Stephen Hawking (26 page)

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 and still exist. 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 okay for science to investigate events that happened 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 spacetime 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 spacetime. In Hawking's models, a complete curved spacetime 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 spacetime 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 around 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 spacetime, the general rules of the sum-over-histories approach as applied to families of curved spacetimes 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 farther and farther 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 superdense 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 spacetime. 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 that 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 gotten to be as big as it is today had itself loomed larger and larger during the 1970s. When everybody thought 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 re-collapsing, 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).