From the action on vectorfields 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 (105
) and taking appropriate traces allows us to write it as two
equations,
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Conformal Infinity
Jörg Frauendiener http://www.livingreviews.org/lrr-2000-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |