Cycles of Time: An Extraordinary New View of the Universe

by Roger Penrose

For a system of d degrees of freedom, the configuration space would be a d-dimensional space. For example, if the system consisted of q point particles p1, p2, …, pq (each without any internal degrees of freedom), then the configuration space would have 3q dimensions. This is because each individual particle requires just three coordinates to determine its position, so there are 3q coordinates overall, whereby a single point P of configuration space defines the locations of all of p1, p2,…, pq together (see Fig. 1.4). In more complicated situations, where there are internal degrees of freedom as above, we would have more degrees of freedom for each such particle, but the general idea is the same.

The entropy values that would be attributed to a system would, on the whole, change very little as a result of redrawing the boundaries of the coarse-graining regions in this kind of way, as might result from improved technology. We must indeed bear in mind that there is likely to be always some measure of subjectivity in the precise value of the entropy that one might assign to a system, on account of the precision that might be available in measuring devices at any one time, but we should not adopt the point of view that the entropy is not a physically useful concept for that kind of reason. In practice, this subjectivity will, in normal circumstances, amount to a very small factor. The reason for this is that the coarse-graining regions tend to have volumes that are absolutely stupendously different from one another, and the detailed redrawing of their boundaries will normally make no discernible difference to the entropy values that are assigned.

In my opinion, there is a ‘can of worms’ here, if one demands that there should be a precise objective definition of physical entropy, applicable in all circumstances, with respect to which the Second Law is to be universally valid. I do not see why one should demand that there always be a well-defined, physically precise notion of ‘entropy’, that is entirely objective and consequently ‘out there’ in Nature, in some absolute sense,[1.11] where this ‘objective entropy’ almost never decreases as time progresses. Must there always be an actual entropy concept that applies to the slightly tinted viscous fluid between the cylinders, or to the configurations of nuclear spins that had appeared to become totally disorganized, though retaining a precise ‘memory’ of the order that they had had before? I do not see that this need be the case. Entropy is clearly an extremely useful physical concept, but I do not see why it need be assigned a truly fundamental and objective role in physics. Indeed, it seems reasonable to me that the usefulness of the physical notion of entropy has its origin largely in the fact that, for systems that we tend to encounter in the actual universe, it turns out that the normal measures of ‘macroscopic’ quantities give rise to coarse-graining volumes that do in fact differ from one another by stupendously large factors. There is a profound issue, however, as to why, in the universe that we know, they should differ by such enormous factors. These enormous fa...

Why has our reasoning gone so sadly astray—this being apparently just the same reasoning that seemed convincingly to lead us to expect that the Second Law, with overwhelming probability, must hold for the future evolution of an ordinary physical system? The trouble with the reasoning, as I have provided it, lies in the assumption that the evolution can be regarded as effectively ‘random’ in relation to the coarse-graining regions. Of course it is not really random, as noted above, since it is precisely determined by the dynamical (e.g. Newton’s) laws. But we have taken it that there is no particular bias in this dynamical behaviour, in relation to these coarse-graining regions, and this supposition seemed to be fine for the future evolution. When we consider the past evolution, however, we find that this is manifestly not the case. There is a great deal of bias, for example, in the past-evolved behaviour of the egg, where it would appear to be guided inexorably—if viewed from a time-reversed perspective—from an original messy broken state, through exceptionally improbable actions albeit all in accordance with the dynamical laws, to the exceedingly improbable state of being balanced, complete and unbroken, at the edge of the table. If such behaviour were to be observed in future-directed behaviour it would be regarded as an impossible form of teleology or magic. Why do we regard such clearly focused behaviour as being perfectly acceptable if it is directed towards the past,...

So long as we may take it that the dynamical laws are such that there is an appropriate degree of continuity in the way that the entropy of the universe behaves, as noted above, then we need merely suppose that the universe’s initial state—what we call the Big Bang—had, for some reason, an extraordinarily tiny entropy (a tininess which, as we shall be seeing in the next part, has a rather subtle character). The appropriate continuity of entropy would then imply a relatively gradual increase of the universe’s entropy from then on (in the normal time-direction), giving us some kind of theoretical justification of the Second Law. So the key issue is indeed the specialness of the Big Bang, and the extraordinary minuteness of the initial coarse-graining region that represents the nature of this special initial state. The issue of the Big-Bang specialness is central to the arguments of this book.

In the first place, it has occasionally been argued that the existence of a Second Law holds no mystery, for our experience of the passage of time is dependent upon an increasing entropy as part of what constitutes our conscious feeling of the passage of time; so whatever time-direction we believe to be the ‘future’ must be that in which entropy increases. According to this argument, had the entropy been decreasing with respect to some time-parameter t, then our conscious feelings of temporal flow would project in the reverse direction, so that we would regard the small values of t to lie in what we think of as our ‘future’ and the large values in our ‘past’. We would therefore take the parameter t to be the reverse of a normal time parameter, so that the entropy would still be increasing into what we experience as being the future. Thus, so the argument goes, our psychological experiences of the passage of time would always be such that the Second Law holds true, irrespective of the physical direction of the progression of entropy. However, even leaving aside the very dubious nature of any such argument from our ‘experience of time progression’—when we know almost nothing about what physical prerequisites might be required for ‘conscious experience’—this argument misses the crucial point that the very usefulness of the notion of entropy depends upon our universe being enormously far from thermal equilibrium, so that coarse-graining regions that are far smaller than max ar...

Sometimes the argument is made (perhaps in conjunction with the above) that the presence of a Second Law is an essential prerequisite for life, so that living beings like ourselves could only exist in a universe (or a universe epoch) in which the Second Law holds true, this law being a necessary ingredient of natural selection, etc. This is an example of ‘anthropic reasoning’ and I shall be returning briefly to this general issue in §3.2 (end) and §3.3. Whatever value this type of argument may have in other contexts, it is next to useless here. Again there is the very dubious aspect of such reasoning that we do not have a great deal more understanding of the physical requirements for life than we do for consciousness. But even apart from this, and even assuming that natural selection is indeed an essential prerequisite for life, and that it does require the Second Law, this still provides no explanation for the fact that the same Second Law operative here on Earth appears to hold everywhere in the observable universe to distances far beyond those of any relevance to local conditions, such as in galaxies thousands of millions of light years distant, and to times far earlier than the beginnings of life on Earth.

If we do not assume the Second Law, or that the universe originated in some extraordinarily special initial state, or something else of this general nature, then we cannot use the ‘improbability’ of the existence of life as a premise for a derivation of a Second Law that is operative at times earlier than the present. No matter how curious and non-intuitive it may seem, the production of life would (if we do not make a prior assumption of the Second Law) be far less probable to come about by natural means—be it by natural selection or any by other seemingly ‘natural’ process—than by a ‘miraculous’ creation simply out of random collisions of the constituent particles!

Accordingly, the mere existence of life provides, in itself, no argument whatever for the full validity of the Second Law.

Fig. 2.3 The three basic kinds of uniform plane geometry as illustrated by Maurits C. Escher: (a) elliptic (positive, K>0); (b) Euclidean (flat, K=0); (c) hyperbolic (negative, K < 0). Copyright M. C. Escher Company (2004).

evidence has mounted which strongly indicates that the expansion of the universe in its later stages does not actually match the evolution rates predicted from the standard Friedman cosmologies that are illustrated in Fig. 2.2. Instead, it appears that our universe has begun to accelerate in its expansion, at a rate that would be explained if we are to include into Einstein’s equations a cosmological constant Λ, with a small positive value. These, and later observations of various kinds,[2.11] have provided fairly convincing evidence of the beginnings of the exponential expansion characteristic of a Friedmann model with Λ>0. This exponential expansion occurs not only with the cases K≤0 which in any case expand indefinitely in their remote futures even when Λ=0, but also in the spatially closed case K>0, provided that Λ is large enough to overcome the tendency for recollapse that the closed Friedmann model possesses. Indeed, the evidence does indicate the presence of a large-enough Λ, so that the value (sign) of K has become more-or-less irrelevant to the expansion rate, where the (positive) value of Λ that appears actually to be present in Einstein’s equations would then dominate the late-time behaviour, providing an exponential expansion independently of the value of K within the observationally acceptable range.

many cosmologists are of the opinion that the viewpoint provided by cosmic inflation does provide a powerful reason for believing that the spatial geometry of the spatial universe must be (apart from relatively small local deviations) actually flat (K=0), so they are pleased by this observational closeness to flatness. Cosmic inflation is a proposal that, within a very tiny time-period somewhere between around 10–36 and 10–32 seconds after the Big Bang, the universe underwent an exponential expansion, increasing its linear dimension by an enormous factor of around 1030 or 1060 (or even 10100) or so.

The circular boundary itself represents infinity for this geometry, and it is this conformal representation of infinity as a smooth finite boundary that I am pointing out here to the reader, as it will be playing a central role in the ideas that we shall be coming to later

the CMB that we now ‘see’ is not really the ‘actual Big Bang’, however, as these photons come to us directly from what is called the ‘surface of last scattering’ which occurred some 379 000 years following the moment of the Big Bang (i.e. when the universe was about 1/36 000 of its present age). Earlier than this, the universe was opaque to electromagnetic radiation because it would have been inhabited by large numbers of separate charged particles—mainly protons and electrons—milling around separately from each other, constituting what is referred to as a ‘plasma’. Photons would have scattered many times in this material, being absorbed and created copiously, and the universe would have been very far from transparent.

The continuous line is Planck’s ‘black-body curve’, which is given by a specific formula,[2.18] and it is what quantum mechanics tells us is the radiation spectrum of thermal equilibrium, for any particular temperature T.

the CMB provides us with the most precise agreement between an observed intensity spectrum and the calculated Planck black-body curve that is known in observational science.

It appears to tell us that what we are looking at comes from a state that must effectively be thermal equilibrium. But what does ‘thermal equilibrium’ actually mean? I refer the reader back to Fig. 1.15, where we find the words ‘thermal equilibrium’ labelling the coarse-graining region of phase space which is (by far) the largest of all. In other words, this is the region representing maximum entropy. But we must recall the thrust of the arguments given in §1.6. These arguments told us that the whole basis of the Second Law must be explained by the fact that the initial state of the universe—which we evidently must take as being the Big Bang—must be a (macroscopic) state of extraordinarily tiny entropy. What we appear to have found is essentially the complete opposite, namely a (macroscopic) state of maximum entropy!

the formation of a black hole represents an enormous increase in the entropy. Indeed, at the present epoch of the universe’s evolution, the greatest entropy contribution, by far, lies in large black holes, like the one at the centre of our own Milky Way galaxy, with a mass of around 4 000 000 times the mass of our Sun. The total entropy in such objects completely swamps that in the CMB, which had previously been thought to represent the dominant contribution to the entropy present in the universe. Thus, the entropy has greatly increased via gravitational condensation from what it was at the creation of the CMB.

Photons arriving at the Earth’s surface from the Sun have higher energy (shorter wavelength) than those returned to space by the Earth. Given an overall energy balance (the Earth does not get hotter over time), there must be more photons leaving than arriving; that is, the energy arriving has lower entropy than that departing.

Now, on Earth, the green plants have, by the process of photosynthesis, found a way of converting the relatively high-frequency photons coming from the Sun to photons of a lower frequency, using this gain in low entropy to build up their substance by extracting carbon from CO2 in the air and returning it as O2. When animals eat plants (or eat other animals that eat plants), they use this source of low entropy, and the O2, to keep down their own entropy.

Accordingly, it is the potential for stars to form (albeit via somewhat complicated processes in appropriate regions in space), through the relentless entropy-raising process of gravitational clumping, from initial material that started off in a very uniform gravitationally low-entropy state. This all comes about, ultimately, from the fact that we have been presented with a Big Bang of a very special nature, the extreme (relative) lowness of its entropy being manifested in the fact that its gravitational degrees of freedom were indeed not initially activated. This is a curiously lop-sided situation,

Thus, in Minkowski’s 4-geometry, we are not now to think of the space-time as being simply built out of a succession of 3-surfaces, each representing what we think of as ‘space’ at various different times (Fig. 2.10). For that interpretation, each of these 3-surfaces would describe a family of events all of which would be taken to be simultaneous with one another. In special relativity, the notion of ‘simultaneous’ for spatially separated events does not have an absolute meaning. Instead, ‘simultaneity’ would depend upon some arbitrarily chosen observer’s velocity.

According to relativity, the notion of ‘simultaneous’, for distant events, is not an absolute thing, but depends upon some observer’s velocity to be specified, so the slicing of space-time into a family of simultaneous 3-spaces is subjective in the sense that for a different observer velocity we get a different slicing. What Minkowski’s space-time achieves is to provide an objective geometry, that is not dependent on some arbitrary observer’s view of the world, and which does not have to change when one observer is replaced by another. In a certain sense, what Minkowski did was to take the ‘relativity’ out of special relativity theory, and to present us with an absolute picture of spatio-temporal activity.

Fig. 2.12 (a) Null cone at p in Minkowski’s 4-space; (b) 3-space description of the future cone as an expanding succession of concentric spheres originating at p.

Einstein’s theory tells us that the speed of any massive particle must always be less than that of light. In space-time terms, this means that the world-line of such a particle—this being the locus of all the events that constitute the particle’s history—must be directed within the null cone at each of its events. See Fig. 2.13. A particle may have a motion that is accelerated at some places along its world-line, whence its world-line need not be straight, the acceleration being expressed, in space-time terms, as a curvature of the world-line. Where the world-line is curved, it is the tangent vector to the world-line that must lie within the null cone. If the particle is massless,[2.28] such as a photon, then its world-line must lie along the null cone at each of its events, since its speed at every one of its events is indeed taken to be light speed. Fig. 2.13 Null cones in , uniformly arranged. World lines of massive particles are directed within the cones and of massless ones along the cones.

The null cones also tell us about causality, which is the issue of determining which events are to be regarded as being able to influence which other events. One of the tenets of (special) relativity theory is the assertion that signals shall not be allowed to propagate faster than light. Accordingly, in terms of the geometry of , we say that an event p would be permitted to have a causal influence on event q if th...

conformal geometry.

Conformal structure does not fix length measure, but it does fix angles via the ratio of length measures in different directions at any point. Length measure can be rescaled up or down at different points without affecting the conformal structure.

the signature of the metric. In algebraic terms, this simply refers to a few + signs being changed to – signs, and it basically tells us how many of a set of n mutually orthogonal directions, for an n-dimensional space, are to be considered as ‘timelike’ (within the null cone) and how many ‘spacelike’ (outside the null cone). In Euclidean geometry, and its curved version known as Riemannian geometry, we think of all directions as being spacelike. The usual idea of ‘space-time’ involves only 1 of the directions being timelike, in such an orthogonal set, the rest being spacelike. We call it Minkowskian if it is flat and Lorentzian if it is curved. In the ordinary type of (Lorentzian) space-time that we are considering here, n=4, and the signature is ‘1+3’ separating our 4 mutually orthogonal directions into 1 timelike direction and 3 spacelike ones. ‘Orthogonality’ between spacelike directions (and between timelike ones, had we had more than 1 of them) means simply ‘at right angles’, whereas between a spacelike and a timelike direction it looks geometrically more like the situation depicted in Fig. 2.18, the orthogonal directions being symmetrically related to the null direction between them. Physically, an observer whose world-line is in the timelike direction regards events in an orthogonal spacelike direction to be simultaneous.

This illustrates the key fact about the metric of space-time, namely that it is really something that has much more directly to do with the measurement of time rather than distance. Instead of providing a length measurement for curves, it directly provides us with a time measurement. Moreover, it is not all curves that are assigned a time measure: it is for the curves referred to as causal, that could be the world-lines of particles, these curves being everywhere either timelike (with tangent vectors within the null cones, achieved by massive particles) or null (with tangent vectors along the null cones, achieved by massless particles). What the space-time metric g does is to assign a time measure to any finite segment of a causal curve (the contribution to the time measure being zero for any portion of the curve which is null). In this sense, the ‘geometry’ that the metric of space-time possesses should really be called ‘chronometry’, as the distinguished Irish relativity theorist John L. Synge has suggested.

If m is the particle’s mass (assumed to be constant), then we find that it has a rest energy[2.33] E given by Einstein’s famous formula E=mc2, which is fundamental to relativity theory. The other, almost equally famous formula—fundamental to quantum theory—is Max Planck’s E=hv (h being Planck’s constant), telling us that this particle’s rest energy defines for it a particular frequency v of quantum oscillation (see Fig. 2.21). In other words, any stable massive particle behaves as a very precise quantum clock, which ‘ticks away’ with the specific frequency in exact proportion to its mass, via the constant (fundamental) quantity c2/h.

But the key point is still that to build a clock we do need mass. Massless particles (e.g. photons) alone cannot be used to make a clock, because their frequencies would have to be zero; a photon would take until eternity before its internal ‘clock’ gets even to its first ‘tick’! This fact will be of great significance for us later.

a geodesic, for a timelike curve, has the physical interpretation as the world-line of a massive particle in free motion under gravity.

A timelike geodesic line l is characterized by the fact that for any two points p and q on l, not too far apart, the longest local curve from p to q is in fact a portion of l.

if we trace back (in time) a light ray that enters the eye of an external observer, situated at a safe distance from the hole, looking towards it, we find that this ray cannot pass backwards across the event horizon into its interior, but hovers just above the surface, to meet the body of the star just a moment before it plunged beneath the horizon. This would theoretically be the case no matter how long the external observer waits (i.e. no matter how far up the picture we place the observer’s eye), but in practice the image perceived by the observer would become highly red-shifted and very rapidly fade from view the later in time the observer is situated, so that in short order the image of the star would become blackness—in accordance with the terminology ‘black hole’.

Whereas the Big Bang is seen as the beginning of time, the singularities in black holes present themselves as representing the end of time—at least as far as the fate of that material that has, at some stage, fallen into the hole is concerned. In this sense, we may regard the problem presented by black-hole singularities to be the time-reverse of that presented by the Big Bang.

singularity. It would therefore appear that—apart from the black-hole case being more local—the two situations are, in effect, time-reverses of one another. Yet, our considerations of the Second Law might well suggest to us that this cannot altogether be the case. The Big Bang must be something of extraordinarily low entropy, in comparison with the situation to be encountered in a black hole.

This also serves to explain the rather curious-seeming fact that spatial infinity for is conformally just the single point i0

we can consider the entire manifold as made up, conformally, of an infinite succession of spaces , where the of each one is joined on to the , of the next,

Big Bang that was so extraordinarily special—yet special in what appears to have been a very peculiar way where, with regard to gravity, its entropy was enormously low in comparison with what it might have been, but the entropy was close to maximum in every other respect.

The kinetic energy in the motions of particles around at that time would have been so enormous as to have completely overwhelmed the particles’ comparatively tiny rest energies (E=mc2, for a particle of rest-mass m). Thus, the rest-mass of the particles would have been effectively irrelevant—as good as zero as far as the relevant dynamical processes are concerned. The contents of the universe, at extremely early times, would have consisted of effectively massless particles.

Massless particles, as we may recall from §2.3, do not appear to be particularly concerned with the full metric nature of space-time, respecting merely its conformal (or null-cone) structure.

In any case, even if it turns out (for example when detailed results from the LHC become known and understood) that things are not quite as current theory suggests, it still remains probable that when energies get higher and higher, rest-masses become more and more irrelevant, physical processes becoming dominated by conformally invariant laws.

When the universe enters this apparently final stage—what one might well call the ‘very boring era’—nothing of great interest seems to be left for it to do. The most exciting events prior to this were the final ‘pops’ of the last tiny remnants of black holes, eventually disappearing (it is supposed) after they had very gradually lost all their mass via the painfully slow process of Hawking radiation. One is left with the dreadful thought of a seemingly interminable boredom confronting the final stages of our great universe—a universe which would have once seemed so exciting, teeming with fascinating activity of hugely different kinds—most of this activity occurring within beautiful galaxies, with a wonderful variety of stars and often attendant planets, among which would be those supporting life of some kind, with its exotic plants and animals, some of whom having the capabilities of deep knowledge and understanding, and profound capabilities of artistic creation. Yet all this will eventually die away. The final dregs of excitement will have to be the waiting, and the waiting, and waiting, for maybe 10100 years or more, for that final pop—perhaps of about the violence of a small artillery shell followed by nothing but further exponential expansion, thinning it out and cooling and emptying and cooling, and thinning out … until eternity. Does that picture present all that our universe has ultimately in store for it? But after I had been depressing myself with such thoughts, ...

I am suggesting ‘the next best thing’, which is to propose that there is a physically real region of space-time prior to which is the remote future of some previous universe phase, and that there is also a physically real universe phase that extends beyond our to become a big bang for a new universe phase. In accordance with this proposal, I shall refer to the phase beginning with our and extending to our as the present aeon, and I am suggesting that the universe as a whole is to be seen as an extended conformal manifold consisting of a (possibly infinite) succession of aeons, each appearing to be an entire expanding universe history. See Fig. 3.3. The ‘’ of each is to be identified with the ‘’ of the next, where the continuation of each aeon to the next is achieved so that, as a conformal space-time structure, the join is perfectly smooth.

The quantity Λg is often interpreted by quantum field theorists as the energy of the vacuum (see §3.5). For reasons to do with relativity, it is argued that this ‘vacuum energy’ ought to be a -tensor proportional to g, but the proportionality factor comes out as something larger than the observed value of Λ by a factor of around 10120, so something is clearly missing from this idea![3.24]

Λ’s observed tiny value is just such as to be starting to have effects on the expansion of the universe that are comparable with the particular totality of attraction due to matter in the universe now, which was enormously greater in the past and which will become enormously smaller in the future, and this seems to be an odd coincidence.

It has been frequently argued that there are many curious coincidences in the relations between the constants of Nature upon which life on Earth seems to depend. Some of these might be readily dismissed as being of value only to certain kinds of life we are familiar with, like the parameters determining the delicate fact that as ice forms from water, it is anomalous in being less dense than the water, so that life can persist in water remaining unfrozen under a protective surface layer of ice even when the external temperature drops below freezing. Others seem to present a more problematic challenge, such as the threat that the whole of chemistry would have been impossible had not the neutron been just marginally more massive than the proton, a fact which leads to a whole variety of different kinds of stable nuclei—these underlying all the different chemical elements—that would not otherwise have come about. One of the most striking of such apparent coincidences was revealed with William Fowler’s confirmation of Fred Hoyle’s remarkable prediction of the existence of a particular energy level of carbon which, had it not existed, would have meant that the production of heavy elements in stars would not have been able to proceed beyond carbon, leaving the planets devoid of nitrogen, oxygen, chlorine, sodium, sulphur, and numerous other elements. (Fowler shared in the 1982 Nobel Prize for this with Chandrasekhar but, strangely, Hoyle was passed over.) The term ‘anthropic princ...

Lee Smolin in his 1997 book Life of the Cosmos.[3.33] Smolin makes the tantalizing suggestion that when black holes form, their internal collapsing regions—through unknown quantum-gravity effects—become converted to expanding ones by some kind of ‘bounce’, each one providing the seed of a new expanding universe phase. Each new ‘baby universe’ then expands to a ‘full-grown’ one with its own black holes, etc., etc. See Fig. 3.12. This collapseexpansion procedure would clearly have to be quite unlike the kind of conformally smooth transition involved in CCC (see Fig. 3.2), and its relation to the Second Law is obscure. Nevertheless the model has the virtue that it can be studied from the point of view of the biological principle of natural selection, and it is not entirely without significant statistical predictions. Smolin makes worthy attempts at such predictions and provides comparisons with observational statistics of black holes and neutron stars. The role of Wheeler’s idea here is that the dimensionless constants might change only moderately in each collapseexpansion process, so that there would be some kind of ‘inheritance’ in the propensity to form new black holes, this being subject to the influences of a kind of natural selection.

there is a conundrum to be faced. It is a conundrum that appears to confront us irrespective of CCC. The issue has to do with the evident fact that the entropy of our universe—or the current aeon, if we are considering CCC—seems to be vastly increasing, despite the fact that the very early universe and the very remote future appear to be uncomfortably similar to one another. Of course they are not really similar in the sense of being nearly identical, but they are alarmingly ‘similar’ according to the usage of that word commonly applied in Euclidean geometry, namely that the distinction between the two seems to be basically just an enormous scale change. Moreover, any overall change of scale is essentially irrelevant to measures of entropy—where that quantity is defined by Boltzmann’s marvellous formula (given in §1.3)—because of the important fact, noted at the end of §3.1, that phase-space volumes are unaltered by conformal scale changes.[3.40] Yet, the entropy does seem to increase, vastly, in our universe, through the effects of gravitational clumping. Our conundrum is to understand how these apparent facts are to square with one another.