Preface

ONE of the deepest mysteries of our universe is the puzzle of whence it came.

When I entered Cambridge University as a mathematics graduate student, in the early 1950s, a fascinating cosmological theory was in the ascendant, known as the steady-state model. According to this scheme, the universe had no beginning, and it remained more-or-less the same, overall, for all time. The steady-state universe was able to achieve this, despite its expansion, because the continual depletion of material arising from the universe’s expansion is taken to be compensated by the continual creation of new material, in the form of an extremely diffuse hydrogen gas. My friend and mentor at Cambridge, the cosmologist Dennis Sciama, from whom I learnt the thrill of so much new physics, was at that time a strong proponent of steady-state cosmology, and he impressed upon me the beauty and power of that remarkable scheme of things.

Yet this theory has not stood the test of time. About 10 years after I had first entered Cambridge, and had become well acquainted with the theory, Arno Penzias and Robert Wilson discovered, to their own surprise, an all-pervading electromagnetic radiation, coming in from all directions, now referred to as the cosmic microwave background or CMB. This was soon identified, by Robert Dicke, as a predicted implication of the ‘flash’ of a Big-Bang origin to the universe, now presumed to have taken place some 14 thousand million years ago—an event that had been first seriously envisaged by Monsignor Georges Lemaître in 1927, as an implication of his work on Einstein’s 1915 equations of general relativity and early observational indications of an expansion of the universe. With great courage and scientific honesty (when the CMB data became better established), Dennis Sciama publicly repudiated his earlier views and strongly supported the idea of the Big Bang origin to the universe from then on.

Since that time, cosmology has matured from a speculative pursuit into an exact science, and intense analysis of the CMB—coming from highly detailed data, generated by numerous superb experiments—has formed a major part of this revolution. However, many mysteries remain, and much speculation continues to be part of this endeavour. In this book, I provide descriptions not only of the main models of classical relativistic cosmology but also of various developments and puzzling issues that have arisen since then. Most particularly, there is a profound oddness underlying the Second Law of thermodynamics and the very nature of the Big Bang. In relation to this, I am putting forward a body of speculation of my own, which brings together many strands of different aspects of the universe we know.

My own unorthodox approach dates from the summer of 2005, though much of the detail is more recent. This account goes seriously into some of the geometry, but I have refrained from including, in the main body of the text, anything serious in the way of equations or other technicalities, all these being banished to the Appendices. The experts, only, are referred to those parts of the book. The scheme that I am now arguing for here is indeed unorthodox, yet it is based on geometrical and physical ideas which are very soundly based. Although something entirely different, this proposal turns out to have strong echoes of the old steady-state model!

I wonder what Dennis Sciama would have made of it.

Prologue

WITH his eyelids half closed, as the rain pelted down on him and the spray from the river stung his eyes, Tom peered into the swirling torrents as the water rushed down the mountainside. ‘Wow’, he said to his Aunt Priscilla, an astrophysics professor from the University of Cambridge, who had taken him to this wonderful old mill, preserved in excellent working order, ‘is it always like this? No wonder all that old machinery can be kept buzzing around at such great speed.’

‘I don’t think it’s always this energetic’, said Priscilla, standing next to him behind the railing at the side of the river, and raising her voice somewhat, so as to be heard over the noise of the rushing water. ‘The water’s much more violent than usual, today, because of all this wet weather. You can see down there that a good portion of the water has had to be diverted away from the mill. Usually they would not do this, because they would have to make the most of a much more sedate flow. But now there’s far more energy in the flow than is needed for the mill.’

Tom stared for some minutes into the wildly tumbling water and admired the patterns it made as it was flung into the air in sprays and convoluted surfaces. ‘I can see there’s a lot of power in that water, and I know that a couple of centuries ago the people were clever enough to see how all this energy could be used to drive these machines—doing the work of many human beings and making all that great woollen cloth. But where did the energy come from that got all that water high up on the mountain in the first place?’

‘The heat of the Sun caused the water in the oceans to evaporate and rise up into the air, so it would eventually come back down again in all this rain. So a good proportion of the rain would be deposited up high into the mountains’, replied Priscilla. ‘It’s really the energy from the Sun that is being harnessed to run the mill.’

Tom felt a little puzzled by this. He was often puzzled by the things that Priscilla told him, and was by nature often quite sceptical. He could not really see how just heat could lift water up into the air. And if there was all that heat around, why did he feel so cold now? ‘It was rather hot yesterday’, he grudgingly agreed. Though, still uneasy, he commented, ‘but I didn’t feel the Sun trying to lift me up into the air then, any more than I do now.’

Aunt Priscilla laughed. ‘No. it’s not really like that. It’s the tiny little molecules in the water in the oceans that the Sun’s heat causes to be more energetic. So these molecules then rush randomly around faster than they would otherwise, and a few of these “hot” molecules will move so fast that they break loose from the surface of the water and are flung into the air. And although there are only a relatively few molecules flung out at one time, the oceans are so vast that there would really be a lot of water flung up into the air altogether. These molecules go to make the clouds and eventually the water molecules fall down again as rain, a lot of which falls high in the mountains.’

Tom was still rather troubled, but at least the rain had now tapered off somewhat. ‘But this rain doesn’t feel at all hot to me.’

‘Think of the Sun’s heat energy first getting converted into the energy of rapid random motion of the water molecules. Then think of this rapid motion resulting in a small proportion of the molecules going so fast that they are flung high in the air in the form of water vapour. The energy of these molecules gets converted into what’s called gravitational potential energy. Think of throwing a ball up into the air. The more energetically you throw it the higher it goes. But when it reaches its maximum height, it stops moving upwards. At that point its energy of motion has all been converted into this gravitational potential energy in its height above the ground. It’s the same with the water molecules. Their energy of motion—the energy that they got from the Sun’s heat—is converted into this gravitational potential energy, now at the top of the mountain, and when it runs down, this is converted back again into the energy in its motion, which is used to run the mill.’

‘So the water isn’t hot at all when it’s up there?’ asked Tom.

‘Exactly, my dear. By the time that these molecules get very high in the sky, they slow down and often actually get frozen into tiny ice crystals—that’s what most clouds are made of—so the energy goes into their height above the ground rather than into their heat motion. Accordingly, the rain won’t be hot at all up there, and it’s still quite cold even when it finally works its way down again, slowed down by the resistance of the air.’

‘That’s amazing!’

‘Yes, indeed’, and encouraged by the boy’s interest, Aunt Priscilla eagerly took advantage of the opportunity to say more. ‘You know, it’s a curious fact that even in the cold water in this river there is still much more heat energy in the motion of the individual molecules running around randomly at great speed than there is in the swirling currents of water rushing down the mountainside!’

‘Goodness. I’m supposed to believe that, am I?’

Tom thought for a few minutes, somewhat confused at first, but then rather attracted by what Priscilla said, remarked excitedly: ‘Now you’ve given me a great idea! Why don’t we build a special kind of mill that just directly uses all that energy of the motion of water molecules in some ordinary lake? It could use lots of tiny little windmill things, maybe like those things that spin in the wind, with little cups on the ends so that they twirl round in the wind no matter which direction the wind is coming from. Only they’d be very tiny and in the water, so that the speed of the water molecules would spin them around, and you could use these to convert the energy in the motion in the water molecules to drive all sorts of machinery.’

‘What a wonderful idea, Tom darling, only unfortunately it wouldn’t work! That’s because of a fundamental physical principle known as the Second Law of thermodynamics, which more or less says that things just get more and more disorganized as time goes on. More to the point, it tells you that you can’t get useful energy out of the random motions of a hot—or cold—body, just like that. I’m afraid what you’re suggesting is what they call a “Maxwell’s demon”.’

‘Don’t you start doing that! You know that Grandpa always used to call me a “little demon” whenever I had a good idea, and I didn’t like it. And, that Second Law thing’s not a very nice kind of law’, Tom complained grumpily. Then his natural scepticism returned: ‘And I’m not sure I can really believe in it anyway.’ Then he continued ‘I think laws like that just need clever ideas to get around them. In any case, I thought you said that it’s the heat of the Sun that’s responsible for heating the oceans and that it’s that random energy of motion that flings it to the top of the mountain, and that’s what’s running the mill.’

‘Yes, you’re right. So the Second Law tells us that actually the heat of the Sun all by itself wouldn’t work. In order to work, we also need the colder upper atmosphere, so that the water vapour can condense up above the mountain. In fact, the Earth as a whole doesn’t get energy from the Sun overall.’

Tom looked at his aunt with a quizzical expression. ‘What does the cold upper atmosphere have to do with it? Doesn’t “cold” mean not so much energy as “hot”? How does a bit of “not-so-much energy” help? I don’t get what you are saying at all. Anyway, I think you are contradicting yourself’, said Tom, gaining confidence in himself. ‘First you tell me that the Sun’s energy runs the mill, and now you tell me that the Sun doesn’t give energy to the Earth after all!’

‘Well, it doesn’t. If it did, then the Earth would just keep on getting hotter and hotter as it gained energy. The energy that the Earth gets from the Sun in the daytime has all to go back into space eventually, which it does because of the cold night sky—except, I suppose, that with global warming, a little part of it does get held back by the Earth. It’s because the Sun is a very hot spot in an otherwise cold dark sky …’

Tom began to lose the thread of what she was saying and his mind began to wander. But he heard her say, ‘… so it’s the manifest organization in the Sun’s energy that enables us to keep the Second Law at bay.’

Tom looked at Aunt Priscilla, almost totally bemused. ‘I don’t think I really understand all that,’ he said, ‘and I don’t see why I need to believe that “Second Law” thing in any case. Anyway, where does all that organization in the Sun come from? Your Second Law should be telling us that the Sun’s getting more disorganized as time goes on, so it would have to have been enormously organized when it was first formed, since all the time it’s sending out organization. Your “Second Law” thing tells us that its organization keeps getting lost.’

‘It has to do with the Sun being such a hot spot in a dark sky. This extreme temperature imbalance provided the needed organization.’

Tom stared at Aunt Priscilla, with little comprehension, and now not really properly believing anything she was telling him. ‘You tell me that counts as organization; well, I don’t see why it should. All right, let’s pretend it somehow does—but then you still haven’t told me where that funny kind of organization comes from.’

‘From the fact that the gas that the Sun condensed from was previously spread uniformly, so that gravity could cause it to form clumps which condensed gravitationally into stars. A very long time ago, the Sun did just this; it condensed from this initially spread-out gas, getting hotter and hotter in the process.’

‘You’ll keep telling me one thing after another, going way back in time, but where does this thing you call “organization”, whatever it is, originally come from?’

‘Ultimately it comes from the Big Bang, which was what started the whole universe off with an utterly stupendous explosion.’

‘A thing like a big walloping explosion doesn’t sound like something organized. I don’t get it at all.’

‘You aren’t the only one! You’re in good company not to get it. Nobody really gets it. It’s one of the biggest puzzles of cosmology where the organization comes from, and in what way the Big Bang really represents organization in any case.’

‘Maybe there was something more organized before the Big Bang? That might do it.’

‘People have actually tried suggesting things like that for some while. There are theories in which our presently expanding universe had a previous collapsing phase which “bounced” to become our Big Bang. And there are other theories where little bits of a previous phase of the universe collapsed into things we call black holes, and these bits “bounced”, to become the seeds of lots and lots of new expanding universes, and there are others where new universes sprang out of things called “false vacuums”…’

‘That all sounds pretty crazy to me,’ Tom said.

‘And, oh yes, there’s another theory that I heard about recently …’

Part 1

The Second Law and its underlying mystery

1.1 The relentless march of randomness

1.2 Entropy, as state counting

1.3 Phase space, and Boltzmann’s definition of entropy

1.4 The robustness of the entropy concept

1.5 The inexorable increase of entropy into the future

1.6 Why is the past different?

1.1 The relentless march of randomness

THE SECOND LAW of thermodynamics—what law is this? What is its central role in physical behaviour? And in what way does it present us with a genuinely deep mystery? In the later sections of this book, we shall try to understand the puzzling nature of this mystery and why we may be driven to extraordinary lengths in order to resolve it. This will lead us into unexplored areas of cosmology, and to issues which I believe may be resolved only by a very radical new perspective on the history of our universe. But these are matters that will be our concern later. For the moment let us restrict our attention to the task of coming to terms with what is involved in this ubiquitous law.

Usually when we think of a ‘law of physics’ we think of some assertion of equality between two different things. Newton’s second law of motion, for example, equates the rate of change of momentum of a particle (momentum being mass times velocity) with the total force acting upon it. As another example, the law of conservation of energy asserts that the total energy of an isolated system at one time is equal to its total energy at any other time. Likewise, the law of conservation of electric charge, of momentum, and of angular momentum, each asserts a corresponding equality for the total electric charge, for the total momentum, and for total angular momentum. Einstein’s famous law E=mc² asserts that the energy of a system is always equal to its mass multiplied by the square of the speed of light. As yet another example, Newton’s third law asserts that the force exerted by a body A on a body B, at any one time, is always equal and opposite to the force acting on A due to B. And so it is for many of the other laws of physics.

These are all equalities—and this applies also to what is called the First Law of thermodynamics, which is really just the law of conservation of energy again, but now in a thermodynamic context. We say ‘thermodynamic’ because the energy of the thermal motions is now being taken into account, i.e. of the random motions of individual constituent particles. This energy is the heat energy of a system, and we define the system’s temperature to be this energy per degree of freedom (as we shall be considering again later). For example, when the friction of air resistance slows down a projectile, this does not violate the full conservation law of energy (i.e. the First Law of thermodynamics)—despite the loss of kinetic energy, due to the projectile’s slowing—because the air molecules, and those in the projectile, become slightly more energetic in their random motions, from heating due to the friction.

However, the Second Law of thermodynamics is not an equality, but an inequality, asserting merely that a certain quantity referred to as the entropy of an isolated system—which is a measure of the system’s disorder, or ‘randomness’—is greater (or at least not smaller) at later times than it was at earlier times. Going along with this apparent weakness of statement, we shall find that there is also certain vagueness or subjectivity about the very definition of the entropy of a general system. Moreover, in most formulations, we are led to conclude that there are occasional or exceptional moments at which the entropy must be regarded as actually (though temporarily) reducing with time (in a fluctuation) despite the general trend being that the entropy increases.

Yet, set against this seeming imprecision inherent in the Second Law (as I shall henceforth abbreviate it), this law has a universality that goes far beyond any particular system of dynamical rules that one might be concerned with. It applies equally well, for example, to relativity theory as it does to Newtonian theory, and also to the continuous fields of Maxwell’s theory of electromagnetism (that we shall be coming to briefly in §2.6, §3.1 and §3.2, and rather more explicitly in Appendix A1) just as well as it does to theories involving only discrete particles. It applies also to hypothetical dynamical theories that we have no good reason to believe have relevance to the actual universe that we inhabit, although it is most pertinent when applied to realistic dynamical schemes, such as Newtonian mechanics, which have a deterministic evolution and are reversible in time, so that for any allowed evolution into the future, reversing the time direction gives us another equally allowable evolution according to the dynamical scheme.

To put things in familiar terms, if we have a moving-picture film depicting some action that is in accordance with dynamical laws—such as Newton’s—that are reversible in time, then the situation depicted when the film is run in reverse will also be in accordance with these dynamical laws. The reader might well be puzzled by this, for whereas a film depicting an egg rolling off a table, falling to the ground, and smashing would represent an allowable dynamical process, the time-reversed film—depicting the smashed egg, originally as a mess on the floor, miraculously assembling itself from the broken pieces of shell, with the yolk and albumen separately joining up to become surrounded by the self-assembling shell, and then jumping up on to the table—is not an occurrence that we expect ever to see in an actual physical process (Fig. 1.1). Yet the full Newtonian dynamics of each individual particle, with its accelerated response (in accordance with Newton’s second law) to all forces acting upon it, and the elastic reactions involved in any collision between constituent particles, is completely reversible in time. This also would be the case for the refined behaviour of relativistic and quantum-mechanical particles, according to the standard procedures of modern physics—although there are some subtleties arising from the black-hole physics of general relativity, and also with regard to quantum mechanics, that I do not wish to get embroiled in just yet. Some of these subtleties will actually be crucially important for us later, and will be considered particularly in §3.4. But for the moment, an entirely Newtonian picture of things will suffice.

We have to accustom ourselves to the fact that the situations that are depicted by both directions of film-running are consistent with Newtonian dynamics, but the one showing the self-assembling egg depicts an occurrence that is inconsistent with the Second Law, and would be such an enormously improbable sequence of events that we can simply reject it as a realistic possibility. What the Second Law indeed states, roughly speaking, is that things are getting more ‘random’ all the time. So if we set up a particular situation, and then let the dynamics evolve it into the future, the system will evolve into a more random-looking state as time progresses. Strictly, we should not say that it will evolve into a more random-looking state but that, in accordance with what has been said above, it is (something like) overwhelmingly likely to evolve into such a more random state. In practice, we must expect that, according to the Second Law, things are indeed getting progressively more and more random with time, but that this represents merely an overwhelming probability, not quite an absolute certainty.

Nevertheless we can assert, with a considerable amount of confidence, that what we shall experience will be an entropy increase—in other words an increase in randomness. Stated that way, the Second Law sounds perhaps like a council of despair, for it tells us that things are just getting more and more disorganized as time progresses. This does not sound like any kind of a mystery, however, as the title of Part 1 seems to be suggesting that it should. It’s just an obvious feature of the way things would behave if left entirely to themselves. The Second Law appears to be just expressing an inevitable and perhaps depressing feature of everyday existence. Indeed, from this point of view, the Second Law of thermodynamics is one of the most natural things imaginable, and certainly something that reflects a completely commonplace experience.

Some might worry that the emergence of life on this Earth, with its seemingly unbelievable sophistication, represents a contradiction with this increase of disorder that the Second Law demands. I shall be explaining later (see §2.2) why there is in fact no contradiction. Biology is, as far as we know, entirely consistent with the overall entropy increase that the Second Law demands. The mystery referred to in the title of Part 1 is a mystery of physics of an entirely different order of scale. Although it has some definite relation to that mysterious and puzzling organization that we are continually being presented with through biology, we have good reason to expect that the latter presents no paradox with regard to the Second Law.

One thing should be made clear, however, with regard to the Second Law’s physical status: it represents a separate principle that must be adjoined to the dynamical laws (e.g. to Newton’s laws), and is not to be regarded as a deduction from them. The actual definition of the entropy of a system at any one moment is, however, symmetrical with regard to the direction of time (so we get the same entropy definition, for our filmed falling egg, at any one moment, irrespective of the direction in which the film is shown), and if the dynamical laws are also symmetrical in time (as is indeed the case with Newtonian dynamics), the entropy of a system being not always constant in time (as is clearly so with the smashing egg), then the Second Law cannot be a deduction from these dynamical laws. For if the entropy is increasing in a particular situation (e.g. egg smashing), this being in accordance with the Second Law, then the entropy must be decreasing in the reversed situation (egg miraculously assembling), which is in gross violation of the Second Law. Since both processes are nevertheless consistent with the (Newtonian) dynamics, we conclude that the Second Law cannot simply be a consequence of the dynamical laws.

1.2 Entropy, as state counting

BUT HOW DOES the physicist’s notion of ‘entropy’, as it appears in the Second Law, actually quantify this ‘randomness’, so that the self-assembling egg can indeed be seen to be overwhelmingly improbable, and thereby rejected as a serious possibility? In order to be a bit more explicit about what the entropy concept actually is, so that we can make a better description of what the Second Law actually asserts, let us consider a physically rather simpler example than the breaking egg. The Second Law tells us, for example, that if we pour some red paint into a pot and then some blue paint into the same pot and give the mixture a good stir, then after a short period of such stirring the different regions of red and of blue will lose their individuality, and ultimately the entire contents of the pot will appear to have the colour of a uniform purple. It seems that no amount of further stirring will convert the purple colour back to the original separated regions of red and blue, despite the time-reversibility of the submicroscopic physical processes underlying the mixing. Indeed, the purple colour should eventually come about spontaneously, even without the stirring, especially if we were to warm the paint up a little. But with stirring, the purple state is reached much more quickly. In terms of entropy, we find that the original state, in which there are distinctly separated regions of red and blue paint, will have a relatively low entropy, but that the pot of entirely purple paint that we end up with will have a considerably larger entropy. Indeed, the whole stirring procedure provides us with a situation that is not only consistent with the Second Law, but which begins to give us a feeling of what the Second Law is all about.

Let us try to be more precise about the entropy concept, so that we can be more explicit about what is happening here. What actually is the entropy of a system? Basically, the notion is a fairly elementary one, although involving some distinctly subtle insights, due mainly to the great Austrian physicist Ludwig Boltzmann, and it has to do just with counting the different possibilities. To make things simple, let us idealize our pot of paint example so that there is just a (very large) finite number of different possibilities for the locations of each molecule of red paint or of blue paint. Let us think of these molecules as red balls or blue balls, these being allowed to occupy only discrete positions, centred within N³ cubical compartments, where we are thinking of our paint pot as an enormously subdivided N×N×N cubical crate composed of these compartments (see Fig. 1.2), where I am assuming that every compartment is occupied by exactly one ball, either red or blue (represented as white and black, respectively, in the figure).