In the early 1960s, astronomers already knew that any star which contains more than about three times as much matter as our Sun ought to end its life by collapsing inward to form what is now known as a black hole. More than twenty years previously, researchers had used Einstein’s equations of general relativity to calculate that such an object would bend space-time completely round upon itself, cutting the central mass off from the rest of the Universe. Light rays passing near such an object would be deflected so much that even photons would orbit around the central “star” in closed loops and could never escape into the Universe outside. Obviously, since it could emit no light, such an object would be black, which is why the American relativist John Wheeler dubbed them “black holes” in 1969.

But although it was well known that the general theory made this prediction, at the time Hawking was completing his undergraduate studies and moving on to research no one took the notion of black holes seriously. The reason is that there are very many known stars that have more than three times the mass of our Sun. They do not col-

lapse because nuclear reactions going on inside the stars make them hot. The heat creates an outward pressure that holds the star up against the pull of gravity. Astronomers knew that when such stars run out of nuclear “fuel” they explode, blasting away their outer layers into space. As recently as thirty years ago, astronomers assumed that such an explosion would always blow away so much matter that the core left behind would have less than three times the mass of our Sun—or, perhaps, that some as yet undiscovered pressure would come into play as the remnant of star stuff began to shrink.

This prejudice was reinforced by the fact that astronomers had indeed discovered many old, dead stars. These stellar cinders all had a bit less than the mass of our Sun, but compressed into a volume only about as big as that of the Earth. Such planet-sized stars are known as white dwarfs. They are held up against the inward pull of gravity by the pressure of the electrons associated with the atoms of which they are made, acting like a kind of electron gas. A white dwarf is so dense that each cubic centimeter of the star contains a million grams of material. Before 1967 these were the densest known objects in the Universe.

But although astronomers did not seriously believe that anything denser than a white dwarf could exist, a few mathematicians enjoyed playing with Einstein’s equations to work out what would happen to matter if it were squeezed to still greater densities. The equations said that if three times as much matter as our Sun contains were squeezed until it occupied a spherical region with a radius of just under 9 kilometers, space-time in its vicinity would be so distorted that not even light could escape. Because nothing can travel faster than light, this meant that nothing at all could ever escape from such an object, which the mathematicians sometimes referred to as a “collapsar” (from collapsed star). It would have become the ultimate bottomless pit into which anything could fall but from which nothing could ever emerge. And the density inside

the collapsar would be greater than the density of the nucleus of an atom; this, theorists of the time thought, was clearly impossible.

In fact, they did consider (but not too seriously) the possibility of stars as dense as the nucleus of an atom. By the 1930s, physicists knew that the nucleus of an atom is made of closely packed particles called protons and neutrons. The protons each carry one unit of positive charge; the neutrons, as their name suggests, are electrically neutral, but each has about the same mass as a proton. In everyday atoms, like the ones this book is made of, each nucleus is surrounded by a cloud of electrons. Each electron carries one unit of negative charge, and there is the same number of electrons as protons, so the atom as a whole is electrically neutral.

But an atom is largely empty space. The nucleus is tiny but very dense, and the cloud of electrons is (by comparison) huge and insubstantial. In proportion to the size of a whole atom, the nucleus is like a grain of sand in the middle of a concert hall. In white dwarf stars, some of the electrons are knocked off their atoms by the high prevailing pressure, and the nuclei are embedded in a sea of electrons that belong to the whole star, not to any particular nucleus. But there is still a lot of space between the nuclei, even though that space contains electrons. Each nucleus has positive charge, and like charges repel, so the nuclei keep their distance from each other.

But quantum theory said that there is a way to make a star denser than a white dwarf. If the star were squeezed even more by gravity, the electrons could be forced to combine with protons to make more neutrons. The result would be a star made entirely of neutrons, and these could be packed together as closely as the protons and neutrons in an atomic nucleus. This would be a neutron star.

Calculations suggested that this ought to happen for any dead star with a mass more than 20 percent larger than that of our Sun (that is, more than 1.2 solar masses). A neutron star would have that much mass packed within a radius of about 10 kilometers, no

bigger than many mountains on Earth. The density of the matter in a neutron star, in grams per cubic centimeter, would be 10¹⁴—that is, 1 followed by 14 zeros, or one hundred thousand billion. Even an object this dense would not be a black hole, though, for light could still escape from its surface into the Universe at large.

Making a black hole from a dead star would require, as the theorists of the early 1960s were well aware, crushing even neutrons out of existence. The quantum equations said, in fact, that there was no way that even neutrons could hold up the weight of a dead star of 3 solar masses or more and that, if any such object were left over from the explosive death throes of a massive star, it would collapse inward completely, shrinking to a mathematical point called a singularity. Long before the collapsing star could reach this state of zero volume and infinite density, it would have wrapped space-time around itself, cutting off the collapsar from the outside Universe.

Indeed, the equations said that if you squeezed any collection of matter hard enough it would collapse in this way. The special feature of objects of more than 3 solar masses is that they will collapse anyway, under their own weight. But if it were possible to squeeze our own Sun down into a sphere with a radius of about 3 kilometers, it would become a black hole. So would the Earth, if it were squeezed down to about a centimeter. In each case, once the object had been squeezed down to the critical size, gravity would take over, closing space-time around the object while it continued to shrink away into the infinite density singularity inside the black hole. But notice that it is much easier to make a black hole if you have a lot of mass. The critical size is not simply proportional to the amount of mass you have; the density at which a black hole forms is larger if you have less mass to squeeze.

For any mass there is a critical radius, called the Schwarzschild radius, at which this will occur. As these examples indicate, the Schwarzschild radius is smaller for less massive objects—you have

to squeeze the Earth harder than the Sun, and the Sun harder than a more massive star, in order to make a black hole. Once it had formed, there would be a surface around the hole (a bit like the surface of the sea) marking the boundary between the Universe at large and the region of highly distorted space-time from which nothing could escape. It would be a one-way horizon (unlike the surface of the sea!) across which both radiation and material particles could happily travel inward, tugged by gravity to join the accumulating mass of the singularity, but across which nothing at all, not even light, could travel outward.

Some mathematicians worried, thirty years ago, about the prediction that black holes must contain singularities. The notion of a point of infinite density made them uneasy. But most astronomers were more pragmatic. First of all, they doubted whether black holes could really exist at all. Probably, they thought, some law of physics would prevent any dead star from having enough leftover mass to collapse in this way. And even if black holes did exist, by their very nature they would keep the singularities at their hearts locked away from sight or investigation. Did it really matter, after all, if theory said that points of infinite density could exist, if the same theory said that such singularities were safely locked away behind uncrossable horizons?

One thing, however, should have worried those astronomers, even in the early 1960s. Just as you need to squeeze a small mass hard to make a black hole, a larger mass needs less of a squeeze to do the same trick. Indeed, a mass of about 4.5 billion solar masses would form a black hole if it were all contained within a sphere only twice the diameter of our Solar System. That mass sounds ludicrous at first. But remember that there are a hundred billion stars in our Milky Way Galaxy. If just 5 percent of the total mass were involved, such a supermassive black hole could indeed form. And the density of such an object would be nothing like the density of the nucleus of an atom or a neutron star. It would be just 1 gram per cubic

centimeter—the same density as water. You could actually make a black hole out of water, if you had enough of it!

One way to understand how this can happen is by an analogy with running tracks. The important thing about a black hole is that it bends space-time completely around itself, so that light rays at the horizon would circle endlessly around the central singularity. But the photon “orbits” can be either very tight or follow a gentle curve. Indoor running tracks are usually tightly curved, to make them fit into the space available. Outdoor running tracks are more gently curved and take up more space. But in both cases, if you run round the track you get back to where you started from—you follow a closed loop. Similarly, a black hole can be very small, with space-time tightly folded around itself, or very large, with light rays following gradual curves around the horizon (or, indeed, they can be any size in between).

Very slowly, during the 1960s, the implications of this began to dawn on cosmologists. The whole Universe, they realized, might behave in some ways like the biggest black hole of them all, with everything in the Universe held together by gravity, and all of space-time forming a self-contained, closed entity that folded round on itself with the ultimate in gradual curvature. But there is one big difference—black holes pull matter inward, toward the singularity; the Universe expands, outward from the Big Bang. The Universe is like a black hole inside out.

Einstein’s equations—the general theory of relativity—said that the Universe could not be static, but must be either expanding or contracting. Observations showed that the Universe is, indeed, expanding. So what did Einstein’s equations say about conditions long ago, when galaxies were packed tightly together, and before? Taken at face value, the equations said that the Universe must have emerged from a point of infinite density, a singularity, about 15 billion years ago. “Obviously” (to astronomers of the 1940s and

1950s, that is) that was ridiculous. The fact that the equations predicted a singularity must mean that they were flawed in some way; no doubt in due course somebody would come up with a better theory, one that didn’t make such extreme predictions. But meanwhile it seemed fairly reasonable to take the equations at face value as far as they applied to conditions that bore some resemblance to those we observe today.

The densest form of matter familiar to us today is nuclear matter: protons and neutrons packed together in the hearts of atoms. So a few brave souls were prepared to contemplate the possibility that the general theory might provide a good guide to how the Universe had evolved from a state in which the overall density was as great as that of the nucleus of an atom, a “primeval atom,” if you like, containing all the mass of the Universe in a kind of neutron superstar.

What came “before” that? How did this primeval superdensity—sometimes referred to as the “cosmic egg”—come into being? Nobody knew; they could only make guesses. Perhaps the cosmic egg had existed for all eternity, until something triggered it into expansion. Or perhaps there had been a previous phase of the Universe in which space-time was collapsing, in line with Einstein’s equations. Such a contracting universe might compress itself to nuclear densities and then “bounce” outward again, into a phase of expansion, without encountering the troublesome singularity.

The notion of the primeval atom, or cosmic egg, emerged in the early 1930s and was refined over the next couple of decades. Even at the beginning of the 1960s, however, this was all still just a mathematical game played by a few experts, as much as anything for their own amusement. The notion of a super-dense cosmic egg, only about thirty times bigger than our Sun but containing everything, which had burst asunder to create the expanding Universe, fitted Einstein’s equations and the observations. But nobody seems to have felt, deep down in their hearts, that their equations described

the Universe. Nobody would have been too worried if it had turned out that the whole idea of the cosmic egg was wrong.

You can get a feel for the way people regarded the idea in the 1950s from their own shorthand terms for describing their work. The equations of the general theory of relativity actually allow for more than one possible description of the overall behavior of space-time. As we have mentioned, either expansion or contraction (but not stasis) is allowed by the equations. Obviously, the Universe we live in cannot be expanding and contracting at the same time; the two solutions to the equations cannot both apply to the Universe today. So the solutions are called models. A cosmological model is a set of equations that describes how a universe (with a small “u”) might behave. The equations have to obey the known laws of physics, but they do not necessarily purport to describe the actual behavior of the real Universe (with a capital “U”). Both the expanding and the contracting solutions to Einstein’s equations describe model universes, intriguing mathematical toys; the expanding solution might describe the real Universe. At the beginning of the 1960s, however, most cosmologists would have preferred to call even the expanding solution simply a model universe.

But during the 1960s the whole notion of the Big Bang, as the theory was known, firmed up. Cosmologists began to believe, as more evidence came in confirming the accuracy of the predictions implicit in the general theory of relativity, that their equations really did describe what was going on out there in the real Universe. This encouraged more theoretical calculations, leading to new predictions, and more observations, in a self-stimulating upward spiral that led to a dramatic revolution in our understanding of the birth of the Universe. By 1976 the Big Bang theory was so well established that American physicist Steven Weinberg was able to write a best-selling popular book, The First Three Minutes, describing the early stages of the Big Bang, how the Universe had emerged from

the super-dense state of the cosmic egg. Although written in the 1970s, the book encapsulated what was essentially the 1960s understanding of the Big Bang; we will not be getting too far ahead of our story if we give a brief résumé of that understanding now.

One of the strangest things to grasp about all these descriptions of the Universe—the relativistic cosmological models—is that the Big Bang does not consist of a huge primeval atom sitting in empty space and then exploding outward. Many people still have this image, in which the galaxies are like fragments of an exploding bomb, hurtling outward through space. But it is wrong.

What Einstein’s equations tell us is that it is space itself that expands, taking galaxies along for the ride. Galaxies were closer together long ago, when the Universe was younger, because the distances between them were more compressed than they are today. You can see this by imagining two spots of paint on a strip of elastic or on a rubber band. When you pull on the ends of the strip, it stretches, and the two paint spots move apart, but they do not move through the material the strip is made of.

So in the very early Universe, at the time of the explosion of the primeval atom, there was no “outside” for the fragments of the explosion to move into. Space was tightly wrapped around itself, so that the cosmic egg was a completely self-contained ball of matter, energy, space, and time. It was, indeed, a super-dense black hole. It still is a black hole—the only difference is that, by expanding, it has become a very low density black hole, in which light now follows very gently curving orbits at the horizon.

We live inside a black hole, but one so huge that the bending of space-time within it is too small to be measured by any astronomical instruments on Earth. The “explosion” of the Big Bang stretched space, literally creating more room in which the material components of the cosmic egg could move. Starting out very hot and dense,

the fireball thinned and cooled as the space available expanded. The process is exactly the same as the way the fluid in the pipes of your refrigerator keeps the fridge cool. In the fridge, fluid expands into a large chamber and cools; at the back of the fridge, it is squeezed into a smaller space and gets hot, but the heat escapes from the piping on the outside before the fluid goes back into the fridge to repeat the cycle. Like that fluid being squeezed, or like air being compressed in a bicycle pump when we use it to inflate a tire, the Universe must have been much hotter when it was more compressed.

How much hotter? If you run your cosmological model all the way back to the singularity predicted by Einstein’s equations, you would be dealing with infinite temperatures, as well as infinite density. But nobody in the 1960s went to that extreme. The infinities were still taken as indicating a breakdown in the general theory of relativity, but even so the moment at which the infinities occurred in the models could be used as a marker for the beginning of time (at least until someone came up with a better theory).

Although the physics of the 1960s could not say what went on during the split second following that beginning of time, it could describe in great detail everything that had happened to the Universe in the 15 billion years beginning just one-tenth of a second later. To an increasing number of cosmologists, the general theory did not really seem such a bad description of the Universe, if it could explain everything that has happened in the past 15 billion years except for the very first one-tenth of a second. This is what it told them.

One-tenth of a second after the “beginning” (or after the “bounce,” as many cosmologists of the 1960s would have argued), the density of the Universe was 30 million times greater than the density of water. The temperature was 30 billion degrees,* and the

Universe consisted of a mixture of very high energy radiation (photons) and material particles, including neutrons, protons, and electrons but also more exotic, unstable particles created ephemerally out of pure radiation. This is the ultimate example of the equivalence between mass and energy, expressed in Einstein’s famous equation E = mc². On the Earth, in an atomic bomb, and inside the Sun, where nuclear reactions take place, tiny amounts of matter (m) are converted into large amounts of energy (E), because c is the speed of light, which is 300,000 kilometers a second, and c² is a very large number indeed. But if you had enough energy to play with, you could actually make matter out of energy; and there was ample energy available to do the trick in the Big Bang—even if many of the particles created in this way were unstable, destined to disappear again in a puff of radiation in far less than the blink of an eye.

One second later, 1.1 seconds after the beginning, the Universe had cooled dramatically—all the way down to ten billion K. At that time, the density was just 380,000 times the density of water, and from then on the reactions between particles were very similar to the nuclear reactions that go on inside the Sun and other stars today.

At a temperature of three billion K, just under 14 seconds from the beginning, the first nuclei of deuterium could form, albeit temporarily. Hydrogen is the simplest atom, with just a single proton in its nucleus and one electron orbiting outside the nucleus. (In a sense, lone protons can be regarded as nuclei of hydrogen atoms.) The next most complicated atom is deuterium, which has a nucleus composed of one proton and one neutron, still with a single electron orbiting around it. Atoms that have the same number of electrons as each other but different numbers of neutrons still have identical

chemical properties and are known as isotopes; deuterium is an isotope of hydrogen and is often known as “heavy hydrogen.”

Temperature is a measure of how fast, on average, the particles that make up matter are moving (which is why there can be no temperature colder than -273°C, at which atomic motion stops), and at temperatures above three billion K, protons and neutrons move too fast to do anything except bounce off each other. Some particles move faster than the average for a particular temperature and some slower, although most have speeds close to the average. So as the temperature fell below that value, some protons and neutrons were moving slowly enough to stick together when they collided. The thing that makes them stick is an attraction known as the strong force. As its name suggests, this is a powerful force of attraction that operates between all protons and neutrons. But it has a very short range, and fast-moving particles brush past or bounce off each other too quickly for the strong force to take hold of them during the brief time they are in range. At first, most of the deuterium nuclei produced in this way were knocked apart by collisions with faster-moving particles, but as the fireball cooled still further the deuterium nuclei had a better and better chance of survival.

Just 3 minutes and 2 seconds after the beginning, the temperature had cooled to below one billion K—the entire Universe was then only seventy times as hot as the heart of the Sun is today. At this point, almost all the deuterium nuclei were able to combine in pairs to form nuclei of helium. These helium nuclei each contain two protons and two neutrons, making four “nucleons” in all, so they are known as helium-4 nuclei (and helium atoms, of course, each have two electrons orbiting around the nucleus).

It happens that helium-4 nuclei are particularly stable. But there are no stable nuclei containing five nucleons (such as you might expect to get if you added a proton or a neutron to a nucleus of helium-4) or eight nucleons (such as you might expect to get if you

stuck two helium-4 nuclei together). So the process of “nucleosynthesis” in the Big Bang stopped with the production of helium-4. Less than 4 minutes after the beginning, matter had settled down into a mixture of about 75 percent hydrogen nuclei and 25 percent helium, intermingling with fast-moving electrons and bathed in a sea of hot radiation.

Half an hour later, 34 minutes after the beginning, the temperature was down to 300 million K, and the density of the Universe was only 10 percent of the density of water. But it took a further 700,000 years for the Universe to cool enough to allow electrons to become attached to the nuclei and form stable atoms. Before then, as soon as a positively charged nucleus tried to latch on to a negatively charged electron, the electron would have been knocked away by an energetic photon. After 700,000 years, however, the temperature of the Universe had fallen to about 4,000 K (roughly the temperature at the surface of the Sun today), and nuclei and electrons were at last able to hold together to form stable atoms.

For most of the past 15 billion years, protons, neutrons, and electrons have been bound up in stars and galaxies formed out of this primeval stuff as gravity pulled clouds of gas together in space. The radiation left over from the Big Bang had nothing more to do with the matter, once it was no longer hot enough to separate electrons from their atomic nuclei and simply cooled steadily as the Universe expanded. But as we shall see, that background radiation, the echo of creation, had a key role to play in persuading cosmologists that one of their “model universes” might actually be telling them something deeply significant about the real Universe. And all this was happening while the person who was to become a key player in taking cosmology that step further in the 1970s, back to the beginning itself, was experiencing upheavals of his own, both personal and professional.