We examine the geometry in the neighbourhood of a crossover 3-surface k, in accordance with the ideas of Part 3, where it is assumed that there is a collar V, of smooth conformal space-time containing k, which extends both to the past and to the future of k, within which only massless fields are present within V prior to the crossover U We choose a smooth metric tensor gab in this collar, consistent with the given conformal structure—locally at least and in an initially somewhat arbitrary way. Let Einstein's physical metric in the 4-region VA, just prior to k be gab, and in the 4-region Vv immediately following k be gab, where gab = n2gab and gab = W2gab.
(Note that these are not quite the conventions used in §3.2, since there the 'unhatted' gab was used for Einstein's physical metric. The explicit formulae given in Appendix A remain valid here as they stand, however.) As a 'mnemonic', we may relate the symbols 'A' and 'v' to the respective portions of the null cones at points of k. In each of these two regions we are to assume that Einstein's equations hold, with a fixed cosmological constant A, and that all gravitational sources in the earlier region VA are taken to be massless, so that their total energy tensor Tab is trace-free
For reasons that will emerge later, I shall use a different letter Uab for the energy tensor in Vv, and it turns out, for consistency with the formalism, that this tensor actually has to acquire a small trace iJaa = /, so that a rest-mass component to the energy tensor begins to emerge in Vv. It may be conjectured that this has something to do with the emergence of rest-mass in accordance with the Higgs mechanism,[B1] but this idea is not explored here. (It should be noted that the 'hatted' quantities such as Tab have their indices raised and lowered respectively by gah and gab or, correspondingly, TAB, ea'b', ¡Tab, and ea-b-, whereas the 'reverse-hatted' quantities such as Uab would use gab, gab, eab, ea'b', Tab, and ea-b-). The Einstein equations hold in the respective regions V A and V v, so we have 'hatted' and 'reverse-hatted' versions holding:
Eab = 8nGTab + Agab, Eab = 8nGUab + Agab, where I assume that the same[B.2] cosmological constant holds in the two regions, so that
For the moment, the metric gab, which straddles the cross-over 3-surface k is chosen completely freely, but smoothly and consistently with the given conformal structures of V A and Vv. Later, I provide a proposal which appears to fix a unique scaling for gab in a canonical and appropriate way, so that ultimately a specific choice of gab is provided, for which I propose the notation 'gab' in standard italic type. I also use standard italic type for the curvature quantities Rabcd, etc. whether or not the specialization of gab to gab is made.
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