B12 Gravitational radiation at k

One feature of the infinite conformai rescaling of the metric, as we pass from VA (with metric gab) to Vv (with metric gab) via k(with metric gab) is the way in which gravitational degrees of freedom, initially present and described in the g-metric by tyabcd (usually non-zero at k), become transferred to other quantities in the g-metric. Whereas we have (A9, P&R 6.8.4)

^ ABCD = ABCD = ^ ABCD = O(w), the conformai behaviour

tells us that tyABCD = O(W2), so that gravitational radiation is very greatly suppressed in the big bang.

However, the degrees of freedom in the gravitational radiation described by ty abcd in VA do make their mark on the early stages of Vv. To see this, we note that differentiating the relation

gives us

VeeWabcd = — V EE'(WtyABCD) = — NEE'tyABCD — W>VEE^ABCD, so that whereas the Weyl curvature vanishes on k its normal derivative provides a measure of the gravitational radiation (free gravitons) out at

WABCD = 0, N'Ve^ABCD = -NNetyABCD = — ^AtyABCD on k

Also, from the Bianchi identities (A5, P&R4.10.7, P&R4.10.8)

V§Wabcd=V§'<Pcdab' and Vca'Ocdab-=0, so we have

VÊ&CDA B' = — Nb tyABCD on k from which it follows that

The operator nb^vB) acts tangentially along k (since Nbcb0NB= 0), so this equation represents a constraint on how cdab behaves on k We also note that

from which it follows that the electric part the normal derivative of the Weyl tensor at on k

of ^abcd on kis basically the quantity

NaV[bOc]d on k while the magnetic part iN^NllpABCD -iNjNl XpA'B'C'D'

which is basically

(£abcd being a skew-symmetrical Levi-Civita tensor), this being the Cotton(-York) tensor which describes the intrinsic conformal curvature of k .[B.13]

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