Seeing Further (41 page)

At first, it was hoped that one of these theories would turn out to be special and attention would then narrow down to reveal it to be the true theory of everything. Unfortunately, things were not so simple. Progress was slow and unremarkable until Edward Witten, at Princeton, discovered that these different string theories are not really different. They are linked to one another by mathematical transformations that amount to exchanging large distances for small ones, and vice versa in a particular way. Nor were these string theories fundamental. Instead, they were each limiting situations of another deeper, as yet unfound, TOE which lives in eleven dimensions of space and time. That theory became known as ‘M-theory’, where M has been said to be an abbreviation for Mystery, Matrix, or Millennium, just as you like.
⁴

Do these ‘extra’ dimensions of space really exist? This is a key question for all these new theories of everything. In most versions, the other dimensions are so small (10
^–33
cm) that no direct experiment will ever see them. But, in some variants, they can be much bigger. The interesting feature is that only the force of gravity will ‘feel’ these extra dimensions and be modified by their presence. In these cases the extra dimensions could be up to one hundredth of a millimetre in extent and they would alter the form of the law of gravity over these and smaller distances. This gives experimental physicists a wonderful challenge: test the law of gravity on submillimetre scales. More sobering still is the fact that all the observed constants of Nature, in our three dimensions, are not truly fundamental, and need not be constant in time or space:
⁵
they are just shadows of the true constants that live in the full complement of dimensions. Sometimes simplicity can be complex too.

The fact that Nature displays populations of identical elementary particles is its most remarkable property. It is the ‘fine tuning’ that surpasses all others. In the nineteenth century another of the Royal Society’s greatest Fellows, James Clerk Maxwell, first stressed that the physical world was composed of identical atoms which were not subject to evolution. Today, we look for some deeper explanation of the sub-atomic particles of Nature from our TOE. One of the most perplexing discoveries by experimentalists has been that such ‘elementary’ particles appear to be extremely numerous. They were supposed to be an exclusive club, but they have ended up with an embarrassingly large clientele.

String theories offered another route to solving this problem. Instead of a TOE containing a population of elementary point-like particles, string theories introduce basic entities that are loops (or lines) of energy which have a tension. As the temperature rises the tension falls and the loops vibrate in an increasingly stringy fashion, but as the temperature falls the tension increases and the loops contract to become more and more point-like. So, at low energies the strings behave like points and allow the theory to make the successful predictions about what we should see there as the intrinsically point-like theories do. However, at high energies, things are different. The hope is that it will be possible to determine the principal energies of vibration of the superstrings. All strings, even guitar strings, have a collection of special vibrational energies that they naturally take up when disturbed. If we could calculate these special energies for super-strings, then they would (by virtue of Einstein’s famous mass-energy equivalence – E = mc
²
) correspond to the masses of the ‘particles’ that we call elementary. So far, these energies have proved too hard to calculate. However, one of them has been found: it corresponds to a particle with zero mass and two units of a quantum attribute called ‘spin’. This spin value ensures that it mediates attractions between all masses. It is the particle we call the ‘graviton’ and it is responsible for mediating the force of gravity. Its appearance shows that string theory necessarily includes gravity and, moreover, its behaviour is described by the equations of general relativity at low energies – a remarkable and compelling feature since earlier candidates for a TOE all failed miserably to include gravity in the unification story at all.

This reflection on the symmetries behind the laws of Nature also tells us why mathematics is so useful in practice. Mathematics is simply the catalogue of all possible patterns. Some of those patterns are especially attractive and are studied or used for decoration, others are patterns in time or in chains of logic. Some are described solely in abstract terms, while others can be drawn on paper or carved in stone. Viewed in this way, it is inevitable that the world is described by mathematics. We could not exist in a universe in which there was neither pattern nor order. The description of that order, and all the other sorts that we can imagine, is what we call mathematics. Yet, although the fact
that mathematics describes the world is not a mystery, the exceptional utility of mathematics is. It could have been that the patterns behind the world were of such complexity that no simple algorithms could approximate them. Such a universe would ‘be’ mathematical, but we would not find mathematics terribly useful. We could prove ‘existence’ theorems about what structures exist but we would be unable to predict the future using mathematics in the way that NASA’s mission control does.

Seen in this light, we recognise that the great mystery about mathematics and the world is that such
simple
mathematics is so far reaching. Very simple patterns, described by mathematics that is easily within our grasp, allow us to explain and understand a huge part of the universe and the happenings within it.

It is often said with hindsight that Nicholas Copernicus taught us not to assume that our position in the universe is special in
every
way. Of course, this does not mean that it cannot be special in
any
way, simply because life is only possible in certain places.
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Once we start distinguishing between the laws of Nature and their outcomes we should also bring this Copernican view to bear upon the laws of Nature as well as their outcomes.

Universal laws of Nature should be just that – universal – they should not just exist in special forms for some privileged observers at special locations, or who are moving in particular ways, in the universe. Alas, Newton’s laws do not have this democratic property. They only have simple forms for privileged observers who are moving in a special way, neither rotating nor accelerating with respect to the distant ‘fixed’ stars. So there were privileged observers in Newton’s universe for whom all the laws of motion look simple.

Newton’s first law of motion demands that bodies acted upon by no forces do not accelerate: they remain at rest or move with constant speed. However, this law of motion will only be observed by a special class of
observers who are neither accelerating nor rotating relative to the fixed stars. The appearance of these special observers for whom all the laws of motion look simpler violates the Copernican principle.

Imagine that you are located inside a spaceship through whose windows you can see the far distant stars. Put the spaceship in a spin. Through the windows you will see the distant stars accelerating past in the opposite sense to the spin, even though they are not acted upon by any forces. Newton’s first law is not true for a spinning observer – a much more complicated law holds. This undemocratic situation signalled that there was something incomplete and unsatisfactory about Newton’s formulation of the laws of motion. One of Einstein’s great achievements was to create a new theory of gravity in which all observers, no matter how they move,
do
find the laws of gravity and motion to take the same form.
⁷
By incorporating this principle of ‘general covariance’, Einstein’s theory of general relativity completed the extension of the Copernican principle from outcomes to laws.

The simplicity and economy of the laws and symmetries that govern Nature’s fundamental forces are not the end of the story. When we look around us we do not observe the laws of Nature; rather, we see the
outcomes
of those laws. The distinction is crucial. Outcomes are much more complicated than the laws that govern them because they do not have to respect the symmetries displayed by the laws. By this subtle interplay, it is possible to have a world which displays an unlimited number of complicated asymmetrical structures yet is governed by a few, very simple, symmetrical laws. This is one of the secrets of the universe.

Suppose we balance a ball at the apex of a cone. If we were to release the ball, then the law of gravitation will determine its subsequent motion. Gravity has no preference for any particular direction in the universe; it is entirely democratic in that respect. Yet, when we release the ball, it will
always fall in some particular direction, either because it was given a little push in one direction, or as a result of quantum fluctuations which do not permit an unstable equilibrium state to persist. So here, in the outcome of the falling ball, the directional symmetry of the law of gravity is broken. This teaches us why science is often so difficult. As observers, we see only the broken symmetries manifested as the outcomes of the laws of Nature; from them, we must work backwards to unmask the hidden symmetries behind the appearances.

We can now understand the answers that we obtained from the different scientists we originally polled about the simplicity of the world. The particle physicist works closest to the laws of Nature themselves, and so is especially impressed by their unity, simplicity and symmetry. But the biologist, the economist, or the meteorologist is occupied with the study of the complex outcomes of the laws, rather than with the laws themselves. As a result, it is the complexities of Nature, rather than her laws, that impress them most.

One of the most important developments in fundamental physics and cosmology over the past twenty years has been the steady dissolution of the divide between laws and outcomes. When the early quest for a theory of everything began many thought that such a theory would uniquely and completely specify all the constants of physics and the structural features of the universe. There would be no room left for wondering about ‘other’ universes, or hypothetical changes to the structure of our observed universe. Remarkably, things did not turn out like that. Candidate theories of everything revealed that many of the features of physics and the universe which we had become accustomed to think of as programmed into the universe from the start in some unalterable way, were nothing of the sort. The number of forces of Nature, their laws of interaction, the populations of elementary particles, the values of the so-called constants of Nature, the
number of dimensions of space, and even whole universes, can all arise in quasi-random fashion in these theories. They are elaborate outcomes of processes that can have many different physically self-consistent results. There are fewer unalterable laws than we might think.

This means that we have to take seriously the possibility that some features of the universe which we call fundamental may not have explanations in the sense that had always been expected. A good example is the value of the infamous cosmological constant which appears to drive the acceleration of the universe today. Its numerical value is very strange. It cannot so far be explained by known theories of physics. Some physicists hope that there will ultimately be a single theory of everything which will predict the exact numerical value of the cosmological constant that the astronomers need to explain their observations. Others recognise that there may not be any explanation of that sort to be found. If the value of the cosmological constant is a random outcome of some exotic symmetry-breaking process near the beginning of the universe’s expansion then all we can say is that it falls within the range of values that permit life to evolve and persist. This is a depressing situation to those who hoped to explain its value. However, it would be a strange (non-Copernican) universe that allowed us to determine everything that we want about it. We may just have to get used to the fact that there are some things we can predict and others that we can only measure. Here is a little piece of science faction to illustrate the point.

Imagine someone in 1600 trying to convince Johannes Kepler that a theory of the solar system won’t be able to predict the number of planets in the solar system. Kepler would have had none of it. He would have been outraged. This would have constituted an admission of complete failure. He believed that the beautiful Platonic symmetries of mathematics required the solar system to have a particular number of planets. For Kepler this would have been the key feature of such a theory. He would have rejected the idea that the number of planets had no part to play in the ultimate theory.

Today, no planetary astronomer would expect any theory of the origin of the solar system to predict the number of planets. It would make no sense. This number is something that falls out at random as a result of a chaotic sequence of formation events and subsequent mergers between embryonic planetesimals. It is simply not a predictable outcome. We concentrate instead on predicting other features of the solar system so as to test the theory of its origin. Perhaps those who are resolutely opposed to the idea that quantities like the cosmological constant might be randomly determined, and hence unpredictable by the theory of everything, might consider how strange Kepler’s views about the importance of the number of planets now seem.