Stephen Hawking, science''s first real rock star, may be the
least-read bestselling author in history--it''s no secret that many
people who own A Brief History of Time have never finished it.
Hawking''s The Universe in a Nutshell aims to remedy the situation,
with a plethora of friendly illustrations to help readers grok some
of the most brain-bending ideas ever conceived.
Does it succeed? Yes and no. While Hawking offers genuinely
accessible context for such complexities as string theory an
關於作者：
Stephen Hawking is Lucasian Professor of Mathematics at the
University of Cambridge and is regarded as one of the most
brilliant theoretical physicists since Einstein.
內容試閱：
CHAPTER 2
THE SHAPE OF TIME
EINSTEIN?S GENERAL RELATIVITY GIVES TIME A SHAPE.
HOW THIS CAN BE RECONCILED WITH QUANTUM THEORY.
What is time? Is it an ever-rolling stream that bears all our
dreams away, as the old hymn says? Or is it a railroad track? Maybe
it has loops and branches, so you can keep going forward and yet
return to an earlier station on the line.
The nineteenth-century author Charles Lamb wrote: ?Nothing
puzzles me like time and space. And yet nothing troubles me less
than time and space, because I never think of them.? Most of us
don?t worry about time and space most of the time, whatever that
may be; but we all do wonder sometimes what time is, how it began,
and where it is leading us.
Any sound scientific theory, whether of time or of any other
concept, should in my opinion be based on the most workable
philosophy of science: the positivist approach put forward by Karl
Popper and others. According to this way of thinking, a scientific
theory is a mathematical model that describes and codifies the
observations we make. A good theory will describe a large range of
phenomena on the basis of a few simple postulates and will make
definite predictions that can be tested. If the predictions agree
with the observations, the theory survives that test, though it can
never be proved to be correct.
On the other hand, if the observations disagree with the
predictions, one has to discard or modify the theory. At least,
that is what is supposed to happen. In practice, people often
question the accuracy of the observations and the reliability and
moral character of those making the observations. If one takes the
positivist position, as I do, one cannot say what time actually is.
All one can do is describe what has been found to be a very good
mathematical model for time and say what predictions it
makes.
Isaac Newton gave us the first mathematical model for time and
space in his PRINCIPIA MATHEMATICA, published in 1687. Newton
occupied the Lucasian chair at Cambridge that I now hold, though it
wasn?t electrically operated in his time. In Newton?s model, time
and space were a background in which events took place but which
weren?t affected by them. Time was separate from space and was
considered to be a single line, or railroad track, that was
infinite in both directions. Time itself was considered eternal, in
the sense that it had existed, and would exist, forever.
By contrast, most people thought the physical universe had been
created more or less in its present state only a few thousand years
ago. This worried philosophers such as the German thinker Immanuel
Kant. If the universe had indeed been created, why had there been
an infinite wait before the creation? On the other hand, if the
universe had existed forever, why hadn?t everything that was going
to happen already happened, meaning that history was over? In
particular, why hadn?t the universe reached thermal equilibrium,
with everything at the same temperature?
Kant called this problem an ?antimony of pure reason,? because it
seemed to be a logical contradiction; it didn?t have a resolution.
But it was a contradiction only within the context of the Newtonian
mathematical model, in which time was an infinite line, independent
of what was happening in the universe. However, as we saw in
Chapter 1, in 1915 a completely new mathematical model was put
forward by Einstein: the general theory of relativity. In the years
since Einstein?s paper, we have added a few ribbons and bows, but
our model of time and space is still based on what Einstein
proposed. This and the following chapters will describe how our
ideas have developed in the years since Einstein?s revolutionary
paper. It has been a success story of the work of a large number of
people, and I?m proud to have made a small contribution.
General relativity combines the time dimension with the three
dimensions of space to form what is called spacetime. The theory
incorporates the effect of gravity by saying that the distribution
of matter and energy in the universe warps and distorts spacetime,
so that it is not flat. Objects in this spacetime try to move in
straight lines, but because spacetime is curved, their paths appear
bent. They move as if affected by a gravitational field.
As a rough analogy, not to be taken too literally, imagine a
sheet of rubber. One can place a large ball on the sheet to
represent the Sun. The weight of the ball will depress the sheet
and cause it to be curved near the Sun. If one now rolls little
ball bearings on the sheet, they won?t roll straight across to the
other side but instead will go around the heavy weight, like
planets orbiting the Sun.
The analogy is incomplete because in it only a two-dimensional
section of space the surface of the rubber sheet is curved, and
time is left undisturbed, as it is in Newtonian theory. However, in
the theory of relativity, which agrees with a large number of
experiments, time and space are inextricably tangled up. One cannot
curve space without involving time as well. Thus time has a shape.
By curving space and time, general relativity changes them from
being a passive background against which events take place to being
active, dynamic participants in what happens. In Newtonian theory,
where time existed independently of anything else, one could ask:
What did God do before He created the universe? As Saint Augustine
said, one should not joke about this, as did a man who said, ?He
was preparing Hell for those who pry too deep.? It is a serious
question that people have pondered down the ages. According to
Saint Augustine, before God made heaven and earth, He did not make
anything at all. In fact, this is very close to modern ideas.
In general relativity, on the other hand, time and space do not
exist independently of the universe or of each other. They are
defined by measurements within the universe, such as the number of
vibrations of a quartz crystal in a clock or the length of a ruler.
It is quite conceivable that time defined in this way, within the
universe, should have a minimum or maximum value?in other words, a
beginning or an end. It would make no sense to ask what happened
before the beginning or after the end, because such times would not
be defined.
It was clearly important to decide whether the mathematical model
of general relativity predicted that the universe, and time itself,
should have a beginning or end. The general prejudice among
theoretical physicists, including Einstein, held that time should
be infinite in both directions. Otherwise, there were awkward
questions about the creation of the universe, which seemed to be
outside the realm of science. Solutions of the Einstein equations
were known in which time had a beginning or end, but these were all
very special, with a large amount of symmetry. It was thought that
in a real body, collapsing under its own gravity, pressure or
sideways velocities would prevent all the matter falling together
to the same point, where the density would be infinite. Similarly,
if one traced the expansion of the universe back in time, one would
find that the matter of the universe didn?t all emerge from a point
of infinite density. Such a point of infinite density was called a
singularity and would be a beginning or an end of time.
In 1963, two Russian scientists, Evgenii Lifshitz and Isaac
Khalatnikov, claimed to have proved that solutions of the Einstein
equations with a singularity all had a special arrangement of
matter and velocities. The chances that the solution representing
the universe would have this special arrangement were practically
zero. Almost all solutions that could represent the universe would
avoid having a singularity of infinite density:
Before the era during which the universe has been expanding,
there must have been a previous contracting phase during which
matter fell together but missed colliding with itself, moving apart
again in the present expanding phase. If this were the case, time
would continue on forever, from the infinite past to the infinite
future.
Not everyone was convinced by the arguments of Lifshitz and
Khalatnikov. Instead, Roger Penrose and I adopted a different
approach, based not on a detailed study of solutions but on the
global structure of spacetime. In general relativity, spacetime is
curved not only by massive objects in it but also by the energy in
it. Energy is always positive, so it gives spacetime a curvature
that bends the paths of light rays toward each other.
Now consider our past light cone, that is, the paths through
spacetime of the light rays from distant galaxies that reach us at
the present time. In a diagram with time plotted upward and space
plotted sideways, this is a cone with its vertex, or point, at
us.
As we go toward the past, down the cone from the vertex, we see
galaxies at earlier and earlier times. Because the universe has
been expanding and everything used to be much closer together, as
we look back further we are looking back through regions of higher
matter density. We observe a faint background of microwave
radiation that propagates to us along our past light cone from a
much earlier time, when the universe was much denser and hotter
than it is now. By tuning receivers to different frequencies of
microwaves, we can measure the spectrum the distribution of power
arranged by frequency of this radiation. We find a spectrum that
is characteristic of radiation from a body at a temperature of 2.7
degrees above absolute zero. This microwave radiation is not much
good for defrosting frozen pizza, but the fact that the spectrum
agrees so exactly with that of radiation from a body at 2.7 degrees
tells us that the radiation must have come from regions that are
opaque to microwaves.
Thus we can conclude that our past light cone must pass through a
certain amount of matter as one follows it back. This amount of
matter is enough to curve spacetime, so the light rays in our past
light cone are bent back toward each other.
As one goes back in time, the cross sections of our past light
cone reach a maximum size and begin to get smaller again. Our past
is pear-shaped.
As one follows our past light cone back still further, the
positive energy density of matter causes the light rays to bend
toward each other more strongly. The cross section of the light
cone will shrink to zero size in a finite time. This means that all
the matter inside our past light cone is trapped in a region whose
boundary shrinks to zero. It is therefore not very surprising that
Penrose and I could prove that in the mathematical model of general
relativity, time must have a beginning in what is called the big
bang. Similar arguments show that time would have an end, when
stars or galaxies collapse under their own gravity to form black
holes. We had sidestepped Kant?s antimony of pure reason by
dropping his implicit assumption that time had a meaning
independent of the universe. Our paper, proving time had a
beginning, won the second prize in the competition sponsored by the
Gravity Research Foundation in 1968, and Roger and I shared the
princely sum of $300. I don?t think the other prize essays that
year have shown much enduring value.
There were various reactions to our work. It upset many
physicists, but it delighted those religious leaders who believed
in an act of creation, for here was scientific proof. Meanwhile,
Lifshitz and Khalatnikov were in an awkward position. They couldn?t
argue with the mathematical theorems that we had proved, but under
the Soviet system they couldn?t admit they had been wrong and
Western science had been right. However, they saved the situation
by finding a more general family of solutions with a singularity,
which weren?t special in the way their previous solutions had been.
This enabled them to claim singularities, and the beginning or end
of time, as a Soviet discovery.
Most physicists still instinctively disliked the idea of time
having a beginning or end. They therefore pointed out that the
mathematical model might not be expected to be a good description
of spacetime near a singularity. The reason is that general
relativity, which describes the gravitational force, is a classical
theory, as noted in Chapter 1, and does not incorporate the
uncertainty of quantum theory that governs all other forces we
know.
This inconsistency does not matter in most of the universe most
of the time, because the scale on which spacetime is curved is very
large and the scale on which quantum effects are important is very
small. But near a singularity, the two scales would be comparable,
and quantum gravitational effects would be important. So what the
singularity theorems of Penrose and myself really established is
that our classical region of spacetime is bounded to the past, and
possibly to the future, by regions in which quantum gravity is
important. To understand the origin and fate of the universe, we
need a quantum theory of gravity, and this will be the subject of
most of this book.
Quantum theories of systems such as atoms, with a finite number
of particles, were formulated in the 1920s, by Heisenberg,
Schr?dinger, and Dirac. Dirac was another previous holder of my
chair in Cambridge, but it still wasn?t motorized. However, people
encountered difficulties when they tried to extend quantum ideas to
the Maxwell field, which describes electricity, magnetism, and
light.
One can think of the Maxwell field as being made up of waves of
different wavelengths the distance between one wave crest and the
next. In a wave, the field will swing from one value to another
like a pendulum.
According to quantum theory, the ground state, or lowest energy
state, of a pendulum is not just sitting at the lowest energy
point, pointing straight down. That would have both a definite
position and a definite velocity, zero. This would be a violation
of the uncertainty principle, which forbids the precise measurement
of both position and velocity at the same time. The uncertainty in
the position multiplied by the uncertainty in the momentum must be
greater than a certain quantity, known as Planck?s constant?a
number that is too long to keep writing down, so we use a symbol
for it:
So the ground state, or lowest energy state, of a pendulum does
not have zero energy, as one might expect. Instead, even in its
ground state a pendulum or any oscillating system must have a
certain minimum amount of what are called zero point fluctuations.
These mean that the pendulum won?t necessarily be pointing straight
down but will also have a probability of being found at a small
angle to the vertical. Similarly, even in the vacuum or lowest
energy state, the waves in the Maxwell field won?t be exactly zero
but can have small sizes. The higher the frequency the number of
swings per minute of the pendulum or wave, the higher the energy
of the ground state.
Calculations of the ground state fluctuations in the Maxwell and
electron fields made the apparent mass and charge of the electron
infinite, which is not what observations show. However, in the
1940s the physicists Richard Feynman, Julian Schwinger, and
Shin?ichiro Tomonaga developed a consistent way of removing or
?subtracting out? these infinities and dealing only with the finite
observed values of the mass and charge. Nevertheless, the ground
state fluctuations still caused small effects that could be
measured and that agreed well with experiment. Similar subtraction
schemes for removing infinities worked for the Yang-Mills field in
the theory put forward by Chen Ning Yang and Robert Mills.
Yang-Mills theory is an extension of Maxwell theory that describes
interactions in two other forces called the weak and strong nuclear
forces. However, ground state fluctuations have a much more serious
effect in a quantum theory of gravity. Again, each wavelength would
have a ground state energy. Since there is no limit to how short
the wavelengths of the Maxwell field can be, there are an infinite
number of different wavelengths in any region of spacetime and an
infinite amount of ground state energy.
Because energy density is, like matter, a source of gravity, this
infinite energy density ought to mean there is enough gravitational
attraction in the universe to curl spacetime into a single point,
which obviously hasn?t happened.
One might hope to solve the problem of this seeming contradiction
between observation and theory by saying that the ground state
fluctuations have no gravitational effect, but this would not work.
One can detect the energy of ground state fluctuations by the
Casimir effect. If you place a pair of metal plates parallel to
each other and close together, the effect of the plates is to
reduce slightly the number of wavelengths that fit between the
plates relative to the number outside. This means that the energy
density of ground state fluctuations between the plates, although
still infinite, is less than the energy density outside by a finite
amount. This difference in energy density gives rise to a force
pulling the plates together, and this force has been observed
experimentally.
Forces are a source of gravity in general relativity, just as
matter is, so it would not be consistent to ignore the
gravitational effect of this energy difference.
Another possible solution to the problem might be to suppose
there was a cosmological constant such as Einstein introduced in an
attempt to have a static model of the universe. If this constant
had an infinite negative value, it could exactly cancel the
infinite positive value of the ground state energies in free space,
but this cosmological constant seems very ad hoc, and it would have
to be tuned to extraordinary accuracy.
Fortunately, a totally new kind of symmetry was discovered in the
1970s that provides a natural physical mechanism to cancel the
infinities arising from ground state fluctuations.
Supersymmetry is a feature of our modern mathematical models that
can be described in various ways. One way is to say that spacetime
has extra dimensions besides the dimensions we experience. These
are called Grassmann dimensions, because they are measured in
numbers known as Grassmann variables rather than in ordinary real
numbers. Ordinary numbers commute; that is, it does not matter in
which order you multiply them: 6 times 4 is the same as 4 times 6.
But Grassmann variables anticommute: x times y is the same as ?y
times x.
Supersymmetry was first considered for removing infinities in
matter fields and Yang-Mills fields in a spacetime where both the
ordinary number dimensions and the Grassmann dimensions were flat,
not curved. But it was natural to extend it to ordinary numbers and
Grassmann dimensions that were curved. This led to a number of
theories called supergravity, with different amounts of
supersymmetry. One consequence of supersymmetry is that every field
or particle should have a ?superpartner? with a spin that is either
12 greater than its own or 12 less.
The ground state energies of bosons, fields whose spin is a whole
number 0, 1, 2 , etc., are positive. On the other hand, the
ground state energies of fermions, fields whose spin is a half
number 12, 32 , etc., are negative. Because there are equal
numbers of bosons and fermions, the biggest infinities cancel in
supergravity theories.
There remained the possibility that there might be smaller but
still infinite quantities left over. No one had the patience needed
to calculate whether these theories were actually completely
finite. It was reckoned it would take a good student two hundred
years, and how would you know he hadn?t made a mistake on the
second page? Still, up to 1985, most people believed that most
supersymmetric supergravity theories would be free of
infinities.
Then suddenly the fashion changed. People declared there was no
reason not to expect infinities in supergravity theories, and this
was taken to mean they were fatally flawed as theories. Instead, it
was claimed that a theory named supersymmetric string theory was
the only way to combine gravity with quantum theory. Strings, like
their namesakes in everyday experience, are one-dimensional
extended objects. They have only length. Strings in string theory
move through a background spacetime. Ripples on the string are
interpreted as particles.
If the strings have Grassmann dimensions as well as their
ordinary number dimensions, the ripples will correspond to bosons
and fermions. In this case, the positive and negative ground state
energies will cancel so exactly that there will be no infinities
even of the smaller sort. Superstrings, it was claimed, were the
TOE, the Theory of Everything.
Historians of science in the future will find it interesting to
chart the changing tide of opinion among theoretical physicists.
For a few years, strings reigned supreme and supergravity was
dismissed as just an approximate theory, valid at low energy. The
qualification ?low energy? was considered particularly damning,
even though in this context low energies meant particles with
energies of less than a billion billion times those of particles in
a TNT explosion. If supergravity was only a low energy
approximation, it could not claim to be the fundamental theory of
the universe. Instead, the underlying theory was supposed to be one
of five possible superstring theories. But which of the five string
theories described our universe? And how could string theory be
formulated, beyond the approximation in which strings were pictured
as surfaces with one space dimension and one time dimension moving
through a flat background spacetime? Wouldn?t the strings curve the
background spacetime?
In the years after 1985, it gradually became apparent that string
theory wasn?t the complete picture. To start with, it was realized
that strings are just one member of a wide class of objects that
can be extended in more than one dimension. Paul Townsend, who,
like me, is a member of the Department of Applied Mathematics and
Theoretical Physics at Cambridge, and who did much of the
fundamental work on these objects, gave them the name ?p-branes.? A
p-brane has length in p directions. Thus a p=1 brane is a string, a
p=2 brane is a surface or membrane, and so on.There seems no reason
to favor the p=1 string case over other possible values of p.
Instead, we should adopt the principle of p-brane democracy: all
p-branes are created equal.
All the p-branes could be found as solutions of the equations of
supergravity theories in 10 or 11 dimensions. While 10 or 11
dimensions doesn?t sound much like the spacetime we experience, the
idea was that the other 6 or 7 dimensions are curled up so small
that we don?t notice them; we are only aware of the remaining 4
large and nearly flat dimensions.
I must say that personally, I have been reluctant to believe in
extra dimensions. But as I am a positivist, the question ?Do extra
dimensions really exist?? has no meaning. All one can ask is
whether mathematical models with extra dimensions provide a good
description of the universe. We do not yet have any observations
that require extra dimensions for their explanation. However, there
is a possibility we may observe them in the Large Hadron Collider
in Geneva. But what has convinced many people, including myself,
that one should take models with extra dimensions seriously is that
there is a web of unexpected relationships, called dualities,
between the models. These dualities show that the models are all
essentially equivalent; that is, they are just different aspects of
the same underlying theory, which has been given the name M-theory.
Not to take this web of dualities as a sign we are on the right
track would be a bit like believing that God put fossils into the
rocks in order to mislead Darwin about the evolution of life.
These dualities show that the five superstring theories all
describe the same physics and that they are also physically
equivalent to supergravity. One cannot say that superstrings are
more fundamental than supergravity, or vice versa. Rather, they are
different expressions of the same underlying theory, each useful
for calculations in different kinds of situations. Because string
theories don?t have any infinities, they are good for calculating
what happens when a few high energy particles collide and scatter
off each other. However, they are not of much use for describing
how the energy of a very large number of particles curves the
universe or forms a bound state, like a black hole. For these
situations, one needs supergravity, which is basically Einstein?s
theory of curved spacetime with some extra kinds of matter. It is
this picture that I shall mainly use in what follows.
To describe how quantum theory shapes time and space, it is
helpful to introduce the idea of imaginary time. Imaginary time
sounds like something from science fiction, but it is a
well-defined mathematical concept: time measured in what are called
imaginary numbers. One can think of ordinary real numbers such as
1, 2, -3.5, and so on as corresponding to positions on a line
stretching from left to right: zero in the middle, positive real
numbers on the right, and negative real numbers on the left.
Imaginary numbers can then be represented as corresponding to
positions on a vertical line: zero is again in the middle, positive
imaginary numbers plotted upward, and negative imaginary numbers
plotted downward. Thus imaginary numbers can be thought of as a new
kind of number at right angles to ordinary real numbers. Because
they are a mathematical construct, they don?t need a physical
realization; one can?t have an imaginary number of oranges or an
imaginary credit card bill.
One might think this means that imaginary numbers are just a
mathematical game having nothing to do with the real world. From
the viewpoint of positivist philosophy, however, one cannot
determine what is real. All one can do is find which mathematical
models describe the universe we live in. It turns out that a
mathematical model involving imaginary time predicts not only
effects we have already observed but also effects we have not been
able to measure yet nevertheless believe in for other reasons. So
what is real and what is imaginary? Is the distinction just in our
minds?
Einstein?s classical i.e., nonquantum general theory of
relativity combined real time and the three dimensions of space
into a four-dimensional spacetime. But the real time direction was
distinguished from the three spatial directions; the world line or
history of an observer always increased in the real time direction
that is, time always moved from past to future, but it could
increase or decrease in any of the three spatial directions. In
other words, one could reverse direction in space, but not in
time.
On the other hand, because imaginary time is at right angles to
real time, it behaves like a fourth spatial direction. It can
therefore have a much richer range of possibilities than the
railroad track of ordinary real time, which can only have a
beginning or an end or go around in circles. It is in this
imaginary sense that time has a shape.
To see some of the possibilities, consider an imaginary time
spacetime that is a sphere, like the surface of the Earth. Suppose
that imaginary time was degrees of latitude. Then the history of
the universe in imaginary time would begin at the South Pole. It
would make no sense to ask, ?What happened before the beginning??
Such times are simply not defined, any more than there are points
south of the South Pole. The South Pole is a perfectly regular
point of the Earth?s surface, and the same laws hold there as at
other points. This suggests that the beginning of the universe in
imaginary time can be a regular point of spacetime, and that the
same laws can hold at the beginning as in the rest of the universe.
The quantum origin and evolution of the universe will be discussed
in the next chapter.
Another possible behavior is illustrated by taking imaginary time
to be degrees of longitude on the Earth. All the lines of longitude
meet at the North and South Poles. Thus time stands still there, in
the sense that an increase of imaginary time, or of degrees of
longitude, leaves one in the same spot. This is very similar to the
way that ordinary time appears to stand still on the horizon of a
black hole. We have come to recognize that this standing still of
real and imaginary time either both stand still or neither does
means that the spacetime has a temperature, as I discovered for
black holes.
Not only does a black hole have a temperature, it also behaves as
if it has a quantity called entropy. The entropy is a measure of
the number of internal states ways it could be configured on the
inside that the black hole could have without looking any
different to an outside observer, who can only observe its mass,
rotation, and charge. This black hole entropy is given by a very
simple formula I discovered in 1974. It equals the area of the
horizon of the black hole: there is one bit of information about
the internal state of the black hole for each fundamental unit of
area of the horizon. This shows that there is a deep connection
between quantum gravity and thermodynamics, the science of heat
which includes the study of entropy. It also suggests that
quantum gravity may exhibit what is called holography.
Information about the quantum states in a region of spacetime may
be somehow coded on the boundary of the region, which has two
dimensions less. This is like the way that a hologram carries a
three-dimensional image on a two-dimensional surface. If quantum
gravity incorporates the holographic principle, it may mean that we
can keep track of what is inside black holes. This is essential if
we are to be able to predict the radiation that comes out of black
holes. If we can?t do that, we won?t be able to predict the future
as fully as we thought. This is discussed in Chapter 4. Holography
is discussed again in Chapter 7. It seems we may live on a
3-brane?a four-dimensional three space plus one time surface that
is the boundary of a five-dimensional region, with the remaining
dimensions curled up very small. The state of the world on a brane
encodes what is happening in the five-dimensional region.
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