Hawking had not been wrong to be dissatisfied with Linde’s ‘new inflation’. Linde had suggested a solution to the problem of the bubbles of broken symmetry not being able to join with one another: suppose the bubbles were large enough so that what would later evolve into our region of the universe could be all inside one bubble. In order for this to be possible, the change from symmetry to broken symmetry would have had to happen much more slowly inside the bubble. Hawking objected that Linde’s ‘new inflation’ bubbles would have had to be too large – larger than the entire universe was at the time all this was happening. The theory also still predicted much larger variations in the temperature of the microwave background radiation than had been observed.
Not long after the Moscow conference Hawking made a trip to Philadelphia. In an address there, as recipient of the Benjamin Franklin Medal in Physics from the Franklin Institute, he departed from a strictly scientific topic to talk about something that had been of serious concern to both himself and Jane since the early years of their marriage – the dangerous threat that the USSR’s and the United States’ escalating nuclear stockpiles were posing to life on Earth. Back in 1962, Diana King had mentioned in Jane Wilde’s hearing: ‘He goes on Ban the Bomb marches.’ Hawking was still marching.
Soon after his return to Cambridge, however, Hawking was back in the inflation discussion. He received in his post a letter from Physics Letters asking him to review Linde’s inflation paper for publication.14 Hawking recommended that they publish the paper,15 in spite of the fact that both he and Linde recognized that there were flaws. The paper was significant and deserved to be widely read, and if Linde were to make the necessary revisions, they would be delayed too long passing through Soviet censorship. At the same time Hawking and a graduate student, Ian Moss, submitted a paper of their own suggesting what they thought was a more satisfactory way to end the inflationary period: if the symmetry had broken (still slowly as Linde had proposed) not just inside bubbles but everywhere at the same time, the result would be the uniform universe that we live in.16 With all these ideas in the air, Hawking and his DAMTP colleague Gary Gibbons decided to organize a workshop, focusing mainly on inflation, to take place the following summer. Hawking’s super-capable secretary, Judy Fella, got to work.
In January 1982 Hawking turned forty, a birthday he had not expected to reach. There was more to celebrate. Hawking was on the New Year’s Honours list, made a Commander of the British Empire. At the investiture ceremony at Buckingham Palace on 23 February, Robert was his father’s assistant. Now Hawking could put CBE after his name.
Making Inflation Work
From 21 June to 9 July 1982, the wizards of inflation finally put their heads together in Cambridge at the Nuffield Workshop on the Very Early Universe. Andrei Linde came over from Russia. Alan Guth was there, and so was Paul Steinhardt, a physicist at the University of Pennsylvania, who with his colleague Andreas Albrecht had come up with a ‘new inflation’ theory independently of Linde and very similar to his, at about the same time.fn2 Hawking’s contribution to the meeting was to show how the temperature of the universe during the inflationary period could inevitably lead to small density perturbations.17
Later in the summer Hawking flew to California again, this time to Santa Barbara, to spend some weeks at the University of California’s new Institute of Theoretical Physics. This was Jim Hartle’s home territory, and while Hawking was there the two of them discussed the idea that he had introduced at the Vatican in September 1981.
The ‘no-boundary proposal’ had taken something of a back seat during all the discussion of inflation theory, but Hawking had not stopped thinking seriously about it. Over the next two years, he and Hartle worked out that proposal.
fn1 Hawking’s comment in A Brief History of Time that the Pope had said scientists ‘should not inquire into the moment of Creation’ was either a misquotation or a mistranslation of the Pope’s words.
fn2 Linde, Steinhardt and Albrecht now are given joint credit for the ‘new inflation model’.
10
‘In all my travels, I have not managed to fall off the edge of the world’
HAWKING’S SUGGESTION THAT black holes emit radiation had been greeted at first with scepticism in 1974, but we’ve seen that most physicists soon came to agree it wasn’t nonsense after all. Black holes must radiate as any hot body does if our other ideas about general relativity and quantum mechanics aren’t badly out. No one has found a primordial black hole, but if one were discovered, physicists would be shocked to find it not emitting a shower of gamma rays and X-rays.
Go back to thinking about the particles that are emitted by a black hole in Hawking radiation. A pair of particles appears at the event horizon. The particle with negative energy falls into the black hole. The fact that its energy is negative means we have energy subtracted from the black hole. What happens to that energy? (Recall that we don’t think energy can simply disappear from the universe.) It is carried off into space with the positive energy particle (see Chapter 6).
The upshot, you’ll remember, is that the black hole loses mass and its event horizon shrinks. For a primordial black hole the whole story may end with the black hole’s disappearing completely, probably with an impressive fireworks display. How can something escape from a black hole if nothing can escape from a black hole? It really was one of the great ‘locked room mysteries’ of all time, solved by ‘S. H.’
The idea that matter in a black hole didn’t necessarily reach the absolute end of time at a singularity had triggered suspicions about another singularity: the singularity Hawking had decided earlier was at the absolute beginning of time. Quantum theory offered a fresh possibility: maybe the Big Bang singularity is, as Hawking terms it, ‘smeared away’. Maybe the door isn’t slammed in our faces after all.
Hawking points to a similar problem which quantum theory solved early in our century, a problem related to Rutherford’s model of the atom: ‘There was a problem with the structure of the atom, which was supposed to consist of a number of electrons orbiting around the central nucleus, like the planets around the Sun’ (look back at Figure 2.1). ‘The previous classical theory predicted that each electron would radiate light waves because of its motion. The waves would carry away energy and so would cause the electrons to spiral inwards until they collided with the nucleus.’1 Something had to be wrong with this picture, because atoms don’t collapse in this manner.
Quantum mechanics, with the uncertainty principle, came to the rescue. We can’t know simultaneously both a definite position and a definite momentum for an electron. ‘If an electron were to sit on the nucleus, it would have both a definite position and a definite velocity,’ Hawking points out. ‘Instead, quantum mechanics predicts that the electron does not have a definite position but that the probability of finding it is spread out over some region around the nucleus.’ The electrons don’t spiral inwards and hit the nucleus. Atoms don’t collapse.
According to Hawking, ‘The prediction of classical theory [that we will find the electrons at the nucleus] is rather similar to the prediction of classical general relativity that there should be a Big Bang singularity of infinite density.’2 Knowing that everything is at one point of infinite density at the Big Bang or in a black hole is too precise a measurement to be allowed by the uncertainty principle. To Hawking’s way of thinking this principle should ‘smear out’ the singularities predicted by general relativity, just as it smeared out the positions of electrons. There is no collapse of the atom, and he suspected that there was no singularity at the beginning of the universe or inside a black hole. Space would be very compressed there, but probably not to a point of infinite density.
The theory of general relativity had predicted that inside a black hole and at the Big Bang the curvature of spacetime becomes infinite. If that doesn’t happen, then Hawking wanted to work out ‘what shape space and time may adopt instead of the point of infinite curvature’.3
When Time is Time an
d Space is Space
If you find the following discussion difficult, don’t hesitate to skim over it. It isn’t necessary to understand every word to appreciate Hawking’s theory, but it’s more interesting if you can. Of course, the maths Hawking uses to describe it, and that you and I would need in order to understand him completely, is much more complicated than the simple maths you will find here.
Relativity theory links space and time in four-dimensional spacetime: three dimensions of space and one of time. Take a look at what a spacetime diagram is like. Overleaf is one that I once drew showing my daughter Caitlin on her way from her classroom at school to the room where the children ate their lunch. The vertical line on the left represents the passage of time. The horizontal line at the bottom represents all the space dimensions. Any single point on our spacetime diagram represents a position in space and a moment in time. Let’s see how this works.
The diagram (Figure 10.1) begins with Caitlin at her desk in her classroom, at 12.00 noon. She sits still, moving forward in time but going nowhere in space. On the diagram a little band of ‘Caitlin’ moves forward in time. At 12.05 the bell rings. Caitlin moves towards the lunch room. (Her desk still moves forward in time but goes nowhere in space.) Caitlin moves in both time and space. At 12.07 she pauses to retie her trainer. For one minute she moves forward in time but does not move in space. At 12.08 she’s off again towards the lunch room, walking a little faster than before so that the food won’t all be gone by the time she gets there. At 12.15 she arrives at the lunch room. A physicist would say we have traced Caitlin’s ‘world-line’.
That spacetime diagram was a very sketchy affair. When physicists draw a spacetime diagram, they often use a common unit for both space and time. They might, for instance, use one yard as the unit of both space and time. (One yard of time is very small, only billionths of a second. It’s the time it takes a photon, which moves at the speed of light, to travel one yard.) In such a spacetime diagram, if something moves four yards in space and four yards in time, its world-line traces a 45-degree angle. That’s the world-line for something moving at the speed of light, a photon, for instance (Figure 10.2).
Figure 10.1. Caitlin in Spacetime
If something moves three yards in space and four in time, it’s moving at three-fourths the speed of light (Figure 10.3a). If something moves four yards in space and three in time, it’s exceeding the speed of light, which isn’t allowed (Figure 10.3b).
Figure 10.2. A spacetime diagram using one yard as the unit of both space and time. If something travels four yards in space and four yards in time, its ‘world-line’ traces a 45-degree angle on a spacetime diagram. That’s the world-line for a photon, or anything else moving at the speed of light.
The next diagram (Figure 10.4) shows two events happening simultaneously. They have no way of knowing about each other at the instant they happen because, for them to do so, the in formation would have to have a world-line running at a 90-degree angle from the time-line. Travelling such a world-line would require faster-than-light travel. Nothing can travel faster than light and light can’t manage anything greater than a 45-degree angle on the diagram.
Figure 10.3. (a) A spacetime diagram showing the world-line traced by something moving three yards in space and four yards in time: three-quarters the speed of light.
Figure 10.3. (b) World-line traced by something moving four yards in space and three yards in time. When the distance travelled is greater in space than in time, as in this case, the object is exceeding the speed of light (not allowed!).
Figure 10.4. A spacetime diagram showing two events (X and Y) that occur simultaneously but at a distance from each other in space. They can’t know about each other at the exact moment they occur, because that would require information from one to the other to travel a world-line running at greater than a 45-degree angle from the time-line. A world-line running at greater than a 45-degree angle requires faster-than-light travel. That’s not allowed in our universe.
Now we’ll talk about the ‘length’ of a world-line. How shall we say what the length of a world-line is – a ‘length’ that takes into account all four dimensions?
Let’s examine the world-line of something that moves a lot faster than Caitlin. The object in Figure 10.5 moves four yards in space and five in time: four-fifths the speed of light. Think of the distance it moves in the ‘space’ direction on the diagram as one side of a triangle (side A). Think of the distance it moves in the ‘time’ direction on the diagram as a second side (side B). That makes two sides of a right-angled triangle. The world-line of the moving object is the hypotenuse of that triangle (side C).
Most of us have learned that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. The square of 4 (side A) is 16. The square of 5 (side B) is 25. The sum of 16 and 25 is 41. The length of side C, the hypotenuse, would be the square root of 41.
Never mind trying to find that square root. That is another issue. It would be our next challenge if we were working with our familiar school geometry. However, for spacetime things work differently. The square of the hypotenuse (side C) is not equal to the sum of the squares of the other two sides. It’s equal to the difference between the squares of the two other sides. Our object travels four yards in space (side A of the triangle) and five yards in time (side B). The square of 4 is 16; the square of 5 is 25. The difference between 25 and 16 is 9. The square root of 9 is 3. So we know that the third side of the triangle, side C, the world-line of our travelling object, is three yards in length in spacetime.
Figure 10.5. A right-angled triangle, using the distance travelled in space as side A, the distance travelled in time as side B, and the world-line travelled in spacetime as side C, the hypotenuse.
Let’s say, just for the fun of it, that the object is someone wearing a watch. The watch will show that length (three yards) as ‘time’. In Figure 10.6 (see p. 206) Lauren remains stationary in space and measures five hours on her watch. Her twin brother Tim, moving at four-fifths the speed of light, meanwhile measures only three hours on his. Tim turns around and returns, again measuring three hours while Lauren measures five. Tim is slightly younger than Lauren when next they meet. This is one of the remarkable, unbelievable things that Einstein taught us about the universe.
Now let’s consider spacetime diagrams and world-lines of some smaller objects, elementary particles.
‘Sums-Over-Histories’, or: The Likelihood of Visiting Venus
Remember the smeared-out positions of electrons in the model of the atom we talked about earlier. Their positions were ‘smeared out’ because we couldn’t measure simultaneously both the position and the momentum of any one of them very precisely. Richard Feynman had a way of dealing with this problem which we now call ‘sums-over-histories’.
Imagine that you are considering all the different routes Red Riding Hood might take from home to her grandmother’s cottage – not just the quickest way as the crow flies or the safest route avoiding as much as possible the wolf-infested woods, but every possible route she might take. There are billions and billions of possible routes. You ultimately get a gigantic fuzzy picture of her making the journey by all these routes at once. However, some are certainly more likely than others. If you study the probabilities of her taking the various routes, you conclude that she is very unlikely at any time between home and grandmother to be found on the planet Venus, for instance. But according to Feynman you must not completely rule out her passing through there. The probability for that route is extremely low, but not zero.
In a similar manner, with sums-over-histories, physicists work out every possible path in spacetime that could have been travelled by a certain particle, all the possible ‘histories’ the particle could have had. It’s possible then to calculate the probability of a particle’s having passed through a particular point, something like calculating how likely Red Riding Hood is to travel by way of the planet Venus. (You don’t wa
nt to get the idea, however, that particles choose a path. That would be carrying the analogy too far.)
Figure 10.6. The ‘twin paradox’
Hawking decided to put sums-over-histories to another use, to study all the different histories the universe could have and which are more probable than others.
As we continue, you’ll need to know that even though the theory of relativity taught us to think of three dimensions of space and one of time as four dimensions of spacetime, there are still physical differences between space and time. One of these differences has to do with the way we measure the four-dimensional distance between two points in spacetime: the hypotenuse of the aforementioned triangle.
Stephen Hawking, His Life and Work Page 16
