Spacetime Dimensions and the Weyl Tensor
The algebraic identities
(8.1)
and
(8.2)
lead to the following special cases for the curvature tensor:
We get this from the symmetry relation in the form
or in the following alternate form straight from the definition
(8.3) 


So in 2D the Riemann tensor is proportional to the Ricci scalar.
(8.4)
Note that a 3D space where necessarily makes the Riemann tensor zero in 3D. As we will see later a zero Ricci tensor in 4D general relativity does not imply and this in turn implies the existence of a nonzero gravitational field. Hence, from the above relation we have obtained the result that in 3D, a zero Ricci tensor condition does imply that and that therefore the 2D gravitational field must be zero.
(8.5)
This last equation can be generalized to ndimensions when . This generalization gives the following result
(8.6) 


The Weyl tensor (or conformal tensor) is defined to be the tensor . In ndimensions, with , the Weyl tensor can be written as follows.
(8.7) 


In four dimensions, we have
(8.8) 


It is straightforward to show that the Weyl tensor possesses the same symmetries as the Riemann tensor, namely,
(8.9)
and
(8.10)
However, it possesses an important extra symmetry
(8.11) 


Combining this result with the previous symmetries, it then follows that the Weyl tensor is tracefree, in other words, it vanishes for any pair of contracted indices. One can think of the Weyl tensor as that part of the curvature tensor for which all contractions vanish.
Two metrics and said to be conformal to each other or (conformally related ) if


where is a nonzero differentiable function. Given a manifold with two metrics defined on it, which are conformal, then it is straightforward from (8.12) to show that angles between vectors are the same for each metric. This is shown as follows. Let and be two rankone tensors. The angle between the vectors is defined through the relation
(8.13)
Using (8.12) gives the following
(8.14)
Ratios of magnitudes of vectors also remain invariant under conformal transformations. Moreover, the null geodesics of one metric coincide with the null geodesics of the other (exercise). The metrics also possess the same Weyl tensor, i.e.


Any quantity that satisfies a relationship like (8.15) is called conformally invariant (gab, , and are examples of quantities which are not conformally invariant). A metric is said to be conformally flat if it can be reduced to the form
(8.16) 


where is a flat metric (the special relativity 'Minkowski' metric).
We end this section by quoting two results concerning conformally flat metrics.
Theorem: A necessary and sufficient condition for a metric to be conformally flat is that its Weyl tensor vanishes everywhere.
Theorem: Any twodimensional Riemannian manifold is conformally flat.
Notes on the Weyl Tensor
The Weyl tensor in General Relativity provides curvature to the spacetime when the Ricci tensor is zero. In General Relativity the source of the Ricci tensor is the energymomentum of the local matter distribution. If the matter distribution is zero then the Ricci tensor will be zero. However the spacetime is not necessarily flat in this case since the Weyl tensor contributes curvature to the Riemann curvature tensor and so the gravitational field is not zero in spacetime void situations. This term allows gravity to propagate in regions where there is no matter/energy source.
07/02/2005 4:54 PM