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
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
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
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
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, Schrvdinger, 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
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.
Excerpted from "The Universe in a Nutshell" by Stephen William Hawking. Copyright © 2001 by Stephen William Hawking. Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher. Excerpts are provided solely for the personal use of visitors to this web site.