## VK o

then these scalings lead us to[319]

This has some curious consequences, which are of considerable importance for CCC. As we approach \+ from its past, we need to use a conformal factor n which tends to zero smoothly,[320] but with a nonzero normal derivative. The geometrical meaning of this is illustrated in Fig. 3.5. The conformal invariance of the wave-propagation equation for K implies that it attains finite (and usually non-zero) values on \+, these values determining the strength (and polarization) of the gravitational radiation—the gravitational analogue of light—as it continues out to infinity and thereby makes its mark on J+. See Fig. 3.6. The same applies to the values of F on J+, determining the strength and polarization of the electromagnetic radiation field (light). But because of the fact that H becomes zero at J+, the displayed equation above, rewritten as C=HK, tells us that the finiteness of K implies that the conformal tensor C must itself become zero on J+ (where we use a metric g finite at J+). Since C provides a direct measure of conformal geometry at J+, the demand of CCC that the conformal geometry be smooth over the crossover 3-surface from each aeon to the next tells us that conformal curvature must also be zero at the big-bang surface U- of the subsequent aeon. Accordingly, CCC actually provides a stronger version of Weyl curvature hypothesis (WCH, see §2.6) than the condition that the conformal curvature merely be finite (which was what Tod's proposal gave directly), namely that this conformal curvature really does vanish at the U- of each aeon, in accordance with the original idea of WCH.

 crossover conformal factor (called H in §3.2, but called ft-1 = -w in Appendix B)

Fig. 3.5 The conformal scale factor goes cleanly from positive to negative at crossover, the curve having a slope that is neither horizontal nor vertical. Here 'conformal time' just refers to 'height' in a suitable conformal diagram.

Fig. 3.5 The conformal scale factor goes cleanly from positive to negative at crossover, the curve having a slope that is neither horizontal nor vertical. Here 'conformal time' just refers to 'height' in a suitable conformal diagram.

Fig. 3.6 The gravitational field is measured by the tensor K, propagates according to a conformally invariant equation, and so generally attains finite non-zero values at \f+

Fig. 3.6 The gravitational field is measured by the tensor K, propagates according to a conformally invariant equation, and so generally attains finite non-zero values at \f+

On the other side of the crossover surface, i.e. just following the U- of the subsequent aeon, we find a conformal factor which becomes infinite at U-, but in just such a way as to make ft-1 behave smoothly at U-.[3 21] Thus, it appears to be the case that ft has to be able to be continued somehow over the crossover 3-surface to become, suddenly, its reciprocal! A way to handle this situation mathematically is to encode the essential information of ft in a way that does not distinguish it from its reciprocal ft-1. This can be done by considering the [?]-tensor n (a 1-form), that mathematicians would write as[322]