A Choice of Universes
Before Hubble demonstrated that the universe is expanding, belief in a static universe (one that isn’t changing in size) was very strong, so much so that when Einstein produced his theory of general relativity in 1915, and that theory predicted the universe was not static, Einstein was so sure it was static that he revised his theory. He put in a ‘cosmological constant’ to balance gravity. Without this cosmological constant the theory of general relativity predicted what we now know to be true: the universe is changing in size.
A Russian physicist, Alexander Friedmann, decided to take Einstein’s theory at face value without the cosmological constant. Doing so, he predicted what Hubble would discover in 1929: The universe is expanding.
Friedmann began with two assumptions: (1) the universe looks much the same in whatever direction you look (except for nearby things like the shape of our Milky Way galaxy and our solar system); (2) the universe looks like this from wherever you are in the universe. In other words, no matter where you travel in space, the universe still looks much the same in whatever direction you look.
Friedmann’s first assumption is fairly easy to accept. The second isn’t. We don’t have any scientific evidence for or against it. Hawking says, ‘We believe it only on grounds of modesty: it would be most remarkable if the universe looked the same in every direction around us, but not around other points in the universe!’3 Perhaps remarkable, but not impossible, you may argue. Modesty seems no more logical a reason for believing something than pride. However, physicists tend to agree with Friedmann.
In Friedmann’s model of the universe all the galaxies move away from one another. The farther apart two galaxies are, the more rapidly they move away from one another. This agrees with what Hubble observed. According to Friedmann, wherever you travel in space you’ll still find all the galaxies moving away from you. In order to understand this, imagine an ant crawling on a balloon that has evenly spaced dots painted on it. You have to pretend the ant can’t see the dimension that would allow it to look ‘out’ from the surface. Nor is it aware that the balloon has an interior. The ant’s universe involves only the surface of the balloon. It looks the same in any direction. No matter where the ant crawls on the balloon, it sees as many dots ahead of it as behind. If the balloon is getting larger, the ant sees all the dots move away, no matter where it stands on the surface. The balloon ‘universe’ fits Friedmann’s two assumptions: it looks the same in all directions. It looks the same no matter where you are in it.
What else can we say about the balloon universe? It isn’t infinite in size. The surface has dimensions we can measure, like the surface of the Earth. No one would suggest that the surface of the Earth is infinite in size. However, it also has no boundaries, no ends. Regardless of where the ant crawls on the surface, it never comes up against any barrier, finds any end to the surface, or falls off an edge. It eventually gets back to where it started.
In Friedmann’s original model, space is like that, with three dimensions rather than two. Gravity bends space around on to itself. The universe is not infinite in size, but neither does it have any end, any boundary. A spaceship will never get to a place in space where the universe ends. That may be difficult to understand, because we tend to think of infinite as meaning ‘having no end’. The two do not have the same meaning.
Hawking points out that although the idea of circumnavigating the universe and ending up where you started makes great science fiction, it doesn’t work, at least with this Friedmann model. You’d have to break the speed limit of the universe (the speed of light) – which isn’t allowed – to get all the way around before the universe ends. It’s an extremely large balloon. We are extremely small ants.
Time in this Friedmann model, like space, isn’t infinite. It can be measured. Time, unlike space, does have boundaries: a beginning and an end. Look at Figure 6.1a. The distance between two galaxies at the beginning of time is zero. They move apart. The expansion is slow enough and there is enough mass in the universe so that eventually gravitational attraction stops the expansion and causes the universe to contract. The galaxies move towards each other again. At the end of time the distance between them is once again zero. That may be what our universe is like.
Figures 6.1b and 6.1c show two other possible models that would also obey Friedmann’s assumptions (that the universe looks the same in every direction and that it looks the same from wherever you are in the universe). In Figure 6.1b, the expansion is much more rapid. Gravity can’t stop it, though it does slow it a little. In Figure 6.1c, the universe is expanding just fast enough not to collapse, but not as fast as in Figure 6.1b. The speed at which galaxies are moving apart grows smaller and smaller, but they always continue to move apart. If the universe is like either of these two models, space is infinite. It doesn’t curve back around on to itself.
Figure 6.1. Three models that obey Friedmann’s assumptions that the universe looks much the same in any direction we look, and the universe looks much the same from wherever you might be in the universe.
Which model fits our universe? Will the universe collapse some day or go on expanding for ever? It depends on how much mass there is in the universe: how many votes there are in the entire democracy. It will take much more mass than we presently observe to close the universe. That is a very simple statement of a more complicated issue, as you’ll see later.
Penrose’s theory about stars collapsing and becoming singularities only worked with a universe infinite in space that will go on expanding for ever (as in Figures 6.1b and 6.1c), not collapse (as in Figure 6.1a). Hawking first set out to prove that a universe infinite in space not only would have singularities in black holes but also must have begun as a singularity. He was confident enough by the time he finished his thesis to write: ‘There is a singularity in our past.’4
In 1968, Hawking’s and Penrose’s essay on the beginning of time won second place in the Gravity Research Foundation Awards, but the question still hung in the balance: what if Friedmann’s first model was correct, where the universe is not infinite in space and eventually recollapses (Figure 6.1a)? Must that sort of universe also have begun as a singularity? By 1970 Hawking and Penrose were able to show that it must have. Their definitive statement on the subject, in the 1970 Proceedings of the Royal Society,5 was a joint paper proving that if the universe obeys general relativity and fits any of the Friedmann models, and if there is as much matter in the universe as we observe, the universe must have begun as a singularity, where all the mass of the universe was compressed to infinite density, where spacetime curvature was infinite.
Physical theories can’t really work with infinite numbers. When the theory of general relativity predicts a singularity of infinite density and infinite spacetime curvature, it’s also predicting its own breakdown. In fact, all our scientific theories break down at a singularity. We lose our ability to predict. We can’t use the laws of physics to say what would emerge from the singularity. It could be any sort of universe. And what about the question of what happened before the singularity? It’s not even clear that this question has any meaning.
A singularity at the beginning of the universe would mean that the beginning of the universe is beyond our science, beyond anything that claims to be a Theory of Everything. We would simply have to say, time began, because we observe that it did, and that in itself is a very big arbitrary element. A singularity is a door slammed in our faces.
Bedtime Story
Physicists are notorious for being eternally preoccupied with their physics. Even more than most of his colleagues, and partly as a consequence of his disability, Hawking was able to take his work with him anywhere, any time, because it was almost all done in his head. As Kip Thorne described it, he had developed a very unusual ability to manipulate mental images of objects, curves, surfaces, shapes, not merely in three dimensions but in spacetime’s four dimensions.6
Typical of Hawking’s work mode was a bedtime discovery he des
cribed in his book A Brief History of Time: ‘One evening in November of 1970, shortly after the birth of my daughter, Lucy, I started to think about black holes as I was getting into bed. My disability makes this rather a slow process, so I had plenty of time.’7 Another physicist might have nipped over to his desk and scribbled some notes and equations, but Hawking made one of the most significant discoveries of his career in his head, got into bed, and lay awake for the rest of the night, eager for dawn to break so that he could phone Penrose and tell him about his new insight. Penrose, Hawking insists, had thought of it but had not realized the implications.
The idea that had struck Hawking was that a black hole can never get smaller because the area of an event horizon (the radius-of-no-return where escape velocity becomes greater than the speed of light) can never decrease.
To review briefly, a collapsing star reaches a radius where escape velocity is the speed of light. What happens to photons emitted by the star as it collapses past that radius? Gravity there is too strong to allow them to escape, but not strong enough to pull them into the black hole. They stay there, hovering. That radius is the event horizon. After that as the star continues to shrink, any photons it emits are drawn back in.
What Hawking realized was that the paths of light rays hovering at the event horizon cannot be paths of light rays that are approaching one another. Paths of light rays that approach one another would bash into one another and fall into the black hole, not hover. In order for the area of the event horizon to get smaller (and the black hole to get smaller), paths of light rays in the event horizon would have to approach one another. But, if they did, they would fall in, and the event horizon would not get smaller.
Another way of thinking about this is to realize that a black hole can get larger. The size of a black hole is determined by its mass, so a black hole gets larger any time anything new falls in and adds to that mass. If nothing can get out of a black hole, its mass can’t possibly decrease. A black hole can’t get smaller.
Hawking’s discovery became known as the second law of black hole dynamics: the area of the event horizon (the border of the black hole) can stay the same or increase but never decrease. If two or more black holes collide and form one black hole, the area of the new event horizon is as big as or bigger than the previous event horizons added together. A black hole can’t get smaller or be destroyed or divided into two black holes, no matter how hard it might get zapped. Hawking’s discovery had a familiar ring to it. It resembled another ‘second law’ in physics: the second law of thermodynamics, which is about entropy.
Entropy is the amount of disorder there is in a system. Disorder always increases, never decreases. An assembled jigsaw puzzle put carefully in a box might get jostled, mixing the pieces and spoiling the picture. But it would be very surprising if any jostling of the box caused a mess of unassembled pieces to fall into place and complete the puzzle picture. In our universe, entropy (disorder) always increases. Broken teacups never reassemble themselves. A messy room never straightens itself up.
Suppose you patch the teacup or tidy up the room. Something does become more ordered. Does entropy decrease? No. The mental and physical energy you burn in the process converts energy to a less useful form. That represents a decrease in the amount of order in the universe which outbalances any increase of order you achieved.
There’s another way in which entropy resembles the event horizon of a black hole. When two systems join, the entropy of the combined system is as great as or greater than the entropy of the two systems added together. A familiar example describes gas molecules in a box. Think of them as little balls bouncing off one another and off the walls of the box. There’s a partition down the centre of the box. Half the box (one side of the partition) is filled with oxygen molecules. The other half is filled with nitrogen molecules. Remove the partition, and oxygen and nitrogen molecules start to mix. Soon there’s a fairly uniform mixture throughout both halves of the box, but that’s a less ordered state than when the partition was in place: entropy – disorder – has increased. (The second law of thermodynamics doesn’t always hold: there is the tiniest of chances, one in many millions of millions, that at some point the nitrogen molecules will be back in their half of the box and the oxygen molecules in the other.)
Suppose you toss the box of mixed-up molecules or anything else that has entropy into a convenient black hole. So much for that bit of entropy, you might think. The total amount of entropy outside the black hole is less than it was before. Have you managed to violate the second law? Someone might argue that the whole universe (inside and outside black holes) hasn’t lost any entropy. But the fact is that anything going into a black hole is just plain lost to our universe. Or is it?
One of John Wheeler’s graduate students at Princeton, Demetrios Christodoulou, pointed out that according to the second law of thermodynamics, the entropy (disorder) of a closed system always increases, never decreases, and that similarly the ‘irreducible mass’ (Christodoulou’s name for a mathematical combination between a black hole’s mass and its speed of rotation) never decreases, no matter what happens to the black hole. Was this resemblance only a coincidence? What connection could Christodoulou’s idea or Hawking’s more general and powerful statement8 (the never-decreasing area of the event horizon) have with entropy and the second law of thermodynamics?
Escape from a Black Hole?
In his first announcement to the scientific community of his idea about the event horizon of a black hole never getting smaller, in December 1970 at the Texas Symposium of Relativistic Astrophysics,9 Hawking insisted that though an increase in the area of the event horizon did indeed resemble an increase of entropy, this was only an analogy.
Another of Wheeler’s graduate students at Princeton, Jacob Bekenstein, begged to differ. Bekenstein insisted that the area of the event horizon of a black hole isn’t only like entropy; it is entropy.10 When you measure the area of the event horizon, you’re measuring the entropy of the black hole. You don’t destroy entropy if you toss it into a black hole. The black hole already has entropy. You only increase it. When something falls into a black hole, such as a box of molecules, it adds to the mass of the black hole, and the event horizon gets larger. It also adds to the entropy of the black hole.
All of this brings us to a puzzling point. If something has entropy, that means it has a temperature. It is not totally cold. If something has a temperature, it has to be radiating energy. If something is radiating energy, you can’t say that nothing is coming out. Nothing was supposed to come out of black holes.
Hawking thought Bekenstein was mistaken. He was irritated by what he thought was Bekenstein’s misuse of his discovery that event horizons never decrease. In 1972 and 1973 he joined forces with two other physicists, James Bardeen and Brandon Carter, and seemed to tread close to a concession by coming up with no fewer than four laws of black hole mechanics that appeared to be almost identical to the four well-known laws of thermodynamics ‘if one only replaced the phrase “horizon area” with “entropy,” and the phrase “horizon surface gravity” with “temperature”’.11 Nevertheless the three authors continued to stress that these were only analogies, and in the final version of their paper12 reiterated that their four laws of black hole mechanics were similar to, but distinct from, the four laws of thermodynamics. Although there were many similarities between entropy and the area of the event horizon, a black hole could not have entropy because it could not emit anything. That was an argument that Bekenstein could not gainsay, but even though he was a graduate student up against a trio of established physicists, he was not convinced. It turned out that Hawking, Bardeen and Carter were wrong. It would fall to Hawking himself to show how.
In 1962 when Hawking had begun his graduate studies at Cambridge, he’d chosen cosmology, the study of the very large, rather than quantum mechanics, the study of the very small. Now, in 1973, he decided to shift ground and look at black holes through the eyes of quantum mechanics. It was
to be the first serious, successful attempt by anybody to fuse the two great theories of the twentieth century: relativity and quantum mechanics. Such a fusion, you’ll remember from Chapter 2, is a difficult hurdle on the road to a Theory of Everything.
In January 1973, Hawking was thirty-one years old. The new year brought the publication of his first full-length book, co-authored with George Ellis and dedicated to Denis Sciama. Hawking describes The Large Scale Structure of Space-Time as ‘highly technical, and quite unreadable’.13 It still appears on the shelves of academic bookshops, and if you pull it down, and are not an accomplished physicist, you will probably agree with him. Although it will never match the sales of A Brief History of Time, it has become a classic in the field.
In August and September of that year, during Cambridge’s long vacation, Stephen and Jane Hawking travelled to Warsaw for the celebration of the 500th anniversary of the birth of Nicolaus Copernicus, and continued east from there to Moscow. They asked Kip Thorne to go with them, because he had been carrying on joint research with Soviet physicists for five years and knew the ropes in the Soviet Union. Hawking wanted to confer with Yakov Borisovich Zel’dovich and Zel’dovich’s graduate student Alexander Starobinsky. These two Russian physicists had been able to show that the uncertainty principle meant that rotating black holes would create and emit particles, produced by the hole’s rotational energy. The radiation would come from just outside the event horizon and would slow down the black hole’s rotation until the rotation stopped and the radiation ceased. Hawking thought Zel’dovich and Starobinsky were on to something, but he wasn’t satisfied with their calculations. After the visit he returned to Cambridge determined to devise a better mathematical treatment.
Stephen Hawking, His Life and Work Page 9
