# The Science of Hawking

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The universe explodes. All of it’s energy is focused on this singular act of creation, and it grows at a rate that has no parallel in our existence. During the first few seconds, there is nothing that we would recognize as matter yet. Space literally seethes with energy. This energy is carried by particles that we consider the elementary building blocks – quarks and leptons – and by particles that are responsible for the four forces that we see at play today. As the universe expands, it cools. After a few years, more complicated particles such as protons and neutrons begin to exist. Sometime later, maybe several thousand years, the universe has cooled further so that atoms can form. First the very simplest, hydrogen, made up of jus two particles. Later, heavier elements are created. And even later still, more complicated structures begin to form through the force of gravity. This is the “big bang,” and history – at least the timeline that we are following – has started.

And this is the history that Stephen Hawking, Lucasian Chair of Mathematics at Cambridge University, has spent his life unraveling. Although Hawking has not been alone in this endeavour - he is one of perhaps several hundred cosmologists and astrophysicists – he is without question the very best known, both in scientific circles and with the public. This point was made in vivid colors last spring when, as a guest of a group of undergraduate students, he paid a visit to the University of Toronto. His public lecture was sold out in fifteen minutes, with people from all walks of life coming to hear him talk about his quest to discover literally how time began.

Hawking’s thinking and writings have communicated, better than any other contemporary scholar, the history of our world to the man in the street. He is often compared to Albert Einstein, the white-haired savant who early in this century discovered that to understand gravity means to understand that we live in a truly four-dimensional world. Hawking is the author of “The Brief History of Time,” [1] a book without equations that describes in laymen’s terms how we understand the evolution of the universe from the big bang on. He has even appeared on the science-fiction TV series “Star Trek – The Next Generation,” playing himself as the scientific giant of the latter twentieth century.

Much has been written about the man [2], but what is Hawking’s science? What have been the contributions that have brought him this stature? And what can we expect to hear from him in the years ahead? The first question is easy to answer – at least for a physicist. The second question can be answered on several levels – his contributions to the creation of knowledge itself, and his role as being one of the preeminent spokespersons of his science to western society. The last question is one that I suspect Hawking himself would love to know the answer to. In addressing these questions, I’m going to embed in the story some of the basic physics that you need to appreciate to understand Hawking’s science. It is a story about black holes, creation and time’s arrow. I hope they will help enlighten those readers who wouldn’t know a black hole if they met one in a back alley.

## A Primer on General Relativity and Big Bang Cosmology

Hawking has spent his scientific life studying what can be described as the structure of our universe and its evolution. A simple view of our world is that it is three-dimensional space filled with different kinds of matter that interact through several different forces(1). The concept we call time can be viewed as parametrizing changes in this three-dimensional space. For example, a motionless ball is simply a certain amount of matter at rest relative to the observer. We describe it as motionless because as time advances, its position in three-dimensional space relative to the observer does not change. Dropping the ball in a gravitational field causes the ball to accelerate – uniformly if the gravitational force is constant – and the motion can be parametrized as a function of time by a simple quadratic formula. This simple view suffices for most people, and it is the world-view that we impart to our children all through school. It is the world view that the natural philosopher, Sir Isaac Newton, is given most credit for. His three force laws are based on this model of our universe. Newton founded his theory of gravitation on this world view, a view that survived for over two centuries.

The truth about our world only begins to be revealed in advanced university physics courses. The true nature of our world, discovered in large part by Einstein, is one where the three space dimensions and the time dimension must be treated in an even-handed manner. Our world is in fact four-dimensional, with one of those dimensions representing time. The same motionless ball would now represent a line in this space-time, a line because its three spatial coordinates would remain constant, whereas the fourth coordinate, time, takes on all values from the point the ball is placed at rest to the time it is subsequently moved.

This change in our view of the relationship between space and time arose from the discovery that the speed of light was a universal constant, something that did not change, no matter how fast we would move relative to the source of the light. Einstein’s revelation was that in a four-dimensional space-time, one could both accommodate this remarkable fact and still have a world where everything else appeared to work like normal. One should not be too critical of Newton – in his time, there wasn’t any hint that the speed of light would create troubles. By Einstein’s time, this had become a unrefutable fact, and it was one of many observations that he was trying to account for. Einstein was able to first solve the problems presented by the constant speed of light by introducing space-time in his Special Theory of Relativity. It took him another ten years to solve what he considered the real mystery – gravity – by a model called the General Theory of Relativity.

In Einstein’s space-time, the effect of the gravitational force is really a consequence of the geometry of this space. Wait! What do you mean by geometry? Isn’t space just space, and objects have a specific geometry? A simple analogy is often used to help explain what I mean. There are many different two-dimensional spaces (we would call them surfaces) that we can imagine if we allow ourselves to think three-dimensionally. For example, the painter’s mat (the canvas stretched on a rectangular frame) is something we would call “flat” (we often call this Euclidean space, named after the famous geometer Euclid). If I rolled a marble across this surface, it would travel in a straight line. If I took the same mat and twisted the frame, stretching the canvas as I did so, the canvas would no longer be flat but have curves and ripples in it. Mathematically, it would now have non-zero curvature. The same marble would now have a trajectory that, if I weren’t aware of the geometry of the canvas, would appear to be influenced by some “force” tugging away at it as it meandered up and around the ripples and curves in the canvas.

Think what would happen if our four-dimensional world had similar curvature in space-time. If I now sent a marble off in a specific direction, it would not necessarily follow a straight path. It’s path could be quite complex, depending on how complex the geometry was. We would call such a space-time non-Euclidean, and that is exactly the picture that Einstein suggested as the one for our universe. But he went further and suggested that the curvature arose from the mass or mass-energy density in our space-time(2). The larger the energy density, the more curved space-time. What this means to our ball is that as it passes by a massive object, it will be deflected towards the object due to the space-time curvature. The more massive the object, the more it will be deflected.

This geometric interpretation of the force of gravity results in a description of the ball’s motion that is almost the same as what Newton’s law of gravity gives us. The two theories make virtually identical predictions in what we call the “weak field” approximation, where the force of gravity is modest. Gravitationally speaking, the environment in which we live is always within this weak field limit. That is why it took more than 200 years of observation and experimentation before there was a strong enough case to overthrow Newton’s law of gravity. The places where his theory did not work had to do with such subtle effects as the rotation of Mercury’s elliptical orbit around the sun and the bending of light, effects that required difficult and patient measurements.

General relativity is what we use to describe the large-scale structure of our universe. It has been verified as being, if not the right model, then such a good approximation that it’s failures are quite subtle indeed. It is also the framework that predicts a host of phenomena that have stretched our imagination. It predicts that if there is enough mass in one region, the force of gravity will be so strong that light itself will be unable to escape its pull, creating what we call a “black hole.” It also is the framework used to determine the past history of the universe. Measurements of the velocity of very distant objects show that our universe is currently expanding rapidly. The farther away the object, the larger its recession velocity. This implies that, at some point in the past, all matter in our universe resulted from some massive initial explosion – the big bang.

It is important to note that the big bang is not simply matter exploding outward to fill the universe. It is the entire universe that is exploding outward – in effect, the size of our universe is increasing. As we turn the clock back, we can see that matter and energy in our universe is getting more squished together, getting more dense, and is getting hotter. General relativity does not predict the big bang – it allows for a universe that is unchanging with time, or what we call the “steady state” universe. The need to invent the big bang model arises from a large body of observations (the recessional velocity of distance galaxies is just one) all pointing to the same model.

The most significant shortcoming of general relativity is that we do not have a way of integrating the theory with understanding of our world at the most microscopic level. At the same time that Einstein was developing his geometric model of gravity, other physicists were discovering that energy appeared to be doled out in small but well defined chunks, or “quanta.” When they looked at materials that gave off light, they found that each light particle, or photon, had an energy that was some multiple of this fundamental quantum. This resulted in yet another revolution in our understanding of our world and gave birth to a new theory called “quantum mechanics.” It predicted that an object could be found in one of a number of distinct states, each with a unique energy, and for an object to change its quantum state, it would have to emit or absorb some number of these quanta of energy.

Quantum mechanics remains the only theory that successfully describes how atoms and molecules work, and is the basis for such common-place devices as transistors and lasers. The implications of quantum mechanics are enormous, yet Einstein was never able to find a way to rationalize it with his theory of gravity(3). We were left with two physical models, one that described gravity and that had been tested at distance scales ranging from about one metre to many light-years, and the other that discussed what happened at the atomic level, but that had few manifestations at distance scales larger than about a hundredth of a micron(4). Physicists had attempted to bridge the gap between these two theories with little success until the late 1960’s and early 1970’s.

## Hawking and Black Holes

Enter Stephen Hawking. He was introduced to theoretical physics at Cambridge University by his Ph.D. advisor, David Sciama, an expert in quantum theories of elementary particles and cosmology. As a graduate student in the early 1960’s, he became fascinated by Einstein’s theory and its implications, in part inspired by the work of a British mathematician, Roger Penrose. As would any successful scientist or problem solver, he chose to pick away at specific aspects of the theory, developing as he went a deep intuition for how the theory works. He first major exploration was into the behaviour of a funny, theoretical object called a black hole.

What is a black hole? Before I can offer an answer, we first need to recall that gravity is a force that is always attractive. If one throws a bunch of matter out in space, the various pieces will not stay put. Gravity will act on each chunk of matter and attempt to draw the pieces together. As a result, any collection of objects that interact gravitationally is inherently unstable(5).

Now, suppose we had a large amount of mass sitting in one place. Gravity would pull all this matter toward the centre of the mass. The more mass, the stronger the pull. At some point, the force of gravity would be so strong that it would overwhelm even photons, the particles that make up light, so that any nearby photon would be sucked in. That is what we call a black hole.

In other words, a black hole is an object that has so much mass concentrated in one location that the force of gravity near the object is ineluctable. Nothing – not even light – can escape its gravitational pull. Since the black hole absorbs all light, it is completely black. The amount of mass require to create a black hole is surprisingly small on an astrophysical scale. All you need is about 3 times the mass of our sun and you have the makings of a black hole sometime in the far future.

By the early 1970’s, Hawking became one of the leading experts in black holes, discovering a host of properties for these strange objects. Black holes had been considered cosmic vacuum cleaners, since they suck in anything that come too close to them. There is a specific radius from the centre of the black hole, known as the “event horizon,” that signifies the point of no return. Once you step inside the event horizon, you are trapped in the black hole for eternity. Hawking discovered that quantum mechanics plays a certain game right at the event horizon. It allows matter and antimatter particles to be created. Quantum mechanics plays this game all the time, usually with no real effect – a particle and its antimatter equivalent (for example, an electron and a positron, the antimatter version of an electron) will be created, survive for an extremely brief instant of time, and then will vanish, leaving things exactly the way they were before the particle-antiparticle pair was created. While the pair exists, nature plays with the law of the conservation of energy. One particle has positive energy and the other has negative energy. We would normally call such particle-antiparticle pairs as “virtual,” since quantum mechanics allows these to exist for only a short time. Hawking realized that when this game takes place at the event horizon of a black hole, it is possible for the negative energy particle to be kicked across the horizon into the black hole, never to be heard from again, and for the positive energy particle to escape the black hole by being kicked in the opposite direction, in effect being a “real” particle.

This has many consequences. A black hole is not truly black, since it can have a glow due to the particles that escape in this manner. This glow is, aptly enough, called “Hawking radiation.” A second consequence is that since the black hole radiates energy, it could literally evaporate away! This also means that a black hole has a temperature, just like any other glowing object. Hawking along with others quickly discovered that they could even quantify the entropy, or level of disorder, of a black hole (the entropy of a black hole is proportional to the surface area of the event horizon).

The discovery of Hawking radiation started a flurry of theoretical work to understand all the quantum mechanical implications of black holes, and propelled Hawking to the forefront of research into gravity and quantum mechanics. His next major research topic was somewhat more ambitious.

## Hawking and the History of the Universe

Hawking turned his attention to an even more fundamental cosmological problem by the early 1980’s. Most of us have a picture of the big bang starting with an incredibly hot, dense universe filled with swarms of matter and antimatter particles. As the universe expanded and cooled, these particles begin to clump together to form the elementary particles we now see around us. Eventually, these formed into stable, neutral atoms such as hydrogen and helium. As further cooling occurred, the atoms form molecules, which aggregated into gas clouds, stars, galaxies, and other even larger scale structures. But what happened to the antimatter? For every atom of normal matter, we should be left with a corresponding antimatter equivalent, or the energy that results when the matter and antimatter annihilate each other. We see no evidence for this antimatter in our local cluster of galaxies. Where did it go?

This simple notion of the big bang has other shortcomings. It doesn’t tell us why our universe looks the way it does, with large clusters of galaxies separated from each other by even larger voids. It also does not tell us whether the universe will continue to expand forever, or eventually reach some maximum size and begin to contract.

One possible explanation was suggested by the Soviet scientist, Andre Linde. He noted that it was conceivable that the antimatter could be converted into matter if a very unusual set of conditions arose during the very first instance after the big bang. These conditions required that the universe undergo what physicists would call a “phase transition” – move from one type of quantum mechanical state to a second, more stable state. In addition, this phase transition had to allow the universe to grow very rapidly (or “inflate”) at the same time. During this inflationary period, what is normally a subtle effect called “CP-violation” (CP stands for charge-parity) would cause the antimatter to convert into matter, leaving us with the matter-dominated universe we see around is today.

This model, appropriately known as “inflation,” was in large part the brainchild of Alan Guth, a theoretical physicist who had worried about this matter-no antimatter problem. Together with a number of other cosmologists, Hawking took this idea and filled in many of the details. What ingredient in the theory ensured that the universe developed into what we see around is today? What is it in the theory that really gives us an arrow of time in this theory? What conditions had to be placed on the initial universe in order for the expansion to work out just right.

The first question is considered the “boundary condition problem,” and it formed a conceptual and practical roadblock to making predictions with the theory. The basic problem is that quantum mechanics in principle would allow an infinite number of different universes to exist. However, if we want to ask questions about one of them – ours in this case – we have to determine what specific conditions must be obeyed in order to get our universe out of the theory. Hawking proposed that you just avoid asking this question altogether by assuming that the structure of space-time at the moment of the big bang was such that there was no “past” but only future events. In effect, he and his collaborators conjectured that one did not have to worry about what was going on with the universe at a specific “boundary” in space-time, but that once you defined the type of universe you were in, you just had to let the universe evolve according to the laws of physics. Small, unpredictable quantum fluctuations during inflation then gave us the universe we see today.

This view allowed Hawking to consider the universe as a single quantum mechanical system, a view that he has maintained in his subsequent work. Based on this, he also tackled the question of whether the arrow of time, as we understand it, always points in the same direction, even if the universe contracts. Hawking himself waffled on this philosophical issue before being able to argue persuasively that, in a well-defined sense, the arrow of time would continue to point in the same direction.

The idea of inflation was completely new in the early 1980’s, and it required a great deal of fine-tuning to get it to agree with what we observe in the world around us. Now, two decades later, there is still some dispute about how well it describes the world. But it has been an extremely productive theory. It has led cosmologists to ask a number of key questions and prompt the right measurements. The recent mapping of the cosmic microwave background radiation performed by the COBE satellite is an example of the sorts of observations that have been prompted by predictions made by inflation. Hawking has continued to be at the forefront of this very exciting effort.

## Returning to the Big Problem

Over the last 15 years, Hawking has continued to work on the original “big problem” that formed part of Einstein’s legacy: How do we marry quantum mechanics and general relativity? Our observations of the world have given us few clues, mainly because gravity is such a weak force that probing it at a quantum scale has not been possible. We only have precise tests of gravity and of general relativity at the scale of about a centimeter, much larger than the atomic scale where quantum effects become important and easily observed.

Actually, it is much worse than that, because we do know comes from extrapolating our ideas back to the very earliest times in the universe and asking questions such as “What did gravity look like then? What would have been its effects?” Although we do not have definitive answers, the work of Hawking and others has shown that the nature of space-time only begins to illustrate quantum effects when we get down to an extraordinarily small distance, known as the Planck scale. This scale is so small it is even hard for a cosmologist to comprehend. The size of the atom (about 10^{-8} m) is about 1 part in 1026 of the size of the visible universe. The Planck scale is to the atom, as the atom is to the visible universe.

Hawking is currently trying to understand what the geometry of the universe looked like at the moment of the big bang. He has advocated the idea that one form of a mathematical solution to general relativity, known as the instanton, describes the state of the universe at the big bang, but that this instanton was not uniform or spherical, but shaped like a four-dimensional pea. Along with a Neil Turok, collaborator at Cambridge, he has developed this idea to show that it is compatible with his “no boundary” proposal. It complies with the concept of inflation to give us something like the universe we see today.

These ideas are imaginative and controversial. They show that Hawking continues to be a strong influence in the search for an understanding of how our universe evolved.

## Hawking Today and Tomorrow

Stephen Hawking’s relationship to the scientific community is first and foremost defined by his science, but it has also been shaped by Hawking’s complex personality.

Part of the problem is self-inflicted, due to his own hubris. He has been involved in several high-profile disputes with other scientist regarding priority, which are conflicts over who should be given credit for a scientific discovery. These disputes have at times been divisive, and in at least one case Hawking has had difficulty graciously admitting defeat. These have revealed a man who can be quite stubborn and unwilling to admit to error.

But part of his complex relationship with other researchers is due to his public appeal as a scientist who has captured the imagination of the person on the street. I believe there is distrust and at times even contempt among some in the scientific fraternity toward attempts to reach out to the public. Some of this results from the need to water down science to present it to a non-expert. However, the importance of such outreach in communicating the excitement of the research and the possibilities the work holds far outweighs any possible distortions that can occur.

Nonetheless, Hawking remains a historic figure, perhaps the most imminent physicist of his time. He also will continue to epitomize to the public the true character of the scientist struggling to understand how our world really works.

The universe continues to unfold today. And what is remarkable is that its future will be uncovered by understanding our past -- a past into which Hawking has perhaps the strongest insights and vision. I strongly recommend continuing to tune in to what he says. For this remarkable mind may one day give birth to the complete history of the physical universe.

## Notes

(1) There are four known forces or interactions. In increasing order of strength, they are gravity, the weak force, electromagnetism and the strong force. Gravity is an attractive force between any objects with mass. The weak force, as its name suggests, is a subtle force that we find at play in the nuclei of atoms – it is responsible for most radioactivity. Electromagnetism is the force that influences objects with electric charge. The strong force is indeed the strongest of the four forces. It holds together the protons and neutrons that make up atomic nuclei. However, its range or the distance over which it interacts is limited to the atomic scale.

(2) The identification of mass as one form of energy, E=mc^{2}, was another consequence of Einstein’s Special Theory of Relativity.

(3) Einstein found the whole idea of quantum mechanics repugnant, though he realized that he could not ignore it either. The comment “God does not play with dice” is widely attributed to Einstein, who found the probabilistic nature of quantum mechanics to be one of its most serious defects.

(4) This corresponds to about 10^{-5} m, or about the size of a good size molecule or a small virus.

(5) There are systems that interact gravitationally that appear stable, such as our solar system. Gravity does allow some forms of motion that are dynamically stable, such as the orbit of a planet about a star. However, if you peer more closely, you will see evidence of various levels of instability – take the rings of Saturn, for example, where each ring has formed in a chaotic manner.

## References

[1] Stephen Hawking, “A Brief History of Time”, New York, Bantam, 1988.

[2] See, for example, Michael White and John Gribbin, “Stephen Hawking: A Life in Science,” London, England, Viking, 1992.