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. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor.

In general relativity, the Weyl curvature is the only part of the curvature that exists in free space—a solution of the vacuum Einstein equation—and it governs the propagation of gravitational waves through regions of space devoid of matter. More generally, the Weyl curvature is the only component of curvature for Ricci-flat manifolds and always governs the characteristics of the field equations of an Einstein manifold.

:<math>C_{abcd} = R_{abcd} - \frac{2}{n - 2}\left(g_{a[c}R_{d]b} - g_{b[c}R_{d]a}\right) + \frac{2}{(n - 1)(n - 2)}R~g_{a[c}g_{d]b}</math>

where <math>R_{abcd}</math> is the Riemann tensor, <math>R_{ab}</math> is the Ricci tensor, <math>R</math> is the Ricci scalar (the scalar curvature) and brackets around indices refers to the antisymmetric part. Equivalently,

:<math>{C_{ab^{cd} = {R_{ab^{cd} - 4S_{[a}^{[c}\delta_{b]}^{d]}</math>

where S denotes the Schouten tensor.

Properties

Conformal rescaling

The Weyl tensor has the special property that it is invariant under conformal changes to the metric. That is, if <math>g_{\mu\nu}\mapsto g'_{\mu\nu} = f g_{\mu\nu}</math> for some positive scalar function <math>f</math> then the (1,3) valent Weyl tensor satisfies <math>{C'}^{a}_{\ \ bcd} = C^{a}_{\ \ bcd}</math>. For this reason the Weyl tensor is also called the conformal tensor. It follows that a necessary condition for a Riemannian manifold to be conformally flat is that the Weyl tensor vanish. In dimensions ≥ 4 this condition is sufficient as well. In dimension 3 the vanishing of the Cotton tensor is a necessary and sufficient condition for the Riemannian manifold being conformally flat. Any 2-dimensional (smooth) Riemannian manifold is conformally flat, a consequence of the existence of isothermal coordinates.

Indeed, the existence of a conformally flat scale amounts to solving the overdetermined partial differential equation

:<math>Ddf - df\otimes df + \left(|df|^2 + \frac{\Delta f}{n - 2}\right)g = \operatorname{Ric}.</math>

In dimension ≥ 4, the vanishing of the Weyl tensor is the only integrability condition for this equation; in dimension 3, it is the Cotton tensor instead.

Symmetries

The Weyl tensor has the same symmetries as the Riemann tensor. This includes:

:<math>\begin{align}

C(u, v) &= -C(v, u) \\

\langle C(u, v)w, z \rangle &= -\langle C(u, v)z, w \rangle \\

C(u, v)w + C(v, w)u + C(w, u)v &= 0.

\end{align}</math>

In addition, of course, the Weyl tensor is trace free:

:<math>\operatorname{tr} C(u, \cdot)v = 0</math>

for all u, v. In indices these four conditions are

:<math>\begin{align}

C_{abcd} = -C_{bacd} &= -C_{abdc} \\

C_{abcd} + C_{acdb} + C_{adbc} &= 0 \\

{C^a}_{bac} &= 0.

\end{align}</math>

Bianchi identity

Taking traces of the usual second Bianchi identity of the Riemann tensor eventually shows that

:<math>\nabla_a {C^a}_{bcd} = 2(n - 3)\nabla_{[c}S_{d]b}</math>

where S is the Schouten tensor. The valence (0,3) tensor on the right-hand side is the Cotton tensor, apart from the initial factor.

See also

  • Curvature of Riemannian manifolds
  • Christoffel symbols provides a coordinate expression for the Weyl tensor.
  • Lanczos tensor
  • Peeling theorem
  • Petrov classification
  • Plebanski tensor
  • Weyl curvature hypothesis
  • Weyl scalar

Notes

References

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