There is of course the awkwardness, in our particular situation here, that in describing the transition fromVA to Vv we do not have a smoothly varying quantity in either H or w which describes the scaling back to both Einstein metrics gab and gab in a uniform way. But an appropriate proposal for addressing this issue does indeed appear to be to adopt the reciprocal proposal w=- H-1, referred to above, and it is then convenient to consider the 1-form n, defined by n dH dw n2-1 1-w2 i.e.
van van n=
since this 1-form is then finite and smooth across kso long as we adhere to the assumptions that are implicit in the above reciprocal proposal. The quantity n encodes the information of the metric scaling of the spacetime, albeit in a (necessarily) slightly ambiguous way.[B7] We can integrate to obtain a parameter t, so that n = dT, - coth t = H (t<0), tanh t=w (t>0).
We notice that even here there is the awkward issue of a sign change, because although n is insensitive to the replacement of H by H-1, or to the replacement of w by w-1 there is a change of sign when we pass from H-1 to w. We might take the view that the sign of the conformal factor is irrelevant, in any case, since in the rescalings of the metric gab=H2gab and gab=w2gab, the conformal factors H and w appear squared, so that adopting the positive rather than the negative value of each of these conformal factors might be regarded as purely conventional. However, as we recall from Appendix A, there are numerous quantities that scale with H (or w) unsquared, most notably there being the discrepancy between the scalings Vabcd=Vabcd and ¡¡tabcd=H-1 i/iabcd, leading to
in the space V A, since there the Einstein physical metric is gab, which gives us
(this convention differing from what was adopted in §3.2 since it is now the hatted metric in which Einstein's equations hold). Thus, when considering the smooth behaviour of quantities across k where both n and œ change sign (through ^ and 0, respectively), we must exercise care in keeping track of the physical significance of these signs.
The specific reciprocal relationship between n and œ that is being invoked here is, however, dependent upon a restriction in the choice of scaling for the gab metric, namely that the condition
holds, in conjunction with R = 4A=R(see B1). This scaling is easy to arrange, locally at least, by simply choosing a new (local) metric gab for V, to be gab = n2gab where n is some smooth solution of the ra-equation over the crossover. This g-metric is not yet the unique g-metric that we are looking for to cover the cross-over in a canonical way, however, since there are many possible solutions n of the ra-equation that could be chosen. We shall come to some further requirements shortly, that our canonical metric gab might be required to satisfy. For the moment, let us simply assume that our metric gab has been chosen to have R = 4A (i.e. we relabel the above gab as our new choice of gab). Without such a restriction as R = 4A, this reciprocal relationship between n and œ could not be precise, although for the type of conformal factor œ that we expect to find with Tod's proposal[B8] (see end of §2.6, and §3.1, §3.2), the behaviour of the conformal factor for the case of a big bang with pure radiation as the gravitational sources, as with the Tolman radiation-filled solutions[B 9] (see §3.3), indeed behaves, in the past limit as the big bang is approached, as though it were proportional to the reciprocal of the smooth continuation of an H scale factor for a previous aeon. The choice of R = 4A for the metric of V, at k, is what fixes this proportionality factor to be (-)1. This is illustrated in the fact that the somewhat remarkable relation (coming about when we act on na with the divergence operator Va and then apply the ra-equation for H)
a -2nb nb, which arises when this restriction is placed on R, the specific choice of form n (rather than some more general form dH/(H2-A), say), depends upon this restriction that the conformal factor H is to propagate to become (minus) its inverse w = - 1/H, rather than to, say, - A/H. Note that at k, where H=<», we must have nnb=§a and that at kwe also have na = Vaw = Na, the normal vector to k, of length ^aa as noted earlier (P&R9.6.17).
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