by Peter Coles
While Einstein’s equations also seem quite accurate for most purposes, it is similarly natural to attempt the construction of a quantum theory of gravity. Einstein himself always believed that his theory was incomplete in this sense, and would eventually need to be replaced by a more complete theory. By analogy with the breakdown of classical electromagnetism, one can argue that this should happen when gravitational fields are very strong, or on length scales that are extremely short. Attempts to build such a theory have so far been unsuccessful.
Although there is nothing resembling a complete picture of what a quantum theory of gravity might involve, there are some interesting speculative ideas. For example, since general relativity is essentially a theory of space-time, space and time themselves must become quantized in quantum gravity theories. This suggests that, although space and time appear continuous and smooth to us, on minuscule scales of the Planck length (around 10–33 cm), space is much more lumpy and complicated, perhaps consisting of a foam-like topology of bubbles connected by tunnels called wormholes that are continually forming and closing in the Planck time, which is 10–43 seconds. It also seems to make sense to imagine that quantized gravitational waves, or gravitons, might play the role of the gauge bosons in other fundamental interactions, such as the photons in the theory of quantum electrodynamics. As yet, there is no concrete evidence that these ideas are correct.
The tiny scales of length and time involved in quantum gravity demonstrate why this quantum gravity is a field for theorists rather than experimentalists. No device has yet been built capable of forcing particles into a region of order the Planck length or less. The enormous energies required to do this are needed to reveal the quantum nature of gravity. But that is precisely why so many theoreticians have turned away from particle experiments as such, and towards cosmology. The Big Bang must have involved phenomena on the Planck scale, so it may in principle be possible to learn about fundamental physics from cosmology.
The beginning of time
The presence of a singularity at the very beginning of the Universe is very bad news for the Big Bang model. Like the black hole singularity, it is a real singularity where the temperature and density become truly infinite. In this respect the Big Bang can be thought of as a kind of time-reverse of the gravitational collapse that forms a black hole. As was the case with the Schwarzschild solution, many physicists thought that the initial cosmological singularity could be a consequence of the special form of the solutions of Einstein’s equations used to model the Big Bang, but this is now known not to be the case. Hawking and Penrose generalized Penrose’s original black hole theorems to show that a singularity invariably exists in the past of an expanding Universe in which certain very general conditions apply. Physical theory completely fails us at the instant of the Big Bang where the nasty infinities appear.
29. Space-time foam. One of the ideas associated with quantum gravity is that space-time itself may turn into a seething mass of bubbles and tubes all springing into existence and disappearing on a timescale comparable to the Planck time.
So is it possible to avoid this singularity? And if so, how? It is most likely that the initial cosmological singularity might well just be a consequence of extrapolating deductions based on the classical theory of general relativity into a situation where this theory is no longer valid. This is what Einstein says in the paragraph quoted in Chapter 3 during the discussion of black holes. What is needed is quantum gravity, but we don’t have such a theory and, since we don’t have it we don’t know whether it would solve the riddle of the Universe’s apparently pathological birth.
There are, however, ways of avoiding the initial singularity in classical general relativity without appealing to quantum effects. Firstly, one could try to avoid the singularity by proposing an equation of state for matter in the very early Universe that does not obey the conditions laid down by Hawking and Penrose. The most important of these conditions is a restriction on the behaviour of matter at high energies called the strong energy condition. There are various ways in which this condition might indeed be violated. In particular, it is violated during the accelerated expansion predicted in theories of cosmic inflation. Models in which this condition is violated right at the very beginning can have a ‘bounce’ rather than a singularity. Running the clock back, the Universe reaches a minimum size and then expands again.
Whether the singularity is avoidable or not remains an open question, and the issue of whether we can describe the very earliest phases of the Big Bang, before the Planck time, will remain open at least until a complete theory of quantum gravity is constructed.
Time’s arrow
The existence of a singularity at the beginning of the Universe calls into question the very nature of space, and particularly of time, at the instant of creation. It would be nice to include at this point a clear definition of what time actually is. Everyone is familiar with what time does, and how events tend to be ordered in sequences. We are used to describing events that invariably follow other events in terms of a chain of cause and effect. But we can’t get much further than these simple ideas. In the end, the best statement of what is time is that time is whatever it is that is measured by clocks.
Einstein’s theories of relativity effectively destroyed the Newtonian concepts of absolute space and absolute time. Instead of having three spatial dimensions and one time dimension which are absolute and unchanging regardless of the motions of particles or experimenters, relativistic physics merges these together in a single four-dimensional entity called space-time. For many purposes, time and space can be treated as mathematically equivalent in these theories: different observers generally measure different time intervals between the same two events, but the four-dimensional space-time interval is always the same.
However, the successes of Einstein’s theoretical breakthroughs tend to mask the fact that we all know from everyday experience that time and space are essentially different. We can travel north or south, east and west, but we can only go forwards in time to the future, not backwards in time to the past. And we are quite happy with the idea that both London and New York exist at a given time at different spatial locations. But nobody would say that the year 5001 exists in the same way that we think the present exists. We are also happy to say that what we do now causes things to happen in the future, but do not consider events at the same time in two locations as causing each other. Space and time really are quite different.
On a cosmological level, the Big Bang certainly appears to have a preferred direction. But the equations describing it are again time-symmetric. Our universe happens to be expanding rather than contracting, but it could have been collapsing and described by the same laws. Or could it be that the directionality of time that we observe is somehow singled out by the large-scale expansion of the Universe? It has been speculated, by Hawking and others, that if we lived in a closed Universe that eventually stopped expanding and began to contract, then time would effectively run backwards during the contraction phase. In fact, if this happened we would not be able to tell the difference between a contracting Universe with time running backwards and an expanding Universe with time running forwards. Hawking was convinced for a time that this had to be the case, but later changed his mind.
A more abstract problem stems from the fact that Einstein’s theory is fully four-dimensional: the entire world-line of a particle, charting the whole history of its motions in space-time, can be calculated from the theory. A particle which exists at different times exists in the same way two particles might exist at the same time in different places. This is strongly at odds with our ideas of free will. Does our future exist already? Are things really predetermined in this way?
These questions are not restricted to relativity theory and cosmology. Many physical theories are symmetric between past and future in the same way as they are symmetric between different spatial locations. The question of how the perceived asymmetry of time can be reconciled with these theorie
s is a deep philosophical puzzle. There are at least two other branches of physical theory in which raise the question of the arrow of time, as it is sometimes called.
One emerges directly from a seemingly omnipotent physical principle, called the Second Law of Thermodynamics. This states that the entropy of a closed system never decreases. The entropy is a measure of the disorder of a system, so this law means that the degree of disorder of a system always tends to increase. I have verified this experimentally many times through periodic observation of my office. The second law is a macroscopic statement; it deals with big things like steam engines, but it arises from a microscopic description of atoms and energy states provided by detailed physical theories. The laws governing these microstates are all entirely reversible with respect to time. So how can an arrow of time emerge?
Laws similar to the classical laws of thermodynamics have also been constructed to describe the properties of black holes and of gravitational fields in general. Although the definition of the entropy associated with gravitational fields is difficult to define, these laws seem to indicate that the arrow of time persists even in a collapsing Universe. It was for this reason that Hawking abandoned his time-reversal idea.
Another arrow-of-time problem emerges from quantum mechanics, which is again time-symmetric, but in which weird phenomena occur such as the collapse of the wavefunction when an experiment is performed. Wavefunctions appear to do this only in one direction of time and not the other but, as I have hinted above, this may well just be a conceptual difficulty arising from the interpretation of quantum mechanics itself.
The no-boundary hypothesis
Space and time are very different concepts to us, living as we do in a low-energy world far removed from the Big Bang. But does that mean that space and time were always different? Or in a quantum theory of gravity could they really be the same? In classical relativity theory, space-time is a four-dimensional construction wherein the three dimensions of space and one dimension of time are welded together. But space and time are not equivalent. One idea associated with quantum cosmology, developed by Hawking together with Jim Hartle, is that the characteristic signature of time may be erased when the gravitational field is very strong. The idea is based on an ingenious use of the properties of imaginary numbers. (Imaginary numbers are all multiples of the number i which is defined to be the square root of minus one.) This tinkering with the nature of time is part of the no boundary hypothesis of quantum cosmology due to Hartle and Hawking. Since, in this theory, time loses the characteristics that separate it from space, the concept of a beginning in time becomes meaningless. Space-times with this signature therefore have no boundary. There is no Big Bang, no singularity, because there is no time, just another direction of space.
This view of the Big Bang is one in which there is no creation, because the word creation implies some kind of ‘before and after’. If there is no time then the Universe has no beginning. Asking what happened before the Big Bang is like asking what is further north than the North Pole. The question is meaningless.
I should stress that the no-boundary conjecture is not accepted by all quantum cosmologists: other ways of understanding the beginning (or lack of it) have been proposed. The Russian physicist Alexander Vilenkin proposed an alternative treatment of quantum cosmology in which there is a definite creation, through which the universe emerges by a process of quantum tunnelling out of nothing.
Theories of everything
I have tried to describe just a few of the areas in which particle physicists and cosmologists have been attempting to weld together quantum physics and gravity theory. This is one step in the direction of what many physicists feel is the ultimate goal of science: to write the mathematical laws describing all known forces of nature in the form of one equation that you might, perhaps if you have no dress-sense, wear on your T-shirt.
The laws of physics, sometimes also called the laws of nature, are the basic tools of physical science. They comprise mathematical equations that govern the behaviour of matter (in the form of elementary particles) and energy according to the various fundamental interactions described above. Sometimes experimental results obtained in the laboratory or observations of natural physical processes are used to infer mathematical rules which describe these data. Other times a theory is created first as the result of a hypothesis or physical principle which receives experimental confirmation only at a later stage. As our understanding evolves, seemingly disparate physical laws become unified in a single overarching theory. The examples given above show how influential this theme has been over the past hundred years or so.
But there are deep philosophical questions lying below the surface of all this activity. For example, what if the laws of physics were different in the early Universe? Could one still carry out this work? The answer to this is that modern physical theories actually predict that the laws of physics do change. As one goes to earlier and earlier stages in the Big Bang, for example, the nature of the electromagnetic and weak interactions changes so that they become indistinguishable at sufficiently high energies. But this change in the law is itself described by another law: the so-called electroweak theory. Perhaps this law itself is modified at scales where grand unified theories take precedence, and so on right back to the very beginning of the Universe.
Whatever the fundamental rules are, however, physicists have to assume that they apply for all times since the Big Bang. It is merely the low-energy outcomes of these fundamental rules that change with time. Making this assumption, they are able to build a coherent picture of the thermal history of the Universe which does not seem to be in major conflict with the observations. This makes this assumption reasonable, but does not prove it to be correct.
Another set of important questions revolves around the role of mathematics in physical theory. Is nature really mathematical? Or are the rules we devise merely a kind of shorthand to enable us to describe the Universe on as few pieces of paper as possible? Do we discover laws of physics or do we invent them? Is physics simply a map, or is it the territory itself?
There is also another deep issue connected with the laws of physics pertaining to the very beginning of space and time. In some versions of quantum cosmology, for example, one has to posit the existence of physical laws that exist, as it were, in advance of the physical universe they are supposed to describe. This has caused many theoreticians to adopt a philosophical approach that mirrors the ideas of Plato. In the Platonic tradition, true existence belongs to the idealized world of form rather than our imperfect world of the senses. To the Neoplatonic cosmologists, what really exists are the mathematical equations of the (yet unknown) theory of everything, rather than the physical world of matter and energy. On the other hand, not all cosmologists get carried away in this manner. To those of a more pragmatic disposition the laws of physics are simply a neat description of our Universe whose significance lies simply in their usefulness.
There have been many attempts to produce theories of everything, involving such exotic ideas as supersymmetry and string theory (or even a combination of the two known as superstring theory). In superstring theory, particles are not treated as particles at all but as oscillations in one-dimensional entities called strings. The different modes of vibration string loops correspond to different particles. The strings themselves live in a space of ten or twenty-six dimensions. Our space-time has only four dimensions (three space and one time), so the extra dimensions must be hidden. Perhaps they are wrapped up so small that they cannot be observed. After much excitement in the 1980s this idea went out of fashion, largely due to the technical complexity involved in handling such complicated multidimensional objects. More recently, these ideas have experienced a kind of renaissance, with the generalization of the concept of strings into ‘branes’, higher-dimensional objects whose name derives from ‘membrane’, and the realization that there is, in effect, a single theory (called ‘M-theory’) describing all versions of this kind of approach. These a
re exciting ideas but they are relatively undeveloped; string theory has not yet made any clear predictions that have impacted on cosmology. It remains to be seen whether the grander-than-grand unification to which these approaches aspire can actually be realized.
The search for a theory of everything also raises interesting philosophical questions. Some physicists, Hawking among them, would regard the construction of a theory of everything as being, in some sense, reading the mind of God, or at least unravelling the inner secrets of physical reality. Others simply argue that a physical theory is just a description of reality, rather like a map. A theory might be good for making predictions and understanding the outcomes of observation or experiment but it is no more than that. At the moment we use a different map for gravity from the one we use for electromagnetism or for weak nuclear interactions. This may be cumbersome, but it is not disastrous. A theory of everything would simply be a single map, rather than a set of different ones that one uses in different circumstances. This latter philosophy is pragmatic. We use theories for the same reasons that we use maps: because they are useful. The famous London Underground map is certainly useful, but it is not a particularly accurate representation of physical reality. Nor does it need to be.
And in any case one has to worry about the nature of explanation afforded by a theory of everything. How will it explain, for example, why the theory of everything is what it is and not some other theory? To my mind, this is the biggest problem of all. Can any theory based on quantum mechanics be complete in any sense, when quantum theory is in its nature indeterministic? Moreover, developments in mathematical logic have cast doubt on the ability of any theory to be completely self-contained. The logician Kurt Gödel has proved a theorem, known as the incompleteness theorem, that shows that any mathematical theory will always contain things that can’t be proved within the theory.
