
7 Appendix: Conformal
Rescalings and Curvature
We compile here some formulae that are helpful for performing
conformal rescalings. Suppose we rescale the given metric
to a new metric
. Define
. The Levi-Civita connection
of the new metric is given in terms of the
Levi-Civita connection
of
by its action on an arbitrary vector field
,
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
The tensor
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
, can be uniquely expressed in terms of
the Ricci tensor,
The tensor
is proportional to the trace-free part
of the Ricci tensor, while
is a multiple of the scalar curvature.
Under the conformal rescaling
, the different parts of the
curvature transform as follows:
Thus, the Weyl tensor is invariant under conformal rescalings. When
is expressed entirely in terms of the transformed
quantities we get the relation
from which we can deduce (note that the contractions are performed
with the transformed metric)
Next consider the Bianchi identity
.
Inserting the decomposition (108) and taking
appropriate traces allows us to write it as two equations,
