CCC and quantum gravity

The scheme of CCC provides us with a different outlook on various intriguing issues—in addition to the Second Law—that have confronted cosmology for many years. In particular, there is the question of how we are to view the singularities that arise in the classical theory of general relativity, and of how quantum mechanics enters into this picture. We find that CCC has something particular to say, not only about the nature of the Big-Bang singularity, but also about what happens when we try to propagate our physics, as we know it, into the future as far as it will go, where it apparently either terminates irretrievably at a singularity in a black hole, or else continues into the indefinite future, to be reborn, according to CCC, in the big bang of a new aeon.

Let me begin this section by examining again the situation in the very remote future, in order to raise an issue that I had left aside in the previous section. When, in §3.4, I addressed the matter of the increasing of entropy into the very remote future, I argued that, in accordance with CCC, by far the major entropy-raising processes are the formation (and congealing) of large black holes, followed by their eventual evaporation away through Hawking radiation after the CMB cools to lower than the Hawking temperature of the holes. Yet, as we have seen, CCC's requirement that the initial phase-space coarse-graining region (§1.3, §3.4) must be capable of actually matching the final one, despite the enormous increase in entropy, can be satisfied if we accept a huge 'information loss' in black holes (as Hawking originally argued for but later retracted), this allowing the phase space to become enormously 'thinned down' owing to a vast loss of phase-space dimensionality from the swallowing, and subsequent destruction, of degrees of freedom by the holes. Once the black holes have all evaporated away, we find that the zero of the entropy measure must be reset, because of this great loss of degrees of freedom, which means, in effect, that a very large number gets subtracted from the entropy value, and the allowable states in the ensuing big bang for the following aeon find themselves greatly restricted, so as to satisfy a 'Weyl curvature hypothesis', this providing the potential for gravitational clumping in the succeeding aeon.

There is, however, another ingredient to this discussion, at least in the opinion of a good many cosmologists, which I have ignored, despite its having some definite relevance to our central topic (see the end of the first paragraph of §3.4). This is the issue of 'cosmological entropy', arising from the existence of cosmological event horizons when A > 0. In Fig. 2.42(a),(b), I illustrated the idea of a cosmological event horizon, which occurs when there is the spacelike future conformal boundary that arises if there is a positive cosmological constant A. We recall that a cosmological event horizon is the past light cone of the ultimate endpoint o+ (on \+) of the 'immortal' observer O of §2.5; see Fig. 3.17. If we take the view that such event horizons should be treated in the same way as black-hole event horizons, then the same Bekenstein-Hawking formula for black-hole entropy (Sbh=-A; see §2.6) should be applied also to a cosmological event horizon. This gives us an ultimate 'entropy' value

Sa = - Aa, in Planck units, where Aa is the area of spatial cross-section of the horizon in the remote future limit. In fact, we find (see Appendix B5) that this area is exactly in Planck units, so this proposed entropy value would be which depends solely on the value of A, and has nothing to do with any of the details of what has actually happened in the universe (where I assume that A is indeed a cosmological constant). In conjunction with this, if we accept the validity of the analogy, we would expect a tempera-ture,[3 60]which is argued to be

With the observed value of A, the temperature Ta would have the absurdly tiny value ~ 10-30K, and the entropy Sa would have the huge value ~ 3x 10122.

It should be pointed out that this entropy value is far larger than what we would expect could be achieved through the formation and final evaporation of black holes in the presently observable universe as we find it, which could hardly be expected to reach more than around 10115. These would be black holes in the region within our present particle horizon (§2.5). But we should ask what region of the universe is the entropy Sa supposed to refer to? One's first reaction might be to think that it is intended to refer to the ultimate entropy of the whole universe, since it is just a single number, precisely determined by the value of the cosmo-logical constant A, and independent not only of any of the detailed activity going on within the universe, but also of the choice of eternal observer O, who provides for us the particular future end-point o+ on J*. However, this viewpoint will not work, particularly because the universe might be spatially infinite, with an indefinite number of black holes within it altogether, in which case the present entropy of the universe could easily exceed Sa, in contradiction with the Second Law. A more appropriate interpretation of Sa might well be that it is the ultimate entropy of that portion of our universe encompassed by some cosmological event horizon—the past light cone of some arbitrarily chosen o* on J*. The material involved in this entropy would be the portion lying within the particle horizon of o+ (see Fig. 3.17).

Degree Freedom Black Hole
Fig. 3.17 In the current picture of our universe/aeon, our present particle horizon has a radius of roughly % of that where our ultimate particle horizon is expected to reside.

As we shall be seeing in §3.6, by the time o+ is reached, the universe within its particle horizon should, according to the evolution of the universe that standard cosmology predicts,[3 61] then be about (|)3~3.4 times more than the amount of material within our present particle horizon, so if that material were all to be collected into a single black hole we would get an entropy of around (j)6~11.4 times 10124, where 10124 was cited in §2.6 as a rough upper limit for the entropy attainable by the material within our present observable universe. Thus we get a possible black hole with an entropy of around 10125. If that entropy were in principle attainable within a universe with our observed value of A, then we should have a gross violation of the Second Law (since 10125»3 x 10122). However, if we accept the above value of Ta for an irreducible ambient temperature of a universe, for the observed value of A, then such an enormous black hole would always remain cooler than this ambient temperature, so it would never evaporate away by Hawking radiation. This still causes a problem, because we could choose o+ to be a point on outside this monstrous black hole, whose past light cone nevertheless encounters that hole (in the same sense that an external past light cone might ever be considered to encounter a back hole), so it seems that its entropy ought to be included —see

Fig. 3.18, and again we appear to get a vast contradiction with the Second Law.

Fig. 3.18 The past light-cone of any 'observer' whether or not at 'encounters' a black hole by engulfing it, rather than by intersecting its horizon.

Moreover, we have a certain leeway here, and can consider that this amount of material—around 1081 baryons' worth (3.4 times the 1080 within our presently observable universe, times about 3 because there is about this much more dark matter than baryonic matter)—can be separated into 100 separate regions of the mass of 1079 protons each. If each of these formed a black hole, its temperature should remain greater than Ta and it would evaporate away having reached an entropy of ~ 10119. Having 100 of these, we get a total entropy of ~ 10121 which, being larger than 3x 10120, seems still to violate the second Law, but not by so very much. These figures are perhaps too rough for a definitive conclusion to be deduced from them. But, in my view, they provide some initial evidence for caution concerning the physical interpratation of Sa as an actual entropy and, correspondingly, of Ta as an actual temperature.

I am inclined to be sceptical about Sa representing a true entropy in any case, for at least two further reasons. In the first place, if A really is a constant, so Sa is just a fixed number, then A does not give rise to any actually discernable degrees of freedom. The relevant phase space is no bigger because of the presence of A than it would be without it.

Fig. 3.18 The past light-cone of any 'observer' whether or not at 'encounters' a black hole by engulfing it, rather than by intersecting its horizon.

From the perspective of CCC, this is particularly clear, because when we match the available freedom at the \+ of the previous aeon with that at the U- of the succeeding aeon, we find absolutely no room for the huge number of putative discernable degrees of freedom that could provide the enormous cosmological entropy Sa. Moreover, it seems clear to me that this comment applies also if we do not assume CCC, because of the remark made early in §3.4 about the invariance of the volume measure under conformal scale change.[3 62]

However, we must consider the possibility that 'A' is not really a constant, but is some strange kind of matter: a 'dark-energy scalar field', as favoured by some cosmologists. Then one might consider that the huge Sa entropy comes from the degrees of freedom in this A-field. Personally, I am very unhappy with this sort of proposal, as it raises many more difficult questions than it answers. If A is to be regarded as a varying field, on a par with other fields such as electromagnetism, then instead of calling Ag just a separate 'A-term' in the Einstein field equation

(in Planck units)—as given towards the end of §2.6—we say that there is no 'A-term' in the Einstein field equation, as such, but instead take the view that the A-field has an energy tensor T(A) which (when multiplied by 8n) is closely equal to Ag

8nT(A) = Ag, this being now regarded as a contribution to the total energy tensor, which now becomes T+T(A), and we now think of Einstein's equation as written without A-term:

But Ag is a very strange form for (8n x ) an energy tensor to have, being quite unlike that of any other field. For example, we think of energy as being basically the same as mass (Einstein's 'E=mc2'), and so it should have an attractive influence on other matter, whereas this 'A-field' would have a repulsive effect on other matter, despite its energy being positive. Even more serious, in my view, is that the weak-energy condition referred to in §2.4 (which is only marginally satisfied by the exact term Ag) will almost certainly be grossly violated as soon as the A-field is allowed to vary in a serious way.

Personally, I would say that an even more fundamental objection to referring to Sa = ^ as an actual objective entropy is that here, as opposed to the case of a black hole, there is not the physical justification of absolute information loss at a singularity. People have tended to make the argument that the information is 'lost' to an observer once it goes past the observer's event horizon. But this is just an observer-dependent notion; if we take a succession of spacelike surfaces like those in Fig. 3.19, we see that nothing is actually 'lost' with regard to the universe as a whole that could be associated with the cosmolog-ical entropy, since there is no space-time singularity (apart from those already present inside individual black holes).[3 63] Moreover, I am not aware of any clear physical argument to justify the entropy Sa, like the Bekenstein argument for black-hole entropy alluded to earlier in this section.[3 64]

Fig. 3.19 For a cosmological event horizon there is no information loss (unlike the case of a black hole), as is evident from the all-encompassing nature of a family of global time-slices.

Perhaps my difficulty with this is made clearer in the case of the cosmological 'temperature' Ta since this temperature has a strongly observer-dependent aspect to it. In the case of a black hole, the Hawking temperature is provided by what is called the 'surface gravity' which has to do with the accelerating effect felt by an observer supported in a stationary configuration close to the hole, where 'stationary' refers to the relation between the observer and a reference frame held fixed at infinity. If the observer falls freely into the hole, on the other hand, then the local Hawking temperature would not be felt.[365] The Hawking temperature thus has this subjective aspect to it, and may be regarded as an instance of what is referred to as the Unruh effect that a rapidly accelerating observer would feel even in flat Minkowski space M. When we come to consider the cosmological temperature of de Sitter space D, we would expect, by the same token, that it would be an accelerating observer who should feel this temperature, not one who is in free fall (i.e. in geodesic motion; see end of §2.3). An observer moving freely in a de Sitter background would be unaccelerated in these terms, and, so it seems, should not experience the temperature Ta.

The main argument for cosmological entropy seems to be an elegant but purely formal mathematical procedure based upon analytic continuation (§3.3). The mathematics is certainly enticing, but objections can be raised to its general relevance since, technically, it applies only to exactly symmetrical space-times (like de Sitter space D).[3 66] Again there is the subjective element of the observer's state of acceleration, arising because D possesses many different symmetries, corresponding to different states of observer acceleration.

This issue is brought into better focus if we look more carefully at the Unruh effect, within Minkowski space M. In Fig. 3.20, I have tried to indicate a family of uniformly accelerating observers—referred to as Rindler observers[367]—who, according to the Unruh effect, would experience a temperature (extremely tiny, for any achievable acceleration) even though they move through a complete vacuum. This arises through considerations of quantum field theory. The future 'horizon' ^o for these observers associated with this temperature is also shown, and we may well take the view that there should be an entropy associated with ^o, for consistency with this temperature and with the Bekenstein-Hawking discussion of black holes. Indeed, if we imagine what goes on in a small region in the close neighbourhood of the horizon of a very large black hole, then the situation would be very closely approximated by that shown in Fig. 3.21, where ^0 locally coincides with the black-hole horizon and where the Rindler observers would now be the 'observers supported in a stationary configuration close to the hole' considered above. These observers are the ones who 'feel' the local Hawking temperature, whereas an observer who falls freely directly into the hole, being analogous to an inertial (unaccelerated) observer in M, would not experience this local temperature. The entire entropy associated with ^0 would, however, have to be infinite, if we carry this picture in M right out to infinity, which illustrates the fact that the full discussion of blackhole entropy and temperature actually involves some non-local considerations.

Fig. 3.20 Rindler (uniformly accelerating) observers, feeling the Unruh temperature.

A cosmological event horizon ^a arising when A > 0, as considered above, has a strong resemblance to a Rindler horizon ^o.[3 68] Indeed, on taking the limit A^0, we find that ^a actually becomes a Rindler horizon—but now globally. This would be consistent with the entropy

Fig. 3.20 Rindler (uniformly accelerating) observers, feeling the Unruh temperature.

Rindler observers formula Sa = 3n/A leading to S0=«», but it also leads us to question an assignment of objective reality to this entropy, since this infinite entropy seems to make little objective sense in the case of Minkowski space.[3 69]

Fig. 3.21 Observers supported in a stationary configuration near a black hole's horizon feel a strong acceleration and the Hawking temperature. The situation, locally, is just like that of Fig. 3.20.

I believe that it has been worth raising these matters at some length here, because an assignment of a temperature and entropy to the vacuum is an issue of quantum gravity that is deeply related to the concept referred to as 'vacuum energy'. According to our current understandings of quantum field theory, the vacuum is not something totally devoid of activity, but consists of a seething bustle of processes going on at a very tiny scale, where what are called virtual particles and their anti-particles momentarily appear and disappear in 'vacuum fluctuations'. Such vacuum fluctuations, at the Planck scale lp, would be expected to be dominated by gravitational processes, and the performing of the necessary calculations that would be needed for obtaining this vacuum energy is something far beyond the scope of currently understood mathematical procedures. Nevertheless, general arguments of symmetry, to do with the requirements of relativity, tell us that a good overall descrip black-hc horizo black-hc horizo

Fig. 3.21 Observers supported in a stationary configuration near a black hole's horizon feel a strong acceleration and the Hawking temperature. The situation, locally, is just like that of Fig. 3.20.

tion of this vacuum energy ought to be by an energy tensor Tv of the form

Tv = Ag for some A. This looks like exactly the kind of energy term T(A) that would be provided by a cosmological constant, as we have seen above, so it is frequently argued that a natural interpretation of the cosmolog-ical constant is that it is this vacuum energy, where

The point of view would appear to be to regard the 'degrees of freedom' responsible for the large cosmological entropy Sa to be those in the 'vacuum fluctuations'. These are not what I have referred to above as 'discernable' degrees of freedom since, if they count at all towards phasespace volume, they do this uniformly throughout space-time, and constitute merely a background, to which normal physical activity going on within the space-time appears not to contribute.

Perhaps even more seriously, a trouble with this interpretation appears to be that when attempted calculations are made for obtaining the actual value of A, the answer comes out as

A = o>, or A = 0, or A « tp-2, tp being the Planck time, see §3.2. The first of these answers is the most honest (and a common kind of conclusion that the direct application of the rules of quantum field theory tends to yield!), but it is also the most wrong. The second and the third are basically guesses as to what the answer should come out as after the application of one or another of the standard procedures of 'getting rid of infinities' is applied (such procedures, when applied with appropriate skill, often providing superbly accurate answers in non-quantum-gravity circumstances). The answer A=0 seems to have been the favoured one so long as it had been believed that A=0 fitted the observational facts. But since the supernova observations referred to in §2.1 indicated that it is more probable that A > 0, and later observations supported this conclusion, a non-zero value for A has become favoured. If the cosmological constant really is vacuum energy in this sense of gravitational 'quantum fluctuations', then the only scale available is the Planck scale, which is why it seems that tP (or equivalently lP), or some reasonably small multiple of it, ought to provide the needed scale for A. For dimensional reasons, A has to be the inverse square of a distance, so a rough answer A^tp-2 is to be expected. However, as we have seen in §2.1, the observed value of A is more like

so something is clearly seriously wrong, either with this interpretation (A = A/8^) or with the calculation!

Our understanding of these matters has not settled down to being free of contention, so it is perhaps of some interest to see what CCC has to say about them. The physical status of Sa and Ta does not crucially affect CCC, because even if the entropy Sa and the temperature Ta are to be regarded as physically 'true', this would not need to alter the picture presented by CCC. No black hole, that we expect to arise in the universe we know, would reach anything like the size at which Ta would seriously affect its evolution. As for Sa, it does not really appear to help with the conundrum of §3.4, since the issue there concerned discernable degrees of freedom (i.e. degrees of freedom that relate to actual dynamical processes) and simply introducing an 'entropy' with the fixed value 3n/A does not really change anything. We can simply ignore it, since it seems to play no role in the dynamics, and even if considered 'real' it appears to correspond to no physically discernable degrees of freedom. Either way, my personal position will be to ignore both Sa and Ta and to proceed without them.

The scheme of CCC does, on the other hand, provide a clear, but unconventional, perspective on how quantum gravity would affect the classical space-time singularities. The inevitability of space-time singularities in classical general relativity (§2.4, §2.6, §3.3) had led physicists to turn to some form of quantum gravity, in order to understand the physical consequences of the extraordinarily large space-time curvatures that are expected to arise in the vicinity of such singularities. But there has been very little agreement on how quantum gravity might alter these classically singular regions. There is, indeed, very little agreement about what 'quantum gravity' actually ought to be, in any case.

Nevertheless, theorists had learned to take the view that, so long as radii of space-time curvature are very large in comparison with the Planck length lp (see §3.2), then a reasonably 'classical' picture of space-time can be maintained, perhaps only with tiny 'quantum corrections' to the standard classical equations of general relativity. But when space-time curvatures get extremely large, the radii of curvature getting down to the absurdly tiny scale of lp (of some 20 orders of magnitude smaller than the classical radius of a proton), then even the standard picture of a smoothly continuous space would seem to have to be completely abandoned, and replaced by something radically different from the smooth space-time picture that we are used to.

Moreover, as had been argued strongly by John Wheeler and others, even the ordinary closely flat space-time of our experiences, if it were to be examined at the minute Planck scale, would be found to have a turbulent chaotic character, or perhaps a discrete granular one—or have some other kind of unfamiliar structure better described in some other way. Wheeler presented the case for quantum effects of gravity causing the space-time at the Planck level to curl up into topological complications that he viewed as a kind of 'quantum foam' of 'wormholes'.[370] Others have suggested that some kind of discrete structure might manifest itself (like entangled, knotted 'loops',[371] spin foams,[372] lattice-like structure,[373] causal sets,[374] polyhedral structure/3751 etc.[376]), or that a mathematical structure, modelled on quantum-mechanical ideas, referred to as 'non-commutative geometry'[377] might become relevant, or that higher-dimensional geometry might play a role, involving string-like or membrane-like ingredients,[378] or even that spacetime itself might fade away completely, where our normal macroscopic picture of space-time arises only as a useful notion derived from a different more primitive geometric structure (as happens with 'Machian'[3.79] theories and with 'twistor' theory[3 80]). It is clear from this multitude of very different alternative suggestions that there is no agreement whatsoever as to what might actually be going on in 'space-time' at the Planck scale.

However, according to CCC, we find at the Big Bang something very different from such wild or revolutionary suggestions. We are provided with a much more conservative picture, were we have a perfectly smooth space-time, differing from that of Einstein only in that there is no conformal scaling provided, and where time evolution can be treated by conventional mathematical procedures. In CCC, the singularities occurring deep within black holes, on the other hand, would have a very different kind of structure from the Big-Bang singularity and we would have to consider some kind of exotic information-destroying physics which might indeed have to incorporate quantum-gravity ideas differing very much from the notions of space-time used in today's physics, and which might well have to incorporate some wild or revolutionary idea among those mentioned above.

For many years, it has been my own view that these two different singular ends of time appear to have very distinct characters. This is in keeping with the Second Law, where for some reason gravitational degrees of freedom would have to be greatly suppressed at the initial end, though not at the final end. I had always found it extremely puzzling why quantum gravity should appear to treat these two occurrences of spacetime singularities in so different a way. Yet I had imagined, in accordance with what seems now to be the prevailing view, that it should be some form of quantum gravity that governs the kind of geometrical structure that we find close to both of these types of singular space-time geometry. However, apparently at variance with the common view, I had taken the position that the true 'quantum gravity' must be a grossly time-asymmetric scheme, involving whatever modification to the standard present-day rules of quantum mechanics might be required—in accordance with aspirations I have mooted, towards the end of §3.4.

What I had not anticipated, before turning to the point of view that CCC provides, is that the Big Bang should be treated as part of an essentially classical evolution, in which deterministic differential equations like those of standard general relativity govern behaviour. The question was: how could CCC escape the conclusion that enormous space-time curvatures, with radii down to the level of the Planck scale lp near the Big Bang, ought to imply that quantum gravity enters the scene, with all the chaos this entails? CCC's answer is that there is curvature and there is curvature; or, to be more precise, there is Weyl curvature C and Einstein curvature E (the latter being equivalent to Ricci curvature; see §2.6 and Appendix A). The point of view of CCC is to agree that when radii of curvature approach the Planck scale, the madness of quantum gravity (whatever it is) must indeed begin to take over, but the curvature in question must be Weyl cuvature, as described by the conformal curvature tensor C. Accordingly, the radii of curvature involved in the Einstein tensor E can become as small as they like, and the space-time geometry will still remain essentially classical and smooth so long as the Weyl cuvature radii are large on the Planck scale (Fig. 3.22).

Fig. 3.22 A 'radius of curvature' is a reciprocal measure of curvature, which is small when curvature is large and large when curvature is small. Quantum gravity is commonly argued to become dominant when space-time curvature radii approach the Planck length, but CCC maintains this applies only to Weyl curvature.

In CCC we find that C=0 at the Big Bang (whence infinite radii of Weyl curvature), so we are justified in thinking that essentially classical considerations should suffice. Thus, the detailed nature of the big bang of each aeon is completely determined by what happened in the remote future of the prior aeon, and this should lead to observational consequences, some being considered in §3.6. Here, classical equations continue the evolution of the massless fields that were present in the very remote future of the immediately preceding aeon into the big bang of the next. On the other hand, currently standard approaches to the very early universe assume that quantum gravity should be what determines behaviour at the Big Bang. In essence, this is the kind of way (though in terms of the 'inflaton field') that inflationary cosmology would decree how the slight deviations in the CMB temperature (of a few parts in 105) over the sky come about, initially, from 'quantum fluctuations'. However, CCC provides a completely different perspective on this, as we shall be seeing in the next section.

large curvature means small radius of curvature

Fig. 3.22 A 'radius of curvature' is a reciprocal measure of curvature, which is small when curvature is large and large when curvature is small. Quantum gravity is commonly argued to become dominant when space-time curvature radii approach the Planck length, but CCC maintains this applies only to Weyl curvature.

large curvature means small radius of curvature small curvature means large radius of curvature

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