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.
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 end our work
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 small-scale 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
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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,
