(from the Emperorís New Mind, Penrose, pp 339-345 copyright 1989, Penguin Books)
How special was the big bang?
Let us try to understand just how much of a constraint a condition such as WEYL = 0 at the big bang was. For simplicity (as with the above discussion) we shall suppose that the universe is closed. In order to be able to work out some clear-cut figures, we shall assume, furthermore, that the number B of baryons-that is, the number of protons and neutrons, taken together-in the universe is roughly given by
B = 10^80.
(There is no particular reason for this figure, apart from the fact that, observationally B must be at least as large as this; Eddington once claimed to have calculated B exactly, obtaining a figure which was close to the above value! No-one seems to believe this particular calculation any more, but the value 10^80 appears to have stuck.) If B were taken to be larger than this (and perhaps, in actual fact, B = infinity) then the figures that we would obtain would be even more striking than the extraordinary figures that we shall be arriving at in a minute! Try to imagine the phase space (cf. p. 177) of the entire universe! Each point in this phase space represents a different possible way that the universe might have started off. We are to picture the Creator, armed with a `pin' which is to be placed at some point in the phase space (Fig. 7.19 not shown). Each different positioning of the pin provides a different universe. Now the accuracy that is needed for the Creator's aim depends upon the entropy of the universe that is thereby created. It would be relatively `easy' to produce a high entropy universe, since then there would be a large volume of the phase space available for the pin to hit. (Recall that the entropy is proportional to the logarithm of the volume of the phase space concerned.) But in order to start off the universe in state of low entropy-so that there will indeed be a second law of thermodynamics-the Creator must aim for a much tinier volume of the phase space. How tiny would this region be, in order that a universe closely resembling the one in which we actually live would be the result? In order to answer this question, we must first turn to a very remarkable formula, due to Jacob Bekenstein (1972) and Stephen Hawking (1975), which tells us what the entropy of a black hole must be.
Consider a black hole, and suppose that its horizon's surface area is A. The Bekenstein-Hawking formula for the black hole's entropy is the:
Sbh = A/4 + (kc^3 / Gh)
where k is Boltzmann's constant, c is the speed of light, G is Newton's gravitational constant, and h is Planck's constant over 2pi. The essential part of this formula is the A/4. The part in parentheses merely consists of the appropriate physical constants. Thus, the entropy of a black hole is proportional to its surface area. For a spherically symmetrical black hole, this surface area turns out to be proportional to the square of the mass of the hole
A = m^2 x 8pi(G^2/c^4).
Putting this together with the Bekenstein-Hawking formula, we find that the entropy of a black hole is proportional to the square of its mass:
Sbh = m^2 x 2pi (kG/hc)
Thus, the entropy per unit mass of a black hole is proportional to its mass, and so gets larger and larger for larger and larger black holes. Hence, for a given amount of mass-or equivalently, by Einstein's E = mc^2, for a given amount of energy-the greatest entropy is achieved when the material has all collapsed into a black hole! Moreover, two black holes gain (enormously) in entropy when they mutually swallow one another up to produce a single united black hole! Large black holes, such as those likely to be found in galactic centres, will provide absolutely stupendous amounts of entropy-far and away larger than the other kinds of entropy that one encounters in other types of physical situation.There is actually a slight qualification needed to the statement that the greatest entropy is achieved when all the mass is concentrated in a black hole. Hawking's analysis of the thermodynamics of black holes, shows that there should be a non-zero temperature also associated with a black hole. One implication of this is that not quite all of the mass-energy can be contained within the black hole, in the maximum entropy state, the maximum entropy being achieved by a black hole in equilibrium with a `thermal bath of radiation'. The temperature of this radiation is very tiny indeed for a black hole of any reasonable size. For example, for a black hole of a solar mass, this temperature would be about 10^-7 K, which is somewhat smaller than the lowest temperature that has been measured in any laboratory to date, and very considerably lower than the 2.7 K temperature of intergalactic space. For larger black holes, the Hawking temperature is even lower!
The Hawking temperature would become significant for our discussion only if either: (i) much tinier black holes, referred to as mini-black holes, might exist in our universe; or (ii) the universe does not recollapse before the Hawking evaporation time-the time according to which the black hole would evaporate away completely. With regard to (i), mini-black holes could only be produced in a suitably chaotic big bang. Such mini-black holes cannot be very numerous in our actual universe, or else their effects would have already been observed; moreover, according to the viewpoint that I am expounding here, they ought to be absent altogether. As regards (ii), for a solar-mass black hole, the Hawking evaporation time would be some 10^54 times the present age of the universe, and for larger black holes, it would be considerably longer. It does not seem that these effects should substantially modify the above arguments.
To get some feeling for the hugeness of black-hole entropy, let us consider what was previously thought to supply the largest contribution to the entropy of the universe, namely the 2.7 K black-body background radiation. Astrophysicists had been struck by the enormous amounts of entropy that this radiation contains, which is far in excess of the ordinary entropy figures that one encounters in other processes (e.g. in the sun). The background radiation entropy is something like 10^8 for every baryon (where I am now choosing `natural units', so that Boltzmann's constant, is unity). (In effect, this means that there are 10^8 photons in the background radiation for every baryon.) Thus, with 10^88 baryons in all, we should have a total entropy of
for the entropy in the background radiation in the universe. Indeed, were it not for the black holes, this figure would represent the total entropy of the universe, since the entropy in the background radiation swamps that in all other ordinary processes. The entropy per baryon in the sun, for example, is of order unity. On the other hand, by black-hole standards, the background radiation entropy is utter `chicken feed'. For the Bekenstein-Hawking formula tells us that the entropy per baryon in a solar mass black hole is about 10^20, in natural units, so had the universe consisted entirely of solar mass black holes, the total figure would have been very much larger than that given above, namely
Of course, the universe is not so constructed, but this figure begins to tell us how `small' the entropy in the background radiation must be considered to be when the relentless effects of gravity begin to be taken into account. Let us try to be a little more realistic. Rather than populating our galaxies entirely with black holes, let us take them to consist mainly of ordinary stars-some 10^11 of them-and each to have a million (i.e. 10^6) solar-mass black-hole at its core (as might be reasonable for our own Milky Way galaxy). Calculation shows that the entropy per baryon would now be actually somewhat larger even than the previous huge figure, namely now 10^21, giving a total entropy, in natural units, of
We may anticipate that, after a very long time, a major fraction of the galaxies' masses will be incorporated into the black holes at their centres. When this happens, the entropy per baryon will be 10^31, giving a monstrous total of
However, we are considering a closed universe so eventually it should recollapse; and it is not unreasonable to estimate the entropy of the final crunch by using the Bekenstein-Hawking formula as though the whole universe had formed a black hole. This gives an entropy per baryon of 10^43, and the absolutely stupendous total, for the entire big crunch would be
This figure will give us an estimate of the total phase-space volume V available to the Creator, since this entropy should represent the logarithm of the volume of the (easily) largest compartment. Since 10^123 is the logarithm of the volume, the volume must be the exponential of 10^123, i.e.
V = 10^10^123.
in natural units! (Some perceptive readers may feel that I should have used the figure e^10^123, but for numbers of this size, the a and the 10 are essentially interchangeable!) How big was the original phase-space volume W that the Creator had to aim for in order to provide a universe compatible with the second law of thermodynamics and with what we now observe? It does not much matter whether we take the value
W = 10^10^101 or W = 10^10^88
given by the galactic black holes or by the background radiation, respectively, or a much smaller (and, in fact, more appropriate) figure which would have been the actual figure at the big bang. Either way, the ratio of V to W will be, closely
V/W = 10^10^123.
This now tells us how precise the Creator's aim must have been: namely to an accuracy of one part in 10^10^123.
This is an extraordinary figure. One could not possibly even write the number down in full, in the ordinary denary notation: it would be `1' followed by 10^123 successive `0 's! Even if we were to write a `0' on each separate proton and on each separate neutron in the entire universe-and we could throw in all the other particles as well for good measure-we should fall far short of writing down the figure needed. The precision needed to set the universe on its course is seen to be in no way inferior to all that extraordinary precision that we have already become accustomed to in the superb dynamical equations (Newton's, Maxwell's, Einstein's) which govern the behaviour of things from moment to moment. But why was the big bang so precisely organized, whereas the big crunch (or the singularities in black holes) would be expected to be totally chaotic? It would appear that this question can be phrased in terms of the behaviour of the WEYL part of the space-time curvature at space-time singularities. What we appear to find is that there is a constraint
WEYL = 0
(or something very like this) at initial space-time singularities-but not at final singularities-and this seems to be what confines the Creator's choice to this very tiny region of phase space. The assumption that this constraint applies at any initial (but not final) space-time singularity, I have termed The Weyl Curvature Hypothesis. Thus, it would seem, we need to understand why such a time-asymmetric hypothesis should apply if we are to comprehend where the second law has come from.
How can we gain any further understanding of the origin of the second law? We seem to have been forced into an impasse. We need to understand why space-time singularities have the structures that they appear to have; but space-time singularities are regions where our understanding of physics has reached its limits. The impasse provided by the existence of space-time singularities is sometimes compared with another impasse: that encountered by physicists early in the century, concerning the stability of atoms (cf. p. 228). In each case, the well-established classical theory had come up with the answer `infinity', and had thereby proved itself inadequate for the task. The singular behaviour of the electromagnetic collapse of atoms was forestalled by quantum theory; and likewise it should be quantum theory that yields a finite theory in place of the `infinite' classical space-time singularities in the gravitational collapse of stars. But it can be no ordinary quantum theory. It must be a quantum theory of the very structure of space and time. Such a theory, if one existed, would be referred to as `quantum gravity'. Quantum gravity's lack of existence is not for want of effort, expertise, or ingenuity on the part of the physicists. Many first-rate scientific minds have applied themselves to the construction of such a theory, but Without success. This is the impasse to which we have been finally led in our attempts to understand the directionality and the flow of time.
The reader may well be asking what good our journey has done us. In our quest for understanding as to why time seems to flow in just one direction and not in the other, we have had to travel to the very ends of time, and where the very notions of space have dissolved away. What have we learnt from all this? We have learnt that our theories are not yet adequate to provide answers, but what good does this do us in our attempts to understand the mind? Despite the lack of an adequate theory, I believe that there are indeed important lessons that we can learn from our journey. We must now head back for home. Our return trip will be more speculative than was the outward one, but in my opinion, there is no other reasonable route back!