not gone down – but since there is no way to look inside the black
hole, we cannot see how much entropy the matter inside it has. It would be
nice, then, if there was some feature of the black hole by which observers
outside the black hole could tell its entropy, and which would increase
whenever matter carrying entropy fell into the black hole. Following the
discovery, described above, that the area of the event horizon increased
whenever matter fell into a black hole, a research student at Princeton named
Jacob Bekenstein suggested that the area of the event horizon was a measure
of the entropy of the black hole. As matter carrying entropy fell into a black
hole, the area of its event horizon would go up, so that the sum of the
entropy of matter outside black holes and the area of the horizons would
never go down. This suggestion seemed to prevent the second law of
thermodynamics from being violated in most situations. However, there was
one fatal flaw. If a black hole has entropy, then it ought to also have a
temperature. But a body with a particular temperature must emit radiation at
a certain rate. It is a matter of common experience that if one heats up a
poker in a fire it glows red hot and emits radiation, but bodies at lower
temperatures emit radiation too; one just does not normally notice it because
the amount is fairly small. This radiation is required in order to prevent
violation of the second law. So black holes ought to emit radiation. But by
their very definition, black holes are objects that are not supposed to emit
anything. It therefore seemed that the area of the event horizon of a black
hole could not be regarded as its entropy. In 1972 I wrote a paper with
Brandon Carter and an American colleague, Jim Bardeen, in which we
pointed out that although there were many similarities between entropy and
the area of the event horizon, there was this apparently fatal difficulty. I must
admit that in writing this paper I was motivated partly by irritation with
Bekenstein, who, I felt, had misused my discovery of the increase of the area
of the event horizon. However, it turned out in the end that he was basically
correct, though in a manner he had certainly not expected. In September
1973, while I was visiting Moscow, I discussed black holes with two leading
Soviet experts, Yakov Zeldovich and Alexander Starobinsky. They
convinced me that, according to the quantum mechanical uncertainty
principle, rotating black holes should create and emit particles. I believed
their arguments on physical grounds, but I did not like the mathematical way
in which they calculated the emission. I therefore set about devising a better
mathematical treatment, which I described at an informal seminar in Oxford
at the end of November 1973. At that time I had not done the calculations to
find out how much would actually be emitted. I was expecting to discover
just the radiation that Zeldovich and Starobinsky had predicted from rotating
black holes. However, when I did the calculation, I
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found, to my surprise and annoyance, that even non-rotating black holes
should apparently create and emit particles at a steady rate. At first I thought
that this emission indicated that one of the approximations I had used was
not valid. I was afraid that if Bekenstein found out about it, he would use it
as a further argument to support his ideas about the entropy of black holes,
which I still did not like. However, the more I thought about it, the more it
seemed that the approximations really ought to hold. But what finally
convinced me that the emission was real was that the spectrum of the
emitted particles was exactly that which would be emitted by a hot body, and
that the black hole was emitting particles at exactly the correct rate to
prevent violations of the second law. Since then the calculations have been
repeated in a number of different forms by other people. They all confirm
that a black hole ought to emit particles and radiation as if it were a hot body
with a temperature that depends only on the black hole’s mass: the higher the
mass, the lower the temperature.
How is it possible that a black hole appears to emit particles when we know
that nothing can escape from within its event horizon? The answer, quantum
theory tells us, is that the particles do not come from within the black hole,
but from the “empty” space just outside the black hole’s event horizon! We
can understand this in the following way: what we think of as “empty” space
cannot be completely empty because that would mean that all the fields,
such as the gravitational and electromagnetic fields, would have to be
exactly zero. However, the value of a field and its rate of change with time
are like the position and velocity of a particle: the uncertainty principle
implies that the more accurately one knows one of these quantities, the less
accurately one can know the other. So in empty space the field cannot be
fixed at exactly zero, because then it would have both a precise value (zero)
and a precise rate of change (also zero). There must be a certain minimum
amount of uncertainty, or quantum fluctuations, in the value of the field. One
can think of these fluctuations as pairs of particles of light or gravity that
appear together at some time, move apart, and then come together again and
annihilate each other. These particles are virtual particles like the particles
that carry the gravitational force of the sun: unlike real particles, they cannot
be observed directly with a particle detector. However, their indirect effects,
such as small changes in the energy of electron orbits in atoms, can be
measured and agree with the theoretical predictions to a remarkable degree
of accuracy. The uncertainty principle also predicts that there will be similar
virtual pairs of matter particles, such as electrons or quarks. In this case,
however, one member of the pair will be a particle and the other an
antiparticle (the antiparticles of light and gravity are the same as the
particles).
Because energy cannot be created out of nothing, one of the partners in a
particle/antiparticle pair will have positive energy, and the other partner
negative energy. The one with negative energy is condemned to be a shortlived virtual particle because real particles always have positive energy in
normal situations. It must therefore seek out its partner and annihilate with it.
However, a real particle close to a massive body has less energy than if it
were far away, because it would take energy to lift it far away against the
gravitational attraction of the body. Normally, the energy of the particle is
still positive, but the gravitational field inside a black hole is so strong that
even a real particle can have negative energy there. It is therefore possible, if
a black hole is present, for the virtual particle with negative energy to fall
into the black hole and become a real particle or antiparticle. In this case it
no longer has to annihilate with its partner. Its forsaken partner may fall into
the black hole as well. Or, having positive energy, it might also escape from
the vicinity of the black hole as a real particle or antiparticle Figure 7:4.
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Figure 7:4
To an observer at a distance, it will appear to have been emitted from the
black hole. The smaller the black hole, the shorter the distance the particle
with negative energy will have to go before it becomes a real particle, and
thus the greater the rate of emission, and the apparent temperature, of the
black hole. The positive energy of the outgoing radiation would be balanced
by a flow of negative energy particles into the black hole. By Einstein’s
equation E = mc2 (where E is energy, m is mass, and c is the speed of light),
energy is proportional to mass. A flow of negative energy into the black hole
therefore reduces its mass. As the black hole loses mass, the area of its event
horizon gets smaller, but this decrease in the entropy of the black hole is
more than compensated for by the entropy of the emitted radiation, so the
second law is never violated. Moreover, the lower the mass of the black
hole, the higher its temperature. So as the black hole loses mass, its
temperature and rate of emission increase, so it loses mass more quickly.
What happens when the mass of the black hole eventually becomes
extremely small is not quite clear, but the most reasonable guess is that it
would disappear completely in a tremendous final burst of emission,
equivalent to the explosion of millions of H-bombs. A black hole with a
mass a few times that of the sun would have a temperature of only one ten
millionth of a degree above absolute zero. This is much less than the
temperature of the microwave radiation that fills the universe (about
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2.7º above absolute zero), so such black holes would emit even less than
they absorb. If the universe is destined to go on expanding forever, the
temperature of the microwave radiation will eventually decrease to less than
that of such a black hole, which will then begin to lose mass. But, even then,
its temperature would be so low that it would take about a million million
million million million million million million million million million years
(1 with sixty-six zeros after it) to evaporate completely. This is much longer
than the age of the universe, which is only about ten or twenty thousand
million years (1 or 2 with ten zeros after it). On the other hand, as mentioned
in Chapter 6, there might be primordial black holes with a very much smaller
mass that were made by the collapse of irregularities in the very early stages
of the universe. Such black holes would have a much higher temperature and
would be emitting radiation at a much greater rate. A primordial black hole
with an initial mass of a thousand million tons would have a lifetime roughly
equal to the age of the universe. Primordial black holes with initial masses
less than this figure would already have completely evaporated, but those
with slightly greater masses would still be emitting radiation in the form of
X rays and gamma rays. These X rays and gamma rays are like waves of
light, but with a much shorter wavelength. Such holes hardly deserve the
epithet black: they really are white hot and are emitting energy at a rate of
about ten thousand megawatts.
One such black hole could run ten large power stations, if only we could
harness its power. This would be rather difficult, however: the black hole
would have the mass of a mountain compressed into less than a million
millionth of an inch, the size of the nucleus of an atom! If you had one of
these black holes on the surface of the earth, there would be no way to stop it
from falling through the floor to the center of the earth. It would oscillate
through the earth and back, until eventually it settled down at the center. So
the only place to put such a black hole, in which one might use the energy
that it emitted, would be in orbit around the earth – and the only way that
one could get it to orbit the earth would be to attract it there by towing a
large mass in front of it, rather like a carrot in front of a donkey. This does
not sound like a very practical proposition, at least not in the immediate
future. But even if we cannot harness the emission from these primordial
black holes, what are our chances of observing them? We could look for the
gamma rays that the primordial black holes emit during most of their
lifetime. Although the radiation from most would be very weak because they
are far away, the total from all of them might be detectable. We do observe
such a background of gamma rays: Figure 7:5 shows how the observed
intensity differs at different frequencies (the number of waves per second).
However, this background could have been, and probably was, generated by
processes other than primordial black holes. The dotted line in Figure 7:5
shows how the intensity should vary with frequency for gamma rays given
off by primordial black holes, if there were on average 300 per cubic lightyear. One can therefore say that the observations of the gamma ray
background do not provide any positive evidence for primordial black holes,
but they do tell us that on average there cannot be more than 300 in every
cubic light-year in the universe. This limit means that primordial black holes
could make up at most one millionth of the matter in the universe.
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Figure 7:5
With primordial black holes being so scarce, it might seem unlikely that
there would be one near enough for us to observe as an individual source of
gamma rays. But since gravity would draw primordial black holes toward
any matter, they should be much more common in and around galaxies. So
although the gamma ray background tells us that there can be no more than
300 primordial black holes per cubic light-year on average, it tells us nothing
about how common they might be in our own galaxy. If they were, say, a
million times more common than this, then the nearest black hole to us
would probably be at a distance of about a thousand million kilometers, or
about as far away as Pluto, the farthest known planet. At this distance it
would still be very difficult to detect the steady emission of a black hole,
even if it was ten thousand megawatts. In order to observe a primordial black
hole one would have to detect several gamma ray quanta coming from the
same direction within a reasonable space of time, such as a week. Otherwise,
they might simply be part of the background. But Planck’s quantum
principle tells us that each gamma ray quantum has a very high energy,
because gamma rays have a very high frequency, so it would not take many
quanta to radiate even ten thousand megawatts. And to observe these few
coming from the distance of Pluto would require a larger gamma ray
detector than any that have been constructed so far. Moreover, the detector
would have to be in space, because gamma rays cannot penetrate the
atmosphere.
Of course, if a black hole as close as Pluto were to reach the end of its life
and blow up, it would be easy to detect the final burst of emission. But if the
black hole has been emitting for the last ten or twenty thousand million
years, the chance of it reaching the end of its life within the next few years,
rather than several million years in the past or future, is really rather small!
So in order to have a reasonable chance of seeing an explosion before your
research grant ran out, you would have to find a way to detect any
explosions within a distance of about one light-year. In fact bursts of
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gamma rays from space have been detected by satellites originally
constructed to look for violations of the Test Ban Treaty. These seem to
occur about sixteen times a month and to be roughly uniformly distributed in
direction across the sky. This indicates that they come from outside the Solar
System since otherwise we would expect them to be concentrated toward the
plane of the orbits of the planets. The uniform distribution also indicates that
the sources are either fairly near to us in our galaxy or right outside it at
cosmological distances because otherwise, again, they would be
concentrated toward the plane of the galaxy. In the latter case, the energy
required to account for the bursts would be far too high to have been
produced by tiny black holes, but if the sources were close in galactic terms,
it might be possible that they were exploding black holes. I would very much
like this to be the case but I have to recognize that there are other possible
explanations for the gamma ray bursts, such as colliding neutron stars. New
observations in the next few years, particularly by gravitational wave
detectors like LIGO, should enable us to discover the origin of the gamma
ray bursts.
Even if the search for primordial black holes proves negative, as it seems it
may, it will still give us important information about the very early stages of
the universe. If the early universe had been chaotic or irregular, or if the
pressure of matter had been low, one would have expected it to produce
many more primordial black holes than the limit already set by our
observations of the gamma ray background. Only if the early universe was
very smooth and uniform, with a high pressure, can one explain the absence
of observable numbers of primordial black holes. The idea of radiation from
black holes was the first example of a prediction that depended in an
essential way on both the great theories of this century, general relativity and
quantum mechanics. It aroused a lot of opposition initially because it upset
the existing viewpoint: “How can a black hole emit anything?” When I first
announced the results of my calculations at a conference at the RutherfordAppleton Laboratory near Oxford, I was greeted with general incredulity. At
the end of my talk the chairman of the session, John G. Taylor from Kings
College, London, claimed it was all nonsense. He even wrote a paper to that
effect. However, in the end most people, including John Taylor, have come
to the conclusion that black holes must radiate like hot bodies if our other
ideas about general relativity and quantum mechanics are correct. Thus, even
though we have not yet managed to find a primordial black hole, there is
fairly general agreement that if we did, it would have to be emitting a lot of
gamma rays and X rays. The existence of radiation from black holes seems
to imply that gravitational collapse is not as final and irreversible as we once
thought. If an astronaut falls into a black hole, its mass will increase, but
eventually the energy equivalent of that extra mass will be returned to the
universe in the form of radiation. Thus, in a sense, the astronaut will be
“recycled.” It would be a poor sort of immortality, however, because any
personal concept of time for the astronaut would almost certainly come to an
end as he was torn apart inside the black hole! Even the types of particles
that were eventually emitted by the black hole would in general be different
from those that made up the astronaut: the only feature of the astronaut that
would survive would be his mass or energy. The approximations I used to
derive the emission from black holes should work well when the black hole
has a mass greater than a fraction of a gram. However, they will break down
at the end of the black hole’s life when its mass gets very small. The most
likely outcome seems to be that the black hole will just disappear, at least
from our region of the universe, taking with it the astronaut and any
singularity there might be inside it, if indeed there is one. This was the first
indication that quantum mechanics might remove the singularities that were
predicted by general relativity. However, the methods that I and other people
were using in 1974 were not able to answer questions such as whether
singularities would occur in quantum gravity. From 1975 onward I therefore
started to develop a more powerful approach to quantum gravity based on
Richard Feynrnan’s idea of a sum over histories. The answers that this
approach suggests for the origin and fate of the universe and its contents,
such as astronauts, will be de-scribed in the next two chapters. We shall see
that although the uncertainty principle places limitations on the accuracy of
all our predictions, it may at the same time remove the fundamental
unpredictability that occurs at a space-time singularity.
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CHAPTER 8
THE ORIGIN AND FATE OF THE UNIVERSE
Einstein’s general theory of relativity, on its own, predicted that space-time
began at the big bang singularity and would come to an end either at the big
crunch singularity (if the whole universe recollapsed), or at a singularity
inside a black hole (if a local region, such as a star, were to collapse). Any
matter that fell into the hole would be destroyed at the singularity, and only
the gravitational effect of its mass would continue to be felt outside. On the
other hand, when quantum effects were taken into account, it seemed that
the mass or energy of the matter would eventually be returned to the rest of
the universe, and that the black hole, along with any singularity inside it,
would evaporate away and finally disappear. Could quantum mechanics
have an equally dramatic effect on the big bang and big crunch
singularities? What really happens during the very early or late stages of the
universe, when gravitational fields are so strong that quantum effects cannot
be ignored? Does the universe in fact have a beginning or an end?
And if so, what are they like?
Throughout the 1970s I had been mainly studying black holes, but in 1981
my interest in questions about the origin and fate of the universe was
reawakened when I attended a conference on cosmology organized by the
Jesuits in the Vatican. The Catholic Church had made a bad mistake with
Galileo when it tried to lay down the law on a question of science, declaring
that the sun went round the earth. Now, centuries later, it had decided to
invite a number of experts to advise it on cosmology. At the end of the
conference the participants were granted an audience with the Pope. He told
us that it was all right to study the evolution of the universe after the big
bang, but we should not inquire into the big bang itself because that was the
moment of Creation and therefore the work of God. I was glad then that he
did not know the subject of the talk I had just given at the conference – the
possibility that space-time was finite but had no boundary, which means
that it had no beginning, no moment of Creation. I had no desire to share
the fate of Galileo, with whom I feel a strong sense of identity, partly
because of the coincidence of having been born exactly 300 years after his
death!
In order to explain the ideas that I and other people have had about how
quantum mechanics may affect the origin and fate of the universe, it is
necessary first to understand the generally accepted history of the universe,
according to what is known as the “hot big bang model.” This assumes that
the universe is described by a Friedmann model, right back to the big bang.
In such models one finds that as the universe expands, any matter or
radiation in it gets cooler. (When the universe doubles in size, its
temperature falls by half.) Since temperature is simply a measure of the
average energy – or speed – of the particles, this cooling of the universe
would have a major effect on the matter in it. At very high temperatures,
particles would be moving around so fast that they could escape any
attraction toward each other due to nuclear or electromagnetic forces, but as
they cooled off one would expect particles that attract each other to start to
clump together. Moreover, even the types of particles that exist in the
universe would depend on the temperature. At high enough temperatures,
particles have so much energy that whenever they collide many different
particle/antiparticle pairs would be produced – and although some of these
particles would annihilate on hitting antiparticles, they would be produced
more rap-idly than they could annihilate. At lower temperatures, however,
when colliding particles have less energy, particle/antiparticle pairs would
be produced less quickly – and annihilation would become faster than
production.
At the big bang itself the universe is thought to have had zero size, and so to
have been infinitely hot. But as the universe expanded, the temperature of
the radiation decreased. One second after the big bang, it would have fallen
to about ten thousand million degrees. This is about a thousand times the
temperature at the center of the sun, but temperatures as high as this are
reached in H-bomb explosions. At this time the universe would have
contained mostly photons, electrons, and neutrinos (extremely light
particles that are affected only by the weak force and gravity) and their
antiparticles, together with some protons and neutrons. As the universe
continued to expand and the temperature to drop, the rate at which
electron/antielectron pairs were being produced in collisions would have
fallen below the rate at which they were being destroyed by annihilation. So
most of the electrons and antielectrons would have annihilated with each
other to produce more photons, leaving only a few electrons left over. The
neutrinos and antineutrinos, however, would not have annihilated with each
other, because these particles interact with themselves and with other
particles only very weakly. So they should still be around today. If we could
observe them, it would provide a good test of this picture of a very hot early
stage of the universe. Unfortunately, their energies nowadays would be too
low for us to observe them directly. However, if neutrinos are not massless,
but have a small mass of their own, as suggested by some recent
experiments, we might be able to detect them indirectly: they could be a
form of “dark matter,” like that mentioned earlier, with sufficient
gravitational attraction to stop the expansion of the universe and cause it to
collapse again.
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About one hundred seconds after the big bang, the temperature would have
fallen to one thousand million degrees, the temperature inside the hottest
stars. At this temperature protons and neutrons would no longer have
sufficient energy to escape the attraction of the strong nuclear force, and
would have started to combine together to produce the nuclei of atoms of
deuterium (heavy hydrogen), which contain one proton and one neutron.
The deuterium nuclei would then have combined with more protons and
neutrons to make helium nuclei, which contain two protons and two
neutrons, and also small amounts of a couple of heavier elements, lithium
and beryllium. One can calculate that in the hot big bang model about a
quarter of the protons and neutrons would have been converted into helium
nuclei, along with a small amount of heavy hydrogen and other elements.
The remaining neutrons would have decayed into protons, which are the
nuclei of ordinary hydrogen atoms.
This picture of a hot early stage of the universe was first put forward by the
scientist George Gamow in a famous paper written in 1948 with a student
of his, Ralph Alpher. Gamow had quite a sense of humor – he persuaded the
nuclear scientist Hans Bethe to add his name to the paper to make the list of
authors “Alpher, Bethe, Gamow,” like the first three letters of the Greek
alphabet, alpha, beta, gamma: particularly appropriate for a paper on the
beginning of the universe! In this paper they made the remarkable
prediction that radiation (in the form of photons) from the very hot early
stages of the universe should still be around today, but with its temperature
reduced to only a few degrees above absolute zero (–273ºC). It was this
radiation that Penzias and Wilson found in 1965. At the time that Alpher,
Bethe, and Gamow wrote their paper, not much was known about the
nuclear reactions of protons and neutrons. Predictions made for the
proportions of various elements in the early universe were therefore rather
inaccurate, but these calculations have been repeated in the light of better
knowledge and now agree very well with what we observe. It is, moreover,
very difficult to explain in any other way why there should be so much
helium in the universe. We are therefore fairly confident that we have the
right picture, at least back to about one second after the big bang.
Within only a few hours of the big bang, the production of helium and other
elements would have stopped. And after that, for the next million years or
so, the universe would have just continued expanding, without anything
much happening. Eventually, once the temperature had dropped to a few
thousand degrees, and electrons and nuclei no longer had enough energy to
overcome the electromagnetic attraction between them, they would have
started combining to form atoms. The universe as a whole would have
continued expanding and cooling, but in regions that were slightly denser
than average, the expansion would have been slowed down by the extra
gravitational attraction. This would eventually stop expansion in some
regions and cause them to start to recollapse. As they were collapsing, the
gravitational pull of matter outside these regions might start them rotating
slightly. As the collapsing region got smaller, it would spin faster – just as
skaters spinning on ice spin faster as they draw in their arms. Eventually,
when the region got small enough, it would be spinning fast enough to
balance the attraction of gravity, and in this way disklike rotating galaxies
were born. Other regions, which did not happen to pick up a rotation, would
become oval-shaped objects called elliptical galaxies. In these, the region
would stop collapsing because individual parts of the galaxy would be
orbiting stably round its center, but the galaxy would have no overall
rotation. As time went on, the hydrogen and helium gas in the galaxies
would break up into smaller clouds that would collapse under their own
gravity. As these contracted, and the atoms within them collided with one
another, the temperature of the gas would increase, until eventually it
became hot enough to start nuclear fusion reactions. These would convert
the hydrogen into more helium, and the heat given off would raise the
pressure, and so stop the clouds from contracting any further. They would
remain stable in this state for a long time as stars like our sun, burning
hydrogen into helium and radiating the resulting energy as heat and light.
More massive stars would need to be hotter to balance their stronger
gravitational attraction, making the nuclear fusion reactions proceed so
much more rapidly that they would use up their hydrogen in as little as a
hundred million years. They would then contract slightly, and as they
heated up further, would start to convert helium into heavier elements like
carbon or oxygen. This, however, would not release much more energy, so a
crisis would occur, as was described in the chapter on black holes. What
happens next is not completely clear, but it seems likely that the central
regions of the star would collapse to a very dense state, such as a neutron
star or black hole. The outer regions of the star may sometimes get blown
off in a tremendous explosion called a supernova, which would outshine all
the other stars in its galaxy. Some of the heavier elements produced near the
end of the star’s life would be flung back into the gas in the galaxy, and
would provide some of the raw material for the next generation of stars.
Our own sun contains about 2 percent of these heavier elements, because it
is a second-or third-generation star, formed some five thousand million
years ago out of a cloud of rotating gas containing the debris of earlier
supernovas. Most of the gas in that cloud went to form the sun or got blown
away, but a small amount of the heavier elements collected together to form
the bodies that now orbit the sun as planets like the earth.
The earth was initially very hot and without an atmosphere. In the course of
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atmosphere from the emission of gases from the rocks. This early
atmosphere was not one in which we could have survived. It contained no
oxygen, but a lot of other gases that are poisonous to us, such as hydrogen
sulfide (the gas that gives rotten eggs their smell). There are, however, other
primitive forms of life that can flourish under such conditions. It is thought
that they developed in the oceans, possibly as a result of chance
combinations of atoms into large structures, called macromolecules, which
were capable of assembling other atoms in the ocean into similar structures.
They would thus have reproduced themselves and multiplied. In some cases
there would be errors in the reproduction. Mostly these errors would have
been such that the new macromolecule could not reproduce itself and
eventually would have been destroyed. However, a few of the errors would
have produced new macromolecules that were even better at reproducing
themselves. They would have therefore had an advantage and would have
tended to replace the original macromolecules. In this way a process of
evolution was started that led to the development of more and more
complicated, self-reproducing organisms. The first primitive forms of life
consumed various materials, including hydrogen sulfide, and released
oxygen. This gradually changed the atmosphere to the composition that it
has today, and allowed the development of higher forms of life such as fish,
reptiles, mammals, and ultimately the human race.
This picture of a universe that started off very hot and cooled as it expanded
is in agreement with all the observational evidence that we have today.
Nevertheless, it leaves a number of important questions unanswered: 1.
Why was the early universe so hot?
2. Why is the universe so uniform on a large scale? Why does it look the
same at all points of space and in all directions? In particular, why is the
temperature of the microwave back-ground radiation so nearly the same
when we look in different directions? It is a bit like asking a number of
students an exam question. If they all give exactly the same answer, you can
be pretty sure they have communicated with each other. Yet, in the model
described above, there would not have been time since the big bang for
light to get from one distant region to another, even though the regions were
close together in the early universe. According to the theory of relativity, if
light cannot get from one region to another, no other information can. So
there would be no way in which different regions in the early universe
could have come to have the same temperature as each other, unless for
some unexplained reason they happened to start out with the same
temperature.
3. Why did the universe start out with so nearly the critical rate of
expansion that separates models that recollapse from those that go on
expanding forever, that even now, ten thousand million years later, it is still
expanding at nearly the critical rate? If the rate of expansion one second
after the big bang had been smaller by even one part in a hundred thousand
million million, the universe would have recollapsed before it ever reached
its present size. 4. Despite the fact that the universe is so uniform and
homogeneous on a large scale, it contains local irregularities, such as stars
and galaxies. These are thought to have developed from small differences in
the density of the early universe from one region to another. What was the
origin of these density fluctuations?
The general theory of relativity, on its own, cannot explain these features or
answer these questions because of its prediction that the universe started off
with infinite density at the big bang singularity. At the singularity, general
relativity and all other physical laws would break down: one couldn’t
predict what would come out of the singularity. As explained before, this
means that one might as well cut the big bang, and any events before it, out
of the theory, because they can have no effect on what we observe. Spacetime w ould have a boundary – a beginning at the big bang.
Science seems to have uncovered a set of laws that, within the limits set by
the uncertainty principle, tell us how the universe will develop with time, if
we know its state at any one time. These laws may have originally been
decreed by God, but it appears that he has since left the universe to evolve
according to them and does not now intervene in it. But how did he choose
the initial state or configuration of the universe? What were the “boundary
conditions” at the beginning of time?
One possible answer is to say that God chose the initial configuration of the
universe for reasons that we cannot hope to understand. This would
certainly have been within the power of an omnipotent being, but if he had
started it off in such an incomprehensible way, why did he choose to let it
evolve according to laws that we could understand?
The whole history of science has been the gradual realization that events do
not happen in an arbitrary manner, but that they reflect a certain underlying
order, which may or may not be divinely inspired. It would be only natural
to suppose that this order should apply not only to the laws, but also to the
conditions at the boundary of space-time that specify the initial state of the
universe. There may be a large number of models of the universe with
different initial conditions that all obey the laws. There ought to be some
principle that picks out one initial state, and hence
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one model, to represent our universe.
One such possibility is what are called chaotic boundary conditions. These
implicitly assume either that the universe is spatially infinite or that there
are infinitely many universes. Under chaotic boundary conditions, the
probability of finding any particular region of space in any given
configuration just after the big bang is the same, in some sense, as the
probability of finding it in any other configuration: the initial state of the
universe is chosen purely randomly. This would mean that the early
universe would have probably been very chaotic and irregular because there
are many more chaotic and disordered configurations for the universe than
there are smooth and ordered ones. (If each configuration is equally
probable, it is likely that the universe started out in a chaotic and disordered
state, simply because there are so many more of them.) It is difficult to see
how such chaotic initial conditions could have given rise to a universe that
is so smooth and regular on a large scale as ours is today. One would also
have expected the density fluctuations in such a model to have led to the
formation of many more primordial black holes than the upper limit that has
been set by observations of the gamma ray background.
If the universe is indeed spatially infinite, or if there are infinitely many
universes, there would probably be some large regions somewhere that
started out in a smooth and uniform manner. It is a bit like the well-known
horde of monkeys hammering away on typewriters – most of what they
write will be garbage, but very occasionally by pure chance they will type
out one of Shakespeare’s sonnets. Similarly, in the case of the universe,
could it be that we are living in a region that just happens by chance to be
smooth and uniform? At first sight this might seem very improbable,
because such smooth regions would be heavily outnumbered by chaotic and
irregular regions. However, suppose that only in the smooth regions were
galaxies and stars formed and were conditions right for the development of
complicated self-replicating organisms like ourselves who were capable of
asking the question: why is the universe so smooth.? This is an example of
the application of what is known as the anthropic principle, which can be
paraphrased as “We see the universe the way it is because we exist.”
There are two versions of the anthropic principle, the weak and the strong.
The weak anthropic principle states that in a universe that is large or infinite
in space and/or time, the conditions necessary for the development of
intelligent life will be met only in certain regions that are limited in space
and time. The intelligent beings in these regions should therefore not be
surprised if they observe that their locality in the universe satisfies the
conditions that are necessary for their existence. It is a bit like a rich person
living in a wealthy neighborhood not seeing any poverty. One example of
the use of the weak anthropic principle is to “explain” why the big bang
occurred about ten thousand million years ago – it takes about that long for
intelligent beings to evolve. As explained above, an early generation of
stars first had to form. These stars converted some of the original hydrogen
and helium into elements like carbon and oxygen, out of which we are
made. The stars then exploded as supernovas, and their debris went to form
other stars and planets, among them those of our Solar System, which is
about five thousand million years old. The first one or two thousand million
years of the earth’s existence were too hot for the development of anything
complicated. The remaining three thousand million years or so have been
taken up by the slow process of biological evolution, which has led from
the simplest organisms to beings who are capable of measuring time back to
the big bang. Few people would quarrel with the validity or utility of the
weak anthropic principle. Some, however, go much further and propose a
strong version of the principle. According to this theory, there are either
many different universes or many different regions of a single universe,
each with its own initial configuration and, perhaps, with its own set of laws
of science. In most of these universes the conditions would not be right for
the development of complicated organisms; only in the few universes that
are like ours would intelligent beings develop and ask the question, “Why is
the universe the way we see it?” The answer is then simple: if it had been
different, we would not be here!
The laws of science, as we know them at present, contain many
fundamental numbers, like the size of the electric charge of the electron and
the ratio of the masses of the proton and the electron. We cannot, at the
moment at least, predict the values of these numbers from theory – we have
to find them by observation. It may be that one day we shall discover a
complete unified theory that predicts them all, but it is also possible that
some or all of them vary from universe to universe or within a single
universe. The remarkable fact is that the values of these numbers seem to
have been very finely adjusted to make possible the development of life.
For example, if the electric charge of the electron had been only slightly
different, stars either would have been unable to burn hydrogen and helium,
or else they would not have exploded. Of course, there might be other forms
of intelligent life, not dreamed of even by writers of science fiction, that did
not require the light of a star like the sun or the heavier chemical elements
that are made in stars and are flung back into space when the stars explode.
Nevertheless, it seems clear that there are relatively few ranges of values for
the numbers that would allow the development of any form of intelligent
life. Most sets of values would give rise to universes that, although they
might be very beautiful, would contain no one able to wonder at that beauty.
One can take this either as evidence of a divine purpose in Creation and the
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laws of science or as support for the strong anthropic principle.
There are a number of objections that one can raise to the strong anthropic
principle as an explanation of the observed state of the universe. First, in
what sense can all these different universes be said to exist? If they are
really separate from each other, what happens in another universe can have
no observable consequences in our own universe. We should therefore use
the principle of economy and cut them out of the theory. If, on the other
hand, they are just different regions of a single universe, the laws of science
would have to be the same in each region, because otherwise one could not
move continuously from one region to another. In this case the only
difference between the regions would be their initial configurations and so
the strong anthropic principle would reduce to the weak one.
A second objection to the strong anthropic principle is that it runs against
the tide of the whole history of science. We have developed from the
geocentric cosmologies of Ptolemy and his forebears, through the
heliocentric cosmology of Copernicus and Galileo, to the modern picture in
which the earth is a medium-sized planet orbiting around an average star in
the outer suburbs of an ordinary spiral galaxy, which is itself only one of
about a million million galaxies in the observable universe. Yet the strong
anthropic principle would claim that this whole vast construction exists
simply for our sake. This is very hard to believe. Our Solar System is
certainly a prerequisite for our existence, hand one might extend this to the
whole of our galaxy to allow for an earlier generation of stars that created
the heavier elements. But there does not seem to be any need for all those
other galaxies, nor for the universe to be so uniform and similar in every
direction on the large scale.
One would feel happier about the anthropic principle, at least in its weak
version, if one could show that quite a number of different initial
configurations for the universe would have evolved to produce a universe
like the one we observe. If this is the case, a universe that developed from
some sort of random initial conditions should contain a number of regions
that are smooth and uniform and are suitable for the evolution of intelligent
life. On the other hand, if the initial state of the universe had to be chosen
extremely carefully to lead to something like what we see around us, the
universe would be unlikely to contain any region in which life would
appear. In the hot big bang model described above, there was not enough
time in the early universe for heat to have flowed from one region to
another. This means that the initial state of the universe would have to have
had exactly the same temperature everywhere in order to account for the
fact that the microwave back-ground has the same temperature in every
direction we look. The initial rate of expansion also would have had to be
chosen very precisely for the rate of expansion still to be so close to the
critical rate needed to avoid recollapse. This means that the initial state of
the universe must have been very carefully chosen indeed if the hot big
bang model was correct right back to the beginning of time. It would be
very difficult to explain why the universe should have begun in just this
way, except as the act of a God who intended to create beings like us.
In an attempt to find a model of the universe in which many different initial
configurations could have evolved to something like the present universe, a
scientist at the Massachusetts Institute of Technology, Alan Guth, suggested
that the early universe might have gone through a period of very rapid
expansion. This expansion is said to be
“inflationary,” meaning that the universe at one time expanded at an
increasing rate rather than the decreasing rate that it does today. According
to Guth, the radius of the universe increased by a million million million
million million (1
with thirty zeros after it) times in only a tiny fraction of a second. Guth
suggested that the universe started out from the big bang in a very hot, but
rather chaotic, state. These high temperatures would have meant that the
particles in the universe would be moving very fast and would have high
energies. As we discussed earlier, one would expect that at such high
temperatures the strong and weak nuclear forces and the electromagnetic
force would all be unified into a single force. As the universe expanded, it
would cool, and particle energies would go down. Eventually there would
be what is called a phase transition and the symmetry between the forces
would be broken: the strong force would become different from the weak
and electromagnetic forces. One common example of a phase transition is
the freezing of water when you cool it down. Liquid water is symmetrical,
the same at every point and in every direction. However, when ice crystals
form, they will have definite positions and will be lined up in some
direction. This breaks water’s symmetry. In the case of water, if one is
careful, one can “supercool” it: that is, one can reduce the temperature
below the freezing point (OºC) without ice forming. Guth suggested that the
universe might behave in a similar way: the temperature might drop below
the critical value without the symmetry between the forces being broken. If
this happened, the universe would be in an unstable state, with more energy
than if the symmetry had been broken. This special extra energy can be
shown to have an antigravitational effect: it would have acted just like the
cosmological constant that Einstein introduced into general relativity when
he was trying to construct a static model of the universe. Since the universe
would already be expanding just as in the hot big bang model, the repulsive
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this cosmological constant would therefore have made the universe expand
at an ever-increasing rate. Even in regions where there were more matter
particles than average, the gravitational attraction of the matter would have
been outweighed by the repulsion of the effective cosmological constant.
Thus these regions would also expand in an accelerating inflationary
manner. As they expanded and the matter particles got farther apart, one
would be left with an expanding universe that contained hardly any
particles and was still in the supercooled state. Any irregularities in the
universe would simply have been smoothed out by the expansion, as the
wrinkles in a balloon are smoothed away when you blow it up. Thus the
present smooth and uniform state of the universe could have evolved from
many different non-uniform initial states.
In such a universe, in which the expansion was accelerated by a
cosmological constant rather than slowed down by the gravitational
attraction of matter, there would be enough time for light to travel from one
region to another in the early universe. This could provide a solution to the
problem, raised earlier, of why different regions in the early universe have
the same properties. Moreover, the rate of expansion of the universe would
automatically become very close to the critical rate determined by the
energy density of the universe. This could then explain why the rate of
expansion is still so close to the critical rate, without having to assume that
the initial rate of expansion of the universe was very carefully chosen.
The idea of inflation could also explain why there is so much matter in the
universe. There are something like ten million million million million
million million million million million million million million million
million (1 with eighty zeros after it) particles in the region of the universe
that we can observe. Where did they all come from? The answer is that, in
quantum theory, particles can be created out of energy in the form of
particle/antiparticle pairs. But that just raises the question of where the
energy came from. The answer is that the total energy of the universe is
exactly zero. The matter in the universe is made out of positive energy.
However, the matter is all attracting itself by gravity. Two pieces of matter
that are close to each other have less energy than the same two pieces a long
way apart, because you have to expend energy to separate them against the
gravitational force that is pulling them together. Thus, in a sense, the
gravitational field has negative energy. In the case of a universe that is
approximately uniform in space, one can show that this negative
gravitational energy exactly cancels the positive energy represented by the
matter. So the total energy of the universe is zero.
Now twice zero is also zero. Thus the universe can double the amount of
positive matter energy and also double the negative gravitational energy
without violation of the conservation of energy. This does not happen in the
normal expansion of the universe in which the matter energy density goes
down as the universe gets bigger. It does happen, however, in the
inflationary expansion because the energy density of the supercooled state
remains constant while the universe expands: when the universe doubles in
size, the positive matter energy and the negative gravitational energy both
double, so the total energy remains zero. During the inflationary phase, the
universe increases its size by a very large amount. Thus the total amount of
energy available to make particles becomes very large. As Guth has
remarked, “It is said that there’s no such thing as a free lunch. But the
universe is the ultimate free lunch.”
The universe is not expanding in an inflationary way today. Thus there has
to be some mechanism that would eliminate the very large effective
cosmological constant and so change the rate of expansion from an
accelerated one to one that is slowed down by gravity, as we have today. In
the inflationary expansion one might expect that eventually the symmetry
between the forces would be broken, just as super-cooled water always
freezes in the end. The extra energy of the unbroken symmetry state would
then be released and would reheat the universe to a temperature just below
the critical temperature for symmetry between the forces. The universe
would then go on to expand and cool just like the hot big bang model, but
there would now be an explanation of why the universe was expanding at
exactly the critical rate and why different regions had the same temperature.
In Guth’s original proposal the phase transition was supposed to occur
suddenly, rather like the appearance of ice crystals in very cold water. The
idea was that “bubbles” of the new phase of broken symmetry would have
formed in the old phase, like bubbles of steam surrounded by boiling water.
The bubbles were supposed to expand and meet up with each other until the
whole universe was in the new phase. The trouble was, as I and several
other people pointed out, that the universe was expanding so fast that even
if the bubbles grew at the speed of light, they would be moving away from
each other and so could not join up. The universe would be left in a very
non-uniform state, with some regions still having symmetry between the
different forces. Such a model of the universe would not correspond to what
we see.
In October 1981, I went to Moscow for a conference on quantum gravity.
After the conference I gave a seminar on the inflationary model and its
problems at the Sternberg Astronomical Institute. Before this, I had got
someone else to give my lectures for me, because most people could not
understand my voice. But there was not time to prepare this seminar, so I
gave it myself, with one of my graduate students repeating my words. It
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much more contact with my audience. In the audience was a young
Russian, Andrei Linde, from the Lebedev Institute in Moscow. He said that
the difficulty with the bubbles not joining up could be avoided if the
bubbles were so big that our region of the universe is all contained inside a
single bubble. In order for this to work, the change from symmetry to
broken symmetry must have taken place very slowly inside the bubble, but
this is quite possible according to grand unified theories. Linde’s idea of a
slow breaking of symmetry was very good, but I later realized that his
bubbles would have to have been bigger than the size of the universe at the
time! I showed that instead the symmetry would have broken everywhere at
the same time, rather than just inside bubbles. This would lead to a uniform
universe, as we observe. I was very excited by this idea and discussed it
with one of my students, Ian Moss. As a friend of Linde’s, I was rather
embarrassed, however, when I was later sent his paper by a scientific
journal and asked whether it was suitable for publication. I replied that there
was this flaw about the bubbles being bigger than the universe, but that the
basic idea of a slow breaking of symmetry was very good. I recommended
that the paper ¿
published as it was because it would take Linde several months to correct it,
since anything he sent to the West would have to be passed by Soviet
censorship, which was neither very skillful nor very quick with scientific
papers. Instead, I wrote a short paper with Ian Moss in the same journal in
which we pointed out this problem with the bubble and showed how it
could be resolved.
The day after I got back from Moscow I set out for Philadelphia, where I
was due to receive a medal from the Franklin Institute. My secretary, Judy
Fella, had used her not inconsiderable charm to persuade British Airways to
give herself and me free seats on a Concorde as a publicity venture.
However, I .was held up on my way to the airport by heavy rain and I
missed the plane. Nevertheless, I got to Philadelphia in the end and received
my medal. I was then asked to give a seminar on the inflationary universe at
Drexel University in Philadelphia. I gave the same seminar about the
problems of the inflationary universe, just as in Moscow. A very similar
idea to Linde’s was put forth independently a few months later by Paul
Steinhardt and Andreas Albrecht of the University of Pennsylvania. They
are now given joint credit with Linde for what is called “the new
inflationary model,” based on the idea of a slow breaking of symmetry.
(The old inflationary model was Guth’s original suggestion of fast
symmetry breaking with the formation of bubbles.) The new inflationary
model was a good attempt to explain why the universe is the way it is.
However, I and several other people showed that, at least in its original
form, it predicted much greater variations in the temperature of the
microwave background radiation than are observed. Later work has also
cast doubt on whether there could be a phase transition in the very early
universe of the kind required. In my personal opinion, the new inflationary
model is now dead as a scientific theory, although a lot of people do not
seem to have heard of its demise and are still writing papers as if it were
viable. A better model, called the chaotic inflationary model, was put
forward by Linde in 1983. In this there is no phase transition or
supercooling. Instead, there is a spin 0 field, which, because of quantum
fluctuations, would have large values in some regions of the early universe.
The energy of the field in those regions would behave like a cosmological
constant. It would have a repulsive gravitational effect, and thus make those
regions expand in an inflationary manner. As they expanded, the energy of
the field in them would slowly decrease until the inflationary expansion
changed to an expansion like that in the hot big bang model. One of these
regions would become what we now see as the observable universe. This
model has all the advantages of the earlier inflationary models, but it does
not depend on a dubious phase transition, and it can moreover give a
reasonable size for the fluctuations in the temperature of the microwave
background that agrees with observation. This work on inflationary models
showed that the present state of the universe could have arisen from quite a
large number of different initial configurations. This is important, because it
shows that the initial state of the part of the universe that we inhabit did not
have to be chosen with great care. So we may, if we wish, use the weak
anthropic principle to explain why the universe looks the way it does now.
It cannot be the case, however, that every initial configuration would have
led to a universe like the one we observe. One can show this by considering
a very different state for the universe at the present time, say, a very lumpy
and irregular one. One could use the laws of science to evolve the universe
back in time to determine its configuration at earlier times. According to the
singularity theorems of classical general relativity, there would still have
been a big bang singularity. If you evolve such a universe forward in time
according to the laws of science, you will end up with the lumpy and
irregular state you started with. Thus there must have been initial
configurations that would not have given rise to a universe like the one we
see today. So even the inflationary model does not tell us why the initial
configuration was not such as to produce something very different from
what we observe. Must we turn to the anthropic principle for an
explanation?
Was it all just a lucky chance? That would seem a counsel of despair, a
negation of all our hopes of understanding the underlying order of the
universe.
In order to predict how the universe should have started off, one needs laws
that hold at the beginning of time. If the classical theory of general
relativity was correct, the singularity theorems that Roger Penrose and I
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the beginning of time would have been a point of infinite density and
infinite curvature of space-time. All the known laws of science would break
down at such a point. One might suppose that there were new laws that held
at singularities, but it would be very difficult even to formulate such laws at
such badly behaved points, and we would have no guide from observations
as to what those laws might be. However, what the singularity theorems
really indicate is that the gravitational field becomes so strong that quantum
gravitational effects become important: classical theory is no longer a good
description of the universe. So one has to use a quantum theory of gravity to
discuss the very early stages of the universe. As we shall see, it is possible
in the quantum theory for the ordinary laws of science to hold everywhere,
including at the beginning of time: it is not necessary to postulate new laws
for singularities, because there need not be any singularities in the quantum
theory. We don’t yet have a complete and consistent theory that combines
quantum mechanics and gravity. However, we are fairly certain of some
features that such a unified theory should have. One is that it should
incorporate Feynman’s proposal to formulate quantum theory in terms of a
sum over histories. In this approach, a particle does not have just a single
history, as it would in a classical theory. Instead, it is supposed to follow
every possible path in space-time, and with each of these histories there are
associated a couple of numbers, one represent-ing the size of a wave and
the other representing its position in the cycle (its phase). The probability
that the particle, say, passes through some particular point is found by
adding up the waves associated with every possible history that passes
through that point. When one actually tries to perform these sums, however,
one runs into severe technical problems. The only way around these is the
following peculiar prescription: one must add up the waves for particle
histories that are not in the “real” time that you and I experience but take
place in what is called imaginary time. Imaginary time may sound like
science fiction but it is in fact a well-defined mathematical concept. If we
take any ordinary (or “real”) number and multiply it by itself, the result is a
positive number. (For example, 2 times 2 is 4, but so is – 2 times – 2.) There
are, however, special numbers (called imaginary numbers) that give
negative numbers when multiplied by themselves. (The one called i, when
multiplied by itself, gives – 1, 2i multiplied by itself gives – 4, and so on.)
One can picture real and imaginary numbers in the following way: The real
numbers can be represented by a line going from left to right, with zero in
the middle, negative numbers like – 1, – 2, etc. on the left, and positive
numbers, 1, 2, etc. on the right. Then imaginary numbers are represented by
a line going up and down the page, with i, 2i, etc. above the middle, and – i,
– 2i, etc. below. Thus imaginary numbers are in a sense numbers at right
angles to ordinary real numbers.
To avoid the technical difficulties with Feynman’s sum over histories, one
must use imaginary time. That is to say, for the purposes of the calculation
one must measure time using imaginary numbers, rather than real ones.
This has an interesting effect on space-time: the distinction between time
and space disappears completely. A space-time in which events have
imaginary values of the time coordinate is said to be Euclidean, after the
ancient Greek Euclid, who founded the study of the geometry of twodimensional surfaces. What we now call Euclidean space-time is very
similar except that it has four dimensions instead of two. In Euclidean
space-time there is no difference between the time direction and directions
in space. On the other hand, in real space-time, in which events are labeled
by ordinary, real values of the time coordinate, it is easy to tell the
difference – the time direction at all points lies within the light cone, and
space directions lie outside. In any case, as far as everyday quantum
mechanics is concerned, we may regard our use of imaginary time and
Euclidean space-time as merely a mathematical device (or trick) to
calculate answers about real space-time.
A second feature that we believe must be part of any ultimate theory is
Einstein’s idea that the gravitational field is represented by curved spacetime: particles try to follow the nearest thing to a straight path in a curved
space, but because space-time is not flat their paths appear to be bent, as if
by a gravitational field. When we apply Feynman’s sum over histories to
Einstein’s view of gravity, the analogue of the history of a particle is now a
complete curved space-time that represents the history of the whole
universe. To avoid the technical difficulties in actually performing the sum
over histories, these curved space-times must be taken to be Euclidean. That
is, time is imaginary and is indistinguishable from directions in space. To
calculate the probability of finding a real space-time with some certain
property, such as looking the same at every point and in every direction, one
adds up the waves associated with all the histories that have that property.
In the classical theory of general relativity, there are many different possible
curved space-times, each corresponding to a different initial state of the
universe. If we knew the initial state of our universe, we would know its
entire history. Similarly, in the quantum theory of gravity, there are many
different possible quantum states for the universe. Again, if we knew how
the Euclidean curved space-times in the sum over histories behaved at early
times, we would know the quantum state of the universe.
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In the classical theory of gravity, which is based on real space-time, there
are only two possible ways the universe can behave: either it has existed for
an infinite time, or else it had a beginning at a singularity at some finite
time in the past. In the quantum theory of gravity, on the other hand, a third
possibility arises. Because one is using Euclidean space-times, in which the
time direction is on the same footing as directions in space, it is possible for
space-time to be finite in extent and yet to have no singularities that formed
a boundary or edge. Space-time would be like the surface of the earth, only
with two more dimensions. The surface of the earth is finite in extent but it
doesn’t have a boundary or edge: if you sail off into the sunset, you don’t
fall off the edge or run into a singularity. (I know, because I have been
round the world!)
If Euclidean space-time stretches back to infinite imaginary time, or else
starts at a singularity in imaginary time, we have the same problem as in the
classical theory of specifying the initial state of the universe: God may
know how the universe began, but we cannot give any particular reason for
thinking it began one way rather than another. On the other hand, the
quantum theory of gravity has opened up a new possibility, in which there
would be no boundary to space-time and so there would be no need to
specify the behavior at the boundary. There would be no singularities at
which the laws of science broke down, and no edge of space-time at which
one would have to appeal to God or some new law to set the boundary
conditions for space-time. One could say: “The boundary condition of the
universe is that it has no boundary.” The universe would be completely selfcontained and not affected by anything outside itself. It would neither be
created nor destroyed, It would just BE. It was at the conference in the
Vatican mentioned earlier that I first put forward the suggestion that maybe
time and space together formed a surface that was finite in size but did not
have any boundary or edge. My paper was rather mathematical, however, so
its implications for the role of God in the creation of the universe were not
generally recognized at the time (just as well for me). At the time of the
Vatican conference, I did not know how to use the “no boundary” idea to
make predictions about the universe. However, I spent the following summer at the University of California, Santa Barbara. There a friend and
colleague of mine, Jim Hartle, worked out with me what conditions the
universe must satisfy if space-time had no boundary. When I returned to
Cambridge, I continued this work with two of my research students, Julian
Luttrel and Jonathan Halliwell.
I’d like to emphasize that this idea that time and space should be finite
“without boundary” is just a proposal: it cannot be deduced from some
other principle. Like any other scientific theory, it may initially be put
forward for aesthetic or metaphysical reasons, but the real test is whether it
makes predictions that agree with observation. This, how-ever, is difficult to
determine in the case of quantum gravity, for two reasons. First, as will be
explained in Chapter 11, we are not yet sure exactly which theory
successfully combines general relativity and quantum mechanics, though
we know quite a lot about the form such a theory must have. Second, any
model that described the whole universe in detail would be much too
complicated mathematically for us to be able to calculate exact predictions.
One therefore has to make simplifying assumptions and approximations –
and even then, the problem of extracting predictions remains a formidable
one.
Each history in the sum over histories will describe not only the space-time
but everything in it as well, including any complicated organisms like
human beings who can observe the history of the universe. This may
provide another justification for the anthropic principle, for if all the
histories are possible, then so long as we exist in one of the histories, we
may use the anthropic principle to explain why the universe is found to be
the way it is. Exactly what meaning can be attached to the other histories, in
which we do not exist, is not clear. This view of a quantum theory of
gravity would be much more satisfactory, however, if one could show that,
using the sum over histories, our universe is not just one of the possible
histories but one of the most probable ones. To do this, we must perform the
sum over histories for all possible Euclidean space-times that have no
boundary. Under the “no boundary” proposal one learns that the chance of
the universe being found to be following most of the possible histories is
negligible, but there is a particular family of histories that are much more
probable than the others. These histories may be pictured as being like the
surface of the earth, with the distance from the North Pole representing
imaginary time and the size of a circle of constant distance from the North
Pole representing the spatial size of the universe. The universe starts at the
North Pole as a single point. As one moves south, the circles of latitude at
constant distance from the North Pole get bigger, corresponding to the
universe expanding with imaginary time Figure 8:1. The universe would
reach a maximum size at the equator and would contract with increasing
imaginary time to a single point at the South Pole. Ever though the universe
would have zero size at the North and South Poles, these points would not
be singularities, any more than the North aid South Poles on the earth are
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Figure 8:1
The history of the universe in real time, however, would look very different.
At about ten or twenty thousand million years ago, it would have a
minimum size, which was equal to the maximum radius of the history in
imaginary time. At later real times, the universe would expand like the
chaotic inflationary model proposed by Linde (but one would not now have
to assume that the universe was created somehow in the right sort of state).
The universe would expand to a very large size Figure 8:1 and eventually it
would collapse again into what looks like a singularity in real time. Thus, in
a sense, we are still all doomed, even if we keep away from black holes.
Only if we could picture the universe in terms of imaginary time would
there be no singularities.
If the universe really is in such a quantum state, there would be no
singularities in the history of the universe in imaginary time. It might seem
therefore that my more recent work had completely undone the results of
my earlier work on singularities. But, as indicated above, the real
importance of the singularity theorems was that they showed that the
gravitational field must become so strong that quantum gravitational effects
could not be ignored. This in turn led to the idea that the universe could be
finite in imaginary time but without boundaries or singularities. When one
goes back to the real time in which we live, however, there will still appear
to be singularities. The poor astronaut who falls into a black hole will still
come to a sticky end; only if he lived in imaginary time would he encounter
no singularities.
This might suggest that the so-called imaginary time is really the real time,
and that what we call real time is just a figment of our imaginations. In real
time, the universe has a beginning and an end at singularities that form a
boundary to space-time and at which the laws of science break down. But in
imaginary time, there are no singularities or boundaries. So maybe what we
call imaginary time is really more basic, and what we call real is just an
idea that we invent to help us describe what we think the universe is like.
But according to the approach I described in Chapter 1, a scientific theory is
just a mathematical model we make to describe our observations: it exists
only in our minds. So it is meaningless to ask: which is real, “real” or
“imaginary” time? It is simply a matter of which is the more useful
description.
One can also use the sum over histories, along with the no boundary
proposal, to find which properties of the universe are likely to occur
together. For example, one can calculate the probability that the universe is
expanding at nearly the same rate in all different directions at a time when
the density of the universe has its present value. In the simplified models
that have been examined so far, this probability turns out to be high; that is,
the proposed no boundary condition leads to the prediction that it is
extremely probable that the present rate of expansion of the universe is
almost the same in each direction. This is consistent with the observations
of the microwave background radiation, which show that it has almost
exactly the same intensity in any direction. If the universe were expanding
faster in some directions than in others, the intensity of the radiation in
those directions would be reduced by an
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additional red shift.
Further predictions of the no boundary condition are currently being
worked out. A particularly interesting problem is the size of the small
departures from uniform density in the early universe that caused the
formation first of the galaxies, then of stars, and finally of us. The
uncertainty principle implies that the early universe cannot have been
completely uniform because there must have been some uncertainties or
fluctuations in the positions and velocities of the particles. Using the no
boundary condition, we find that the universe must in fact have started off
with just the minimum possible non-uniformity allowed by the uncertainty
principle. The universe would have then undergone a period of rapid
expansion, as in the inflationary models. During this period, the initial nonuniformities would have been amplified until they were big enough to
explain the origin of the structures we observe around us. In 1992 the
Cosmic Background Explorer satellite (COBE) first detected very slight
variations in the intensity of the microwave background with direction. The
way these non-uniformities depend on direction seems to agree with the
predictions of the inflationary model and the no boundary proposal. Thus
the no boundary proposal is a good scientific theory in the sense of Karl
Popper: it could have been falsified by observations but instead its
predictions have been confirmed. In an expanding universe in which the
density of matter varied slightly from place to place, gravity would have
caused the denser regions to slow down their expansion and start
contracting. This would lead to the formation of galaxies, stars, and
eventually even insignificant creatures like ourselves. Thus all the
complicated structures that we see in the universe might be explained by the
no boundary condition for the universe together with the uncertainty
principle of quantum mechanics.
The idea that space and time may form a closed surface without boundary
also has profound implications for the role of God in the affairs of the
universe. With the success of scientific theories in describing events, most
people have come to believe that God allows the universe to evolve
according to a set of laws and does not intervene in the universe to break
these laws. However, the laws do not tell us what the universe should have
looked like when it started – it would still be up to God to wind up the
clockwork and choose how to start it off. So long as the universe had a
beginning, we could suppose it had a creator. But if the universe is really
completely self-contained, having no boundary or edge, it would have
neither beginning nor end: it would simply be. What place, then, for a
creator?
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CHAPTER 9
THE ARROW OF TIME
In previous chapters we have seen how our views of the nature of time have
changed over the years. Up to the beginning of this century people believed
in an absolute time. That is, each event could be labeled by a number called
“time” in a unique way, and all good clocks would agree on the time
interval between two events. However, the discovery that the speed of light
appeared the same to every observer, no matter how he was moving, led to
the theory of relativity – and in that one had to abandon the idea that there
was a unique absolute time. Instead, each observer would have his own
measure of time as recorded by a clock that he carried: clocks carried by
different observers would not necessarily agree. Thus time became a more
personal concept, relative to the observer who measured it.
When one tried to unify gravity with quantum mechanics, one had to
introduce the idea of “imaginary” time. Imaginary time is indistinguishable
from directions in space. If one can go north, one can turn around and head
south; equally, if one can go forward in imaginary time, one ought to be
able to turn round and go backward. This means that there can be no
important difference between the forward and backward directions of
imaginary time. On the other hand, when one looks at “real” time, there’s a
very big difference between the forward and backward directions, as we all
know. Where does this difference between the past and the future come
from? Why do we remember the past but not the future?
The laws of science do not distinguish between the past and the future.
More precisely, as explained earlier, the laws of science are unchanged
under the combination of operations (or symmetries) known as C, P, and T.
(C means changing particles for antiparticles. P means taking the mirror
image, so left and right are interchanged. And T means reversing the
direction of motion of all particles: in effect, running the motion backward.)
The laws of science that govern the behavior of matter under all normal
situations are unchanged under the combination of the two operations C and
P on their own. In other words, life would be just the same for the
inhabitants of another planet who were both mirror images of us and who
were made of antimatter, rather than matter.
If the laws of science are unchanged by the combination of operations C
and P, and also by the combination C, P, and T, they must also be
unchanged under the operation T alone. Yet there is a big difference
between the forward and backward directions of real time in ordinary life.
Imagine a cup of water falling off a table and breaking into pieces on the
floor. If you take a film of this, you can easily tell whether it is being run
forward or backward. If you run it backward you will see the pieces
suddenly gather themselves together off the floor and jump back to form a
whole cup on the table. You can tell that the film is being run backward
because this kind of behavior is never observed in ordinary life. If it were,
crockery manufacturers would go out of business. The explanation that is
usually given as to why we don’t see broken cups gathering themselves
together off the floor and jumping back onto the table is that it is forbidden
by the second law of thermodynamics. This says that in any closed system
disorder, or entropy, always increases with time. In other words, it is a form
of Murphy’s law: things always tend to go wrong! An intact cup on the
table is a state of high order, but a broken cup on the floor is a disordered
state. One can go readily from the cup on the table in the past to the broken
cup on the floor in the future, but not the other way round.
The increase of disorder or entropy with time is one example of what is
called an arrow of time, something that distinguishes the past from the
future, giving a direction to time. There are at least three different arrows of
time. First, there is the thermodynamic arrow of time, the direction of time
in which disorder or entropy increases. Then, there is the psychological
arrow of time. This is the direction in which we feel time passes, the
direction in which we remember the past but not the future. Finally, there is
the cosmological arrow of time. This is the direction of time in which the
universe is expanding rather than contracting. In this chapter I shall argue
that the no boundary condition for the universe, together with the weak
anthropic principle, can explain why all three arrows point in the same
direction – and moreover, why a well-defined arrow of time should exist at
all. I shall argue that the psychological arrow is determined by the
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and that these two arrows necessarily always point in the same direction. If
one assumes the no boundary condition for the universe, we shall see that
there must be well-defined thermodynamic and cosmological arrows of
time, but they will not point in the same direction for the whole history of
the universe. However, I shall argue that it is only when they do point in the
same direction that conditions are suitable for the development of
intelligent beings who can ask the question: why does disorder increase in
the same direction of time as that in which the universe expands?
I shall discuss first the thermodynamic arrow of time. The second law of
thermodynamics results from the fact that there are always many more
disordered states than there are ordered ones. For example, consider the
pieces of a jigsaw in a box. There is one, and. only one, arrangement in
which the pieces make a complete picture. On the other hand, there are a
very large number of arrangements in which the pieces are disordered and
don’t make a picture.
Suppose a system starts out in one of the small number of ordered states. As
time goes by, the system will evolve according to the laws of science and its
state will change. At a later time, it is more probable that the system will be
in a disordered state than in an ordered one because there are more
disordered states. Thus disorder will tend to increase with time if the system
obeys an initial condition of high order. Suppose the pieces of the jigsaw
start off in a box in the ordered arrangement in which they form a picture. If
you shake the box, the pieces will take up another arrangement. This will
probably be a disordered arrangement in which the pieces don’t form a
proper picture, simply because there are so many more disordered
arrangements. Some groups of pieces may still form parts of the picture, but
the more you shake the box, the more likely it is that these groups will get
broken up and the pieces will be in a completely jumbled state in which
they don’t form any sort of picture. So the disorder of the pieces will
probably increase with time if the pieces obey the initial condition that they
start off in a condition of high order. Suppose, however, that God decided
that the universe should finish up in a state of high order but that it didn’t
matter what state it started in. At early times the universe would probably
be in a disordered state. This would mean that disorder would decrease with
time. You would see broken cups gathering themselves together and
jumping back onto the table. However, any human beings who were
observing the cups would be living in a universe in which disorder
decreased with time. I shall argue that such beings would have a
psychological arrow of time that was backward. That is, they would
remember events in the future, and not remember events in their past. When
the cup was broken, they would remember it being on the table, but when it
was on the table, they would not remember it being on the floor.
It is rather difficult to talk about human memory because we don’t know
how the brain works in detail. We do, however, know all about how
computer memories work. I shall therefore discuss the psychological arrow
of time for computers. I think it is reasonable to assume that the arrow for
computers is the same as that for humans. If it were not, one could make a
killing on the stock exchange by having a computer that would remember
tomorrow’s prices! A computer memory is basically a device containing
elements that can exist in either of two states. A simple example is an
abacus. In its simplest form, this consists of a number of wires; on each
wire there are a number of beads that can be put in one of two positions.
Before an item is recorded in a computer’s memory, the memory is in a
disordered state, with equal probabilities for the two possible states. (The
abacus beads are scattered randomly on the wires of the abacus.) After the
memory interacts with the system to be remembered, it will definitely be in
one state or the other, according to the state of the system. (Each abacus
bead will be at either the left or the right of the abacus wire.) So the
memory has passed from a disordered state to an ordered one. However, in
order to make sure that the memory is in the right state, it is necessary to
use a certain amount of energy (to move the bead or to power the computer,
for example). This energy is dissipated as heat, and increases the amount of
disorder in the universe. One can show that this increase in disorder is
always greater than the increase in the order of the memory itself. Thus the
heat expelled by the computer’s cooling fan means that when a computer
records an item in memory, the total amount of disorder in the universe still
goes up. The direction of time in which a computer remembers the past is
the same as that in which disorder increases.
Our subjective sense of the direction of time, the psychological arrow of
time, is therefore determined within our brain by the thermodynamic arrow
of time. Just like a computer, we must remember things in the order in
which entropy increases. This makes the second law of thermodynamics
almost trivial. Disorder increases with time
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because we measure time in the direction in which disorder increases You
can’t have a safer bet than that!
But why should the thermodynamic arrow of time exist at all? Or, in other
words, why should the universe be in a state of high order at one end of
time, the end that we call the past? Why is it not in a state of complete
disorder at all times? After all, this might seem more probable. And why is
the direction of time in which disorder increases the same as that in which
the universe expands?
In the classical theory of general relativity one cannot predict how the
universe would have begun because all the known laws of science would
have broken down at the big bang singularity. The universe could have
started out in a very smooth and ordered state. This would have led to welldefined thermodynamic and cosmological arrows of time, as we observe.
But it could equally well have started out in a very lumpy and disordered
state. In that case, the universe would already be in a state of complete
disorder, so disorder could not increase with time. It would either stay
constant, in which case there would be no well-defined thermodynamic
arrow of time, or it would decrease, in which case the thermodynamic
arrow of time would point in the opposite direction to the cosmological
arrow. Neither of these possibilities agrees with what we observe. However,
as we have seen, classical general relativity predicts its own downfall.
When the curvature of space-time becomes large, quantum gravitational
effects will become important and the classical theory will cease to be a
good description of the universe. One has to use a quantum theory of
gravity to understand how the universe began.
In a quantum theory of gravity, as we saw in the last chapter, in order to
specify the state of the universe one would still have to say how the
possible histories of the universe would behave at the boundary of spacetime in the past. One could avoid this difficulty of having to describe what
we do not and cannot know only if the histories satisfy the no boundary
condition: they are finite in extent but have no boundaries, edges, or
singularities. In that case, the beginning of time would be a regular, smooth
point of space-time and the universe would have begun its expansion in a
very smooth and ordered state. It could not have been completely uniform,
because that would violate the uncertainty principle of quantum theory.
There had to be small fluctuations in the density and velocities of particles.
The no boundary condition, however, implied that these fluctuations were
as small as they could be, consistent with the uncertainty principle. The
universe would have started off with a period of exponential or
“inflationary” expansion in which it would have increased its size by a very
large factor. During this expansion, the density fluctuations would have
remained small at first, but later would have started to grow. Regions in
which the density was slightly higher than average would have had their
expansion slowed down by the gravitational attraction of the extra mass.
Eventually, such regions would stop expanding and collapse to form
galaxies, stars, and beings like us. The universe would have started in a
smooth and ordered state, and would become lumpy and disordered as time
went on. This would explain the existence of the thermodynamic arrow of
time. But what would happen if and when the universe stopped expanding
and began to contract? Would the thermodynamic arrow reverse and
disorder begin to decrease with time? This would lead to all sorts of
science-fiction-like possibilities for people who survived from the
expanding to the contracting phase. Would they see broken cups gathering
themselves together off the floor and jumping back onto the table? Would
they be able to remember tomorrow’s prices and make a fortune on the
stock market? It might seem a bit academic to worry about what will
happen when the universe collapses again, as it will not start to contract for
at least another ten thousand million years. But there is a quicker way to
find out what will happen: jump into a black hole. The collapse of a star to
form a black hole is rather like the later stages of the collapse of the whole
universe. So if disorder were to decrease in the contracting phase of the
universe, one might also expect it to decrease inside a black hole. So
perhaps an astronaut who fell into a black hole would be able to make
money at roulette by remembering where the ball went before he placed his
bet. (Unfortunately, however, he would not have long to play before he was
turned to spaghetti. Nor would he be able to let us know about the reversal
of the thermodynamic arrow, or even bank his winnings, because he would
be trapped behind the event horizon of the black hole.)
At first, I believed that disorder would decrease when the universe
recollapsed. This was because I thought that the universe had to return to a
smooth and ordered state when it became small again. This would mean
that the contracting phase would be like the time reverse of the expanding
phase. People in the contracting phase would live their lives backward: they
would die before they were born and get younger as the universe
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contracted.
This idea is attractive because it would mean a nice symmetry between the
expanding and contracting phases. However, one cannot adopt it on its own,
independent of other ideas about the universe. The question is: is it implied
by the no boundary condition, or is it inconsistent with that condition? As I
said, I thought at first that the no boundary condition did indeed imply that
disorder would decrease in the contracting phase. I was misled partly by the
analogy with the surface of the earth. If one took the beginning of the
universe to correspond to the North Pole, then the end of the universe
should be similar to the beginning, just as the South Pole is similar to the
North. However, the North and South Poles correspond to the beginning
and end of the universe in imaginary time. The beginning and end in real
time can be very different from each other. I was also misled by work I had
done on a simple model of the universe in which the collapsing phase
looked like the time reverse of the expanding phase. However, a colleague
of mine, Don Page, of Penn State University, pointed out that the no
boundary condition did not require the contracting phase necessarily to be
the time reverse of the expanding phase. Further, one of my students,
Raymond Laflamme, found that in a slightly more complicated model, the
collapse of the universe was very different from the expansion. I realized
that I had made a mistake: the no boundary condition implied that disorder
would in fact continue to increase during the contraction. The
thermodynamic and psychological arrows of time would not reverse when
the universe begins to recontract, or inside black holes.
What should you do when you find you have made a mistake like that?
Some people never admit that they are wrong and continue to find new, and
often mutually inconsistent, arguments to support their case – as Eddington
did in opposing black hole theory. Others claim to have never really
supported the incorrect view in the first place or, if they did, it was only to
show that it was inconsistent. It seems to me much better and less confusing
if you admit in print that you were wrong. A good example of this was
Einstein, who called the cosmological constant, which he introduced when
he was trying to make a static model of the universe, the biggest mistake of
his life.
To return to the arrow of time, there remains the question: why do we
observe that the thermodynamic and cosmological arrows point in the same
direction? Or in other words, why does disorder increase in the same
direction of time as that in which the universe expands? If one believes that
the universe will expand and then contract again, as the no boundary
proposal seems to imply, this becomes a question of why we should be in
the expanding phase rather than the contracting phase.
One can answer this on the basis of the weak anthropic principle.
Conditions in the contracting phase would not be suitable for the existence
of intelligent beings who could ask the question: why is disorder increasing
in the same direction of time as that in which the universe is expanding?
The inflation in the early stages of the universe, which the no boundary
proposal predicts, means that the universe must be expanding at very close
to the critical rate at which it would just avoid recollapse, and so will not
recollapse for a very long time. By then all the stars will have burned out
and the protons and neutrons in them will probably have decayed into light
particles and radiation. The universe would be in a state of almost complete
disorder. There would be no strong thermodynamic arrow of time. Disorder
couldn’t increase much because the universe would be in a state of almost
complete disorder already. However, a strong thermodynamic arrow is
necessary for intelligent life to operate. In order to survive, human beings
have to consume food, which is an ordered form of energy, and convert it
into heat, which is a disordered form of energy. Thus intelligent life could
not exist in the contracting phase of the universe. This is the explanation of
why we observe that the thermodynamic and cosmological arrows of time
point in the same direction. It is not that the expansion of the universe
causes disorder to increase. Rather, it is that the no boundary condition
causes disorder to increase and the conditions to be suitable for intelligent
life only in the expanding phase.
To summarize, the laws of science do not distinguish between the forward
and backward directions of time. However, there are at least three arrows of
time that do distinguish the past from the future. They are the
thermodynamic arrow, the direction of time in which disorder increases; the
psychological arrow, the direction of time in which we remember the past
and not the future; and the cosmological arrow, the direction of time in
which the universe expands rather than contracts. I have shown that the
psychological arrow is essentially the same as the thermodynamic arrow, so
that the two would always point in the same direction. The no boundary
proposal for the universe predicts the existence of a well-defined
thermodynamic arrow of time because the universe must start off in a
smooth and ordered state. And the reason we observe this thermodynamic
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agree with the cosmological arrow is that intelligent beings can exist only in
the expanding phase. The contracting phase will be unsuitable because it
has no strong thermodynamic arrow of time. The progress of the human
race in understanding the universe has established a small corner of order in
an increasingly disordered universe. If you remember every word in this
book, your memory will have recorded about two million pieces of
information: the order in your brain will have increased by about two
million units. However, while you have been reading the book, you will
have converted at least a thousand calories of ordered energy, in the form of
food, into disordered energy, in the form of heat that you lose to the air
around you by convection and sweat. This will increase the disorder of the
universe by about twenty million million million million units – or about
ten million million million times the increase in order in your brain – and
that’s if you remember everything in this book. In the next chapter but one I
will try to increase the order in our neck of the woods a little further by
explaining how people are trying to fit together the partial theories I have
described to form a complete unified theory that would cover everything in
the universe.
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A Brief History of Time – Stephen Hawking… Chapter 10
CHAPTER 10
WORMHOLES AND TIME TRAVEL
The last chapter discussed why we see time go forward: why disorder
increases and why we remember the past but not the future. Time was
treated as if it were a straight railway line on which one could only go one
way or the other.
But what if the railway line had loops and branches so that a train could
keep going forward but come back to a station it had already passed? In
other words, might it be possible for someone to travel into the future or the
past?
H. G. Wells in The Time Machine explored these possibilities as have
countless other writers of science fiction. Yet many of the ideas of science
fiction, like submarines and travel to the moon, have become matters of
science fact. So what are the prospects for time travel?
The first indication that the laws of physics might really allow people to
travel in time came in 1949 when Kurt Godel discovered a new space-time
allowed by general relativity. Godel was a mathematician who was famous
for proving that it is impossible to prove all true statements, even if you
limit yourself to trying to prove all the true statements in a subject as
apparently cut and dried as arithmetic. Like the uncertainty principle,
Godel’s incompleteness theorem may be a fundamental limitation on our
ability to understand and predict the universe, but so far at least it hasn’t
seemed to be an obstacle in our search for a complete unified theory. Godel
got to know about general relativity when he and Einstein spent their later
years at the Institute for Advanced Study in Princeton. His space-time had
the curious property that the whole universe was rotating. One might ask:
“Rotating with respect to what?” The answer is that distant matter would be
rotating with respect to directions that little tops or gyroscopes point in.
This had the side effect that it would be possible for someone to go off in a
rocket ship and return to earth before he set out. This property really upset
Einstein, who had thought that general relativity wouldn’t allow time travel.
However, given Einstein’s record of ill-founded opposition to gravitational
collapse and the uncertainty principle, maybe this was an encouraging sign.
The solution Godel found doesn’t correspond to the universe we live in
because we can show that the universe is not rotating. It also had a non-zero
value of the cosmological constant that Einstein introduced when he
thought the universe was unchanging. After Hubble discovered the
expansion of the universe, there was no need for a cosmological constant
and it is now generally believed to be zero. However, other more reasonable
space-times that are allowed by general relativity and which permit travel
into the past have since been found. One is in the interior of a rotating black
hole. Another is a space-time that contains two cosmic strings moving past
each other at high speed. As their name suggests, cosmic strings are objects
that are like string in that they have length but a tiny cross section. Actually,
they are more like rubber bands because they are under enormous tension,
something like a million million million million tons. A cosmic string
attached to the earth could accelerate it from 0 to 60 mph in 1/30th of a
second. Cosmic strings may sound like pure science fiction but there are
reasons to believe they could have formed in the early universe as a result
of symmetry-breaking of the kind discussed in Chapter 5. Because they
would be under enormous tension and could start in any configuration, they
might accelerate to very high speeds when they straighten out.
The Godel solution and the cosmic string space-time start out so distorted
that travel into the past was always possible. God might have created such a
warped universe but we have no reason to believe he did. Observations of
the microwave background and of the abundances of the light elements
indicate that the early universe did not have the kind of curvature required
to allow time travel. The same conclusion follows on theoretical grounds if
the no boundary proposal is correct. So the question is: if the universe starts
out without the kind of curvature required for time travel, can we
subsequently warp local regions of space-time sufficiently to allow it?
A closely related problem that is also of concern to writers of science
fiction is rapid interstellar or intergalactic
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travel. According to relativity, nothing can travel faster than light. If we
therefore sent a spaceship to our nearest neighboring star, Alpha Centauri,
which is about four light-years away, it would take at least eight years
before we could expect the travelers to return and tell us what they had
found. If the expedition were to the center of our galaxy, it would be at least
a hundred thousand years before it came back. The theory of relativity does
allow one consolation. This is the so-called twins paradox mentioned in
Chapter 2. Because there is no unique standard of time, but rather observers
each have their own time as measured by clocks that they carry with them,
it is possible for the journey to seem to be much shorter for the space
travelers than for those who remain on earth. But there would not be much
joy in returning from a space voyage a few years older to find that everyone
you had left behind was dead and gone thousands of years ago. So in order
to have any human interest in their stories, science fiction writers had to
suppose that we would one day discover how to travel faster than light.
What most of thee authors don’t seem to have realized is that if you can
travel faster than light, the theory of relativity implies you can also travel
back in the, as the following limerick says: There was a young lady of
Wight
Who traveled much faster than light.
She departed one day,
In a relative way,
And arrived on the previous night
The point is that the theory of relativity says hat there is no unique measure
of time that all observers will agree on Rather, each observer has his or her
own measure of time. If it is possible for a rocket traveling below the speed
of light to get from event A (say, the final of the 100-meter race of the
Olympic Games in 202) to event B
(say, the opening of the 100,004th meeting of the Congress of Alpha
Centauri), then all observers will agree that event A happened before event
B according to their times. Suppose, however, that the spaceship would
have to travel faster than light to carry the news of the race to the Congress.
Then observers moving at different speeds can disagree about whether
event A occurred before B or vice versa. According to the time of an
observer who is at rest with respect to the earth, it may be that the Congress
opened after the race. Thus this observer would think that a spaceship could
get from A to B in time if only it could ignore the speed-of-light speed
limit. However, to an observer at Alpha Centauri moving away from the
earth at nearly the speed of light, it would appear that event B, the opening
of the Congress, would occur before event A, the 100-meter race. The
theory of relativity says that the laws of physics appear the same to
observers moving at different speeds. This has been well tested by
experiment and is likely to remain a feature even if we find a more
advanced theory to replace relativity Thus the moving observer would say
that if faster-than-light travel is possible, it should be possible to get from
event B, the opening of the Congress, to event A, the 100-meter race. If one
went slightly faster, one could even get back before the race and place a bet
on it in the sure knowledge that one would win.
There is a problem with breaking the speed-of-light barrier. The theory of
relativity says that the rocket power needed to accelerate a spaceship gets
greater and greater the nearer it gets to the speed of light. We have
experimental evidence for this, not with spaceships but with elementary
particles in particle accelerators like those at Fermilab or CERN (European
Centre for Nuclear Research). We can accelerate particles to 99.99
percent of the speed of light, but however much power we feed in, we can’t
get them beyond the speed-of-light barrier. Similarly with spaceships: no
matter how much rocket power they have, they can’t accelerate beyond the
speed of light.
That might seem to rule out both rapid space travel and travel back in time.
However, there is a possible way out. It might be that one could warp
space-time so that there was a shortcut between A and B One way of doing
this would be to create a wormhole between A and B. As its name suggests,
a wormhole is a thin tube of space-time which can connect two nearly flat
regions far apart.
There need be no relation between the distance through the wormhole and
the separation of its ends in the nearly Hat background. Thus one could
imagine that one could create or find a wormhole that world lead from the
vicinity of the Solar System to Alpha Centauri. The distance through the
wormhole might be only a few million miles even though earth and Alpha
Centauri are twenty million million miles apart in ordinary space. This
would allow news of the 100-meter race to reach the opening of the
Congress. But then an observer moving toward 6e earth should also be able
to find another wormhole that would enable him to get from the opening of
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the Congress on Alpha Centauri back to earth before the start of the race. So
wormholes, like any other possible form of travel faster than light, would
allow one to travel into the past. The idea of wormholes between different
regions of space-time was not an invention of science fiction writers but
came from a very respectable source.
In 1935, Einstein and Nathan Rosen wrote a paper in which they showed
that general relativity allowed what they called “bridges,” but which are
now known as wormholes. The Einstein-Rosen bridges didn’t last long
enough for a spaceship to get through: the ship would run into a singularity
as the wormhole pinched off. However, it has been suggested that it might
be possible for an advanced civilization to keep a wormhole open. To do
this, or to warp space-time in any other way so as to permit time travel, one
can show that one needs a region of space-time with negative curvature,
like the surface of a saddle. Ordi-nary matter, which has a positive energy
density, gives space-time a positive curvature, like the surface of a sphere.
So what one needs, in order to warp space-time in a way that will allow
travel into the past, is matter with negative energy density. Energy is a bit
like money: if you have a positive balance, you can distribute it in various
ways, but according to the classical laws that were believed at the beginning
of the century, you weren’t allowed to be overdrawn. So these classical laws
would have ruled out any possibility of time travel. However, as has been
described in earlier chapters, the classical laws were superseded by
quantum laws based on the uncertainty principle. The quantum laws are
more liberal and allow you to be overdrawn on one or two accounts
provided the total balance is positive. In other words, quantum theory
allows the energy density to be negative in some places, provided that this
is made up for by positive energy densities in other places, so that the total
energy re-mains positive. An example of how quantum theory can allow
negative energy densities is provided by what is called the Casimir effect.
As we saw in Chapter 7, even what we think of as “empty” space is filled
with pairs of virtual particles and antiparticles that appear together, move
apart, and come back together and annihilate each other. Now, suppose one
has two parallel metal plates a short distance apart. The plates will act like
mirrors for the virtual photons or particles of light. In fact they will form a
cavity between them, a bit like an organ pipe that will resonate only at
certain notes. This means that virtual photons can occur in the space
between the plates only if their wavelengths (the distance between the crest
of one wave and the next) fit a whole number of times into the gap between
the plates. If the width of a cavity is a whole number of wavelengths plus a
fraction of a wave-length, then after some reflections backward and forward
between the plates, the crests of one wave will coincide with the troughs of
another and the waves will cancel out.
Because the virtual photons between the plates can have only the resonant
wavelengths, there will be slightly fewer of them than in the region outside
the plates where virtual photons can have any wavelength. Thus there will
be slightly fewer virtual photons hitting the inside surfaces of the plates
than the outside surfaces. One would therefore expect a force on the plates,
pushing them toward each other. This force has actually been detected and
has the predicted value. Thus we have experimental evidence that virtual
particles exist and have real effects.
The fact that there are fewer virtual photons between the plates means that
their energy density will be less than elsewhere. But the total energy density
in “empty” space far away from the plates must be zero, because otherwise
the energy density would warp the space and it would not be almost flat. So,
if the energy density between the plates is less than the energy density far
away, it must be negative. We thus have experimental evidence both that
space-time can be warped (from the bending of light during eclipses) and
that it can be curved in the way necessary to allow time travel (from the
Casimir effect). One might hope therefore that as we advance in science and
technology, we would eventually manage to build a time machine. But if so,
why hasn’t anyone come back from the future and told us how to do it?
There might be good reasons why it would be unwise to give us the secret
of time travel at our present primitive state of development, but unless
human nature changes radically, it is difficult to believe that some visitor
from the future wouldn’t spill the beans. Of course, some people would
claim that sightings of UFOs are evidence that we are being visited either
by aliens or by people from the future. (If the aliens were to get here in
reasonable time, they would need faster-than-light travel, so the two
possibilities may be equivalent.) However, I think that any visit by aliens or
people from the future would be much more obvious and, probably, much
more unpleasant. If they are going to reveal themselves at all, why do so
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regarded as reliable witnesses? If they are trying to warn us of some great
danger, they are not being very effective.
A possible way to explain the absence of visitors from the future would be
to say that the past is fixed because we have observed it and seen that it
does not have the kind of warping needed to allow travel back from the
future. On the other hand, the future is unknown and open, so it might well
have the curvature required. This would mean that any time travel would be
confined to the future. There would be no chance of Captain Kirk and the
Starship Enterprise turning up at the present time.
This might explain why we have not yet been overrun by tourists from the
future, but it would not avoid the problems that would arise if one were able
to go back and change history. Suppose, for example, you went back and
killed your great-great-grandfather while he was still a child. There are
many versions of this paradox but they are essentially equivalent: one
would get contradictions if one were free to change the past. There seem to
be two possible resolutions to the paradoxes posed by time travel. One I
shall call the consistent histories approach. It says that even if space-time is
warped so that it would be possible to travel into the past, what happens in
space-time must be a consistent solution of the laws of physics. According
to this viewpoint, you could not go back in time unless history showed that
you had already arrived in the past and, while there, had not killed your
great-great-grandfather or committed any other acts that would conflict with
your current situation in the present. Moreover, when you did go back, you
wouldn’t be able to change recorded history. That means you wouldn’t have
free will to do what you wanted. Of course, one could say that free will is
an illusion anyway. If there really is a complete unified theory that governs
everything, it presumably also determines your actions. But it does so in a
way that is impossible to calculate for an organism that is as complicated as
a human being. The reason we say that humans have free will is because we
can’t predict what they will do. However, if the human then goes off in a
rocket ship and comes back before he or she set off, we will be able to
predict what he or she will do because it will be part of recorded history.
Thus, in that situation, the time traveler would have no free will.
The other possible way to resolve the paradoxes of time travel might be
called the alternative histories hypothesis. The idea here is that when time
travelers go back to the past, they enter alternative histories which differ
from recorded history. Thus they can act freely, without the constraint of
consistency with their previous history. Steven Spiel-berg had fun with this
notion in the Back to the Future films: Marty McFly was able to go back
and change his parents’ courtship to a more satisfactory history. The
alternative histories hypothesis sounds rather like Richard Feynman’s way
of expressing quantum theory as a sum over histories, which was described
in Chapters 4 and 8. This said that the universe didn’t just have a single
history: rather it had every possible history, each with its own probability.
However, there seems to be an important difference between Feynman’s
proposal and alternative histories. In Feynman’s sum, each history
comprises a complete space-time and everything in it. The space-time may
be so warped that it is possible to travel in a rocket into the past. But the
rocket would remain in the same space-time and therefore the same history,
which would have to be consistent. Thus Feynman’s sum over histories
proposal seems to support the consistent histories hypothesis rather than the
alternative histories. The Feynman sum over histories does allow travel into
the past on a microscopic scale. In Chapter 9 we saw that the laws of
science are unchanged by combinations of the operations C, P, and T. This
means that an antiparticle spinning in the anticlockwise direction and
moving from A to B can also be viewed as an ordinary particle spinning
clockwise and moving backward in time from B to A. Similarly, an
ordinary particle moving forward in time is equivalent to an antiparticle
moving backward in time. As has been discussed in this chapter and
Chapter 7, “empty” space is filled with pairs of virtual particles and
antiparticles that appear together, move apart, and then come back together
and annihilate each other.
So, one can regard the pair of particles as a single particle moving on a
closed loop in space-time. When the pair is moving forward in time (from
the event at which it appears to that at which it annihilates), it is called a
particle. But when the particle is traveling back in time (from the event at
which the pair annihilates to that at which it appears), it is said to be an
antiparticle traveling forward in time. The explanation of how black holes
can emit particles and radiation (given in Chapter 7) was that one member
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of a virtual particle/ antiparticle pair (say, the antiparticle) might fall into
the black hole, leaving the other member without a partner with which to
annihilate. The forsaken particle might fall into the hole as well, but it
might also escape from the vicinity of the black hole. If so, to an observer at
a distance it would appear to be a particle emitted by the black hole.
One can, however, have a different but equivalent intuitive picture of the
mechanism for emission from black holes. One can regard the member of
the virtual pair that fell into the black hole (say, the antiparticle) as a
particle traveling backward in time out of the hole. When it gets to the point
at which the virtual particle/antiparticle pair appeared together, it is
scattered by the gravitational field into a particle traveling forward in time
and escaping from the black hole. If, instead, it were the particle member of
the virtual pair that fell into the hole, one could regard it as an antiparticle
traveling back in time and coming out of the black hole. Thus the radiation
by black holes shows that quantum theory allows travel back in time on a
microscopic scale and that such time travel can produce observable effects.
One can therefore ask: does quantum theory allow time travel on a
macroscopic scale, which people could use? At first sight, it seems it
should. The Feynman sum over histories proposal is supposed to be over all
histories. Thus it should include histories in which space-time is so warped
that it is possible to travel into the past. Why then aren’t we in trouble with
history? Suppose, for example, someone had gone back and given the Nazis
the secret of the atom bomb?
One would avoid these problems if what I call the chronology protection
conjecture holds. This says that the laws of physics conspire to prevent
macroscopic bodies from carrying information into the past. Like the
cosmic censorship conjecture, it has not been proved but there are reasons
to believe it is true. The reason to believe that chronology protection
operates is that when space-time is warped enough to make travel into the
past possible, virtual particles moving on closed loops in space-time can
become real particles traveling forward in time at or below the speed of
light. As these particles can go round the loop any number of times, they
pass each point on their route many times. Thus their energy is counted over
and over again and the energy density will become very large. This could
give space-time a positive curvature that would not allow travel into the
past. It is not yet clear whether these particles would cause positive or
negative curvature or whether the curvature produced by some kinds of
virtual particles might cancel that produced by other kinds. Thus the
possibility of time travel remains open. But I’m not going to bet on it. My
opponent might have the unfair advantage of knowing the future.
PREVIOUS
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A Brief History of Time – Stephen Hawking… Chapter 11
CHAPTER 11
THE UNIFICATION OF PHYSICS
As was explained in the first chapter, it would be very difficult to construct
a complete unified theory of everything in the universe all at one go. So
instead we have made progress by finding partial theories that describe a
limited range of happenings and by neglecting other effects or
approximating them by certain numbers. (Chemistry, for example, allows us
to calculate the interactions of atoms, without knowing the internal structure
of an atom’s nucleus.) Ultimately, however, one would hope to find a
complete, consistent, unified theory that would include all these partial
theories as approximations, and that did not need to be adjusted to fit the
facts by picking the values of certain arbitrary numbers in the theory. The
quest for such a theory is known as “the unification of physics.” Einstein
spent most of his later years unsuccessfully searching for a unified theory,
but the time was not ripe: there were partial theories for gravity and the
electromagnetic force, but very little was known about the nuclear forces.
Moreover, Einstein refused to believe in the reality of quantum mechanics,
despite the important role he had played in its development. Yet it seems
that the uncertainty principle is a fundamental feature of the universe we
live in. A successful unified theory must, therefore, necessarily incorporate
this principle.
As I shall describe, the prospects for finding such a theory seem to be much
better now because we know so much more about the universe. But we
must beware of overconfidence – we have had false dawns before! At the
beginning of this century, for example, it was thought that everything could
be explained in terms of the properties of continuous matter, such as
elasticity and heat conduction. The discovery of atomic structure and the
uncertainty principle put an emphatic end to that. Then again, in 1928,
physicist and Nobel Prize winner Max Born told a group of visitors to
Gottingen University, “Physics, as we know it, will be over in six months.”
His confidence was based on the recent discovery by Dirac of the equation
that governed the electron. It was thought that a similar equation would
govern the proton, which was the only other particle known at the time, and
that would be the end of theoretical physics. However, the discovery of the
neutron and of nuclear forces knocked that one on the head too. Having said
this, I still believe there are grounds for cautious optimism that we may now
be near the end of the search for the ultimate laws of nature. In previous
chapters I have described general relativity, the partial theory of gravity, and
the partial theories that govern the weak, the strong, and the
electromagnetic forces. The last three may be combined in so-called grand
unified theories, or GUTs, which are not very satisfactory because they do
not include gravity and because they contain a number of quantities, like the
relative masses of different particles, that cannot be predicted from the
theory but have to be chosen to fit observations. The main difficulty in
finding a theory that unifies gravity with the other forces is that general
relativity is a “classical” theory; that is, it does not incorporate the
uncertainty principle of quantum mechanics. On the other hand, the other
partial theories depend on quantum mechanics in an essential way. A
necessary first step, therefore, is to combine general relativity with the
uncertainty principle. As we have seen, this can produce some remarkable
consequences, such as black holes not being black, and the universe not
having any singularities but being completely self-contained and without a
boundary. The trouble is, as explained in Chapter 7, that the uncertainty
principle means that even
“empty” space is filled with pairs of virtual particles and antiparticles.
These pairs would have an infinite amount of energy and, therefore, by
Einstein’s famous equation E = mc2, they would have an infinite amount of
mass. Their gravitational attraction would thus curve up the universe to
infinitely small size. Rather similar, seemingly absurd infinities occur in the
other partial theories, but in all these cases the infinities can be canceled out
by a process called renormalization. This involves canceling the infinities
by introducing other infinities. Although this technique is rather dubious
mathematically, it does seem to work in practice, and has been used with
these theories to make predictions that agree with observations to an
extraordinary degree of accuracy. Renormalization, however, does have a
serious drawback from the point of view of trying to find a complete theory,
because it means that the actual values of the masses and the strengths of
the forces cannot be predicted from the theory, but have to be chosen to fit
the observations. In attempting to incorporate the uncertainty principle into
general relativity, one has only two quantities that can be adjusted: the
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sufficient to remove all the infinities. One therefore has a theory that seems
to predict that certain quantities, such as the curvature of space-time, are
really infinite, yet these quantities can be observed and measured to be
perfectly finite! This problem in combining general relativity and the
uncertainty principle had been suspected for some time, but was finally
confirmed by detailed calculations in 1972. Four years later, a possible
solution, called “supergravity,” was suggested. The idea was to combine the
spin-2 particle called the graviton, which carries the gravitational force,
with certain other particles of spin 3/2, 1, ½, and 0. In a sense, all these
particles could then be regarded as different aspects of the same
“superparticle,” thus unifying the matter particles with spin ½ and 3/2 with
the force-carrying particles of spin 0, 1, and 2. The virtual
particle/antiparticle pairs of spin ½ and 3/2 would have negative energy,
and so would tend to cancel out the positive energy of the spin 2, 1, and 0
virtual pairs. This would cause many of the possible infinities to cancel out,
but it was suspected that some infinities might still remain. However, the
calculations required to find out whether or not there were any infinities left
uncancelled were so long and difficult that no one was prepared to
undertake them. Even with a computer it was reckoned it would take at
least four years, and the chances were very high that one would make at
least one mistake, probably more. So one would know one had the right
answer only if someone else repeated the calculation and got the same
answer, and that did not seem very likely!
Despite these problems, and the fact that the particles in the super-gravity
theories did not seem to match the observed particles, most scientists
believed that supergravity was probably the right answer to the problem of
the unification of physics. It seemed the best way of unifying gravity with
the other forces. However, in 1984
there was a remarkable change of opinion in favor of what are called string
theories. In these theories the basic objects are not particles, which occupy a
single point of space, but things that have a length but no other dimension,
like an infinitely thin piece of string. These strings may have ends (the socalled open strings) or they may be joined up with themselves in closed
loops (closed strings) Figure 11:1 and Figure 11:2.
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Figures 11:1 & 11:2
A particle occupies one point of space at each instant of time. Thus its
history can be represented by a line in space-time (the “world-line”). A
string, on the other hand, occupies a line in space at each moment of time.
So its history in space-time is a two-dimensional surface called the worldsheet. (Any point on such a world-sheet can be described by two numbers,
one specifying the time and the other the position of the point on the string.)
The world-sheet of an open string is a strip: its edges represent the paths
through space-time of the ends of the string Figure 11:1. The world-sheet of
a closed string is a cylinder or tube Figure 11:2: a slice through the tube is a
circle, which represents the position of the string at one particular time. Two
pieces of string can join together to form a single string; in the case of open
strings they simply join at the ends Figure 11:3, while in the case of closed
strings it is like the two legs joining on a pair of trousers Figure 11:4.
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Figure 11:3
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Figure 11:4
Similarly, a single piece of string can divide into two strings. In string
theories, what were previously thought of as particles are now pictured as
waves traveling down the string, like waves on a vibrating kite string. The
emission or absorption of one particle by another corresponds to the
dividing or joining together of strings. For example, the gravitational force
of the sun on the earth was pictured in particle theories as being caused by
the emission of a graviton by a particle in the sun and its absorption by a
particle in the earth Figure 11:5.
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Figures 11:5 & 11:6
In string theory, this process corresponds to an H-shaped tube or pipe
Figure 11:6 (string theory is rather like plumbing, in a way). The two
vertical sides of the H correspond to the particles in the sun and the earth,
and the horizontal crossbar corresponds to the graviton that travels between
them. String theory has a curious history. It was originally invented in the
late 1960s in an attempt to find a theory to describe the strong force. The
idea was that particles like the proton and the neutron could be regarded as
waves on a string. The strong forces between the particles would
correspond to pieces of string that went between other bits of string, as in a
spider’s web. For this theory to give the observed value of the strong force
between particles, the strings had to be like rubber bands with a pull of
about ten tons. In 1974 Joel Scherk from Paris and John Schwarz from the
California Institute of Technology published a paper in which they showed
that string theory could describe the gravitational force, but only if the
tension in the string were very much higher, about a thousand million
million million million million million tons (1 with thirty-nine zeros after
it). The predictions of the string theory would be just the same as those of
general relativity on normal length scales, but they would differ at very
small distances, less than a thousand million million million million
millionth of a centimeter (a centimeter divided by 1 with thirty-three zeros
after it). Their work did not receive much attention, however, because at
just about that time most people abandoned the original string theory of the
strong force in favor of the theory based on quarks and gluons, which
seemed to fit much better with observations. Scherk died in tragic
circumstances (he suffered from diabetes and went into a coma when no
one was around to give him an injection of insulin). So Schwarz was left
alone as almost the only supporter of string theory, but now with the much
higher proposed value of the string tension. In 1984 interest in strings
suddenly revived, apparently for two reasons. One was that people were not
really making much progress toward showing that supergravity was finite or
that it could explain the kinds of particles that we observe. The other was
the publication of a paper by John Schwarz and Mike Green of Queen Mary
College, London, that showed that string theory might be able to explain the
existence of particles that have a built-in left-handedness, like some of the
particles that we observe. Whatever the reasons, a large number of people
soon began to work on string theory and a new version was developed, the
so-called heterotic string, which seemed as if it might be able to explain the
types of particles that we observe. String theories also lead to infinities, but
it is thought they will all cancel out in versions like the heterotic string
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(though this is not yet known for certain). String theories, however, have a
bigger problem: they seem to be consistent only if space-time has either ten
or twenty-six dimensions, instead of the usual four! Of course, extra spacetime dimensions are a commonplace of science fiction indeed, they provide
an ideal way of overcoming the normal restriction of general relativity that
one cannot travel faster than light or back in time (see Chapter 10). The idea
is to take a shortcut through the extra dimensions. One can picture this in
the following way. Imagine that the space we live in has only two
dimensions and is curved like the surface of an anchor ring or torus Figure
11:7.
Figure 11:7
If you were on one side of the inside edge of the ring and you wanted to get
to a point on the other side, you would have to go round the inner edge of
the ring. However, if you were able to travel in the third dimension, you
could cut straight across.
Why don’t we notice all these extra dimensions, if they are really there?
Why do we see only three space dimensions and one time dimension? The
suggestion is that the other dimensions are curved up into a space of very
small size, something like a million million million million millionth of an
inch. This is so small that we just don’t notice it: we see only one time
dimension and three space dimensions, in which space-time is fairly flat. It
is like the surface of a straw. If you look at it closely, you see it is twodimensional (the position of a point on the straw is described by two
numbers, the length along the straw and the distance round the circular
direction). But if you look at it from a distance, you don’t see the thickness
of the straw and it looks one-dimensional (the position of a point is
specified only by the length along the straw). So it is with space-time: on a
very small scale it is ten-dimensional and highly curved, but on bigger
scales you don’t see the curvature or the extra dimensions. If this picture is
correct, it spells bad news for would-be space travelers: the extra
dimensions would be far too small to allow a spaceship through. However,
it raises another major problem. Why should some, but not all, of the
dimensions be curled up into a small ball? Presumably, in the very early
universe all the dimensions would have been very curved. Why did one
time dimension and three space dimensions flatten
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out, while the other dimensions remain tightly curled up?
One possible answer is the anthropic principle. Two space dimensions do
not seem to be enough to allow for the development of complicated beings
like us. For example, two-dimensional animals living on a one-dimensional
earth would have to climb over each other in order to get past each other. If
a two-dimensional creature ate something it could not digest completely, it
would have to bring up the remains the same way it swallowed them,
because if there were a passage right through its body, it would divide the
creature into two separate halves: our two-dimensional being would fall
apart Figure 11:8. Similarly, it is difficult to see how there could be any
circulation of the blood in a two-dimensional creature.
Figure 11:8
There would also be problems with more than three space dimensions. The
gravitational force between two bodies would decrease more rapidly with
distance than it does in three dimensions. (In three dimensions, the
gravitational force drops to 1/4 if one doubles the distance. In four
dimensions it would drop to 1/5, in five dimensions to 1/6, and so on.) The
significance of this is that the orbits of planets, like the earth, around the
sun would be unstable: the least disturbance from a circular orbit (such as
would be caused by the gravitational attraction of other planets) would
result in the earth spiraling away from or into the sun. We would either
freeze or be burned up. In fact, the same behavior of gravity with distance
in more than three space dimensions means that the sun would not be able
to exist in a stable state with pressure balancing gravity. It would either fall
apart or it would collapse to form a black hole. In either case, it would not
be of much use as a source of heat and light for life on earth. On a smaller
scale, the electrical forces that cause the electrons to orbit round the nucleus
in an atom would behave in the same way as gravitational forces. Thus the
electrons would either escape from the atom altogether or would spiral into
the nucleus. In either case, one could not have atoms as we know them.
It seems clear then that life, at least as we know it, can exist only in regions
of space-time in which one time dimension and three space dimensions are
not curled up small. This would mean that one could appeal to the weak
anthropic principle, provided one could show that string theory does at least
allow there to be such regions of the universe – and it seems that indeed
string theory does. There may well be other regions of the universe, or other
universes (whatever that may mean), in which all the dimensions are curled
up small or in which more than four dimensions are nearly flat, but there
would be no intelligent beings in such regions to
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observe the different number of effective dimensions.
Another problem is that there are at least four different string theories (open
strings and three different closed string theories) and millions of ways in
which the extra dimensions predicted by string theory could be curled up.
Why should just one string theory and one kind of curling up be picked out?
For a time there seemed no answer, and progress got bogged down. Then,
from about 1994, people started discovering what are called dualities:
different string theories and different ways of curling up the extra
dimensions could lead to the same results in four dimensions. Moreover, as
well as particles, which occupy a single point of space, and strings, which
are lines, there were found to be other objects called p-branes, which
occupied two-dimensional or higher-dimensional volumes in space. (A
particle can be regarded as a 0-brane and a string as a 1-brane but there
were also p-branes for p=2 to p=9.) What this seems to indicate is that there
is a sort of democracy among supergravity, string, and p-brane theories:
they seem to fit together but none can be said to be more fundamental than
the others. They appear to be different approximations to some fundamental
theory that are valid in different situations.
People have searched for this underlying theory, but without any success so
far. However, I believe there may not be any single formulation of the
fundamental theory any more than, as Godel showed, one could formulate
arithmetic in terms of a single set of axioms. Instead it may be like maps –
you can’t use a single map to describe the surface of the earth or an anchor
ring: you need at least two maps in the case of the earth and four for the
anchor ring to cover every point. Each map is valid only in a limited region,
but different maps will have a region of overlap. The collection of maps
provides a complete description of the surface. Similarly, in physics it may
be necessary to use different formulations in different situations, but two
different formulations would agree in situations where they can both be
applied. The whole collection of different formulations could be regarded as
a complete unified theory, though one that could not be expressed in terms
of a single set of postulates. But can there really be such a unified theory?
Or are we perhaps just chasing a mirage? There seem to be three
possibilities:
1. There really is a complete unified theory (or a collection of overlapping
formulations), which we will someday discover if we are smart enough.
2. There is no ultimate theory of the universe, just an infinite sequence of
theories that describe the universe more and more accurately.
3. There is no theory of the universe: events cannot be predicted beyond a
certain extent but occur in a random and arbitrary manner.
Some would argue for the third possibility on the grounds that if there were
a complete set of laws, that would infringe God’s freedom to change his
mind and intervene in the world. It’s a bit like the old paradox: can God
make a stone so heavy that he can’t lift it? But the idea that God might want
to change his mind is an example of the fallacy, pointed out by St.
Augustine, of imagining God as a being existing in time: time is a property
only of the universe that God created. Presumably, he knew what he
intended when he set it up!
With the advent of quantum mechanics, we have come to recognize that
events cannot be predicted with complete accuracy but that there is always
a degree of uncertainty. If one likes, one could ascribe this randomness to
the intervention of God, but it would be a very strange kind of intervention:
there is no evidence that it is directed toward any purpose. Indeed, if it
were, it would by definition not be random. In modern times, we have
effectively removed the third possibility above by redefining the goal of
science: our aim is to formulate a set of laws that enables us to predict
events only up to the limit set by the uncertainty principle. The second
possibility, that there is an infinite sequence of more and more refined
theories, is in agreement with all our experience so far. On many occasions
we have increased the sensitivity of our measurements or made a new class
of observations, only to discover new phenomena that were not predicted
by the existing theory, and to account for these we have had to develop a
more advanced theory. It would therefore not be very surprising if the
present generation of grand unified theories was wrong in claiming that
nothing essentially new will happen between the electroweak unification
energy of about 100 GeV and the grand unification energy of about a
thousand million million GeV. We might indeed expect to find several new
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basic than the quarks and electrons that we now regard as “elementary”
particles. However, it seems that gravity may provide a limit to this
sequence of “boxes within boxes.” If one had a particle with an energy
above what is called the Planck energy, ten million million million GeV (1
followed by nineteen zeros), its mass would be so concentrated that it
would cut itself off from the rest of the universe and form a little black hole.
Thus it does seem that the sequence of more and more refined theories
should have some limit as we go to higher and higher energies, so that there
should be some ultimate theory of the universe. Of course, the Planck
energy is a very long way from the energies of around a hundred GeV,
which are the most that we can produce in the laboratory at the present
time. We shall not bridge that gap with particle accelerators in the
foreseeable future! The very early stages of the universe, however, are an
arena where such energies must have occurred. I think that there is a good
chance that the study of the early universe and the requirements of
mathematical consistency will lead us to a complete unified theory within
the lifetime of some of us who are around today, always presuming we
don’t blow ourselves up first. What would it mean if we actually did
discover the ultimate theory of the universe? As was explained in Chapter
1, we could never be quite sure that we had indeed found the correct theory,
since theories can’t be proved. But if the theory was mathematically
consistent and always gave predictions that agreed with observations, we
could be reasonably confident that it was the right one. It would bring to an
end a long and glorious chapter in the history of humanity’s intellectual
struggle to understand the universe. But it would also revolutionize the
ordinary person’s understanding of the laws that govern the universe. In
Newton’s time it was possible for an educated person to have a grasp of the
whole of human knowledge, at least in outline. But since then, the pace of
the development of science has made this impossible. Because theories are
always being changed to account for new observations, they are never
properly digested or simplified so that ordinary people can understand
them. You have to be a specialist, and even then you can only hope to have
a proper grasp of a small proportion of the scientific theories. Further, the
rate of progress is so rapid that what one learns at school or university is
always a bit out of date. Only a few people can keep up with the rapidly
advancing frontier of knowledge, and they have to devote their whole time
to it and specialize in a small area. The rest of the population has little idea
of the advances that are being made or the excitement they are generating.
Seventy years ago, if Eddington is to be believed, only two people
understood the general theory of relativity. Nowadays tens of thousands of
university graduates do, and many millions of people are at least familiar
with the idea. If a complete unified theory was discovered, it would only be
a matter of time before it was digested and simplified in the same way and
taught in schools, at least in outline. We would then all be able to have
some understanding of the laws that govern the universe and are responsible
for our existence. Even if we do discover a complete unified theory, it
would not mean that we would be able to predict events in general, for two
reasons. The first is the limitation that the uncertainty principle of quantum
mechanics sets on our powers of prediction. There is nothing we can do to
get around that. In practice, however, this first limitation is less restrictive
than the second one. It arises from the fact that we could not solve the
equations of the theory exactly, except in very simple situations. (We cannot
even solve exactly for the motion of three bodies in Newton’s theory of
gravity, and the difficulty increases with the number of bodies and the
complexity of the theory.) We already know the laws that govern the
behavior of matter under all but the most extreme conditions. In particular,
we know the basic laws that underlie all of chemistry and biology. Yet we
have certainly not reduced these subjects to the status of solved problems:
we have, as yet, had little success in predicting human behavior from
mathematical equations! So even if we do find a complete set of basic laws,
there will still be in the years ahead the intellectually challenging task of
developing better approximation methods, so that we can make useful
predictions of the probable outcomes in complicated and realistic situations.
A complete, consistent, unified theory is only the first step: our goal is a
complete understanding of the events around us, and of our own existence.
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A Brief History of Time – Stephen Hawking… Chapter 12
CHAPTER 12
CONCLUSION
We find ourselves in a bewildering world. We want to make sense of what
we see around us and to ask: What is the nature of the universe? What is
our place in it and where did it and we come from? Why is it the way it is?
To try to answer these questions we adopt some “world picture.” Just as an
infinite tower of tortoises supporting the fiat earth is such a picture, so is the
theory of superstrings. Both are theories of the universe, though the latter is
much more mathematical and precise than the former. Both theories lack
observational evidence: no one has ever seen a giant tortoise with the earth
on its back, but then, no one has seen a superstring either. However, the
tortoise theory fails to be a good scientific theory because it predicts that
people should be able to fall off the edge of the world. This has not been
found to agree with experience, unless that turns out to be the explanation
for the people who are supposed to have disappeared in the Bermuda
Triangle!
The earliest theoretical attempts to describe and explain the universe
involved the idea that events and natural phenomena were controlled by
spirits with human emotions who acted in a very humanlike and
unpredictable manner. These spirits inhabited natural objects, like rivers and
mountains, including celestial bodies, like the sun and moon. They had to
be placated and their favor sought in order to ensure the fertility of the soil
and the rotation of the seasons. Gradually, however, it must have been
noticed that there were certain regularities: the sun always rose in the east
and set in the west, whether or not a sacrifice had been made to the sun god.
Further, the sun, the moon, and the planets followed precise paths across the
sky that could be predicted in advance with considerable accuracy. The sun
and the moon might still be gods, but they were gods who obeyed strict
laws, apparently without any exceptions, if one discounts stories like that of
the sun stopping for Joshua.
At first, these regularities and laws were obvious only in astronomy and a
few other situations. However, as civilization developed, and particularly in
the last 300 years, more and more regularities and laws were discovered.
The success of these laws led Laplace at the beginning of the nineteenth
century to postulate scientific determinism; that is, he suggested that there
would be a set of laws that would determine the evolution of the universe
precisely, given its configuration at one time. Laplace’s determinism was
incomplete in two ways. It did not say how the laws should be chosen and it
did not specify the initial configuration of the universe. These were left to
God. God would choose how the universe began and what laws it obeyed,
but he would not intervene in the universe once it had started. In effect, God
was confined to the areas that nineteenth-century science did not
understand. We now know that Laplace’s hopes of determinism cannot be
realized, at least in the terms he had in mind. The uncertainty principle of
quantum mechanics implies that certain pairs of quantities, such as the
position and velocity of a particle, cannot both be predicted with complete
accuracy. Quantum mechanics deals with this situation via a class of
quantum theories in which particles don’t have well-defined positions and
velocities but are represented by a wave. These quantum theories are
deterministic in the sense that they give laws for the evolution of the wave
with time. Thus if one knows the wave at one time, one can calculate it at
any other time. The unpredictable, random element comes in only when we
try to interpret the wave in terms of the positions and velocities of particles.
But maybe that is our mistake: maybe there are no particle positions and
velocities, but only waves. It is just that we try to fit the waves to our
preconceived ideas of positions and velocities. The resulting mismatch is
the cause of the apparent unpredictability.
In effect, we have redefined the task of science to be the discovery of laws
that will enable us to predict events up to the limits set by the uncertainty
principle. The question remains, however: how or why were the laws and
the initial state of the universe chosen?
In this book I have given special prominence to the laws that govern
gravity, because it is gravity that shapes the large-scale structure of the
universe, even though it is the weakest of the four categories of forces. The
laws of gravity were incompatible with the view held until quite recently
that the universe is unchanging in time:
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the fact that gravity is always attractive implies that the universe must be
either expanding or contracting. According to the general theory of
relativity, there must have been a state of infinite density in the past, the big
bang, which would have been an effective beginning of time. Similarly, if
the whole universe recollapsed, there must be another state of infinite
density in the future, the big crunch, which would be an end of time. Even
if the whole universe did not recollapse, there would be singularities in any
localized regions that collapsed to form black holes. These singularities
would be an end of time for anyone who fell into the black hole. At the big
bang and other singularities, all the laws would have broken down, so God
would still have had complete freedom to choose what happened and how
the universe began.
When we combine quantum mechanics with general relativity, there seems
to be a new possibility that did not arise before: that space and time together
might form a finite, four-dimensional space without singularities or
boundaries, like the surface of the earth but with more dimensions. It seems
that this idea could explain many of the observed features of the universe,
such as its large-scale uniformity and also the smaller-scale departures from
homogeneity, like galaxies, stars, and even human beings. It could even
account for the arrow of time that we observe. But if the universe is
completely self-contained, with no singularities or boundaries, and
completely described by a unified theory, that has profound implications for
the role of God as Creator. Einstein once asked the question: “How much
choice did God have in constructing the universe?” If the no boundary
proposal is correct, he had no freedom at all to choose initial conditions. He
would, of course, still have had the freedom to choose the laws that the
universe obeyed. This, however, may not really have been all that much of a
choice; there may well be only one, or a small number, of complete unified
theories, such as the heterotic string theory, that are self-consistent and
allow the existence of structures as complicated as human beings who can
investigate the laws of the universe and ask about the nature of God. Even if
there is only one possible unified theory, it is just a set of rules and
equations. What is it that breathes fire into the equations and makes a
universe for them to describe? The usual approach of science of
constructing a mathematical model cannot answer the questions of why
there should be a universe for the model to describe. Why does the universe
go to all the bother of existing? Is the unified theory so compelling that it
brings about its own existence? Or does it need a creator, and, if so, does he
have any other effect on the universe? And who created him?
Up to now, most scientists have been too occupied with the development of
new theories that describe what the universe is to ask the question why. On
the other hand, the people whose business it is to ask why, the philosophers,
have not been able to keep up with the advance of scientific theories. In the
eighteenth century, philosophers considered the whole of human
knowledge, including science, to be their field and discussed questions such
as: did the universe have a beginning? However, in the nineteenth and
twentieth centuries, science became too technical and mathematical for the
philosophers, or anyone else except a few specialists. Philosophers reduced
the scope of their inquiries so much that Wittgenstein, the most famous
philosopher of this century, said, “The sole remaining task for philosophy is
the analysis of language.” What a comedown from the great tradition of
philosophy from Aristotle to Kant!
However, if we do discover a complete theory, it should in time be
understandable in broad principle by everyone, not just a few scientists.
Then we shall all, philosophers, scientists, and just ordinary people, be able
to take part in the discussion of the question of why it is that we and the
universe exist. If we find the answer to that, it would be the ultimate
triumph of human reason – for then we would know the mind of God.
ALBERT EINSTEIN
Einstein’s connection with the politics of the nuclear bomb is well known:
he signed the famous letter to President Franklin Roosevelt that persuaded
the United States to take the idea seriously, and he engaged in postwar
efforts to prevent nuclear war. But these were not just the isolated actions of
a scientist dragged into the world of politics. Einstein’s life was, in fact, to
use his own words, “divided between politics and equations.”
Einstein’s earliest political activity came during the First World War, when
he was a professor in Berlin. Sickened by what he saw as the waste of
human lives, he became involved in antiwar demonstrations. His
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advocacy of civil disobedience and public encouragement of people to
refuse conscription did little to endear him to his colleagues. Then,
following the war, he directed his efforts toward reconciliation and
improving international relations. This too did not make him popular, and
soon his politics were making it difficult for him to visit the United States,
even to give lectures.
Einstein’s second great cause was Zionism. Although he was Jewish by
descent, Einstein rejected the biblical idea of God. However, a growing
awareness of anti-Semitism, both before and during the First World War,
led him gradually to identify with the Jewish community, and later to
become an outspoken supporter of Zionism. Once more unpopularity did
not stop him from speaking his mind. His theories came under attack; an
anti-Einstein organization was even set up. One man was convicted of
inciting others to murder Einstein (and fined a mere six dollars). But
Einstein was phlegmatic. When a book was published entitled 100 Authors
Against Einstein, he retorted, “If I were wrong, then one would have been
enough!”
In 1933, Hitler came to power. Einstein was in America, and declared he
would not return to Germany. Then, while Nazi militia raided his house and
confiscated his bank account, a Berlin newspaper displayed the headline
“Good News from Einstein – He’s Not Coming Back.” In the face of the
Nazi threat, Einstein renounced pacifism, and eventually, fearing that
German scientists would build a nuclear bomb, proposed that the United
States should develop its own. But even before the first atomic bomb had
been detonated, he was publicly warning of the dangers of nuclear war and
proposing international control of nuclear weaponry. Throughout his life,
Einstein’s efforts toward peace probably achieved little that would last –
and certainly won him few friends. His vocal support of the Zionist cause,
however, was duly recognized in 1952, when he was offered the presidency
of Israel. He declined, saying he thought he was too naive in politics. But
perhaps his real reason was different: to quote him again, “Equations are
more important to me, because politics is for the present, but an equation is
something for eternity.”
GALILEO GALILEI
Galileo, perhaps more than any other single person, was responsible for the
birth of modern science. His renowned conflict with the Catholic Church
was central to his philosophy, for Galileo was one of the first to argue that
man could hope to understand how the world works, and, moreover, that we
could do this by observing the real world.
Galileo had believed Copernican theory (that the planets orbited the sun)
since early on, but it was only when he found the evidence needed to
support the idea that he started to publicly support it. He wrote about
Copernicus’s theory in Italian (not the usual academic Latin), and soon his
views became widely supported outside the universities. This annoyed the
Aristotelian professors, who united against him seeking to persuade the
Catholic Church to ban Copernicanism.
Galileo, worried by this, traveled to Rome to speak to the ecclesiastical
authorities. He argued that the Bible was not intended to tell us anything
about scientific theories, and that it was usual to assume that, where the
Bible conflicted with common sense, it was being allegorical. But the
Church was afraid of a scandal that might undermine its fight against
Protestantism, and so took repressive measures. It declared Copernicanism
“false and erroneous” in 1616, and commanded Galileo never again to
“defend or hold” the doctrine. Galileo acquiesced.
In 1623, a longtime friend of Galileo’s became the Pope. Immediately
Galileo tried to get the 1616 decree revoked. He failed, but he did manage
to get permission to write a book discussing both Aristotelian and
Copernican theories, on two conditions: he would not take sides and would
come to the conclusion that man could in any case not determine how the
world worked because God could bring about the same effects in ways
unimagined by man, who could not place restrictions on God’s
omnipotence. The book, Dialogue Concerning the Two Chief World
Systems, was completed and published in 1632, with the full backing of the
censors – and was immediately greeted throughout Europe as a literary and
philosophical masterpiece. Soon the Pope, realizing that people were seeing
the book as a convincing argument in favor of
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Copernicanism, regretted having allowed its publication. The Pope argued
that although the book had the official blessing of the censors, Galileo had
nevertheless contravened the 1616 decree. He brought Galileo before the
Inquisition, who sentenced him to house arrest for life and commanded him
to publicly renounce Copernicanism. For a second time, Galileo acquiesced.
Galileo remained a faithful Catholic, but his belief in the independence of
science had not been crushed. Four years before his death in 1642, while he
was still under house arrest, the manuscript of his second major book was
smuggled to a publisher in Holland. It was this work, referred to as Two
New Sciences, even more than his support for Copernicus, that was to be the
genesis of modern physics.
ISAAC NEWTON
Isaac Newton was not a pleasant man. His relations with other academics
were notorious, with most of his later life spent embroiled in heated
disputes. Following publication of Principia Mathematica – surely the most
influential book ever written in physics – Newton had risen rapidly into
public prominence. He was appointed president of the Royal Society and
became the first scientist ever to be knighted. Newton soon clashed with the
Astronomer Royal, John Flamsteed, who had earlier provided Newton with
much-needed data for Principia, but was now withholding information that
Newton wanted. Newton would not take no for an answer: he had himself
appointed to the governing body of the Royal Observatory and then tried to
force immediate publication of the data. Eventually he arranged for
Flamsteed’s work to be seized and prepared for publication by Flamsteed’s
mortal enemy, Edmond Halley. But Flamsteed took the case to court and, in
the nick of time, won a court order preventing distribution of the stolen
work. Newton was incensed and sought his revenge by systematically
deleting all references to Flamsteed in later editions of Principia. A more
serious dispute arose with the German philosopher Gottfried Leibniz. Both
Leibniz and Newton had independently developed a branch of mathematics
called calculus, which underlies most of modern physics. Although we now
know that Newton discovered calculus years before Leibniz, he published
his work much later. A major row ensued over who had been first, with
scientists vigorously defending both contenders. It is remarkable, however,
that most of the articles appearing in defense of Newton were originally
written by his own hand – and only published in the name of friends! As the
row grew, Leibniz made the mistake of appealing to the Royal Society to
resolve the dispute. Newton, as president, appointed an “impartial”
committee to investigate, coincidentally consisting entirely of Newton’s
friends! But that was not all: Newton then wrote the committee’s report
himself and had the Royal Society publish it, officially accusing Leibniz of
plagiarism. Still unsatisfied, he then wrote an anonymous review of the
report in the Royal Society’s own periodical. Following the death of
Leibniz, Newton is reported to have declared that he had taken great
satisfaction in “breaking Leibniz’s heart.”
During the period of these two disputes, Newton had already left
Cambridge and academe. He had been active in anti-Catholic politics at
Cambridge, and later in Parliament, and was rewarded eventually with the
lucrative post of Warden of the Royal Mint. Here he used his talents for
deviousness and vitriol in a more socially acceptable way, successfully
conducting a major campaign against counterfeiting, even sending several
men to their death on the gallows.
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A Brief History of Time – Stephen Hawking… Glossary
GLOSSARY
Absolute zero: The lowest possible temperature, at which substances
contain no heat energy. Acceleration: The rate at which the speed of an
object is changing. Anthropic principle: We see the universe the way it is
because if it were different we would not be here to observe it.
Antiparticle: Each type of matter particle has a corresponding antiparticle.
When a particle collides with its antiparticle, they annihilate, leaving only
energy.
Atom: The basic unit of ordinary matter, made up of a tiny nucleus
(consisting of protons and neutrons) surrounded by orbiting electrons.
Big bang: The singularity at the beginning of the universe.
Big crunch: The singularity at the end of the universe.
Black hole: A region of space-time from which nothing, not even light, can
escape, because gravity is so strong.
Casimir effect: The attractive pressure between two flat, parallel metal
plates placed very near to each other in a vacuum. The pressure is due to a
reduction in the usual number of virtual particles in the space between the
plates.
Chandrasekhar limit: The maximum possible mass of a stable cold star,
above which it must collapse into a black hole.
Conservation of energy: The law of science that states that energy (or its
equivalent in mass) can neither be created nor destroyed.
Coordinates: Numbers that specify the position of a point in space and
time. Cosmological constant: A mathematical device used by Einstein to
give space-time an inbuilt tendency to expand.
Cosmology: The study of the universe as a whole.
Dark matter: Matter in galaxies, clusters, and possibly between clusters,
that can not be observed directly but can be detected by its gravitational
effect. As much as 90 percent of the mass of the universe may be in the
form of dark matter.
Duality: A correspondence between apparently different theories that lead
to the same physical results. Einstein-Rosen bridge: A thin tube of spacetime linking two black holes. Also see Wormhole. Electric charge: A
property of a particle by which it may repel (or attract) other particles that
have a charge of similar (or opposite) sign.
Electromagnetic force: The force that arises between particles with electric
charge; the second strongest of the four fundamental forces.
Electron: A particle with negative electric charge that orbits the nucleus of
an atom. Electroweak unification energy: The energy (around 100 GeV)
above which the distinction between the electromagnetic force and the
weak force disappears.
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A Brief History of Time – Stephen Hawking… Glossary
Elementary particle: A particle that, it is believed, cannot be subdivided.
Event: A point in space-time, specified by its time and place.
Event horizon: The boundary of a black hole.
Exclusion principle: The idea that two identical spin-1/2 particles cannot
have (within the limits set by the uncertainty principle) both the same
position and the same velocity.
Field: Something that exists throughout space and time, as opposed to a
particle that exists at only one point at a time.
Frequency: For a wave, the number of complete cycles per second.
Gamma rays: Electromagnetic rays of very short wavelength, produced in
radio-active decay or by collisions of elementary particles.
General relativity: Einstein’s theory based on the idea that the laws of
science should be the same for all observers, no matter how they are
moving. It explains the force of gravity in terms of the curvature of a fourdimensional space-time.
Geodesic: The shortest (or longest) path between two points.
Grand unification energy: The energy above which, it is believed, the
electro-magnetic force, weak force, and strong force become
indistinguishable from each other.
Grand unified theory (GUT): A theory which unifies the electromagnetic,
strong, and weak forces. Imaginary time: Time measured using imaginary
numbers.
Light cone: A surface in space-time that marks out the possible directions
for light rays passing through a given event.
Light-second (light-year): The distance traveled by light in one second
(year). Magnetic field: The field responsible for magnetic forces, now
incorporated along with the electric field, into the electromagnetic field.
Mass: The quantity of matter in a body; its inertia, or resistance to
acceleration. Microwave background radiation: The radiation from the
glowing of the hot early universe, now so greatly red-shifted that it appears
not as light but as microwaves (radio waves with a wavelength of a few
centimeters). Also see COBE, on page 145.
Naked singularity: A space-time singularity not surrounded by a black
hole. Neutrino: An extremely light (possibly massless) particle that is
affected only by the weak force and gravity. Neutron: An uncharged
particle, very similar to the proton, which accounts for roughly half the
particles in an atomic nucleus.
Neutron star: A cold star, supported by the exclusion principle repulsion
between neutrons. No boundary condition: The idea that the universe is
finite but has no boundary (in imaginary time). Nuclear fusion: The
process by which two nuclei collide and coalesce to form a single, heavier
nucleus. Nucleus: The central part of an atom, consisting only of protons
and neutrons, held together by the strong
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A Brief History of Time – Stephen Hawking… Glossary
force.
Particle accelerator: A machine that, using electromagnets, can accelerate
moving charged particles, giving them more energy.
Phase: For a wave, the position in its cycle at a specified time: a measure of
whether it is at a crest, a trough, or somewhere in between.
Photon: A quantum of light.
Planck’s quantum principle: The idea that light (or any other classical
waves) can be emitted or absorbed only in discrete quanta, whose energy is
proportional to their wavelength. Positron: The (positively charged)
antiparticle of the electron. Primordial black hole: A black hole created in
the very early universe. Proportional: ‘X is proportional to Y’ means that
when Y is multiplied by any number, so is X. ‘X is inversely proportional to
Y’ means that when Y is multiplied by any number, X is divided by that
number. Proton: A positively charged particle, very similar to the neutron,
that accounts for roughly half the particles in the nucleus of most atoms.
Pulsar: A rotating neutron star that emits regular pulses of radio waves.
Quantum: The indivisible unit in which waves may be emitted or
absorbed. Quantum chromodynamics (QCD): The theory that describes
the interactions of quarks and gluons. Quantum mechanics: The theory
developed from Planck’s quantum principle and Heisenberg’s uncertainty
principle.
Quark: A (charged) elementary particle that feels the strong force. Protons
and neutrons are each composed of three quarks.
Radar: A system using pulsed radio waves to detect the position of objects
by measuring the time it takes a single pulse to reach the object and be
reflected back.
Radioactivity: The spontaneous breakdown of one type of atomic nucleus
into another. Red shift: The reddening of light from a star that is moving
away from us, due to the Doppler effect. Singularity: A point in space-time
at which the space-time curvature becomes infinite. Singularity theorem:
A theorem that shows that a singularity must exist under certain
circumstances – in particular, that the universe must have started with a
singularity.
Space-time: The four-dimensional space whose points are events. Spatial
dimension: Any of the three dimensions that are spacelike – that is, any
except the time dimension. Special relativity: Einstein’s theory based on
the idea that the laws of science should be the same for all observers, no
matter how they are moving, in the absence of gravitational phenomena.
Spectrum: The component frequencies that make up a wave. The visible
part of the sun’s spectrum can be seen in a rainbow.
Spin: An internal property of elementary particles, related to, but not
identical to, the everyday concept of spin.
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Stationary state: One that is not changing with time: a sphere spinning at a
constant rate is stationary because it looks identical at any given instant.
String theory: A theory of physics in which particles are described as
waves on strings. Strings have length but no other dimension.
Strong force: The strongest of the four fundamental forces, with the
shortest range of all. It holds the quarks together within protons and
neutrons, and holds the protons and neutrons together to form atoms.
Uncertainty principle: The principle, formulated by Heisenberg, that one
can never be exactly sure of both the position and the velocity of a particle;
the more accurately one knows the one, the less accurately one can know
the other.
Virtual particle: In quantum mechanics, a particle that can never be
directly detected, but whose existence does have measurable effects.
Wave/particle duality: The concept in quantum mechanics that there is no
distinction between waves and particles; particles may sometimes behave
like waves, and waves like particles. Wavelength: For a wave, the distance
between two adjacent troughs or two adjacent crests. Weak force: The
second weakest of the four fundamental forces, with a very short range. It
affects all matter particles, but not force-carrying particles.
Weight: The force exerted on a body by a gravitational field. It is
proportional to, but not the same as, its mass. White dwarf: A stable cold
star, supported by the exclusion principle repulsion between electrons.
Wormhole: A thin tube of space-time connecting distant regions of the
universe. Wormholes might also link to parallel or baby universes and could
provide the possibility of time travel.
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A Brief History of Time – Stephen Hawking… Acknowledgments
ACKNOWLEDGMENTS
Many people have helped me in writing this book. My scientific colleagues
have without exception been inspiring. Over the years my principal
associates and collaborators were Roger Penrose, Robert Geroch, Brandon
Carter, George Ellis, Gary Gibbons, Don Page, and Jim Hartle. I owe a lot
to them, and to my research students, who have always given me help when
needed.
One of my students, Brian Whitt, gave me a lot of help writing the first
edition of this book. My editor at Bantam Books, Peter Guzzardi, made
innumerable comments which improved the book considerably. In addition,
for this edition, I would like to thank Andrew Dunn, who helped me revise
the text. I could not have written this book without my communication
system. The software, called Equalizer, was donated by Walt Waltosz of
Words Plus Inc., in Lancaster, California. My speech synthesizer was
donated by Speech Plus, of Sunnyvale, California. The synthesizer and
laptop computer were mounted on my wheelchair by David Mason, of
Cambridge Adaptive Communication Ltd. With this system I can
communicate better now than before I lost my voice.
I have had a number of secretaries and assistants over the years in which I
wrote and revised this book. On the secretarial side, I’m very grateful to
Judy Fella, Ann Ralph, Laura Gentry, Cheryl Billington, and Sue Masey.
My assistants have been Colin Williams, David Thomas, and Raymond
Laflamme, Nick Phillips, Andrew Dunn, Stuart Jamieson, Jonathan
Brenchley, Tim Hunt, Simon Gill, Jon Rogers, and Tom Kendall. They, my
nurses, colleagues, friends, and family have enabled me to live a very full
life and to pursue my research despite my disability.
Stephen Hawking
ABOUT THE AUTHOR
Stephen Hawking, who was born in 1942 on the anniversary of Galileo’s
death, holds Isaac Newton’s chair as Lucasian Professor of Mathematics at
the University of Cambridge. Widely regarded as the most brilliant
theoretical physicist since Einstein, he is also the author of Black Holes and
Baby Universes, published in 1993, as well as numerous scientific papers
and books.