Let us return to the basic question that we have been trying to address in this part, namely the issue of how our universe happened to come about with a 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. This issue tends to be muddied, in most modern cosmological considerations, however, by the popular idea that in the very early stages of the universe's existence it underwent an exponential expansion— referred to as cosmic inflation—during a phase lasting for a tiny time-period somewhere between around 10-36 and 10-32 seconds following the Big Bang, increasing the linear dimension of the universe by some enormous factor of somewhere between 1020 and 1060, or perhaps even 10100 or so. This huge expansion is supposed to explain the uniformity of the early universe (among other things), where practically all early irregularities are taken to have been ironed out simply by the expansion. However, these discussions seem hardly ever to be taken as addressing the fundamental question that I have been concerned with in Part 1, namely the origin of the extraordinary manifest specialness of the Big Bang, which must have been initially present, in order that there be a second law of thermodynamics. In my view, this idea underlying infla-tion—that the uniformity in the universe that we now observe should be the result of (inflationary) physical processes acting in its early evolution—is basically misconceived.

Why do I say that it is misconceived? Let us examine this issue in terms of some general considerations. The dynamics underlying inflation is taken to be governed in the same general way as are other physical processes, where there are time-symmetrical dynamical laws underlying this activity. There is taken to be a particular physical field known as the 'inflaton field' that is held to be responsible for inflation, although the precise nature of the equations governing the inflaton field would generally differ from one version of inflation to another. As part of the inflationary process, there would be some sort of 'phase transition' taking place, which may be thought of in terms of some kind of analogy with the transition between solid and liquid states that occurs with freezing or melting, etc. Such transitions would be regarded as proceeding in accordance with the Second Law, and would normally be accompanied by a raising of entropy. Accordingly, the inclusion of an inflaton field in the dynamics of the universe does not affect the essential arguments that were being put forward in Part 1. We still need to understand the extraordinarily low-entropy start of the universe, and according to the arguments of §2.2 this lowness of entropy lay essentially in the fact that the gravitational degrees of freedom were not excited, at least not nearly to the extent that involved all other degrees of freedom.

It will certainly be helpful to try to understand what a high entropy initial state would be like, when gravitational degrees of freedom are to be taken into consideration. We can get some appreciation of this if we imagine the time-reversed context of a collapsing universe, since this collapse, if taken in accordance with the Second Law, ought to lead us to a singular state of genuinely high entropy. It should be made clear that our mere consideration of a collapsing universe has nothing to do with whether our actual universe will ever re-collapse, like the closed A=0 Friedmann model of Fig. 2.2. This collapse is being taken simply as a hypothetical situation, and it is certainly in accord with Einstein's equations. In a general collapse situation, like the general collapses to a black hole that we considered in §2.4, we may expect all sorts of irregularities to emerge, but when local regions of material get sufficiently concentrated, trapped surfaces are likely to come about and space-time singularities are expected to arise.[2 50] Whatever density irregularities are present initially would intensify greatly, and the final singularity would be expected to be that coming from an extraordinary mess of congealing black holes. It is here that the considerations of Belinski, Khalatnikov and Lifshitz might well come into play. And if the BKL conjecture is correct (see §2.4), then some extremely complicated singularity structure is indeed to be expected.

I shall return to this issue of singularity structure shortly, but for the moment, let us consider the relevance of inflationary physics. Let us focus attention on the state of the universe at, for example, the time of decoupling, when the radiation that we now see as the CMB was produced (see §2.2). In our actual expanding universe, there was a very great uniformity in the matter distribution at that time. This is clearly taken to be a puzzle—for otherwise there would be no point in introducing inflation in order to explain it! Since it is accepted that there is something to explain, we must consider that there might, instead, have been enormous irregularities at that time. The inflationist's claim would have to be that the presence of an inflaton field actually renders such irregularities highly improbable. But is this really the case?

Not at all, for we can imagine this situation of a highly lumpy matter distribution at the time of decoupling, but with time reversed, so that this picture represents a very irregular collapsing universe.[2 51] As our imagined universe collapses inwards, the irregularities will become magnified, and deviations from FLRW symmetry (see §2.1) will become more and more exaggerated. Then, the situation will be so far from FLRW homogeneity and isotropy that the inflationary capabilities of the inflaton field will find no role, and (time-reversed) inflation will simply not take place, since this depends crucially on having an FLRW background (at least with regard to calculations that have actually been carried through).

We are therefore led to the clear implication that our irregular collapsing model will indeed collapse down to a state involving a horrendous mess of congealing black holes, this leading to a highly complicated enormously high-entropy singularity, very possibly of a BKL type, which is quite unlike the highly uniform low-entropy singularity of closely FLRW form that we seem to have had in our actual Big Bang. This would happen quite independently of whether or not an inflaton field is present in the allowed physical processes. Thus, time-reversing our imagined collapsing lumpy universe back again, so as to obtain a possible picture of an expanding universe, we find that it starts with a high-entropy singularity which, it seems, could have been an initial state for our actual universe and, indeed, would be a far more probable initial state (i.e. of much larger entropy) than the Big Bang that actually occurred. The black holes that congeal together in the final stages of our envisaged collapse would, when time-reversed to an expanding universe, provide us with the image of an initial singularity consisting of multiply bifurcating white holes![252] A white hole is the timereverse of a black hole, and I have indicated the sort of situation that this provides us with in Fig. 2.45. It is the total absence of such white-hole singularities that singles out our Big Bang as being so extraordinarily special.

emerging matter emerging matter

In terms of phase-space volume, initial singularities of this nature (with multiply bifurcating white holes) would occupy a stupendously larger region than do those resembling the singularity that gave rise to our actual Big Bang. The mere potential presence of an inflaton field certainly cannot provide the power to 'iron out' the irregularities of such a conglomeration of white-hole singularities. This can be said with confidence quite apart from any detailed considerations of the nature of the inflaton field. It is just an issue of having equations that can be evolved equally in either direction in time, up until a singular state is reached.

But we can certainly say more about the actual enormity of the phasespace volume, if we take into account the entropy values, and therefore the phase-space volumes, that are actually assigned to black holes, according to the well-accepted Bekenstein-Hawking formula for the entropy value of a black hole. For a non-rotating hole of mass M, this entropy is

Sbh =-:-M 2, ch whereas the entropy lies between this value and one-half of it if the hole is rotating, depending on the amount of the rotation. The fraction preceding 'M2' is just a constant, where k, G, and h are the constants of Boltzmann, Newton, and Planck, respectively, c being the speed of light. In fact, we can rephrase this entropy formula in a more general form kc3A SBH=4GF

where A is the surface area of the horizon and h=hJln, this formula being applicable whether or not the hole is rotating. In the Planck units to be introduced at the end of §3.2, we have

Although there is, in my opinion, still no completely satisfactory account of this entropy in terms of the counting of internal black-hole states,[2 53] such an entropy value is nevertheless an essential ingredient to the maintaining of a consistent Second Law in a quantum physical world external to the black hole. As mentioned already in §2.2, easily the largest contribution to the entropy of the present universe comes from the contributions from large black holes in galactic centres. If a total mass consisting of that lying within our present observable universe (that lying within our present particle horizon; see §2.5) were to form a black hole, this would attain an entropy of roughly 10124, and we may consider this to provide a rough lower limit to entropy that would be achievable by our collapsing universe model involving the same amount of material. The phase-space volume corresponding to this would then be something like[2 54]

(because of the logarithm in Boltzmann's entropy formula, given in §1.3), whereas the region of phase space corresponding to the state of the actual observed universe at the time of decoupling, for the same body of matter, namely that in the observed CMB, had a volume no greater than about

The probability of finding ourselves in a universe of such a degree of specialness, if it had come about just by chance,[255] has the utterly absurdly tiny value of around 1/101Q124 irrespective of inflation. This is the kind of figure that needs some completely different kind of theoretical explanation!

There is, however, one further issue that may be considered to have importance here. This is the question of whether an initial singularity of such a complicated white-hole-type structure could reasonably be referred to as an 'instantaneous event'. The question is basically a matter of whether such a singularity, when viewed as some kind of past 'conformal boundary' to the space-time, can be appropriately thought of as 'spacelike'. Such a spacelike initial singularity could then be taken to represent the zero of some cosmic time coordinate and regarded as the 'moment' of such a highly irregular big bang.

In fact, the time-reverse of an Oppenheimer-Snyder collapse indeed has a spacelike initial singularity, as is clear from its strict conformal diagram Fig. 2.46, this being the time-reverse of Fig. 2.38(a). Moreover, it is a feature of general BKL singularities that they seem to have this spacelike character. More generally still, a spacelike nature is expected for generic singularities (allowing for their possibly being null in places) on the basis of strong cosmic censorship,[2 56] a yet unproved conjecture for solutions of Einstein's equations (referred to already in §2.4) which tells us that 'naked singularities' do not occur in generic gravitational collapse, the singularities that result being always hidden from direct observation, as by a black hole's event horizon. Strong cosmic censorship tells us that these singularities ought indeed to be spacelike, at least in general. In accordance with this expectation, it seems to me to be perfectly reasonable to refer to such a white-hole-ridden initial singularity as indeed being an instantaneous event.

An important question now arises: what geometrical criterion distinguishes the kind of 'smooth' singularity that appears to be what characterized the very low-entropy singularity of our Big Bang, from the more general high-entropy type of singularity that arises in the white-hole-ridden time-reversed collapses just considered? We need some clear-cut way of saying that 'the gravitational degrees of freedom were not activated'. But for this, we need to identify the mathematical quantity that actually measures 'gravitational degrees of freedom'.

A good analogy for the gravitational field is the electromagnetic field, which resembles it in many significant ways, although there are, nevertheless, some important differences. The electromagnetic field is described, in relativity physics, by a tensor quantity F, referred to as the

Maxwell field tensor—after the great Scottish scientist James Clerk Maxwell, who first found, in 1861, the equations satisfied by the electromagnetic field, and he showed that these explain the propagation of light. We may recall that in §2.3, we encountered another tensor quantity, namely the metric tensor g. Tensors are essential for general relativity theory, as they provide mathematical descriptions of geometrical or physical entities in ways that are unaffected by (or 'carried along' by) the 'rubber-sheet' deformations (diffeomorphisms) that we considered in §2.3. The tensor F is determined by 6 independent numbers per point (3 for the components of the electric field at that point and 3 more for the magnetic field). The metric tensor g has 10 independent components per point. In standard tensor notation, it is usual to denote the collection of components of the metric by gab, or some such, with two lower indices (and it has a symmetry gab=gba). In the case of Maxwell's tensor F, the collection of components would be denoted by Fab (with the antisymmetry Fab=-Fba). Each of these tensors has a valence [2], which refers to the fact that there are just two lower indices. But tensors with upper indices can occur also, a [^-tensor being described by a collection of components denoted by an entity with p upper indices and q lower indices. There is an algebraic procedure known as contraction (or transvection) which allows us to connect a lower index to an upper one (rather in the manner of chemical bonding), thereby removing these two indices from the final expression—but it is not my purpose here to go into the algebraic operations of the tensor calculus.

The degrees of freedom in the electromagnetic field are indeed measured by the Maxwell tensor F, but in Maxwell theory there is also a source for the electromagnetic field, known as the charge-current vector J. This may be thought of as a [0]-tensor, whose 4 components per point describe the 1 component of electric charge density together with the 3 components of electric current. In a stationary situation, the charge density acts as the source of electric field and the current density as the source for the magnetic field, but things get more complicated when the situation is not stationary.

We now ask for the analogues of F and J in the case of the gravitational field, as described by Einstein's general theory of relativity. In this theory there is a curvature to space-time (which can be calculated once one knows how the metric g varies throughout the space-time), described by a [0]-tensor R, called the Riemann(-Christoffel) tensor, with somewhat complicated symmetries resulting in R having 20 independent components per point. These components can be separated into two parts, constituting a [0]-tensor C, with 10 independent components, called the Weyl conformal tensor, and a symmetric [0]-tensor E, also with 10 independent components, called the Einstein tensor (this being equivalent to a slightly different [0]-tensor referred to as the Ricci tensor[257]). According to Einstein's field equations, it is E that provides the source to the gravitational field. This is normally expressed[2 58]in the form

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