Conformal Infinity6 Appendix: Reduction Of The

7 Appendix: Conformal Rescalings And Curvature 

We compile here some formulae which are helpful for performing conformal rescalings. Suppose we rescale the given metric tex2html_wrap_inline3929 to a new metric tex2html_wrap_inline5528 . Define tex2html_wrap_inline5530 . The Levi-Civita connection tex2html_wrap_inline4099 of the new metric is given in terms of the Levi-Civita connection tex2html_wrap_inline5534 of tex2html_wrap_inline3929 by its action on an arbitrary vectorfield tex2html_wrap_inline5384,

  equation1597

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

  equation1603

The tensor tex2html_wrap_inline4069 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 tex2html_wrap_inline5542, can be uniquely expressed in terms of the Ricci tensor

  equation1620

The tensor tex2html_wrap_inline4340 is proportional to the trace-free part of the Ricci tensor, while tex2html_wrap_inline5546 is a multiple of the scalar curvature.

Under the conformal rescaling tex2html_wrap_inline5548, the different parts of the curvature transform as follows:

  equation1637

  equation1647

Thus, the Weyl tensor is invariant under conformal rescalings. When tex2html_wrap_inline5550 is expressed entirely in terms of the transformed quantities we get the relation

  equation1657

from which we can deduce (note that the contractions are performed with the transformed metric)

  equation1665

  equation1674

Next consider the Bianchi identity tex2html_wrap_inline5552 . Inserting the decomposition (105Popup Equation) and taking appropriate traces allows us to write it as two equations,

  equation1683

  equation1692



Conformal Infinity6 Appendix: Reduction Of The

image Conformal Infinity
Jörg Frauendiener
http://www.livingreviews.org/lrr-2000-4
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