B4 The normal N to k

We observe that Q^^, as J* ( = k) is approached from below, since the role of Q is to scale up the finite g-metric at J* by an infinite amount, to become the remote future of the earlier aeon. However, we find that the quantity to = - Q-1

approaches zero from below, at J+, in a smooth way (the minus sign being needed for what follows), and it does this so that the quantity

Vato=Na is non-zero on the cross-over 3-surface k( = J+), and so provides us, at points of k with a future-pointing timelike 4-vector N normal to k

The idea is to try to arrange things so that this particular 'w' continues smoothly across k, from VA into the Vv region, and with non-zero derivative, so that it actually becomes the same (positive) quantity 'w' as is required for V v's Einstein metric gab = w2gab (and it is for this reason that the minus sign is needed in 'w = - H-1'). It may be remarked that the 'normalization' condition (P&R 9.6.17)

is an automatic general property of conformal infinity (here k) when there are just massless sources for the gravitational field, so that is a unit normal to k, irrespective of the particular choice of conformal factor H.

As an incidental comment, we see that from this we can readily derive the fact, noted in §3.5, that the limiting area of cross-section of any cosmological event horizon must be 12n/A. Any event horizon (taken in the earlier aeon) is the past light cone C of the future end point o+, on k, of some immortal observer in that aeon, as in §2.5 (see Fig. 2.43). Then the limiting area of cross-section of C as o+ is approached from below is 4nr2, where r (in the g-metric) is the spatial radius of the cross-section. In the gab metric, this area becomes 4nr2H2, and we readily find from the above (B4) that Hr approaches (|A)-1/2 in the limit, as the cross-section approaches o+, so that our required event-horizon area is indeed 4n x(3/A) = 12n/A. (Although this argument has been presented in the context of CCC, all that is required for it is a small degree of smoothness for the spacelike conformal infinity which, as the work of Friedrich has shown,[B 6] is a very mild assumption, when A > 0.)

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