There are various aspects of this proposal that require a good deal more detailed attention than I have given above. One key issue concerns what the full contents of the universe might be likely to be in the very remote future. The discussion above concentrated mainly on the considerable background of photons that would be present, from starlight, from the CMB, and from black-hole Hawking evaporation. I have also considered that there would be a significant contribution to this background from gravitons, by which I mean the basic (quantum) constituents of gravitational waves, these waves being 'ripples' of space-time curvature, arising largely from close encounters between extremely large black holes in galactic centres.
Photons and gravitons are both massless, so it seems not unreasonable to adopt a philosophy, relevant to the very remote future, that since, in a very late stage in the universe's history it would in principle be impossible to build a clock out of such material, then the universe itself, in the remote future, would somehow 'lose track of the scale of time' and so the geometry of the physical universe really becomes conformal geometry (i.e. null-cone geometry), rather than the full metric geometry of Einstein's general relativity. In fact, we shall be seeing shortly that there are subtleties in connection with the gravitational field which compel us to moderate this philosophy somewhat. But for the moment, let us confront another difficulty with this philosophical standpoint which needs to be faced.
When considering what the main contents of the universe might be in the late stages of its existence, I have ignored the fact that there would be much material within bodies that do not ever find themselves within a black hole, having been flung out from their parent galaxies through random processes, where in some cases the body would also escape from the galactic cluster within which it had originally resided and where there would, indeed, also be much dark matter that would never fall into a black hole. What, for example, would be the fate of a white dwarf star that had escaped in this way, cooled down to become an invisible black dwarf? It has often been suggested that protons might eventually decay away, though observational limits tell us that the rate at which this could happen would have to be very slow indeed. In any case, there would be decay products of some kind, and although much of the material of the black dwarf might eventually collapse into a black hole via such processes, there would be likely to be many 'rogue' massive particles that had, in some form, escaped from the clusters of galaxies to which they had originally been attached.
My concern is particularly with electrons—and also with their anti-particles, the positrons—because they are the least massive electrically charged particles. It is not a particularly unconventional view that protons, and other charged particles more massive than electrons and positrons, might eventually, after vast periods of time, decay into less massive particles. We might imagine that all protons could ultimately decay in this way, but if we accept the conventional view that electric charge must be absolutely conserved, then the ultimate decay products of a proton must contain a net positive charge, so that at least one positron would be expected to be among the eventual survivors. A similar argument would apply to negatively charged particles, and it is hard to escape the conclusion that there would have to be numerous electrons present as well, to accompany these positrons. There might also be more massive charged particles such as protons and anti-protons, if these do not eventually decay, but the key problem lies with the electrons and positrons.
Why is this a problem? Could there not be another type of charged particle (one both of a positive and of a negative charge) which is actually massless, so that electrons and positrons could eventually decay into these, and the above philosophical standpoint be retained? The answer appears to be 'no'. For the mere existence of such a type of massless charged particle, among the menagerie of particle types taking part in today's physical activities, would have made its presence copiously manifest in numerous particle processes. Yet, these processes are actually seen to take place without the production of such massless charged particles. Consequently, there are no massless charged particles around today. Will the (massive) electrons and positrons then have to be around until eternity, in contradiction with the intended philosophical standpoint?
One possibility for retaining this standpoint is raised by the thought that the remaining electrons and positrons might seek each other out and eventually mutually annihilate one another completely to produce merely photons, which would then be harmless to this philosophy. But, unfortunately, in the extremely remote future, many individual charged particles will find themselves isolated within their cosmo-logical event horizons, as shown in Fig. 3.4 (see also Fig. 2.43 in §2.5), and when that happens—as it sometimes must—it removes any possibility of such an eventual charge annihilation. A possible resolution would be to weaken our philosophical standpoint somewhat, and to argue that the odd electron or positron, trapped within its event horizon, would hardly be of much use for the construction of an actual clock. For my own part, I am dissatisfied with such a line of reasoning, as it seems to me to lack the kind of rigour that physical laws ought to demand.
positron positron electron's event horizon
electron electron's event horizon i positron s+
Fig. 3.4 There will be the occasional 'rogue' electron or positron, ultimately trapped within its horizon and unable to lose its electric charge through pair annihilation.
A more radical resolution might be to suppose that charge conservation is actually not one of Nature's stringent requirements. Accordingly, it might be the case that, at extremely occasional moments, a charged particle might decay into one that is without electric charge, and over the reaches of eternity, all electric charge could, accordingly, eventually vanish away. On this consideration, electrons or positrons might eventually become converted into one of their uncharged siblings, say a neutrino, in which case it would also be a requirement that, among the three known types of neutrino, there is one without rest-mass. Quite apart from there being no evidence whatever for any violation of charge conservation, such a possibility is an extremely unpleasant one, theoretically, and it would also seem to demand that the photon itself acquire a small mass, which would in itself nullify the proposed philosophical standpoint.
The one remaining possibility that occurs to me, and which actually strikes me as something to be considered seriously, not merely the least of all evils, is that the notion of rest-mass is not the absolute constant that we imagine it to be. The idea is that over the reaches of eternity, the surviving massive particles—the electrons, positrons, neutrinos, and also protons and antiprotons, if they do not eventually decay, and moreover whatever might be the constituent of the dark matter (necessarily without charge, but possessing rest-mass)—would find that their very rest-masses would very, very gradually fade away, attaining the value zero in the eventual limit. Again, there is absolutely no observational evidence, as of now, for such a violation of ordinary notions concerning rest-mass, but in this case the theoretical backing of the conventional ideas is far less substantial than for charge conservation. In the case of electric charge, we have an additive quantity, in the sense that the total charge of a system is always the sum of all its individual constituents, but with rest-mass, this is certainly not the case. (Einstein's E=mc2 tells us that the kinetic energy of the motions of the constituents will contribute to the total.) Moreover, although the actual value of the basic electric charge (say that of the anti-down-quark, which is one third of that of the proton) remains a theoretical mystery, the values of all other charges found in the universe are whole-number multiples of this value. Nothing like this appears to be the case for rest-mass, and the underlying reason for the particular values of the rest-masses of individual particle types is completely unknown. So there appears to be still the freedom that the rest-mass of a fundamental particle is not an absolute constant—as indeed it is not, according to standard particle physics, in the very early universe, as remarked above, in §3.1— and that it might indeed fade away to zero in the very remote future.
In relation to this, one final technical comment may be made concerning the status of rest-mass in particle physics. A standard procedure for addressing the idea of an 'elementary particle' is to look for what are termed the 'irreducible representations of the Poincare group'. Any elementary particle is supposed to be described according to such an irreducible representation. The Poincare group is the mathematical structure describing the symmetries of Minkowski space M, and this procedure is a natural one in the context of special relativity and quantum mechanics. The Poincare group possesses two quantities referred to as Casimir operators, these being rest-mass and intrinsic spin, and accordingly the rest-mass and spin are deemed to be 'good quantum numbers', which remain constant so long as the particle is a stable one and does not interact with anything. However, this role of M appears to be less fundamental when there is a positive cosmological constant A present in physical laws (as A=0 for M), and it would seem that, when we are concerned with matters related to cosmology, it should be the symmetry group of de Sitter space-time D, rather than of M, that should ultimately be our concern (see §2.5, Fig. 2.36(a),(b)). However, it turns out that rest-mass is not exactly a Casimir operator of the de Sitter group (there being a small additional term involving A), so that its ultimate status is more questionable in this case, and a very slow decay of rest-mass seems to me to be not out of the question.
The extremely gradual decaying away of rest-mass, according to this proposal, does have its curious implications, however, with regard to the whole scheme of CCC, because it raises a new issue in relation to the measurement of time. We recall that near the end of §2.3 a particle's rest-mass was used to provide a well-defined scale of time, such a scaling being all that is needed so that we may pass from a conformal structure to a full metric. If, as seems to be required from the above discussion, we need particles' masses to decay away, albeit extremely gradually, then we are led into a bit of a quandary. Do we still adopt this idea of using particles' rest-masses for precisely defining our space-time's metric, when massive particles are still around, but with slowly decaying masses? If we try to settle on some particular particle type, say an electron, as providing us with the standard of time, then with the kind of decay rates that would seem to be required in order for electrons to be considered adequately 'massless' when is reached (see Appendix A2), it would turn out that is not at infinity at all, and the universe's expansion, according to this 'electron metric' would either have to slow to a halt or else to reverse into a collapse. It would appear that such behaviour would not be consistent with Einstein's equations. Moreover, if instead of an 'electron metric' we used a 'neutrino metric' or 'proton metric', say, then the detailed geometrical behaviour of the space-time would be likely to differ from the corresponding behaviour that would be obtained by use of electrons (unless the scaling to zero occurs with all mass values retaining exactly their initial proportions). To me, this does not appear very satisfactory.
It seems that in order to preserve some appropriate form of Einstein's equations—with constant A—throughout the entire history of the aeon, we need to use another proposal for scaling for the metric. What we can do, although this would hardly be a 'practical' solution for the purposes of building a clock, would be to use A itself to determine a scale, or, what appears to be closely related to this, we might use the effective value of the gravitational constant G. Then the picture of an evolving and unendingly exponentially expanding universe continuing into its remote future would be retained, but without seriously disturbing the philosophy that, locally, the universe will eventually lose track of the scale of time.
This matter is closely related to another one which I have glossed over until now, namely the fact that whereas there is a conformal invariance for the free gravitational field, as described by the Weyl conformal tensor C (since C indeed describes the conformal curvature), the coupling of the field to its sources is not conformally invariant. This is quite different from what happens in Maxwell's theory, where there is a conformal invariance which holds both for the free electromagnetic field F and for the coupling between F and its sources as described by the charge-current vector J. Thus, again, when we bring gravity into the picture in a serious way, the basic philosophy of CCC gets a little muddied. We must take the view that, in a sense, the philosophy of CCC asserts that it is gravity-free physics (and A-free physics) that loses track of time, not completely physics as a whole.
Let us try to understand the relation of Einstein's theory to conformal invariance. It is a somewhat delicate matter. In the case of electromag-netism, the entire equations are preserved under the conformal rescaling. We are to examine what happens when the space-time metric g is replaced by a conformally related one g by means of a scale factor H, this being a positive number varying smoothly over space-time (see §2.3, §3.1):
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