Squaring the Second

Let us therefore return to the question which started out this whole enterprise, namely the origin of the Second Law. The first point to be made is that 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. Some physicists have argued that the ultimate maximum entropy achieved by our universe will arise not from clumping to black holes, but from the Berkenstein-Hawking entropy of the cosmological event horizon. This possibility will be addressed in §3.5, where I argue that it does not invalidate the discussion of this section.

Let us examine more carefully the likely state of the early universe, where some appropriate condition has been imposed to kill off gravitational degrees of freedom at the Big Bang, so that gravitational entropy is low in what we find in the early universe. Do we need to take into account cosmic inflation? The reader will have realized that I am sceptical of the reality of this presumed process (§2.6), but no matter; in this discussion it makes little difference. We can either ignore the possibility of inflation, or perhaps take the view (see §3.6) that CCC provides merely a different interpretation of inflation where the inflationary phase is the exponentially expanding phase of the previous aeon, or else we can simply consider the situation just following the cosmic 'moment' —at around 10-32s—when inflation is considered just to have turned off.

As I have argued at the beginning of §3.1, it is reasonable to suppose that this early-universe state (say at around 10-32s) would be dominated by conformally invariant physics, and inhabited by effectively massless ingredients. Whether or not Tod's proposal, of §2.6, is correct in all detail, it seems that we do not go too far wrong in taking the early-universe state, in which gravitational degrees of freedom are indeed hugely suppressed, to be one in which a conformal stretching out would provide us with a smooth non-singular state still essentially inhabited by massless entities, perhaps largely photons. We would need to consider also the additional degrees of freedom in the dark matter, also taken to be effectively massless in those early moments.

At the other end of the time scale, we have an ultimate exponentially expanding de Sitter-like universe (§2.5), again largely inhabited by mass-less ingredients (photons). There could well be other stray material consisting, say, of stable massive particles, but the entropy would lie almost entirely within the photons. It would appear that we still do not go far wrong (appealing to the results of Friedrich cited in §3.1) if we assume that we can conformally squash down the remote future to obtain a smooth universe state not at all unlike that we obtained by conformally stretching out the situation close to the Big Bang (say at 10-32s). If anything, there might well be more degrees of freedom activated in the stretched-out Big Bang, because in addition to the degrees of freedom perhaps activated in dark matter, Tod's proposal still allows for the presence of gravitational degrees of freedom in a non-zero (but finite) Weyl tensor C, rather than the requirement C=0 demanded by CCC (see §2.6, §3.2). But if such degrees of freedom are indeed present, this will only make our conundrum more severe, the problem to be faced being that the entropy of the very early universe is seen to be hardly smaller (if not actually larger) than that to be found in the very remote future, despite the fact that there must surely be absolutely enormous increases in entropy taking place between 10-32s and the very remote future.

In order to address this conundrum properly, we need to understand the nature and magnitude of the major contributions to what we expect to be an enormous increase of entropy. At the present time, it appears that easily the major contribution to the entropy of the universe comes from huge black holes at the centres of most (or all?) galaxies. It is hard to find an accurate estimate of the sizes of black holes in galaxies generally. By their very nature, black holes are hard to see! But our own galaxy may well be fairly typical, and it appears to contain a black hole of some 4x 106M© (see §2.4), which by the Bekenstein-Hawking entropy formula provides us with an entropy per baryon for our galaxy of some 1021 (where 'baryon', here means, in effect, a proton or neutron, where for ease of description, I am taking baryon number to be conserved—no violation of this conservation principle having been yet observed). So let us take this figure as a plausible estimate for the current entropy per baryon in the universe generally.[3 41] If we bear in mind that the next largest contribution to the entropy appears to be the CMB, where the entropy per baryon is not more than around 109, we see how stupendously the entropy appears to have increased already, since the time of decoupling—let alone since 10-32s—and it is the entropy of black holes that is basically responsible for this vast entropy increase. To make this more dramatic, let me write this out in more everyday notation. The entropy per baryon in the CMB is around 1000000000, whereas (according to the above estimate), the current entropy per baryon is about

1000 000 000 000 000 000 000, this being mainly in black holes. Moreover, we must expect these black holes, and consequently the entropy in the universe, to grow very considerably in the future, so that even this number will be utterly swamped in the far future. Thus, our conundrum takes the form of the question: how can this be squared with what has been said in the early parts of this section? What will ultimately have happened to all this black-hole entropy?

We must try to understand how the entropy will ultimately appear to have shrunk by such an enormous factor. In order to see where all that entropy has gone, let us recall what, indeed, is supposed to be the fate, in the very remote future, of all those black holes responsible for the vast entropy increase. According to what has been said in §2.5, after around 10100 years or so the holes will all have gone, having evaporated away through the process of Hawking radiation, each presumed to disappear finally with a 'pop'.

We must bear in mind that the raising of entropy due to the swallowing of material by a black hole, and also the hole's eventual reduction in size (and mass) due to its Hawking evaporation, would be fully consistent with the Second Law; not only that, but also these phenomena are direct implications of the Second Law. To appreciate this, in a general way, we do not need to understand the subtleties of Hawking's 1974 initial argument for the temperature and entropy of a black hole (taken to have formed in the distant past from some gravitational collapse). If we are not concerned with the exact coefficient 8kGn2/ch that appears in the Bekenstein-Hawking entropy formula of §2.6, and we would be satisfied with some approximation to it, then we can find justification for the general form of black-hole entropy purely from Bekenstein's orig-inal[342] 1972 demonstration, which was an entirely physical argument based on the Second Law and on quantum-mechanical and general-relativistic principles, as applied to imagined experiments concerned with the lowering of objects into black holes. Hawking's black-hole surface temperature TBH, which for a non-rotating hole of mass M is rr K


(the constant K in fact being given by K = 1/(4n)), then follows from standard thermodynamic principles[3 43] once the entropy formula is accepted. This is the temperature as seen from infinity, and the rate at which a black hole will radiate is then determined by assuming that this temperature is spread uniformly over a sphere whose radius is the Schwarzschild radius (see §2.4) of the black hole.

I am stressing these points here, just to emphasize that black-hole entropy and temperature, and the process of Hawking evaporation of these strange entities, albeit of an unfamiliar character, are nevertheless very much a part of the physics of our universe, fitting in with fundamental principles that we have become accustomed to—most particularly with the Second Law. The enormous entropy that black holes possess is to be expected from their irreversible character and the remarkable fact that the structure of a stationary black hole needs only a very few parameters to characterize its state.[344]Since there must be a vast volume of phase space corresponding to any particular set of values of these parameters, Boltzmann's formula (§1.3) suggests a very large entropy. From the consistency of physics as a whole, we have every reason to expect that our present general picture of the role and the behaviour of black holes will indeed hold true—except that the eventual 'pop' at the end of the black-hole's existence is somewhat conjectural. Nevertheless, it is hard to see what else could happen to it at that stage.

But do we really need to believe in the pop? As long as the spacetime described by the black hole remains a classical (i.e. non-quantum) geometry, the radiation should continue to extract mass/energy from the hole at such a rate that would cause it to disappear in a finite time—this time being ~ 2 x 1067(M/M©)3 years, for a hole of mass M if nothing more falls into the hole.[345] But how long can we expect the notions of classical space-time geometry to provide a reliable picture? The general expectation (just from dimensional considerations) is that only when the hole approaches the absurdly tiny Planck dimension lp of ~ 10-35m (around 10-20 of the classical radius of a proton) do we expect to have to involve some form of quantum gravity, but whatever happens at that very late stage, the only mass left would presumably be somewhere around that of a Planck mass mP, with an energy content of only around a Planck energy Ep, and it is hard to see that it then could last very much longer than around a Planck time tp (see end of §3.2). Some physicists have contemplated the possibility that the end-point might be a stable remnant of mass ~ mp, but this causes some difficulties with quantum field theory.[346] Moreover, whatever the final fate of a black hole might be, its final state of existence seems to be independent of the hole's original size, and has to do with just an utterly minute fraction of the black-hole's mass/energy. There appears to be no complete agreement among physicists about the final state of this tiny remnant of a black hole,[347] but CCC would require that nothing with rest-mass should persist to eternity, so the 'pop' picture (together with the ultimate decaying away of the rest-mass of any massive particle produced in the pop) is very acceptable from CCC's point of view, and it is also consistent with the Second Law.

Yet, despite all this consistency, there is something distinctly odd about a black hole, in that the future evolution of the space-time, seemingly unique among future-evolving physical phenomena, results in an inevitable internal space-time singularity. Although this singularity is a consequence of classical general relativity (§2.4, §2.6), it is hard to believe that this classical description would have to be seriously modified by quantum-gravity considerations, until enormous space-time curvatures are encountered, where the radii of curvature of space-time begin to get down to the extremely tiny scale of the Planck length lp (see end of §3.2). Particularly for a huge galactic-centre black hole, the place where such tiny curvature radii begin to show themselves will be an utterly minute region hugging the singularity in the classical space-time picture. The location referred to as a 'singularity' in classical space-time descriptions should really be thought of as where 'quantum gravity takes over'. But in practice this makes little difference, since there is no generally accepted mathematical structure to replace Einstein's picture of continuous space-time, so we do not ask what happens further, but merely adjoin a singular boundary of wildly diverging curvature, possibly acting in accordance with BKL-type chaotic behaviour (§2.4, §2.6).

To get a better understanding of the role of this singularity in the classical picture, we do well to examine the conformal diagram Fig. 3.13, whose two parts are basically re-drawings of Fig. 2.38(a) and Fig. 2.41, respectively. These pictures, when interpreted as strict conformal diagrams, incorporate exact spherical symmetry, which is unlikely to remain at all accurate whenever irregularities are present in the collapse. However, if we allow ourselves to assume strong cosmic censorship (see end of §2.5, and §2.6) right up until just before the pop,[348] then the singularity should be essentially spacelike, and the pictures of Fig. 3.13 remain qualitatively appropriate as schematic conformal diagrams, despite the extreme irregularity in the space-time geometry near the classical singularity.

Fig. 3.13 Conformal diagrams drawn irregularly (to suggest a lack of symmetry) to indicate (a) gravitational collapse to a black hole; (b) collapse followed by Hawking evaporation. The singularity remains spacelike according to strong cosmic censorship.

The regions where one should expect quantum-gravity effects to invalidate the classical space-time picture would be very close indeed to the singularity, where space-time curvatures begin to reach the extremes where classical space-time physics can no longer be trusted. At this stage, there seems to be very little hope of adopting a standpoint like that involved in the 'crossover 3-surfaces' of CCC, in which the space-time could be extended smoothly through the singularity in order to arrive at some kind of continuation through to 'the other side'. Indeed, Tod's proposal is intended to distinguish the very tame singularity encountered

Fig. 3.13 Conformal diagrams drawn irregularly (to suggest a lack of symmetry) to indicate (a) gravitational collapse to a black hole; (b) collapse followed by Hawking evaporation. The singularity remains spacelike according to strong cosmic censorship.

at the Big Bang from the kind of thing—perhaps of chaotic BKL nature— that one must expect at a black hole's singularity. Despite Smolin's stimulating proposal described in §3.3 (Fig. 3.12), I see little hope of quantum gravity coming to our rescue, so as to allow us to obtain a 'bounce' of some kind, for which the emerging space-time mirrors what came in, in any direct sense, according to some kind of basically time-symmetric fundamental physical processes. If it could, then what emerges would be something of the character of the white hole of Fig. 2.46 or the mess of bifurcating white holes that we contemplated in §2.6 (contrast Fig. 3.2). Such behaviour would certainly be most unlike the kind of situation that we find in the universe with which we are familiar, and would possess nothing resembling the Second Law of our experiences.

Be that as it may, what seems to be happening—at least according to any kind of physical evolution that we seem to be able to contemplate— is that physics comes to an end at such regions. Or, if it does not, then it continues into some kind of universe-structure of a completely foreign character to that of which we have knowledge. Either way, the material encountering the singular region is lost to the universe we know, and it seems that any information carried by that material is also lost. But is it lost? Or can it somehow slither its way out sideways, in the diagram of Fig. 3.13(b), where quantum-gravity distortions of normal ideas of space-time geometry are somehow permitting a kind of seemingly spacelike propagation that would be illegal according to the normal causality rules of §2.3? Even if so, it is hard to see that any of this information could emerge, by such means, very much before the moment of the pop, so that the vast amount of information that was contained in the material that went to form a large black hole, say of many millions of solar masses, could somehow all come flooding out just at around that one moment, and from that tiny region, that constitutes the pop. Personally, I find this very hard to believe. It seems to me to be much more plausible that the information contained in all processes whose future evolution is directed into such a space-time singularity is accordingly destroyed.

However, there is an alternative suggestion,[3.49] frequently argued for, that somehow the information has been 'leaking out' for a long time previously, encoded in what are referred to as 'quantum entanglements', that would be expressed in subtle correlations in the Hawking radiation coming from the hole. On this view, the Hawking radiation would not be exactly 'thermal' (or 'random'), but the full information that would seem to have been irretrievably lost in the singularity is somehow taken fully into account (repeated?) outside the hole. Again I have my severe doubts about any such suggestions. It would seem that, according to proposals of this kind, whatever information finds its way to the vicinity of the singularity, must somehow be 'repeated' or 'copied' as this external entanglement information, which would in itself violate basic quantum principles.[3 50]

Moreover, in his original 1974 argument,[351] demonstrating the presence of thermal radiation emanating from a black hole, Hawking explicitly made use of the fact that information coming in, in the form of a test wave, would have to be shared between what escapes from the hole and what falls into it. It is the assumption that the part that falls into the hole is irretrievably lost that leads to the conclusion that what comes out must have a thermal character, with a temperature that is precisely equal to what we now call the Hawking temperature. This argument depends upon use of the conformal diagram of Fig. 2.38(a), which to me makes it manifestly clear that the incoming information is indeed shared between that falling into the hole and that escaping to infinity, where that falling into the hole is lost—this being an essential part of the discussion. Indeed, for many years, Hawking himself has been one of the strongest proponents of the viewpoint that information is indeed lost in black holes. Yet, at the 17th International Conference on General Relativity and Gravitation, held in Dublin in 2004, Hawking announced that he had changed his mind and, publicly forfeiting a bet that he (and Kip Thorne) had made with John Preskill, argued that he had been mistaken and that he now believed[3.52] that the information must in fact all be retrieved externally to the hole. It is certainly my personal opinion that Hawking should have stuck to his guns, and that his earlier viewpoint was far closer to the truth!

However, Hawking's revised opinion is much more in line with what might be regarded as the 'conventional' viewpoint among quantum field theorists. Indeed, the actual destruction of physical information is not something that appeals to most physicists, the idea that information can be destroyed in a black hole in this way being frequently referred to as the 'black-hole information paradox'. The main reason that physicists have trouble with this information loss is that they maintain a faith that a proper quantum-gravity description of the fate of a black hole ought to be consistent with one of the fundamental principles of quantum theory known as unitary evolution, which is basically a time-symmetric[3 53] deterministic evolution of a quantum system, as governed by the fundamental Schrödinger equation. By its very nature, information cannot be lost in the process of unitary evolution, because of its reversibility. Hence the information loss that seems to be a necessary ingredient of Hawking evaporation of black holes is, in fact, inconsistent with unitary evolution.

I cannot go into the details of quantum theory here,[354] but a brief mention of the rudimentary ideas will be important for our further discussion. The basic mathematical account of a quantum system at a particular time is provided by the quantum state or wavefunction of the system, for which the Greek letter ^ is frequently used. As mentioned above, if left to itself the quantum state ^ evolves with time according to the Schrödinger equation, this being unitary evolution, a deterministic, basically time-symmetric, continuous process for which I use the letter U. However, in order to ascertain what value some observable parameter q might have achieved at some time t, a quite different mathematical process is applied to referred to as making an observation, or measurement. This is described in terms of a certain operation 0 which is applied to providing us with a set of possible alternatives ^1, ^2, ^3, ^4, ..., one for each of the possible outcomes qi, q2, q3, q4, ... of the chosen parameter q, and with respective probabilities P1, P2, P3, P4, . for these outcomes. This entire set of alternatives, with corresponding probabilities, is determined by 0 and ^ by a specific mathematical procedure. To mirror what actually appears to happen in the physical world, upon measurement, we find that ^ simply jumps to one of the given set of alternatives ^1, ^2, ^3, ^4, ..., say to ^j, where this choice appears to be completely random, but with a probability given by the corresponding Pj. This replacement of ^ by the particular choice ^j that Nature comes up with is referred to as the reduction of the quantum state or the collapse of the wavefunction, for which I use the letter R. Following this measurement, which has caused ^ to jump (to j the new wave-function again proceeds according to U until a new measurement is made, and so on.

What is particularly strange about quantum mechanics is this very curious hybrid, whereby the quantum state's behaviour seems to alternate between these two quite different mathematical procedures, the continuous and deterministic U and the discontinuous and probabilistic R. Not surprisingly, physicists are not happy with this state of affairs, and will adopt one or another of a number of different philosophical standpoints. Schrodinger himself is reported (by Heisenberg) to have said, 'If all this damned quantum jumping were really here to stay then I should be sorry I ever got involved in quantum theory.'[355] Other physicists, fully appreciative of the great contribution that Schrodinger made with the discovery of his evolution equation, while agreeing with his distaste for 'quantum jumping' would nevertheless take issue with Schrodinger's standpoint that the full story of quantum evolution has not yet fully emerged. It indeed is a common view that the full story is somehow contained within U, together with some appropriate 'interpretation' of the meaning of ^—and somehow R will emerge from all this, perhaps because the true 'state' involves not just the quantum system under consideration but its complicated environment also, including the measuring device, or perhaps because we, the ultimate observers, are ourselves part of a unitarily evolving state.

I do not wish to enter into all the alternatives or contentions that still thoroughly cloud the U/R issue, but simply state my own position, which is to side basically with Schrodinger himself, and with Einstein, and perhaps more surprisingly with Dirac,[3 56] to whom we owe the general formulation of present-day quantum mechanics,[3 57] and to take the view that present-day quantum mechanics is a provisional theory. This is despite all the theory's marvellously confirmed predictions and the great breadth of observed phenomena that it explains, there being no confirmed observations which tell against it. More specifically, it is my contention that the R phenomenon represents a deviation from the strict adherence of Nature to unitarity, and that this arises when gravity begins to become seriously (even if subtly) involved.[3 58] Indeed, I have long been of the opinion that information loss in black holes, and its consequent violation of U, represents a powerful part of the case that a strict adherence to U cannot be part of the true (still undiscovered) quantum theory of gravity.

I believe that it is this that holds the key to the resolution of the conundrum that confronted us at the beginning of this section. I am thus asking the reader to accept information loss in black holes—and the consequent violation of unitarity—as not only plausible, but a necessary reality, in the situations under consideration. We must re-examine Boltzmann's definition of entropy in the context of black-hole evaporation. What does 'information loss' at the singularity actually mean? A better way of describing this is as a loss of degrees of freedom, so that some of the parameters describing the phase space have disappeared, and the phase space has actually become smaller than it was before. This is a completely new phenomenon when dynamical behaviour is being considered. According to the normal idea of dynamical evolution, as described in §1.3, the phase space T is a fixed thing, and dynamical evolution is described by a point moving in this fixed space, but when the dynamical evolution involves a loss of degrees of freedom at some stage, as appears to be the case here, the phase space actually shrinks as part of the description of this evolution! In Fig. 3.14, I have tried to illustrate how this process would be described, using a low-dimensional analogue.

phase space T* prior phase space T* prior

Black Hole Evolution
Fig. 3.14 Evolution in phase space following black-hole information loss.

In the case of black-hole evaporation, this is a very subtle process, and we should not think of this shrinking as taking place 'suddenly' at any particular time (e.g. at the 'pop'), but surreptitiously. This is all tied up with the fact that, in general relativity, there is no unique 'universal time', and this is of particular relevance in the case of a black hole, where the space-time geometry deviates greatly from spatial homogeneity. This is well illustrated in the Oppenheimer-Snyder collapse picture (§2.4, see Fig. 2.24), with final Hawking evaporation (§2.5, see Fig. 2.40 and Fig. 2.41) where in Fig. 3.15(a) and its strict conformal diagram Fig. 3.15(b) I have drawn, with unbroken lines, one family of spacelike 3-surfaces (constant time slices) where all the information lost in the hole seems to disappear at the 'instant' of the pop, whereas, using broken lines I have drawn a different family of spacelike 3-surfaces for which the information appears to go away gradually, spread out over the entire history of the black hole's existence. Although the pictures strictly refer to spherical symmetry, they still apply in a schematic way so long as strong cosmic censorship is assumed (except, of course, at the pop itself).

Fig. 3.15 A Hawking-evaporating black hole: (a) conventional space-time picture; (b) strict conformal diagram. Loss of internal degrees of freedom may be considered to result only as the 'pop' occurs, this being the picture suggested according to the time-slices given by unbroken lines. Alternatively, according to the time-slices given by the broken lines, the loss occurs gradually over the whole history of the black hole.

Degree Freedom Black Hole

Fig. 3.15 A Hawking-evaporating black hole: (a) conventional space-time picture; (b) strict conformal diagram. Loss of internal degrees of freedom may be considered to result only as the 'pop' occurs, this being the picture suggested according to the time-slices given by unbroken lines. Alternatively, according to the time-slices given by the broken lines, the loss occurs gradually over the whole history of the black hole.

This indifference to when the information loss actually takes place serves to emphasize the fact that its disappearance has no effect on the external (thermo)dynamics, and we may well take the view that the Second Law is proceeding according to its normal practice, where the entropy continues to increase—but we must be careful about what 'entropy' notion we are referring to here. This entropy refers to all the degrees of freedom, including that of all the material that has fallen into the holes. The degrees of freedom referring to what has fallen in, however, will sooner or later have to confront the singularity and will, according to the considerations above, be lost to the system. By the time the black hole has disappeared with a pop, we must have radically reduced the scale of our phase space so that—as in a country experiencing currency devaluation—the phase-space volumes will count, overall, for far less than they did before, though this huge devaluation will not be noticed by the local physics continuing away from the hole in question. Because of the logarithm in Boltzmann's formula, this scaling down of the volumes would simply count as though a large constant had been subtracted from the overall entropy of the universe external to the black hole in question.

We may compare this with the discussion at the end of §1.3 where it was noted that the logarithm in Boltzmann's formula is what gives rise to the additivity of entropy for independent systems. In the foregoing discussion, the degrees of freedom swallowed and finally destroyed by the black holes play the role of the external part of the system under consideration in §1.3, with parameters defining the external phase space X which referred to the Milky Way galaxy external to the laboratory— whereas here it refers to the black holes. See Fig. 3.16. What we are now taking to be the world outside black holes, where we might envisage some experiment being performed, corresponds, in the discussion in §1.3 (Fig. 1.9), to the internal part of the system, defining the phase space P. Just as the removal of degrees of freedom in the Milky Way galaxy of §1.3 (such as some of them being absorbed into the galaxy's central black hole) would make no difference whatever to entropy considerations in the experiment being performed, so also would the information destruction in black holes throughout the universe, finalized as each individually disappears with its pop, yield no effective violation of the Second Law, consistently with what has been emphasized earlier in this section!

Nevertheless, the phase-space volume of the universe as a whole would be very drastically reduced by this information loss,[3 59] and this is basically what we need for the resolution of the conundrum posed at the beginning of this section. It is a subtle matter, and there are many detailed issues of consistency to be satisfied for the reduction in phase-space volume to be adequate for what is required for CCC. In general

Fig. 3.16 The information loss in black holes does not affect local phase space (compare Fig. 1.9), though it contributes to the total, prior to loss.

terms, this consistency seems not unreasonable, since the overall entropy increase that our present aeon will indulge in throughout its entire history is expected to be through the formation (and evaporation) of black holes. Although it is not totally obvious to me how one calculates, with any degree of precision, the effective entropy reduction due to information loss, one could, as a good guess, estimate the Bekenstein-Hawking entropy of the maximum sizes that the black holes would have reached, had it not been for the loss in Hawking radiation, and take the total of this entropy to give the needed scale of reduction in available phase space for the start of the next aeon. Clearly there are many matters in black holes laboratory black holes laboratory

phase space lost in black holes

Fig. 3.16 The information loss in black holes does not affect local phase space (compare Fig. 1.9), though it contributes to the total, prior to loss.

phase space lost in black holes need of more detailed study in order for us to be certain whether CCC is viable in this respect. But I can see no reason to expect that CCC will be contradicted by such considerations.

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