There would also be problems with more than three space dimensions. The gravitational force between two bodies would decrease more rapidly with distance than it does in three dimensions. (In three dimensions, the gravitational force drops to one-quarter as you double the distance. In four dimensions it would drop to one-eighth, in five dimensions to one-sixteenth, and so on.) The significance of this is that the orbits of planets, like the earth, around the sun would be unstable: the least disturbance from a circular orbit (such as would be caused by the gravitational attraction of other planets) would result in the earth spiraling away from or into the sun. We would either freeze or be burned up. In fact, the same behavior of gravity with distance in more than three space dimensions means that the sun would not be able to exist in a stable state, with pressure balancing gravity. The sun would either fall apart or collapse to form a black hole. In either case, it would not be of much use as a source of heat and light for life on earth. On a smaller scale, the electrical forces that cause the electrons to orbit around the nucleus in an atom would behave in the same way as gravitational forces. Thus the electrons would either escape from the atom altogether or would spiral into the nucleus. In either case, there would be no atoms as we know them.
It seems clear then that life, at least as we know it, can exist only in regions of space-time in which one time dimension and exactly three space dimensions are not curled up small. This would mean that we could appeal to the weak anthropic principle, provided we could show that string theory does at least allow there to be such regions of the universe—and it seems that indeed string theory does. There may well be other regions of the universe, or other universes (whatever that may mean), in which all the dimensions are curled up small or in which more than four dimensions are nearly flat, but there would be no intelligent beings in such regions to observe the different number of effective dimensions.
In addition to the question of dimensions, another problem with string theory is that there are at least five different theories (two open-string and three different closed-string theories) and millions of ways in which the extra dimensions predicted by string theory could be curled up. Why should just one string theory and one kind of curling-up be picked out? For a time there seemed no answer, and progress got bogged down. Then, starting in about 1994, people started discovering what are called dualities: different string theories and different ways of curling up the extra dimensions could lead to the same results in four dimensions. Moreover, as well as particles, which occupy a single point of space, and strings, which are lines, there were found to be other objects called p-branes, which occupied two-dimensional or higher-dimensional volumes in space. (A particle can be regarded as a 0-brane and a string as a 1-brane, but there were also p-branes for p = 2 to p = 9. A 2-brane can be thought of as something like a two-dimensional membrane. It is harder to picture the higher-dimensional branes.) What this seems to indicate is that there is a sort of democracy (in the sense of having equal voices) among supergravity, string, and p-brane theories: they seem to fit together, but none can be said to be more fundamental than the others. Instead they all appear to be different approximations to some more fundamental theory, each valid in different situations.
People have searched for this underlying theory, but without any success so far. It is possible that there may not be any single formulation of the fundamental theory any more than, as Gödel showed, one could formulate arithmetic in terms of a single set of axioms. Instead it may be like maps—you can’t use a single flat map to describe the round surface of the earth or the surface of an anchor ring: you need at least two maps in the case of the earth and four for the anchor ring to cover every point. Each map is valid only in a limited region, but different maps will have a region of overlap. The collection of maps provides a complete description of the surface. Similarly, in physics it may be necessary to use different formulations in different situations, but two different formulations would agree in situations where they can both be applied.
The Importance of Being ThreeDimensional
In more than three space dimensions, planetary orbits would be unstable and planets would either fall into the sun or escape its attraction altogether
If this is true, the whole collection of different formulations could be regarded as a complete unified theory, though it would be a theory that could not be expressed in terms of a single set of postulates. But even this may be more than nature allows. Is it possible there is no unified theory? Are we perhaps just chasing a mirage? There seem to be three possibilities:
There really is a complete unified theory (or a collection of overlapping formulations), which we will someday discover if we are smart enough
There is no ultimate theory of the universe, just an infinite sequence of theories that describe the universe more and more accurately but are never exact
There is no theory of the universe events cannot be predicted beyond a certain extent but occur in a random and arbitrary manner
Some would argue for the third possibility on the grounds that if there were a complete set of laws, that would infringe God’s freedom to change His mind and intervene in the world. Yet, since God is all-powerful, couldn’t God infringe on His freedom if He wanted to? It’s a bit like the old paradox: can God make a stone so heavy that He can’t lift it? Actually, the idea that God might want to change His mind is an example of the fallacy, pointed out by St. Augustine, of imagining God as a being existing in time. Time is a property only of the universe that God created. Presumably, He knew what He intended when He set it up!
With the advent of quantum mechanics, we have come to recognize that events cannot be predicted with complete accuracy: there is always a degree of uncertainty. If you like, you could ascribe this randomness to the intervention of God. But it would be a very strange kind of intervention, with no evidence that it is directed toward any purpose. Indeed, if it were, it would bv definition not be random. In modern times, we have effectively removed the third possibility above by redefining the goal of science: our aim is to formulate a set of laws that enables us to predict events only up to the limit set by the uncertainty principle.
The second possibility, that there is an infinite sequence of more and more refined theories, is in agreement with all our experience so far. On many occasions we have increased the sensitivity of our measurements or made a new class of observations, only to discover new phenomena that were not predicted by the existing theory, and to account for these we have had to develop a more advanced theory. By studying particles that interact with more and more energy, we might indeed expect to find new layers of structure more basic than the quarks and electrons that we now regard as “elementary” particles.
Gravity may provide a limit to this sequence of “boxes within boxes.” If we had a particle with an energy above what is called the Planck energy, its mass would be so concentrated that it would cut itself off from the rest of the universe and form a little black hole. Thus it does seem that the sequence of more and more refined theories should have some limit as we study higher and higher energies, so there should be some ultimate theory of the universe. Yet the Planck energy is a very long way from the energies we can produce in the laboratory at the present time. We shall not bridge that gap with particle accelerators in the foreseeable future. The very early stages of the universe, however, are an arena where such energies must have occurred. There is a good chance that the study of the early universe and the requirements of mathematical consistency will lead us to a complete unified theory within the lifetime of some of us who are around today, always presuming we don’t blow ourselves up first!
What would it mean if we actually did discover the ultimate theory of the universe?
As was explained in Chapter 3, we could never be quite sure that we had indeed found the correct theory, since theories can’t be proved. But if the theory was mathematically consistent and always gave predictions that agreed with observations, we could be reasonably confident that i
t was the right one. It would bring to an end a long and glorious chapter in the history of humanity’s intellectual struggle to understand the universe. But it would also revolutionize the ordinary person’s understanding of the laws that govern the universe.
In Newton’s time, it was possible for an educated person to have a grasp of the whole of human knowledge, at least in broad strokes. But since then, the pace of the development of science has made this impossible. Because theories are always being changed to account for new observations, they are never properly digested or simplified so that ordinary people can understand them. You have to be a specialist, and even then you can only hope to have a proper grasp of a small proportion of the scientific theories. Further, the rate of progress is so rapid that what you learn at school or university is always a bit out of date. Only a few people can keep up with the rapidly advancing frontier of knowledge, and they have to devote their whole time to it and specialize in a small area. The rest of the population has little idea of the advances that are being made or the excitement they are generating. On the other hand, seventy years ago, if Eddington is to be believed, only two people understood the general theory of relativity. Nowadays tens of thousands of university graduates do, and many millions of people are at least familiar with the idea. If a complete unified theory is discovered, it will be only a matter of time before it becomes digested and simplified in the same way and taught in schools, at least in outline. We will then all be able to have some understanding of the laws that govern the universe and are responsible for our existence.
Even if we do discover a complete unified theory, though, it would not mean that we would be able to predict events in general, for two reasons. The first is the limitation that the uncertainty principle of quantum mechanics sets on our powers of prediction. There is nothing we can do to get around that. In practice, however, this first limitation is less restrictive than the second one. It arises from the fact that we most likely could not solve the equations of such a theory, except in very simple situations. As we’ve said, no one can solve exactly the quantum equations for an atom consisting of a nucleus plus more than one electron. We can’t even solve exactly the motion of three bodies in a theory as simple as Newton’s theory of gravity, and the difficulty increases with the number of bodies and the complexity of the theory. Approximate solutions usually suffice for applications, but they hardly meet the grand expectations aroused by the term “unified theory of everything”!
Today, we already know the laws that govern the behavior of matter under all but the most extreme conditions. In particular, we know the basic laws that underlie all of chemistry and biology. Yet we have certainly not reduced these subjects to the status of solved problems. And we ha had, as yet, little success in predicting human behavior from mathematical equations! So even if we do find a complete set of basic laws, there will still be in the years ahead the intellectually challenging task of developing better approximation methods so that we can make useful predictions of the probable outcomes in complicated and realistic situations. A complete, consistent, unified theory is only the first step: our goal is a complete understanding of the events around us, and of our own existence.
12
CONCLUSION
WE FIND OURSELVES IN A BEWILDERING world. We want to make sense of what we see around us and to ask: What is the nature of the universe? What is our place in it, and where did it and we come from? Why is it the way it is?
To try to answer these questions, we adopt some picture of the world. Just as an infinite tower of tortoises supporting the flat earth is such a picture, so is the theory of superstrings. Both are theories of the universe, though the latter is much more mathematical and precise than the former. Both theories lack observational evidence: no one has ever seen a giant tortoise with the earth on its back, but then, no one has ever seen a superstring either. However, the tortoise theory fails to be a good scientific theory because it predicts that people should be able to fall off the edge of the world. This has not been found to agree with experience, unless that turns out to be the explanation for the people who are supposed to have disappeared in the Bermuda Triangle!
The earliest theoretical attempts to describe and explain the universe involved the idea that events and natural phenomena were controlled by spirits with human emotions who acted in a very humanlike and unpredictable manner. These spirits inhabited natural objects, such as rivers, mountains, and celestial bodies including the sun and moon. They had to be placated and their favor sought in order to ensure the fertility of the soil and the rotation of the seasons. Gradually, however, it must have been noticed that there were certain regularities: the sun always rose in the east and set in the west, whether or not a sacrifice had been made to the sun god. Further, the sun, the moon, and the planets followed precise paths across the sky that could be predicted in advance with considerable accuracy. The sun and the moon might still be gods, but they were gods who obeyed strict laws, apparently without any exceptions, if one discounts stories such as that of the sun stopping for Joshua.
From Turtles to Curved Space
Ancient and modern views of the universe.
At first, these regularities and laws were obvious only in astronomy and a few other situations. However, as civilization developed, and particularly in the last three hundred years, more and more regularities and laws were discovered. The success of these laws led Laplace at the beginning of the nineteenth century to postulate scientific determinism; that is, he suggested that there would be a set of laws that would determine the evolution of the universe precisely, given its configuration at any one time.
Laplace’s determinism was incomplete in two ways: it did not say how the laws should be chosen, and it did not specify the initial configuration of the universe. These were left to God. God would choose how the universe began and what laws it obeyed, but He would not intervene in the universe once it had started. In effect, God was confined to the areas that nineteenth-century science did not understand.
We now know that Laplace’s hopes of determinism cannot be realized, at least in the terms he had in mind. The uncertainty principle of quantum mechanics implies that certain pairs of quantities, such as the position and velocity of a particle, cannot both be predicted with complete accuracy. Quantum mechanics deals with this situation via a class of quantum theories in which particles don’t have well-defined positions and velocities but are represented by a wave. These quantum theories are deterministic in the sense that they give laws for the evolution of the wave with time. Thus if we know the wave at one time, we can calculate it at any other time. The unpredictable, random element comes in only when we try to interpret the wave in terms of the positions and velocities of particles. But maybe that is our mistake: maybe there are no particle positions and velocities, but only waves. It is just that we try to fit the waves to our preconceived ideas of positions and velocities. The resulting mismatch is the cause of the apparent unpredictability.
In effect, we have redefined the task of science to be the discovery of laws that will enable us to predict events up to the limits set by the uncertainty principle. The question remains, however: how or why were the laws and the initial state of the universe chosen?
This book has given special prominence to the laws that govern gravity, because it is gravity that shapes the large-scale structure of the universe, even though it is the weakest of the four categories of forces. The laws of gravity were incompatible with the view, held until quite recently, that the universe is unchanging in time: the fact that gravity is always attractive implies that the universe must be either expanding or contracting. According to the general theory of relativity, there must have been a state of infinite density in the past, the big bang, which would have been an effective beginning of time. Similarly, if the whole universe collapsed, there must be another state of infinite density in the future, the big crunch, which would be an end of time. Even if the whole universe did not collapse, there would be singularit
ies in any localized regions that collapsed to form black holes. These singularities would be an end of time for anyone who fell into the black hole. At the big bang and other singularities, all the laws would have broken down, so God would still have had complete freedom to choose what happened and how the universe began.
When we combine quantum mechanics with general relativity, there seems to be a new possibility that did not arise before: that space and time together might form a finite, four-dimensional space without singularities or boundaries, like the surface of the earth but with more dimensions. It seems that this idea could explain many of the observed features of the universe, such as its large-scale uniformity and also the smaller-scale departures from homogeneity, including galaxies, stars, and even human beings. But if the universe is completely self-contained, with no singularities or boundaries, and completely described by a unified theory, that has profound implications for the role of God as creator.
Einstein once asked, “How much choice did God have in constructing the universe?” If the no-boundary proposal is correct, God had no freedom at all to choose initial conditions. God would, of course, still have had the freedom to choose the laws that the universe obeyed. This, however, may not really have been all that much of a choice; there may well be only one, or a small number, of complete unified theories, such as string theory, that are self-consistent and allow the existence of structures as complicated as human beings who can investigate the laws of the universe and ask about the nature of God.
A Briefer History of Time Page 12
