A Briefer History of Time

by Stephen Hawking

  A large body, such as the earth or the sun, contains nearly equal numbers of positive and negative charges. Thus, the attractive and repulsive forces between the individual particles nearly cancel each other out, and there is very little net electromagnetic force. However, on the small scales of atoms and molecules, electromagnetic forces dominate. The electromagnetic attraction between negatively charged electrons and positively charged protons in the nucleus causes the electrons to orbit the nucleus of the atom, just as gravitational attraction causes the earth to orbit the sun. The electromagnetic attraction is pictured as being caused by the exchange of large numbers of virtual particles called photons. Again, the photons that are exchanged are virtual particles. However, when an electron changes from one orbit to another one nearer to the nucleus, energy is released and a real photon is emitted—which can be observed as visible light by the human eye, if it has the right wavelength, or by a photon detector such as photographic film. Equally, if a real photon collides with an atom, it may move an electron from an orbit nearer the nucleus to one farther away. This uses up the energy of the photon, so it is absorbed.

  The third category is called the weak nuclear force. We do not come in direct contact with this force in everyday life. It is, however, responsible for radioactivity—the decay of atomic nuclei. The weak nuclear force was not well understood until 1967, when Abdus Salam at Imperial College, London, and Steven Weinberg at Harvard both proposed theories that unified this interaction with the electromagnetic force, just as Maxwell had unified electricity and magnetism about a hundred years earlier. The predictions of the theory agreed so well with experiment that in 1979, Salam and Weinberg were awarded the Nobel Prize for physics, together with Sheldon Glashow, also at Harvard, who had suggested similar unified theories of the electromagnetic and weak nuclear forces.

  The fourth category is the strongest of the four forces, the strong nuclear force. This is another force with which we don’t have direct contact, but it is the force that holds most of our everyday world together. It is responsible for binding the quarks together inside the proton and neutron and for holding the protons and neutrons together in the nucleus of an atom. Without the strong force, the electric repulsion between the positively charged protons would blow apart every atomic nucleus in the universe except those of hydrogen gas, whose nuclei consist of single protons. It is believed that this force is carried by a particle, called the gluon, which interacts only with itself and with the quarks.

  The success of the unification of the electromagnetic and weak nuclear forces led to a number of attempts to combine these two forces with the strong nuclear force into what is called a grand unified theory (or GUT). This title is rather an exaggeration: the resultant theories are not all that grand, nor are they fully unified, as they do not include gravity. They are also not really complete theories, because they contain a number of parameters whose values cannot be predicted from the theory but have to be chosen to fit in with experiment. Nevertheless, they may be a step toward a complete, fully unified theory.

  The main difficulty in finding a theory that unifies gravity with the other forces is that the theory of gravity—general relativity—is the only one that is not a quantum theory: it does not take into account the uncertainty principle. Yet because the partial theories of the other forces depend on quantum mechanics in an essential way, unifying gravity with the other theories would require finding a way to incorporate that principle into general relativity. But no one has yet been able to come up with a quantum theory of gravity.

  The reason a quantum theory of gravity has proven so hard to create has to do with the fact that the uncertainty principle means that even “empty” space is filled with pairs of virtual particles and antiparticles. If it weren’t—if “empty” space were really completely empty— that would mean that all the fields, such as the gravitational and electromagnetic fields, would have to be exactly zero. However, the value of a field and its rate of change with time are like the position and velocity (i.e., change of position) of a particle: the uncertainty principle implies that the more accurately one knows one of these quantities, the less accurately one can know the other. So if a field in empty space were fixed at exactly zero, then it would have both a precise value (zero) and a precise rate of change (also zero), in violation of that principle. Thus there must be a certain minimum amount of uncertainty, or quantum fluctuations, in the value of the field.

  Feynman Diagram of Virtual Particle/Antiparticle Pair

  The uncertainty principle, as applied to the electron, dictates that even in empty space virtual particle/antiparticle pairs appear and then annihilate each other

  One can think of these fluctuations as pairs of particles that appear together at some time, move apart, and then come together again and annihilate each other. They are virtual particles, like the particles that carry the forces: unlike real particles, they cannot be observed directly with a particle detector. However, their indirect effects, such as small changes in the energy of electron orbits, can be measured, and these data agree with the theoretical predictions to a remarkable degree of accuracy. In the case of fluctuations of the electromagnetic field, these particles are virtual photons, and in the case of fluctuations of the gravitational field, they are virtual gravitons. In the case of fluctuations of the weak and strong force fields, however, the virtual pairs are pairs of matter particles, such as electrons or quarks, and their antiparticles.

  The problem is that the virtual particles have energy. In fact, because there are an infinite number of virtual pairs, they would have an infinite amount of energy and, therefore, by Einstein’s equation E=mc 2 (see Chapter 5) they would have an infinite amount of mass. According to general relativity, this means that their gravity would curve the universe to an infinitely small size. That obviously does not happen! Similar seemingly absurd infinities occur in the other partial theories—those of the strong, weak, and electromagnetic forces— but in all these cases a process called renormalization can remov e the infinities, which is why we have been able to create quantum theories of those forces.

  Renormalization involves introducing new infinities that have the effect of canceling the infinities that arise in the theory. However, they need not cancel exactly. We can choose the new infinities so as to leave small remainders. These small remainders are called the renormalized quantities in the theory.

  Although in practice this technique is rather dubious mathematically, it does seem to work, and it has been used with the theories of the strong, weak, and electromagnetic forces to make predictions that agree with observations to an extraordinary degree of accuracy. Renormalization has a serious drawback from the point of view of trying to find a complete theory, though, because it means that the actual values of the masses and the strengths of the forces cannot be predicted from the theory but have to be chosen to fit the observations. Unfortunately, in attempting to use renormalization to remove the quantum infinities from general relativity, we have only two quantities that can be adjusted: the strength of gravity and the value of the cosmological constant, the term Einstein introduced into his equations because he believed that the universe was not expanding (see Chapter 7). As it turns out, adjusting these is not sufficient to remove all the infinities. We are therefore left with a quantum theory of gravity that seems to predict that certain quantities, such as the curvature of space-time, are really infinite—yet these quantities can be observed and measured to be perfectly finite!

  That this would be a problem in combining general relativity and the uncertainty principle had been suspected for some time but was finally confirmed by detailed calculations in 1972. Four years later, a possible solution, called supergravity, was suggested. Unfortunately, the calculations required to find out whether or not there were any infinities left uncanceled in supergravity were so long and difficult that no one was prepared to undertake them. Even with a computer, it was reckoned, it would take many years, and the chances were very high that there wou
ld be at least one mistake, probably more. Thus we would know we had the right answer only if someone else repeated the calculation and got the same answer, and that did not seem very likely! Still, despite these problems, and the fact that the particles in the supergravity theories did not seem to match the observed particles, most scientists believed that the theory could be altered and was probably the right answer to the problem of unifying gravity with the other forces. Then in 1984 there was a remarkable change of opinion in favor of what are called string theories.

  Before string theory, each of the fundamental particles was thought to occupy a single point of space. In string theories, the basic objects are not point particles but things that have a length but no other dimension, like an infinitely thin piece of string. These strings may have ends (so-called open strings) or they may be joined up with themselves in closed loops (closed strings). A particle occupies one point of space at each moment of time. A string, on the other hand, occupies a line in space at each moment of time. Two pieces of string can join together to form a single string; in the case of open strings they simply join at the ends, while in the case of closed strings it is like the two legs joining on a pair of trousers. Similarly, a single piece of string can divide into two strings.

  If the fundamental objects in the universe are strings, what are the point particles we seem to observe in our experiments? In string theories, what were previously thought of as different point particles are now pictured as various waves on the string, like waves on a vibrating kite string. Yet the strings, and the vibrations along it, are so tiny that even our best technology cannot resolve their shape, so they behave, in all of our experiments, as tiny, featureless points. Imagine looking at a speck of dust: up close, or under a magnifying glass, you may find that the fleck has an irregular or even stringlike shape, yet from a distance it looks like a featureless dot.

  In string theory the emission or absorption of one particle by another corresponds to the dividing or joining together of strings. For example, the gravitational force of the sun on the earth was pictured in particle theories as being caused by the emission of the force-carrying particles called gravitons by a matter particle in the sun and their absorption by a matter particle in the earth. In string theory, this process corresponds to an H-shaped tube or pipe (string theory is rather like plumbing, in a way). The two vertical sides of the H correspond to the particles in the sun and the earth, and the horizontal crossbar corresponds to the graviton that travels between them.

  String theory has a curious history. It was originally invented in the late 1960s in an attempt to find a theory to describe the strong force. The idea was that particles such as the proton and the neutron could be regarded as waves on a string. The strong forces between the particles would correspond to pieces of string that went between other bits of string, as in a spiderweb. For this theory to give the observed value of the strong force between particles, the strings had to be like rubber bands with a pull of about ten tons.

  In 1974, Joel Scherk from the École Normale Supérieure in Paris and John Schwarz from the California Institute of Technology published a paper in which they showed that string theory could describe the nature of the gravitational force, but only if the tension in the string was about a thousand million million million million million million tons (1 with thirty-nine zeros after it). The predictions of string theory would be just the same as those of general relativity on normal-length scales, but they would differ at very small distances, less than a thousand million million million million millionth of a centimeter (a centimeter divided by 1 with thirty-three zeros after it). Their work did not receive much attention, however, because at just about that time most people abandoned the original string theory of the strong force in favor of the theory based on quarks and gluons, which seemed to fit much better with observations. Scherk died in tragic circumstances (he suffered from diabetes and went into a coma when no one was around to give him an injection of insulin), so Schwarz was left alone as almost the only supporter of string theory, but now with the much higher proposed value of the string tension.

  Feynman Diagrams in String Theory

  In string theories, long-range forces are viewed as being caused by connecting tubes rather than the interchange of force-carrying particles.

  In 1984, interest in strings suddenly revived, apparently for two reasons. One was that people were not really making much progress toward showing that supergravity was finite or that it could explain the kinds of particles that we observe. The other was the publication of another paper by John Schwarz, this time with Mike Green of Queen Mary College, London. This paper showed that string theory might be able to explain the existence of particles that have a built-in left-handedness, like some of the particles that we observe. (The behavior of most particles would be the same if you changed the experimental setup by reflecting it all in a mirror, but the behavior of these particles would change. It is as if they are left-or right-handed, instead of being ambidextrous.) Whatever the reasons, a large number of people soon began to work on string theory, and a new version was developed that seemed as if it might be able to explain the types of particles that we observe.

  String theories also lead to infinities, but it is thought that in the right version they will all cancel out (though this is not yet known for certain). String theories, however, have a bigger problem: they seem to be consistent only if space-time has either ten or twenty-six dimensions, instead of the usual four! Of course, extra space-time dimensions are a commonplace of science fiction. Indeed, they provide an ideal way of overcoming the normal restriction of general relativity that one cannot travel faster than light or back in time (see Chapter 10). The idea is to take a shortcut through the extra dimensions. You can picture this in the following way. Imagine that the space we live in has only two dimensions and is curved like the surface of an anchor ring or doughnut. If you were on the inside edge of the ring and you wanted to get to a point across the ring on the other side, you would have to move in a circle along the inner edge of the ring until you reached the target point. However, if you were able to travel in the third dimension, you could leave the ring and cut straight across.

  Why don’t we notice all these extra dimensions if they are really there? Why do we see only three space dimensions and one time dimension? The suggestion is that the other dimensions are not like the dimensions we are used to. They are curved up into a space of very small size, something like a million million million million millionth of an inch. This is so small that we just don’t notice it: we see only one time dimension and three space dimensions, in which space-time is fairly flat. To picture how this works, think of the surface of a straw. If you look at it closely, you see the surface is two-dimensional. That is, the position of a point on the straw is described by two numbers, the length along the straw and the distance around the circular dimension. But its circular dimension is much smaller than its dimension of length. Because of that, if you look at the straw from a distance, you don’t see the thickness of the straw and it looks one-dimensional. That is, it appears that to specify the position of a point you need only to give the length along the straw. So it is with space-time, string theorists say: on a very small scale it is ten-dimensional and highly curved, but on bigger scales you don’t see the curvature or the extra dimensions.

  If this picture is correct, it spells bad news for would-be space travelers: the extra dimensions would be far too small to allow a spaceship through. However, it raises a major problem for scientists as well: why should some, but not all, of the dimensions be curled up into a small ball? Presumably, in the very early universe all the dimensions would have been very curved. Why did one time dimension and three space dimensions flatten out, while the other dimensions remain tightly curled up?

  One possible answer is what is called the anthropic principle, which can be paraphrased as “We see the universe the way it is because we exist.” There are two versions of the anthropic principle, the weak and the
strong. The weak anthropic principle states that in a universe that is large or infinite in space and/or time, the conditions necessary for the development of intelligent life will be met only in certain regions that are limited in space and time. The intelligent beings in these regions should therefore not be surprised if they observe that their locality in the universe satisfies the conditions that are necessary for their existence. It is a bit like a rich person living in a wealthy neighborhood not seeing any poverty.

  Some go much further and propose a strong version of the principle. According to this theory, there are either many different universes or many different regions of a single universe, each with its own initial configuration and, perhaps, with its own set of laws of science. In most of these universes the conditions would not be right for the development of complicated organisms; only in the few universes that are like ours would intelligent beings develop and ask the question, “Why is the universe the way we see it?” The answer is then simple: if it had been different, we would not be here!

  Few people would quarrel with the validity or utility of the weak anthropic principle, but there are a number of objections that one can raise to the strong anthropic principle as an explanation of the observed state of the universe. For instance, in what sense can all these different universes be said to exist? If they are really separate from each other, what happens in another universe can have no observable consequences in our own universe. We should therefore use the principle of economy and cut them out of the theory. If, on the other hand, they were just different regions of a single universe, the laws of science would have to be the same in each region, because otherwise we could not move continuously from one region to another. In this case the only difference between the regions would be their initial configurations, so the strong anthropic principle would reduce to the weak one.

  The anthropic principle gives one possible answer to the question of why the extra dimensions of string theory curled up. Two space dimensions do not seem to be enough to allow for the development of complicated beings like us. For example, two-dimensional animals living on a circle (the surface of a two-dimensional earth) would have to climb over each other in order to get past each other. And if a two-dimensional creature ate something it could not digest completely, it would have to bring up the remains the same way it swallowed them, because if there were a passage right through its body, it would divide the creature into two separate halves: our two-dimensional being would fall apart. Similarly, it is difficult to see how there could be any circulation of the blood in a two-dimensional creature.