A9 Weyl tensor conformal scalings

The conformal spinor Wabcd encodes the information of the conformal curvature of space-time, and it is conformally invariant (P&R 6.8.4)

We note the curious (but important) discrepancy between this conformal invariance and that needed to preserve satisfaction of the massless free field equations, where there would be a factor H-1 on the right. To accommodate this discrepancy, we can define a quantity tyabcd which is everywhere proportional to Wabcd, but which scales according to ty ABCD = H~1tyABCD

and we find that our 'Schrodinger equation' for gravitons[A4] (P&R 4.10.9)

in vacuum (Tab=0), is conformally invariant. In §3.2, the above equation is written

Corresponding to the Weyl tensor Cabcd, above (A3, P&R 4.6.41), we can define

and we find the corresponding scalings (written C=H2C and K = HK in §3.2)

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