Three Roads to Quantum Gravity

by Lee Smolin

While this work applies to a much larger class of black holes than can be addressed by string theory, it does have one shortcoming compared with string theory: there is one constant that has to be adjusted to make the entropy and temperature come out right. This constant determines the value of Newton’s gravitational constant, as measured on large scales. It turns out that there is a small change in the value of the constant when one compares its value measured on the Planck scale with the value measured at large distances. This is not surprising. Shifts like this occur commonly in solid state physics, when one takes into account the effect of the atomic structure of matter. This shift is finite, and has to be made just once, for the whole theory. (It is actually equal to the √3/log 2.) Once done it brings the results for all different kinds of black holes in exact agreement with the predictions by Bekenstein and Hawking that we discussed in Chapters 6 to 8.

  Thus, string theory and loop quantum gravity have each added something essential to our understanding of black holes. One may ask whether there is a conflict between the two results. So far none is known, but this is largely because, at the moment, the two methods apply to different kinds of black hole. To be sure, we need to find a way of extending one of the methods so that it covers the cases covered by the other method. When we can do this we will be able to make a clean test of whether the pictures of black holes given by loop quantum gravity and string theory are consistent with each other.

  This is more or less what we have been able to understand so far about black holes from the microscopic point of view. A great deal has been understood, although it must also be said that some very important questions remain unanswered. The most important of these have to do with the interiors of black holes. Quantum gravity should have something to say about the singular region in the interior of a black hole, in which the density of matter and the strength of the gravitational field become infinite. There are speculations that quantum effects will remove the singularity, and that one consequence of this may be the birth of a new universe inside the horizon. This idea has been studied using approximation techniques in which the matter forming the black hole is treated quantum theoretically, but the geometry of spacetime is treated as in the classical theory. The results do suggest that the singularities are eliminated, and one may hope that this will be confirmed by an exact treatment. But, at least so far, neither string theory nor loop quantum gravity, nor any other approach, has been strong enough to study this problem.

  Until 1995 no approach to quantum gravity could describe black holes in any detail. None could explain the meaning of the entropy of a black hole or tell us anything about what black holes look like when probed on the Planck scale. Now we have two approaches that are able to do all these things, at least in some cases. Every time we are able to calculate something about a black hole, in either theory, it comes out right. There are many questions we still cannot answer, but it is difficult to avoid the impression that we are finally understanding something real about the nature of space and time.

  Furthermore, the fact that both string theory and loop quantum gravity both succeed in giving the right answers about quantum black holes is strong evidence that the two approaches may be revealing different sides of a single theory. Like Galileo’s projectiles and Kepler’s planets, there is more and more evidence that we are glimpsing the same world through different windows. To find the relation of his work to Kepler’s, Galileo would only have had to imagine throwing a ball far enough and fast enough that it became a moon. Kepler, from his point of view, could have imagined what a planet orbiting very close to the Sun might have looked like to people living on the Sun. In the present case, we only have to ask whether a string can be woven from a network of loops, or whether, if we look closely enough at a string, we can see the discrete structures of the loops. I personally have little doubt that in the end loop quantum gravity and string theory will be seen as two parts of a single theory. Whether it will take a Newton to find that theory, or whether it is something we mortals can do, is something that only time will tell.

  CHAPTER 14

  WHAT CHOOSES THE LAWS OF NATURE?

  Back in the 1970s there was a simple dream about how physics would end. A unified theory would be found that incorporated quantum theory, general relativity, and the various particles and forces known to us. This would not only be a theory of everything, it would be unique. We would discover that there was only one mathematically consistent quantum theory that unified elementary particle physics with gravity. There could be only one right theory and we would have found it. Because it was unique, this theory would have no free parameters - there would be no adjustable masses or charges. If there was anything to adjust, the theory would then come in different versions, and it would not be unique. There would be only one scale, against which everything was measured, which was the Planck scale. The theory would allow us to calculate the results of any experiment to whatever accuracy we desired. We would calculate the masses of the electron, proton, neutron, neutrinos and all the other particles, and our results would all be in exact agreement with experiments.

  These calculations would have to explain certain very strange features of the observed masses of the particles. For example, why are the masses of the proton and neutron so very small in Planck units? Their masses are of the order of 10-19 Planck masses. Where do such terribly small numbers come from? How could they come out of a theory with no free parameters? If the fundamental atoms of space have the Planck dimensions, then we would expect the elementary particles to have similar dimensions. The fact that protons and neutrons are nearly 20 orders of magnitude lighter than the Planck mass seems very hard to understand. But since the theory would be unique it would have to get this right.

  String theory was invented with the hope that it would be this final theory. It was its potential uniqueness that made it worth studying, even as it became clear that it was not soon going to lead to predictions about the masses of particles or anything else that could be tested experimentally. After all, if there is one unique theory it does not need experiments to verify it - all that is needed is to show that it is mathematically consistent. A unique theory must automatically be proved right by experiments, so it does not matter if a test of the theory is several centuries away. If we accept the assumption that there is one unique theory, then it will pay to concentrate on the problem of testing that theory for mathematical consistency rather than on developing experimental tests for it.

  The problem is that string theory did not turn out to be unique. It was instead found to come in a very large number of versions, each equally consistent. From our present-day perspective, taking into account only the results on the table, it seems that the hope for a unique theory is a false hope. In the current language of string theory, there is no way to distinguish between any of a very large number of theories: they are all equally consistent. Moreover, many of them have adjustable parameters, which could be changed to obtain agreement with experiment.

  Looking back, it is clear that the assumption that a unified theory would be unique was no more than that - an assumption. There is no mathematical or philosophical principle which guarantees there to be only one mathematically consistent theory of nature. In fact, we now know that there can be no such theory. For example, suppose that the world had one or two spatial dimensions, rather than three. For these cases we have constructed lots of consistent quantum theories, including some which have gravity. These were done as warm-up exercises for various research programmes. We keep them around as experimental laboratories in which we can test new ideas in a context where we know we can calculate anything we like. It is always possible that there is only one possible consistent theory to describe worlds that have more than two spatial dimensions. But there is no known reason why this should be true. In the absence of any evidence to the contrary, the fact that there are many consistent theories that describe one- and two-dimensional universes should lead us to doubt the assumption that mathematical consistency in itse
lf allows only one theory of nature.

  Of course, there is a way out, which is the possibility that string theory is not the final theory. Besides the fact that it comes in many versions, there are good reasons to believe this: string theory is background dependent and it is understood only in terms of a certain approximation scheme. A fundamental theory needs to be background independent and capable of being formulated exactly. So most people who work with string theory now believe the M theory conjecture I described in Chapter 11: that there is a single theory, which can be written down exactly and in a way that is independent of any given spacetime, that unifies all the different string theories.

  There is some evidence to support this M theory conjecture. Many physicists, myself included, are now trying to invent the theory. There seem to be three possibilities:1. The correct theory of nature is not a string theory.

  2. The M theory conjecture is false: there is no unified string theory, but one of the string theories will make predictions that agree with experiment.

  3. The M theory conjecture is true: there is a single unified theory, which, however, predicts that the world could come in a great many different physical phases. In these phases the laws of nature appear to be different. Our universe is in one of them.

  If possibility 1 is true, then all we can do is take the story of string theory as a cautionary tale. So let us put this one aside and look at the others. If possibility 2 is true, then we are left with a puzzle: what or who chose which consistent theory applies to our world? Among the list of different possible consistent theories, how was one chosen to apply to our universe?

  There seems to be only one possible answer to this question. Something external to the universe made the choice. If that’s the way things turn out, then this is the exact point at which science will become religion. Or, to put it better, it will then be rational to use science as an argument for religion. One already hears a lot about this in theological circles, as well as from certain scientists, in the form of arguments based on what we might call the anthropic observation. It seems that the universe we live in is very special. For a universe to exist for billions of years and contain the ingredients for life, certain special conditions must be satisfied: the masses of the elementary particles and the strengths of the fundamental forces must be tuned to values very close to the ones actually we observe. If these parameters are outside certain narrow limits, the universe will be inhospitable to life. This raises a legitimate scientific question: given that there seem to be more than one possible consistent set of laws, why is it that the laws of nature are such that the parameters fall within the narrow ranges needed for life? We may call this the anthropic question.

  If there are different possible consistent laws of nature, but no framework which unifies them, then there are only two possible answers to the anthropic question. The first is that we are very lucky indeed. The second is that whatever entity specified the laws did so in order that there would be life. In this case we have an argument for religion. This is of course a version of an argument which is well known to theologians - the God of the Gaps argument. If science raises a question like the anthropic question that cannot be answered in terms of processes that obey the laws of nature, it becomes rational to invoke an outside agency such as God. The scientific version of this argument is called the strong anthropic principle.

  Notice that this argument is valid only if there is no way to explain how the laws of nature might have been chosen except by invoking the action of some entity outside our universe. You may recall the principle with which I started this book: that there is nothing outside the universe. As long as there is a way of answering all our questions without violating this principle, we are doing science and we have no need of any other mode of explanation. So the argument for the strong anthropic principle has logical force only if there is no other possibility.

  But there is another possibility, possibility 3. This is like possibility 2, but with an important difference. If the different string theories describe different phases of a single theory, then it is possible that under the right circumstances there could be a transition from one phase to another. Just as ice melts to water, the universe could ‘melt’ from one phase, in which it is described by one string theory, to another phase, in which it is described by another. We are then still left with the question of why one phase rather than another describes our universe, but this is not so hard to resolve because in this picture the universe is allowed to have changed phase as it evolved in time. There is also the possibility that different regions of the universe exist in different phases.

  Given these possibilities, there are at least two alternatives to the God of the Gaps argument. The first is that there is some process that creates many universes. (Do not worry for the moment about what that process is, for cosmologists have found several attractive ways to make a universe which continually spawns new universes.) The big bang is then not the origin of all that exists, but only a kind of phase transition by which a new region of space and time was created, in a phase different than the one from which it came, and then cooled and expanded. In such a scenario there could be many big bangs, leading to many universes. The astrophysicist Martin Rees has a nice name for this - he calls the whole collection the ‘multiverse’. It is possible that the process creates universes in random phases. Each would then be governed by a different string theory. These universes will have different dimensions and geometries, and they will also have different sets of elementary particles which interact according to different sets of laws. If there are adjustable parameters, it is possible that they are set at random each time a new universe is created.

  So there is a simple answer to the anthropic question. Among all the possible universes, a minority will have the property that their laws are hospitable to life. Since we are alive, we naturally find ourselves in one of them. And since there are a great many universes, we need not worry that the chance of any one of them being hospitable to life is small, because the chance of at least one of them being hospitable to life may not be small. There will then be nothing to explain. Martin Rees likes to put this in the following way: if one finds a bag by the side of the road containing a suit that fits one perfectly, that is something to wonder about. But if one goes into a clothing store and is able to find a suit that fits, there is no mystery because the store carries lots of suits in many different sizes. We may call this the God of The Gap. It is also sometimes called the weak anthropic principle.

  The only problem with this kind of explanation is that it is difficult to see how it could be refuted. As long as your theory yields a very large number of universes, you only need there to be at least one like ours. The theory makes no other predictions apart from the existence of at least one universe like ours. But we already know that, so there is no way to refute this theory. This might seem good, but actually it is not because a theory that cannot be refuted cannot really be part of science. It can’t carry very much explanatory weight, because whatever features our universe has, as long as it can be described by one of the large number of string theories, our theory will not be refuted. Therefore it can make no new predictions about our universe.

  Is it possible to have a theory which gives a scientific answer to the anthropic question? Such a theory may be framed around the possibility that the universe can make a physical transition from one phase to another. If we could look back into the history of the universe to before the big bang, it may be that we would see one or a whole succession of different phases in which the universe had different dimensions and appeared to satisfy different laws. The big bang would then be just be the most recent of a series of transitions the universe has passed through. And even though each phase may be governed by a different string theory, the whole history of the universe would be governed by a single law - M theory. We then need an explanation in physical terms for why the universe ‘chose’ to exist in a phase such as the one in which we find ourselves, which exists for billions of y
ears and is hospitable for life. There are several different possible explanations of this kind, which are described in detail in another book of mine, The Life of the Cosmos, so I shall be brief here.

  One idea is that new universes could form inside black holes. In this case our universe would have a large number of progeny, as it contains at least 1018 black holes. One may also conjecture that the changes in the laws from old universes to new are small, so that the laws in each new universe formed from our own are close to those that hold in our universe. This also means that the laws in the universe from which ours was formed were not very different from those of our own. Given these two assumptions, a mechanism which is formally analogous to natural selection operates, because after many generations those universes that give rise to many black holes will dominate the population of universes. The theory then predicts that a randomly chosen universe will have the property that it will make more black holes than will universes with slightly different values of the parameters. We can then ask whether this prediction is satisfied by our universe. To cut a long story short, up to the present time it seems that it is. The reason is that carbon chemistry is not only good for life, it plays an important role in the processes that make the massive stars that end up as black holes. However, there are several possible observations which could disprove the theory. Thus, unlike the God of the Gaps and the God of The Gap theories, this theory is very vulnerable to being disproved. Of course, this means that it is likely to be disproved.

  The important thing about this theory is that it shows that there are alternatives to both the strong and the weak anthropic principle. And if that is so, then those principles have no logical force. Beyond this, the theory of cosmological natural selection (as it is sometimes called) shows us that physics can learn an important lesson from biology about possible modes of scientific explanation. If we want to stick to our principle that there is nothing outside the universe, then we must reject any mode of explanation in which order is imposed on the universe by an outside agency. Everything about the universe must be explicable only in terms of how the laws of physics have acted in it over the whole span of its history.