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 2-D the Riemann tensor is proportional to the Ricci scalar.

(8.4)

Note that a 3-D space where  necessarily makes the Riemann tensor zero in 3-D. As we will see later a zero Ricci tensor in 4-D 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 3-D, a zero Ricci tensor condition does imply that  and that therefore the 2-D gravitational field must be zero.

(8.5)

This last equation can be generalized to n-dimensions when . This generalization gives the following result

 (8.6)

The Weyl tensor (or conformal tensor) is defined to be the tensor . In n-dimensions, 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 trace-free, 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 non-zero 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 rank-one 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 two-dimensional 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 energy-momentum 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