7 Appendix: Conformal Rescalings and Curvature

We compile here some formulae that are helpful for performing conformal rescalings. Suppose we rescale the given metric &tidle;gab to a new metric gab = 𝜃2g&tidle;ab. Define ϒa = ∇a 𝜃∕𝜃. The Levi–Civita connection ∇a of the new metric is given in terms of the Levi–Civita connection ∇&tidle;a of &tidle;gab by its action on an arbitrary vector field va,

( ) ∇avb = ∇&tidle;avb + Cbacvc = ∇&tidle;avb + δbaϒc + δbcϒa − gbdgacϒd vc. (107 )
From the action on vector fields we can obtain the action on tensors of arbitrary valence in the usual way.

Next, we consider the curvature tensor. It is useful to split the Riemann tensor into several pieces, which transform differently under conformal rescalings. We write

Rab = Cab − 4 δ[aP b] . (108 ) cd cd [c d]
The tensor Cabcd is, of course, Weyl’s conformal tensor, characterised by the property of having the same symmetries as the Riemann tensor with all traces vanishing. The other piece, the tensor Pab, can be uniquely expressed in terms of the Ricci tensor,
1 ( 1 ) Pab = − -- Rab − --gabR = Φab − gabΛ. (109 ) 2 6
The tensor Φab is proportional to the trace-free part of the Ricci tensor, while Λ = 1-R = − 1gabPab 24 4 is a multiple of the scalar curvature.

Under the conformal rescaling 2 &tidle;gab ↦→ gab = 𝜃 &tidle;gab, the different parts of the curvature transform as follows:

&tidle;Cabcd ↦→ Cabcd = C&tidle;abcd, (110 ) 1 &tidle;Pab ↦→ Pab = &tidle;Pab − ∇&tidle;a ϒb + ϒaϒb − -&tidle;gab&tidle;gcdϒcϒd. (111 ) 2
Thus, the Weyl tensor is invariant under conformal rescalings. When P&tidle;ab is expressed entirely in terms of the transformed quantities we get the relation
&tidle; 1- cd Pab = Pab + ∇a ϒb + ϒa ϒb − 2gabg ϒcϒd, (112 )
from which we can deduce (note that the contractions are performed with the transformed metric)
( ) &tidle; 1- 1- Φab = Φab + 𝜃 ∇a∇b 𝜃 − 4 gab□𝜃 , (113 ) Λ&tidle; = 𝜃2Λ − 1-𝜃□ 𝜃 + 1-∇a𝜃 ∇a𝜃. (114 ) 4 2
Next consider the Bianchi identity ∇ [eRab]cd = 0. Inserting the decomposition (108View Equation) and taking appropriate traces allows us to write it as two equations,
∇ Ca + 2∇ P = 0, (115 ) a bcd [c d]b ∇aP ab − ∇bP aa = 0. (116 )

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