In
differential geometry, the
Weyl curvature tensor, named after
Hermann Weyl, is a measure of the
curvature of
spacetime or, more generally, a
pseudo-Riemannian manifold. Like the
Riemann curvature tensor, the Weyl tensor expresses the
tidal force that a body feels when moving along a
geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The
Ricci curvature, or
trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the
traceless component of the Riemann tensor. It is a
tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace-free: metric contraction on any pair of indices yields zero.