A6 Conformal rescalings

In accordance with the conformal rescaling (with H > 0 smoothly varying)

gab ^ gab = H2 gab, we adopt the abstract-index relations gab = H-2 gab, eab = H £ab, sab = H-1£ab £a-b- = H EA'B', Sa'b' = H-1£ab'.

The operator va now must transform va ^ Va so that the action of va on a general quantity written with spinor indices is generated by

Vaa-0 = Vaa-0, Vaa-^b = Vaa'^b - Yba-^a, Vaa-^b^ Vaa'^b'- Yab-^a-, where

Yaa= H-1 VaaH = Valog H, the treatment of a quantity with many lower indices being built up from these rules, one term for each index. (Upper indices have a corresponding treatment, but this will not be needed here.)

We choose the scaling for a massless field (abc..e to be

and then find, applying the above prescriptions, that

so that the vanishing of either side implies the vanishing of the other, whence satisfaction of the massless free-field equations is conformally invariant. In the case of the Maxwell equations with sources, we find that conformal invariance of the whole system VA'Bp\=2nJAA', Vaa-Jaa'= 0 (P&R 5.1.52, P&R 5.1.54 in A2) is preserved with the scalings p ab = ft—1<AB and JAA'= a—4JAA', since we find

Va'b<ab = ft-4 Va b<ab and V7aa'Jaa' = ft-4 Vaa'Jaa-.

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