By the end of 1974, Hawking’s work on black holes had shown that, using the general theory of relativity alone, the equations said that the surface area of a black hole could not shrink—but adding in the quantum rules to the equations revealed that they could not only shrink but would eventually disappear in a puff of gamma radiation. His earlier work with Penrose had shown that, using the general theory of relativity alone, the equations said that the Universe must have been born out of a singularity, a point of infinite density and zero volume, at a time some 15 billion years ago. It was natural that the next scientific question Hawking asked himself was what would happen to this prediction if the quantum rules were added to that set of equations.

This was no easy question to answer. Physicists had been trying to combine quantum theory and relativity theory into one complete, unified theory ever since the quantum revolution in the 1920s; Einstein himself spent the last twenty years of his working life on the problem and failed to come up with a solution. Indeed, a full theory of quantum gravity still eludes the mathematicians. But by

restricting himself to the specific puzzle of how relativity and quantum mechanics interacted at the beginning of time, Hawking was able to make progress, to such an extent that by the early 1980s he was posing the question of whether there ever had been a beginning to time at all. To understand how he arrived at this startling hypothesis, we have to look again at the quantum theory, in a variation developed by the great American physicist Richard Feynman. It is known as the “sum-over-histories” or “path integral” approach.

The essential features of quantum mechanics are demonstrated most clearly in what is known as “the experiment with two holes.” In such an experiment, a beam of light, or a stream of electrons, is directed through two small holes in a wall and on to a screen on the other side. The version using light is known as Young’s experiment and may be familiar from school physics. What happens is that the pattern of light on the screen forms a characteristic arrangement of dark and light stripes, caused when the electromagnetic waves passing through each of the holes interfere with each other. Where the two sets of waves add together, there is a bright stripe; where they cancel each other out, the screen is dark.

This interference is easy to understand in terms of waves. You can get exactly the same effect by making waves in a tank of water and letting them pass through two slits in a barrier. But it is much harder to understand how electrons, which we are used to thinking of as hard particles like tiny snooker balls, can behave in the same way. Yet they do.

What is even stranger is that the same pattern of dark and light stripes slowly builds up on the screen (which can be almost exactly the same as a TV screen) when electrons are fired through the holes one at a time. Why should this be strange? Think about what happens when electrons are fired through just one hole. Instead of a striped pattern on the screen, there is just a bright patch behind the hole. This is indeed what we see if we block off either of the two

holes and fire the electrons through. “Obviously,” each electron can go through only one hole. But when both holes are open, even with electrons fired one at a time through the experiment, we do not see just two patches of brightness behind the holes, but the characteristic stripy pattern of Young’s experiment.

This is the clearest example of the wave-particle duality (see Chapter 2) that lies at the heart of the quantum world. When each electron arrives at the screen, it makes a pinpoint of light, just as you would expect from the arrival of a tiny “snooker ball” particle. But when thousands of those points of light are added together, they produce the striped pattern corresponding to a wave passing through both holes at once. It is as if each individual electron is a wave that passes through both holes simultaneously, interferes with itself, decides which bit of the striped pattern it belongs in, and heads off there to arrive as a particle that makes a pinpoint of light.

Don’t worry if you find this incomprehensible. Niels Bohr, one of the physicists who pioneered the quantum revolution, used to say that “anyone who is not shocked by quantum theory has not understood it,” while Feynman, probably the greatest theoretical physicist since the Second World War, went even further and was fond of saying that nobody understands quantum mechanics. The important thing is not to understand how such a strange behavior as wave-particle duality can occur, but to find a set of equations that describe what is going on and make it possible for physicists to predict how electrons, light waves, and the rest will behave. The sum-over-histories approach was Feynman’s contribution to this more pragmatic form of “understanding” at the quantum level, and in the late 1970s Hawking applied it to the study of the Big Bang.

Feynman said that, instead of thinking of an object such as an electron as a simple particle that follows a single route from A to B (for example, through one of the two holes in Young’s experiment), we have to regard it as following every possible path from A to B

through space-time. It would be easier for a “classical” particle to follow some paths (some “histories”) than others, and this is allowed for in Feynman’s equations by assigning each path a probability, which can be calculated from the quantum rules.

These probabilities can interfere with the probabilities from neighboring “world lines,” as they are called, rather like the way ripples on the surface of a pond interfere with one another. The actual path followed by the particle is then calculated by adding together all the probabilities for individual paths (which is why this is also known as the path integral approach).

In the vast majority of cases, the various probabilities cancel each other out almost entirely, leaving just a few paths, or trajectories, that are reinforced. This is what happens for the trajectories corresponding to an electron moving near the nucleus of an atom. The electron is not allowed to go just anywhere because of the way the probabilities cancel. It is only allowed to move in one of the few orbits around the nucleus where the probabilities reinforce one another.

The experiment with two holes is unusual because it offers the electrons a choice of two equally probable sets of trajectories, one through each hole, and this is why the basic strangeness of the quantum world shows up so clearly in this example. Only Hawking, though, had the chutzpah to apply the path integral approach to calculating the history, not of an individual electron but of the entire Universe; but even he started out in a smaller way, with black hole singularities.

When a black hole evaporates, what happens to the singularity inside it? One simple guess might be that in the final stages of the evaporation the horizon around the hole vanishes, leaving behind the naked singularity that nature is supposed to abhor. In fact, though, the equations developed by Hawking in the early 1970s to

describe exploding black holes could not be pushed to such extremes. Strictly speaking, they could only be applied if the mass of the black hole were still a reasonable fraction of a gram—almost big enough to be weighed on your kitchen scales. The best guess that Hawking, or anyone else, could make in 1974 was that when a black hole has evaporated to this point it would completely disappear, taking the singularity with it. But this was only a guess, based on some general quantum principles.

These principles are aspects of the basic uncertainty principle. Just as there is a fundamental uncertainty about the energy content of the vacuum, so there is a fundamental uncertainty about basic measures such as length and time. The size of these uncertainties is determined by Planck’s constant, which gives us basic “quanta” known as the Planck length and the Planck time.

Both are very small. The Planck length, for example, is 10-35 of a meter, far smaller than the nucleus of an atom. According to the quantum rules, not only is it impossible in principle ever to measure any length more accurately than this (we should be so lucky!), but also there is no meaning to the concept of a length shorter than the Planck length. So if an evaporating black hole were to shrink to the point where it was just one Planck length in diameter, it could not shrink any more. If it lost more energy, it could only disappear entirely. The quantum of time is, similarly, the smallest interval of time that has any meaning. This Planck time is a mere 10-43 of a second, and there is no such thing as a shorter interval of time. (Don’t worry about the exact size of these numbers; what matters is that, although they are exceedingly small, they are not zero.) Quantum theory tells us that we can neither shrink away a black hole to a mathematical point nor look back in time literally to the moment when time “began.” Even if we pushed the Big Bang model to its most extreme limit, we would have to envisage the Universe being created with an “age” equal to the Planck time.

In both cases, quantum mechanics seems to remove the troublesome singularities. If there is no meaning to the concept of a volume with a diameter less than the Planck length, then there is no meaning to the concept of a point of zero volume and infinite density. Quantum theory is telling us that, although the densities reached inside black holes, and at the birth of the Universe, may be staggeringly high by any human measure, they are not infinite. And if the infinities and singularities can be removed, there is at least a hope of finding a set of equations to describe the origin (and, it turns out, the fate) of the Universe. Having started out in 1975 from the puzzle of what happens in the last stages of the evaporation of a black hole, by 1981 Hawking was ready to unveil his new ideas, incorporating Feynman’s sum-over-histories version of quantum mechanics, to explain how the Universe had come into being. The place he chose for the unveiling was—the Vatican.

In fact, the choice of venue was not entirely Hawking’s whim. It happened that the Catholic Church had invited several eminent cosmologists to attend a conference in Rome in 1981, to discuss the evolution of the Universe from the Big Bang onward. By the 1980s, the Church was much more receptive to scientific teaching than it had been in the days of Galileo, and the official view was that it was quite OK for science to investigate events since the Big Bang, leaving the mystery of the moment of creation in the hands of God.

Fortunately, perhaps, Hawking’s investigation of the moment of creation was still couched in rather abstruse mathematical language when he presented it to that conference. Since then, however, he has developed the ideas in a more accessible way (most notably with the help of James Hartle of the University of California). It doesn’t take much intuition to guess that the Pope would probably not approve of the fully developed version of Hawking’s ideas, which seems to do away entirely with a role for God.

What Hawking has tried to do is to develop a sum over histories describing the entire evolution of the Universe. Now this is, of course, impossible. Just one history of this kind would involve working out the trajectory of every single particle through space-time from the beginning of the Universe to the end, and there would be a huge number of such histories involved in the “integration.” But Hawking found that there is a way to simplify the calculations, provided the Universe has a particularly simple form.

Quantum theory comes into the calculations in the form of the sum over histories. General relativity enters in the form of curved space-time. In Hawking’s models, a complete curved space-time that describes the entire history of a model universe is equivalent to a trajectory of a single particle in Feynman’s sum over histories. General relativity allows for the possibility of many different kinds of curvature, and some sorts of curvature turn out to be more probable than others.

If the Universe is like the interior of a black hole, with space-time closed around it, we can imagine, in the standard picture of the Big Bang, that everything (including space) expands outward from the initial singularity, reaches a certain size, and then collapses back into a mirror image of the Big Bang, the so-called “Big Crunch.” In this picture, there is a beginning of time in the initial singularity and an end of time in the final singularity. Hawking calls the beginning and end of time “edges” to this model of the Universe—such a model has no edge in space because space is folded round into a smooth surface like the surface of a balloon, or the surface of the Earth; but there is an edge in time in the beginning, when the Universe appears as a point of zero size.

Hawking wanted to remove the edge in time, as well as the edge in space, to produce a model of the Universe that has no boundaries at all. He found that, without having to go into the detail of calculating every trajectory of every particle through space-time, the gen-

eral rules of the sum-over-histories approach as applied to families of curved space-times said that a certain kind of curvature is much more likely than any other if the no-boundary condition applies.

Hawking stresses that this no-boundary condition is, as yet, just a guess about the nature of the Universe, but it is a guess that leads to a powerful image of reality. This is the cosmological equivalent of saying that the path integral approach tells us that an electron can follow only certain orbits around a nucleus; the Universe has only a limited number of life cycles to choose from, and they all look much the same.

The best way to picture these models is by an extension of the idea of the Universe being represented by the surface of a balloon. In the old picture, this surface represents space, and the evolution of the Universe from bang to crunch is represented by imagining the balloon being first inflated and then deflated. In the new picture, however, the spherical surface represents both space and time, and it stays the same size—much more like the surface of the Earth than the surface of an expanding balloon. So where does the observed expansion of the Universe come into this model?

Now, says Hawking, we have to imagine the Big Bang as corresponding to a point on the surface of the sphere, at the North Pole. A tiny circle drawn around that point (a line of latitude) corresponds to the size of the space occupied by the Universe. As time passes, we have to imagine lines of latitude being drawn further and further away from the North Pole, getting bigger (showing that the Universe expands) all the way to the equator. From the equator down to the South Pole, the lines of latitude get smaller once again, corresponding to the Universe shrinking back to nothing at all as time passes.

We still have an image of the Universe being born in a super-dense state, evolving, and shrinking back into a super-dense state, but there is no longer a discontinuity in time, just as there is no edge

of the world at the North Pole. At the North Pole, there is no direction north, and every direction points south. But this is simply due to the geometry of the curved surface of the Earth. In the same way, at the Big Bang there was no past, and all times lay in the future. And this is simply due to the geometry of curved space-time. The whole package of space and time, matter and energy, is completely self-contained.

A rather nice way to understand what is going on is to imagine you are standing a little way from the North Pole and start to walk due north. Even though you keep walking in a straight line, you will soon find that you are walking due south. In the same way, if you had a working time machine and started traveling backward in time from some moment just after the Big Bang, you would soon find that you were traveling forward in time, even though you had not altered the controls of the time machine. You just cannot get back to a time before the Big Bang (strictly speaking, before the Planck time) because there simply is no “before.”

In A Brief History of Time, Hawking spelled out the implications for religion. He leaves his colleagues in no doubt that he is, at the very least, an agnostic and finds strong support for this belief in his cosmological studies:

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?¹

But even without a creator there were still problems to be solved. Already, in 1981, the attention of Hawking and other theorists was focusing on the next question—how did a tiny seed of a Universe get blown up to the enormous size that we see today?

The puzzle of how the Universe has got to be as big as it is today had itself loomed larger and larger during the 1970s. When everybody thought that the Big Bang theory was just a model to play with, they didn’t worry too much about the details of how it might work. But as evidence built up that this model provides a very good description of the real Universe, it became increasingly important to explain exactly what makes the model, and the Universe, tick.

There were two problems that cosmologists were simply unable to answer in the 1970s. First, why is the Universe so uniform—why does it look the same (on average) in all directions of space, and why, in particular, is the temperature of the microwave background exactly the same in all directions? Secondly, the Universe seems to be delicately balanced on the dividing line between being closed, like a black hole, and open, so that it will expand forever. In terms of the curvature of space, the Universe is remarkably flat. Why is this?

On the basis of general relativity alone, there seems to be no reason why it could not have been, for example, much more tightly curved, in which case the Universe would have expanded only a little way out of the Big Bang before recollapsing, and there would have been insufficient time for stars, planets, and people to evolve. Cosmologists suspected that the smoothness and flatness of the Universe were telling us something fundamental about the nature of the Big Bang, but nobody could see just what that might be until a young researcher at Cornell University, Alan Guth, came up with a new idea.

Guth’s proposal goes by the name “inflation” and stems from quantum physics. He suggested that in the first split second after the beginning, the vacuum of the Universe existed in a highly energetic state, as allowed by the quantum rules, but unstable. The high-energy state is analogous to a container of water cooled, very slowly and carefully, to below 0°C. Such supercooling is possible if the

water is cooled very carefully, but the result is unstable. At a slight disturbance, the water will freeze into ice, and as it does so it gives up energy (exactly the same amount of energy that is needed to melt an ice cube, at 0°C, is released when the same amount of water freezes).

This is where the ice analogy breaks down slightly, for when the Universe cooled from the excited vacuum state to the stable vacuum that we know today, so much energy was released that it became super-hot, not icy, and for a time it expanded super-fast. In a tiny fraction of a second, a region of space far smaller than a proton (but packed full of energy) must have been inflated, according to this theory, into a volume about the size of a grapefruit. At that point the inflation was exhausted, and the grapefruit-sized fireball began the steady expansion associated with the standard model of the Big Bang, growing over the next 15 billion years to become the entire visible Universe.

According to inflation theory, the Universe is so uniform because it has grown out of a seed so small that there was literally no room inside it for irregularities. And the equations also tell us that the inflation process flattened space. The best analogy for how this works is with the wrinkly surface of a prune, which is very far from flat. When you soak the prune in water, it swells up, expanding so that the surface stretches and the wrinkles are smoothed out. Imagine starting out with a prune smaller than a proton and expanding it to the size of a grapefruit, and you can see why space is so very flat today.

The inflationary model has been extensively developed since Guth made the original proposal in 1980. Hawking has been involved in filling in details of this work throughout the 1980s, but the main developments have come from a Soviet researcher, Andrei Linde. Some of Linde’s early contributions were duplicated independently by Paul Steinhardt and Andreas Albrecht, from the

University of Pennsylvania. As we shall see in Chapter 15, the early versions of inflation were overtaken in the 1980s by new insights that provide a spectacular new image of the origin and evolution of not just the Universe but a multiplicity of universes. Hawking played a part in this work, too. From now on, honors and awards would be heaped upon the man to whom the modest recognition offered by the Gravity Research Foundation had been “very welcome” just a short time before.