Weyl Tensor - Definition

Definition

The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a (0,4) valence tensor (by contracting with the metric). The (0,4) valence Weyl tensor is then (Petersen 2006, p. 92)

(1)

where n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor, s is the scalar curvature, and denotes the Kulkarni–Nomizu product of two symmetric (0,2) tensors:

The ordinary (1,3) valent Weyl tensor is then given by contracting the above with the inverse of the metric.

The decomposition (1) expresses the Riemann tensor as an orthogonal direct sum, in the sense that

This decomposition, known as the Ricci decomposition, expresses the Riemann curvature tensor into its irreducible components under the action of the orthogonal group (Singer & Thorpe 1968). In dimension 4, the Weyl tensor further decomposes into invariant factors for the action of the special orthogonal group, the self-dual and antiself-dual parts C+ and C−.

The Weyl tensor can also be expressed using the Schouten tensor, which is a trace-adjusted multiple of the Ricci tensor,

Then

In indices,

where is the Riemann tensor, is the Ricci tensor, is the Ricci scalar (the scalar curvature) and brackets around indices refers to the antisymmetric part. Equivalently,

where S denotes the Schouten tensor.