I shall refer to H, regarded as a particular instance of the massless self-coupled conformally invariant field to, as the phantom field.[B4] It does not provide us with physically independent degrees of freedom; its presence (in the gab-metric) simply allows us the scaling freedom that we need, so that we can rescale the physical metric to obtain a smooth metric gab, conformal to Einstein's physical metric, which smoothly covers each of the joins from one aeon to the next. With the aid of such metrics covering the crossover 3-surfaces, we are enabled to study in detail the specific connections between aeons, in accordance with the requirements of CCC, by using explicit classical differential equations.

The role of the phantom field is just to 'keep track' of Einstein's actual physical metric by telling us how to scale the metric gab back to the physical one (via gab = H2gab). Then we express the satisfaction of Einstein's equations in the pre-crossover space V A, but now written in terms of the g-metric, simply as Tab=Tab[H]; that is to say, Einstein's field equations are expressed in the demand that the total energy tensor Tab of all the physical matter fields in the space-time region VA (assumed massless and having the correct conformal scaling) must be equal to the energy tensor of the phantom field Tab[H]. Whereas this can be regarded as simply a reformulation of Einstein's theory (using gab) within the open region VA, it is actually something more subtle. It allows us to extend our equations up to, and even beyond its future boundary surface \+. But in order to do this effectively, we shall need to look a little more carefully at the relevant equations governing the quantities of interest, and their expected behaviours as kis approached. Moreover, we shall need to understand, and then eliminate, the freedom in the initially some what arbitrary choice of g-metric—i.e. of conformal factor Q—that has been chosen for the 'collar' V that we are concerned with.

There is, indeed, some considerable freedom in Q, as things stand. All that has been required, so far, is that Q be such that the gab, as obtained from Einstein's physical metric gab by gab=Q-2gab is finite, nonzero and smooth across k Even to demand the existence of such an Q may seem like a strong requirement, but there are powerful results due to Helmut Friedrich[B 5] that lead us to expect, when there is a positive cosmological constant A, that full freedom in the massless radiation fields in a fully expanding universe model free of massive sources, is incorporated by a smooth (spacelike) J*. To put this another way, we can expect to find a smooth future conformal boundary J+, to VA, as a more-or-less automatic consequence of the fact that the model is indefinitely expanding, all the gravitational sources being massless fields propagating according to conformally invariant equations. It should be noted that, at this stage, there is no demand that the scalar curvature R of the g-metric even be a constant, let alone that R = 4A, so that the conformal factor Q-1 taking us back to Einstein's gab would not necessarily satisfy the ra-equation (□+|R)ra=fAra3 in the g-metric.

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