An issue that presents itself at this stage is that, according to the requirements of CCC, we want a unique propagation into Vv. This would not be problematic were it not for the unwanted additional freedom arising from an arbitrariness in the conformal factor. As things stand, this freedom provides us with some spurious degrees of freedom, which would inappropriately influence the non-conformally-invariant gravitational dynamics of Vv. These spurious degrees of freedom need to be eliminated in order that the propagation through k be not dependent on this additional data, undetermined by the physics of VA. This spurious 'gauge freedom' in the choice of g-metric can be represented as a conformal factor H that can be applied to gab to provide us with a new metric gab (in accordance with what we had earlier):
gab — gab = H2 gab, and where, as before, we adopt w — w = Hw.
All that we have demanded of H so far is that it be a positive-valued smoothly varying scalar field on V (at least in local patches), which satisfies the ra-equation in the g-metric—this being required in order that the scalar curvature R remain equal to 4A. The ra-equation is a second-order hyperbolic equation of standard type, so we would expect to get a unique solution for H (for a narrow enough collar of k) if the value of H and the value of its normal derivative were both to be specified as smooth functions on k. This would be straightforward if we knew what these values should be chosen to be, in order to achieve some distinctive characterization for the g-metric. So the question arises: what condition on this metric can we demand, in order to eliminate these spurious degrees of freedom?
The kind of thing that we cannot achieve, however, would be some condition on the g-metric (perhaps together with the O-field) that is conformally invariant within the class of rescalings that preserve R=4A. Thus, for a trivial example, we cannot use, as one of our requirements, the demand that the scalar curvature R of the g-metric take any value other than 4A, and the demand that it actually does take the value 4A represents no additional condition whatever on the field, and so cannot be used as a further restriction to reduce the spurious freedom that we wish to eliminate. The same would apply, a little more subtly, to a proposed demand that the squared length gabNaNb of the normal vector Na=VaO to k have some particular value (indices raised and lowered using the g-metric). For if that value were chosen to be anything different from A/3, then (as we saw earlier; P&R 9.6.17) the condition could not be satisfied; whereas if the value is chosen actually to be A/3, then the condition represents no restriction at all on our spurious freedom.
Similar problems would arise also with a demand such as
D abO) = 0, which does not represent any condition on the choice of conformal factor because of the conformal invariance property (noted earlier)
D abO = H DabO, so that DabO = 0 is equivalent to DabO = 0. A condition like DabO = 0 would in any case not do as it stands, because there are several components, and what we require is something that reperesents just two conditions per point of k (like the specification of H and its normal derivative at each point of k ). It may be noted, moreover, that (as we have seen above) DabO necessarily vanishes at kto 3rd order, i.e.
because of the relation DabO = 4nGO3Tab. However, a reasonable-looking condition that could be demanded would be NaNb<fcab = 0 on k More specifically, we could write this suggestion as
We could, in fact, demand that this quantity vanish to 2nd order on k i.e.
which might provide us with a suitable candidate for the required two conditions per point of kthat would be needed in order to fix A, and hence the g-metric via gab = fl2gab. From the definition of Dab, these alternative conditions would be equivalent to demanding that respectively
In tensor notation, the above two different expressions are
NaNb(|gab - §Rab) and NaNb(VaVb - |gab □ )W, where we note (dropping the tildes for the moment) that
We also note that
This rather suggests that a reasonable alternative condition, or pair of conditions, to impose might be, respectively,
NaNbVa Nb = O(w), or O(w2), as this very much simplifies the above condition (where we note that then NbNb—|A vanishes to 2nd or 3rd order, respectively). Conversely, if NbNb—§A vanishes to 2nd order then, on k, then
NaNbVaNb=§NaVa(NbNb)=§NaVa(NbNb—|A) = 0 on k so either of these equivalent conditions (in the form NaN bVaNb = O(w) or NbNb—|A = O(w2)) can be considered alternatively as one of the required restrictions on ft. Note that the above expression ft = Vana/(§A—2nbnb), given in B6, requires a simple pole for ft on k so if the denominator vanishes to 2nd order, the numerator Vana must vanish to 1st order; indeed VafL = O(a) is also a reasonable form of a single condition to be imposed, and we recall from B8 that V(aNb)=-|gabVcNc on k whence 4NaNbVaNb-NaNaVcNc = O(a).
We shall be seeing in B11 below that the energy tensor Uab of Vv necessarily acquires a trace according to the procedures being adopted, this indicating the emergence of gravitational sources with rest-mass. However, we find that this trace vanishes when 3nana=A. One could take the view that the CCC philosophy is best served if the presence of this rest-mass is put off for as long as possible following the big bang. Accordingly, we could well consider that demanding
provides the appropriate two numbers per point of kneeded to fix the g-metric. We shall actually find
2tcGm=a-4 (1 — a2)2 (3nana — A), which becomes infinite at kif the zero in 3nana—A is not at least 4th order. But this is not a problem, because ^ appears only in the g-metric, in which krepresents the singular big bang where other infinite curvature quantities would dominate over ^ if we take the zero in 3nana—A to be 3rd order.
We see that there are several alternative possibilities for the required two conditions per point of k which might suffice to fix ft, and therefore the g-metric, in a unique way. At the time of writing, I have not fully settled on what would appear to be the most appropriate (and which of these conditions are independent of which others). My preference, however, is for the third-order vanishing of 3IIaria-A, as described above.
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