Appendix

Questions about our origins were once regarded as the territory of philosophers and theologians. But gradually the answers have been provided by science; speculations have been replaced by hard facts. Especially in the last two decades, since the 1996 edition of A Brief History of Time, we have made remarkable progress in understanding the genesis and evolution of the universe. Many of the ideas that I originally put forward as hypotheses have now been confirmed. And still other developments have been a complete surprise.

Dark Energy and the Accelerating Expansion of the Universe

For example, in 1998, our picture of the universe’s future was radically revised. Two competing teams using the Hubble Space Telescope independently reached the conclusion that the expansion of our universe is accelerating. The implications for the fate of space were immediate: the eventual re-collapse of the universe (Alexander Friedmann’s big crunch) no longer appears to be an option. Space, it seems, will expand forever.

Why should space expand at an accelerating rate? The cause has become known as “dark energy.” But this is just a name; it doesn’t tell us anything in itself. In fact, Friedmann’s original picture seemed compelling: either gravity is strong enough to pull everything back together and the expansion decelerates over time, or it isn’t strong enough and the expansion coasts along unimpeded. Neither of those scenarios suggested anything about the expansion actually speeding up.

Einstein’s own work holds part of the answer. At one point, he tried modifying his theory of general relativity to make the universe eternal and unchanging—something he was convinced ought to be true—by introducing a so-called cosmological constant  into his equations. This constant plays the role of an “antigravity” force built into the very fabric of space-time. That was in 1917, long before the expansion of the universe was established. Einstein retracted the idea once he realized that Friedmann’s models neatly explained Edwin Hubble’s observations.

The retraction might have been premature. At present, it seems that the acceleration first spotted in 1998 can, in fact, be explained by Einstein’s antigravitation. But that’s not the end of it, because the underlying cosmological constant can be given any value, and therefore can push space apart at any rate. Simple estimates suggest that the acceleration should whip the universe apart long before galaxies could form. So why is the strength of antigravity just as it is?

If the no boundary proposal  is correct, an infinity of universes exist in parallel. Each of these universes might well have a different strength for antigravity, especially if string theory is on the right track to a complete understanding of physics. We would then, naturally, live in one of the universes with a comfortably small dark energy; the anthropic principle  reminds us that, if galaxies had never formed, we would not be here to discuss the matter.

Microwave Background Radiation and the No Boundary Proposal

If the no boundary proposal might be central to understanding these developments, we should examine how it holds up in light of our rapidly improving observational handle on the early cosmos. In particular, we can now understand the origins of structure in our universe using measurements of cosmic microwave background radiation.

As the name suggests, this is made up of microwaves—the kind used by your microwave oven, only much less powerful. They would heat your pizza to only −270:4°C, which isn’t much good for defrosting, let alone cooking. But these ultra-weak microwaves are spectacularly valuable, because there is only one reasonable explanation for their presence: they are radiation left over from an early time when the universe was very hot and dense. As the universe expanded, the radiation cooled until it became just the faint remnant we can detect today.

The existence of this background radiation was established in 1965. Immediately upon its detection, it was seen as powerful direct evidence for predictions based on Einstein’s general relativity. Part of my own PhD thesis work, finished just months before, had been to show that the early hot, dense phase was unavoidable in Einstein’s picture.

But the value of measuring the radiation has become greater still. At first the microwaves seemed to have an identical intensity in every direction. This led to ideas like inflation , which in its initial formulation was intended to explain how the early universe came to be so uniform. On closer inspection, it actually predicted there would be very slight variations from place to place. The deviations from uniformity come about through quantum mechanical uncertainty, which imposes a minimum level of fluctuation.

As successive generations of space telescopes have measured the microwave background radiation with increasing precision—first Cosmic Background Explorer (COBE) in 1992 , then Wilkinson Microwave Anisotropy Probe (WMAP) in 2001, and most recently Planck in 2013—this prediction has proved to be correct. There are indeed changes in the intensity of the radiation, at the level of about 1 part in 100,000. More significantly, we have now determined that the precise pattern of variations agrees with the specific predictions I and others made by combining inflation with the no boundary proposal.

To describe the physical conditions at the big bang, the no boundary proposal combines Einstein’s relativity with quantum theory. It says that when we go back toward the beginning of our universe, space and time become fuzzy and “cap off,” somewhat like the North Pole on the surface of Earth. Asking what came before the big bang is meaningless according to the no boundary proposal, because there is no notion of time available to refer to. It would be like asking what lies north of the North Pole.

With my colleagues James Hartle (with whom I first put forward the no boundary proposal more than thirty years ago) and Thomas Hertog, I have put all this to the test. We calculated what kind of universe would emerge from the big bang according to the no boundary proposal, and compared this prediction with our observations. This confirms that our universe should have come into existence with a burst of inflation.

So the features now measured in the microwave background radiation appear to confirm inflation and the no boundary proposal. But there is one key prediction of the theory that has yet to be verified. According to inflation, a small part of the fluctuations in the microwave radiation can be traced to gravitational waves generated during the phase of rapid expansion. This primordial gravitational radiation is the analogue of the quantum radiation from black holes, and can be regarded as coming from the event horizon of the early inflationary stages of the universe. Its detection would confirm that black holes emit quantum radiation, something almost impossible to confirm directly. I will say more about the detection of gravitational waves below, but those generated in the early universe show up most clearly in the polarization of the radiation. We are only in the early stages of measuring this polarization, and there is real hope that it will provide firm and convincing evidence for our theory of the big bang.

Even without a clear view of the polarization, the cosmic microwave background data are so good that we can now start to fill in some of the blanks. Inflation and the no boundary proposal leave a number of details unspecified: the precise energies involved, for example, and the link to the underlying particle physics. These details subtly change the expected patterns; by carefully studying what is seen, we are now beginning to understand physics near the grand unification energy. To put that in context, it is a million million times higher than can be probed by the very best experimental facility on Earth, the Large Hadron Collider.

Eternal Inflation and the Multiverse

The developments described above mean that in the last two decades, inflation has been transformed from speculation into a cornerstone of modern cosmology. But not everyone likes its conclusions, especially since we now believe inflation likely gives rise to a vast number of universes, known collectively as a multiverse.

As I mentioned above, inflation predicts that the universe will be nearly, but not perfectly, uniform. The deviations from uniformity are imposed by quantum mechanics, and have now been precisely characterized from observations of the cosmic microwave background.

The very same quantum mechanical effect can give rise to the multiverse. Inflation is driven by a strange type of energy that has antigravitational properties; on average, the amount of this energy decreases as inflation proceeds, until there is no longer enough and the accelerated expansion ends. But in some regions of space-time, quantum fluctuations temporarily reverse the overall trend. Such regions gain more energy and consequently inflate for longer.

In 1986, the Russian-American physicist Andrei Linde calculated that if inflation starts at a sufficiently high energy, there will always be someplace where the fluctuations win: the energy remains high, and inflation continues eternally. But there will be other places where the fluctuations lose and the expected trend of decreasing energy takes hold. These patches become entire individual universes such as our own. If we could zoom out far enough, we would see countless other universes, separated by Linde’s regions of the multiverse that are continuing to inflate.

Eternal inflation and the no boundary proposal together predict that our universe is not unique. Instead, from the quantum fuzz at the big bang many different universes emerge, possibly with different local laws of physics and chemistry. We may not live in the most probable of all universes. Rather, we live in one where the conditions are favorable for complexity and the development of life. Even though we cannot go from one universe to another, the successful predictions of the theory for observations in our universe provide support for the worldview predicted by the no boundary proposal.

For a long time, many physicists brushed these arguments to one side. The idea of a multiverse makes some people queasy, and they would rather assume inflation takes place at a lower energy, thereby sidestepping Linde’s argument. However, the latest observations from the Planck satellite make this escapology trick look increasingly implausible.

Gravitational Waves

Using polarization of the cosmic microwave background to show that gravitational waves were produced in the early universe, as mentioned earlier, would be one way to very directly confirm the high energies involved in inflation. I hope that we will not have to wait too long for this development. In the meantime, we have recently seen direct confirmation that gravitational waves  can be produced in the modern universe. Exactly a century after Einstein first predicted the existence of gravitational waves, a worldwide consortium of scientists known as the LIGO Scientific Collaboration announced in 2016 that these waves had been detected for the first time.

The first sixty years were the hardest. During this time there was confusion over the status of the waves: should they exist in practice or are they just a mathematical artifact, unconnected with reality? Even Einstein seemed uncertain, and he came close to publishing an erroneous disproof of their physicality in the 1930s. But over time the physics community settled on the view that the waves should be real. One consequence was that energy would be very slowly lost from orbiting bodies. Until recently, such energy loss was our only evidence for the existence of the waves. This was very convincing, but still indirect.

Actually measuring gravitational waves as they pass through Earth is technologically challenging, which is why it took until 2016. But the decades of technological development have proved worthwhile, because we now have a completely new way to study the universe. Even the first events that LIGO detected—waves resulting from the collision and merging of two black holes—allowed us to confirm our understanding of a process that no traditional telescope has ever been, or will ever be, able to probe.

It was really exciting for me to see observations of colliding black holes. LIGO will observe many such events in the near future. These observations will, I believe, confirm a prediction I made in 1970—that the surface area of the final black hole was greater than the sum of the initial holes’ areas. This “area theorem,” which led to my slightly later realization that black holes will gradually lose their mass over time, was secure on mathematical grounds. But one can never be too sure of an idea until it is tested in nature.

There is a bright future for LIGO and other gravitational wave observatories. We can expect to build up a large catalog of detections that will provide detailed insight into the populations of black holes in our universe. That in turn will allow us to search for even slight deviations from predictions based on Einstein’s theory. As we continue our search for a full quantum theory of gravity, this treasure trove of information about extreme regions of space-time will be immensely valuable.

The Information Paradox

One reason for my excitement over LIGO’s gravitational wave detections is that the area theorem is directly linked to a major controversy surrounding black holes known as the information paradox. Information is a sacred thing in physics; if we are able to describe the entire state of the universe today with a certain amount of information (the positions and speeds of all the particles, for example), we expect to need the same amount of information to describe the entire state of the universe tomorrow. This assumption underlies our ability to make scientific predictions and is quietly built into Newton’s and Einstein’s work; it’s even part of quantum mechanics. One might therefore hope it will remain true when we formulate a final theory of quantum gravity.

When a black hole is formed, information about individual objects that have fallen in (their shapes, sizes, and chemical compositions, for example) becomes obscured. Only a few bits of information are available about what formed it: its total mass, its spin, and its possible electric charge. This is the so-called “no-hair” theorem. This is not too much of a problem, since the objects can just be regarded as hidden away rather than entirely lost. But if, as I showed in a letter published in Nature in 1974, quantum mechanics allows black holes to lose their mass and disappear , there is a difficulty. After the black hole is gone, what has happened to the information?

When I wrote A Brief History of Time, I believed that the information concerning what had fallen into the black hole was truly lost, perhaps residing in a separate universe hived off from our own. In 1997 I even bet Caltech physics professor John Preskill an encyclopedia of his choice that I was right.

It was only later, in 2004, that I realized I had been wrong, after considering what happens to black holes after an infinite amount of time has passed. The amount of information at the start and the end was the same! When I conceded, John asked for an encyclopedia of baseball, which I duly gave him. (My attempt to persuade him that cricket is more interesting was unsuccessful.)

My change of opinion started by considering one of the most remarkable discoveries to arise from string theory: there appears to be an exact correspondence between the behavior of gravity and an obscure branch of physics known as conformal field theory. The details of the link don’t matter for our purposes. All one needs to know is that anything described by conformal field theory—now including black holes—demonstrably preserves information. Very recently, it was realized that the “no-hair” theorem was formulated in too restrictive a way. There is also supertranslation and superrotation hair. It seems that the information about material that formed the black hole remains preserved on the horizon as supertranslation and superrotation hair. We do not yet know if this is enough information to save the principles of quantum mechanics. Neither do we yet know how the information might emerge from the black hole. Even harder questions could then be asked about the fundamental nature of the singularities of space-time that general relativity predicts must exist inside black holes.

Of course, this abstract argument doesn’t tell us exactly how the lost information could manage to leak back out of a black hole in practice.

One must be clear that when the information finally makes its way out of the black-hole-like region, it will emerge in a very hard-to-interpret format. It is like burning a book. The information the book contained is not technically lost, if one keeps the ashes and the smoke—which makes me think again about the baseball encyclopedia I gave John Preskill. I should perhaps have given him its burned remains instead.

Outlook

In the twenty years since the last revision of this book, progress in cosmology has been rapid. Some of the developments, such as the detection of gravitational waves and the steady improvement in our understanding of the early universe, were anticipated; others, like dark energy and the accelerating universe, less so.

Perhaps the most striking trend is one that many find uncomfortable: the no boundary proposal and eternal inflation point increasingly strongly to the idea that our universe is just one of many. Copernicus first suggested in the sixteenth century that we are not placed at the center of even our own universe , yet we are still struggling to accept just how vanishingly small a fragment of reality our familiar world represents. It may not be much longer before the evidence for a multiverse becomes overwhelming.

Despite the vastness of the multiverse, there is a sense in which we remain significant: we can still be proud to be part of a species that is working all this out. With that in mind, the coming years should be just as exciting as the last twenty.

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