Stephen Hawking, His Life and Work

by Kitty Ferguson

  Figure 10.7a (overleaf) shows two separate events (X and Y) on a spacetime diagram. They are connected by a world-line at more than a 45-degree angle from the time-line. No information can pass between these events without exceeding the speed of light. In a case like this, where the distance between two events is greater in space than it is in time, the square on the hypotenuse (side C) of our triangle is a positive number. In the language of physics the square of the ‘four-dimensional separation’ between events X and Y is positive.

  Figure 10.7b also shows two events. The distance between them is greater in time than it is in space. A world-line between these events runs at a less than 45-degree angle from the time-line. Information travelling at less than the speed of light can reach Y from X. When this is true, the square on the hypotenuse (side C) of our triangle is a negative number. Physicists say the square of the four-dimensional separation between X and Y is negative.

  Figure 10.7. A distinction between space and time

  Perhaps you got lost in those last two paragraphs. If you didn’t, a red light may have flashed on in your brain. The square of a number can’t be negative. That doesn’t happen in our mathematics. If the square of a number were a negative number, what number could possibly be its square root? What is the square root, for instance, of minus 9? In our mathematics the square of any number (negative or positive) is always positive: 3 squared (32) is 9; so is minus 3 squared (−32). We can’t possibly arrive at minus 9. It’s impossible for the square of anything to be a negative number.

  Stephen Hawking and other mathematicians and physicists have a way around this problem: imagine that there are numbers that do produce negative numbers when multiplied by themselves, and see what happens. Say that imaginary one, when multiplied by itself, gives minus one. Imaginary two, multiplied by itself, gives minus four. Calculate the sums-over-histories of the particles and sums-over-histories of the universe using imaginary numbers. Calculate them in ‘imaginary’ rather than ‘real’ time. The time it takes to get from point X to point Y in Figure 10.7b is imaginary time – the square root of minus nine – imaginary three.

  Imaginary numbers are a mathematical device (a trick, if you prefer) to help calculate answers that would otherwise be nonsense. ‘Imaginary time’ allows physicists to study gravity on the quantum level in a better way, and it gives them a new way of looking at the early universe.

  Smearing Out the Speed of Light?

  Travelling back into the very early universe, as space becomes more and more compressed, there are fewer possible choices about where a particle is (its position) at a given moment. The position becomes a more and more precise measurement. Because of the uncertainty principle this causes the measurement of the particle’s momentum to become less and less precise.

  First, let’s look at the photon, the particle of light, under more normal circumstances. Photons move at 186,000 miles (300,000 kilometres) per second, making the speed of light 186,000 miles (300,000 kilometres) per second. Now I have to tell you that this might not always be the case. (Having read this far, you are accustomed to such reversals!) We’ve seen that the probability of finding an electron is spread out over some region around the nucleus of an atom: more likely at some distances than others, but definitely a very smeary affair. Photons, like electrons, can’t simultaneously be pinned down precisely as to both position and momentum, because of the uncertainty principle.

  Just so, Richard Feynman and others have told us that the probability that a photon is travelling at 186,000 miles (300,000 kilometres) per second may be spread out over some ‘region’ around that speed. That’s the same as saying that, in one way of thinking about it, the speed of a photon fluctuates more or less around what we call light speed. Over long distances probabilities cancel out, so as to make the speed of a photon 186,000 miles (300,000 kilometres) per second. However, over very small distances, on the quantum level, there’s a possibility that a photon may move at slightly less or slightly more than this speed. These fluctuations won’t be seen directly, but the path of photons on the spacetime diagram, which we’ve drawn as a 45-degree angle, gets a little fuzzy.

  When we’re studying the very early universe, when space is very compressed, that line gets very fuzzy. The uncertainty principle means that the more precisely we measure the position of a photon, the less precisely we’re able to measure its momentum. When we say that in the very early universe everything was packed to near-infinite density (not a singularity, but nearly there), we’re becoming extraordinarily precise about the location of particles such as photons. When we are that precise about position, our imprecision about momentum vastly increases. As we near infinite density we also get near an infinite number of possibilities of what the speed of a photon is. What happens to our spacetime diagram now? Look at Figure 10.8. The world-line of a photon that in more normal circumstances is shown as a 45-degree angle becomes terribly smeared out. It fluctuates and ripples wildly.

  Here is another way of thinking about what causes this ‘rippling’, a way which will link it more clearly with other concepts in this book. Travelling back into the very early universe is like shrinking ourselves to a size so unimaginably tiny that we can see what’s happening on the level of the extremely small. Imagine it like this: if you look at this page, it seems smooth. You can curl the paper a bit, but it’s still smooth. In the same way, although there is some curvature, spacetime around us seems smooth. On the other hand, if you look at this page under a microscope, you see curves and bumps. Similarly, if you look at space-time on the extremely tiny level, billions on billions of times smaller than an atom, you find violent fluctuations in the geometry of spacetime (Figure 10.9) (see here). We’ll discuss this again in Chapter 12 and learn that it might result in something called ‘wormholes’. For the time being, the point is that we would find the same violent fluctuations in the very early universe, where everything was compressed to just such extreme smallness.

  Figure 10.8. The uncertainty principle in the early universe

  How can we explain this violent, chaotic scene? Again we turn to the uncertainty principle. We saw in Chapter 6 that the uncertainty principle also means that a field, such as an electromagnetic field or a gravitational field, can’t have a definite value and a definite rate of change over time. Zero would be a definite measurement, so a field can’t measure zero. All fields would have to measure exactly zero in empty space. So, no zero, no empty space. What do we have instead of empty space? A continuous fluctuation in the value of all fields, a wobbling a bit towards the positive and negative sides of zero so as to average out to zero but not be zero. This fluctuation can be thought of as the pairs of particles in Hawking radiation. Particle pair production is greater where the curvature of spacetime is most severe and changing most quickly. That’s why we expect to find so many of them at the event horizon of a black hole.

  Figure 10.9. The quantum vacuum, as imagined by John Wheeler in 1957, becomes more and more chaotic as you inspect smaller regions of space. At the scale of the atomic nucleus (top), space still looks very smooth. Looking much more closely than that (middle), we see a roughness begin to appear. At a scale 1,000 times smaller still (bottom), the curvature undergoes violent fluctuations.

  In the very early universe we find a situation of extremely great spacetime curvature and rapid change in that curvature. The quantum fluctuations in all fields, including the gravitational field, become very violent. If there are violent fluctuations in a gravitational field, that is the same as saying there are violent fluctuations in the curvature of spacetime. We are not talking about big curves, such as swells on the ocean. We are talking about all sorts of continuously changing crinkles and ripples and swirls. Odd things happen to the world-line of a photon in such a wild and weird environment. Again see Figures 10.8 and 10.9.

  Whichever of these explanations we prefer, the point is that the difference between the time direction and directions in space disappears. When time looks like space, we no longer
have our familiar situation in which the time direction always lies within the 45-degree angle and space directions always lie outside it.

  Hawking sums up what we have just seen: ‘In the very early universe, when space was very compressed, the smearing effect of the uncertainty principle can change the basic distinction between space and time.’ It’s no longer true that if points are farther apart in time than they are in space the square of their separation in four-dimensional spacetime (the square of the hypotenuse of our triangle) is necessarily a negative number. ‘It is possible for the square of [that] separation to become positive under some circumstances. When this is the case, space and time lose their remaining distinction – we might say that time becomes fully spatialized – and it is then more accurate to talk, not of spacetime, but of a four-dimensional space.’4

  When Time Gets Spaced Out

  What would this look like? How is this odd situation of four-dimensional space going to join smoothly with spacetime as we know it, in which time flows as time? Using imaginary time it’s possible to picture four-dimensional space, where time as we know it is non-existent, curving around and forming a closed surface, a surface without any edge or boundary. If you think you can picture this occurring in four dimensions, either you’re mistaken or else you’ve taken a fresh evolutionary step in brain development. Most of us are doomed to think about it in fewer dimensions. It’s easy to picture something with fewer dimensions that hasn’t any edge or boundary: the surface of a ball or the surface of the Earth.

  In the first Friedmann model of the universe, the universe was finite, not infinite, in size. But in that model it was also unbounded. It had no boundaries, no edges in space. It was like the surface of a ball: no edge, but not infinite in size. Hawking thinks the universe may be finite and unbounded in space and time. Time may have no beginning or end. All of it comes around and forms a closed surface, like the surface of the Earth.

  This leaves us fairly helpless. We can picture the surface of the Earth, and we can agree that it is finite and unbounded, but what would a universe that is finite and unbounded in space and time be like? It’s hard to make a mental connection between the shape of a ball and any meaningful concept of a four-dimensional universe. Just trying makes us feel quite blind – groping in the dark. Let’s see what else we can say about it that might be helpful.

  First, we’ll say what it would not be like. There would be no ‘boundary conditions’ – the way things were at the exact point of beginning – because there would be no point of beginning, no boundary, there. The whole thing would just curve around. Hawking suggests we state it exactly like that: the boundary conditions of the universe are that there are no boundaries. There would be no beginning and no end of the universe – anywhere. So don’t even think of asking, But what about before that? That’s like asking what’s south of the South Pole. A signpost pointing ‘south’ has no meaning at the South Pole. An arrow of time indicating ‘this way to the past’ has no meaning when the time dimension has become ‘spacey’.

  If there were no before and after the universe in the time dimension, would there be any ‘elsewhere’, any other place, any outside of such a universe – in space dimensions? Hawking’s model doesn’t say that there isn’t. Can you have an outside when you don’t have any boundary? In the ball model there is a sense in which we do. It’s in the direction the ant on the surface of the balloon in Chapter 6 would see if it could look ‘out’ from the surface – which, you’ll remember, it can’t do. That dimension doesn’t exist for the ant, but that doesn’t necessarily mean it doesn’t exist at all. The idea of having ‘elsewheres’ in space but none in time (no before or after) fits nicely with the idea that the time we live in is only a temporary mutation of what is really a fourth space dimension.

  Since all of this may seem too complicated to be meaningful, let’s look at it in another, more practical, way. Ask again, What would a universe that is finite and unbounded in space and time be like? The calculations are extremely difficult. However, what they appear to be telling us is that a universe like that could be like our own.

  As Hawking described it:

  They predict that the universe must have started out in a fairly smooth and uniform state. It would have undergone a period of what is called exponential or ‘inflationary’ expansion, during which its size would have increased by a very large factor but the density would have remained the same. The universe would then have become very hot and would have expanded to the state that we see it in today, cooling as it expanded. It would be uniform and the same in every direction on very large scales but would contain local irregularities that would develop into stars and galaxies.5

  In real time – and that’s where we live – it would still appear to us that we have singularities at the beginning of the universe and inside black holes.

  Hawking and Jim Hartle presented the physics community with this no-boundary model of the universe in 1983. Hawking liked to emphasize that it was just a proposal. He hadn’t deduced these boundary conditions from some other principle. The model appealed to him. He thought ‘that it really underlies science because it is really the statement that the laws of science hold everywhere’.6 There are no singularities at which they break down. This kind of universe is self-contained. Do we have to explain how it was created? Would it have to be created at all? ‘It would just BE,’ writes Hawking.7

  ‘What Place, Then, for a Creator?’

  This raises some sticky philosophical questions. As Hawking puts it, ‘If the universe has no boundaries but is self-contained … then God would not have had any freedom to choose how the universe began.’8

  Hawking hasn’t said that the no-boundary proposal rules out the existence of God, only that God wouldn’t have had any choice in how the universe began. Other scientists disagree. They don’t think the no-boundary proposal limits God very much. If God had no choice, we still have to wonder who decided that God would have no choice. Perhaps, suggests the physicist Karel Kuchar, that was the choice God made. Don Page, who reviewed A Brief History of Time for the journal Nature in England, has a similar viewpoint. Page, of course, was Hawking’s graduate assistant in the late 1970s. He had moved on to become a professor at the University of Alberta, Edmondton, Canada. He and Hawking were still good friends and were continuing to collaborate on scientific papers, and Hawking was well aware that Page was likely to come up with some arguments to refute the notion that the no-boundary proposal abolished the need for a Creator. Indeed he did.

  To Hawking’s question, ‘What place, then, for a Creator?’, Page answered that in the Judeo-Christian view ‘God creates and sustains the entire Universe rather than just the beginning. Whether or not the Universe has a beginning has no relevance to the question of its creation, just as whether an artist’s line has a beginning and an end, or instead forms a circle with no end, has no relevance to the question of its being drawn.’9 A God existing outside our universe and our time wouldn’t need a ‘beginning’ in order to create, but it could still look to us, from our vantage point in ‘real’ time, as though there had been a ‘beginning’.

  In A Brief History of Time Hawking himself would suggest that there may still be a role for a Creator: ‘Is the unified theory so compelling that it brings about its own existence?’ If not, ‘What is it that breathes fire into the equations and makes a universe for them to describe?’10 In the book A Brief History of Time: A Reader’s Companion, the companion book for the film, he would say that if the no-boundary proposal is correct, he will have succeeded in discovering how the universe began. ‘But I still don’t know why it began.’11 He intended to find out, if he possibly could.

  All of which brings us to a word of caution: although theoretical physicists ask challenging, penetrating questions and present us with mind-blowing proposals and theories, they do not claim to give us ‘ultimate answers’ – even though the subtitle of a later book co-authored by Hawking might suggest that they do. The best science progress
es by suggesting ‘answers’ and then taking apart and disproving those ‘answers’. The most daring and imaginative scientists launch their toy boats and then, it seems, try extremely hard to make them sink.

  Hawking’s work is a prime example. First he proved that the universe had to start as a singularity. Then with his no-boundary proposal he showed us how there might be no singularity after all. In the meantime he told us that black holes could never get smaller, and then he discovered they could. His work on the Big Bang singularity seemed consistent with a biblical view of Creation, but his no-boundary proposal put the Creator out of a job or at least changed the job description. In A Brief History of Time he suggested that we might need the Creator after all, and ‘the ultimate triumph of human reason’ would be to ‘know the mind of God’.12 Hawking is provocative and open-minded in the way that the greatest thinkers have always been. He reaches clearly defined, well-supported conclusions, and then in the next breath he mercilessly questions and breaks down those same conclusions. He doesn’t hesitate to admit that an earlier conclusion was incorrect or incomplete. That’s the way his science – and perhaps all good science – advances, and one of the reasons why physics seems so full of paradoxes.

  In the process, Hawking has supplied eloquent quotations that can be used to support opposing philosophical points of view. He’s been quoted and misquoted by those who believe in God and those who do not. He’s been the hero and villain of both camps. However, those who depend on his statements – or statements of other scientists – to support their belief or unbelief risk having the rug swept out from under them at any moment.

  Meanwhile, although it may appear to us that Hawking completely reversed himself with the no-boundary proposal, he didn’t see it that way. He said that the most important thing about his work on singularities was that it showed that a gravitational field must become so strong that you can’t ignore quantum effects. And when you stop ignoring quantum effects, you find out that the universe could be finite in imaginary time but have no boundaries or singularities.