Indifferential geometry, theWeyl curvature tensor, named afterHermann Weyl,^[1] is a measure of thecurvature ofspacetime or, more generally, apseudo-Riemannian manifold. Like theRiemann curvature tensor, the Weyl tensor expresses thetidal force that a body feels when moving along ageodesic. 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. TheRicci curvature, ortrace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is thetraceless component of the Riemann tensor. Thistensor 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.

Ingeneral relativity, the Weyl curvature is the only part of the curvature that exists in free space—a solution of thevacuum Einstein equation—and it governs the propagation ofgravitational waves through regions of space devoid of matter.^[2] More generally, the Weyl curvature is the only component of curvature forRicci-flat manifolds and always governs thecharacteristics of the field equations of anEinstein manifold.^[2]

In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero. If the Weyl tensor vanishes in dimension ≥ 4, then the metric is locallyconformally flat: there exists alocal coordinate system in which the metric tensor is proportional to a constant tensor. This fact was a key component ofNordström's theory of gravitation, which was a precursor ofgeneral relativity.

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)

${\displaystyle C=R-\frac{1}{n-2}\left(\mathrm{R}\mathrm{i}\mathrm{c}-\frac{s}{n}g\right)\wedge ◯g-\frac{s}{2n(n-1)}g\wedge ◯g}$

wheren is the dimension of the manifold,g is the metric,R is the Riemann tensor,Ric is theRicci tensor,s is thescalar curvature, and${\displaystyle h\wedge ◯k}$ denotes theKulkarni–Nomizu product of two symmetric (0,2) tensors:

In tensor component notation, this can be written as

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 anorthogonaldirect sum, in the sense that

${\displaystyle |R{|}^2=|C{|}^2+{\left|\frac{1}{n-2}\left(\mathrm{R}\mathrm{i}\mathrm{c}-\frac{s}{n}g\right)\wedge ◯g\right|}^2+{\left|\frac{s}{2n(n-1)}g\wedge ◯g\right|}^2.}$

This decomposition, known as theRicci decomposition, expresses the Riemann curvature tensor into itsirreducible components under the action of theorthogonal group.^[3] In dimension 4, the Weyl tensor further decomposes into invariant factors for the action of thespecial orthogonal group, the self-dual and antiself-dual partsC⁺ andC⁻.

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

${\displaystyle P=\frac{1}{n-2}\left(\mathrm{R}\mathrm{i}\mathrm{c}-\frac{s}{2(n-1)}g\right).}$

Then

${\displaystyle C=R-P\wedge ◯g.}$

In indices,^[4]

${\displaystyle C_{abcd}=R_{abcd}-\frac{2}{n-2}\left(g_{a[c}{R}_{d]b}-g_{b[c}{R}_{d]a}\right)+\frac{2}{(n-1)(n-2)}R{g}_{a[c}{g}_{d]b}}$

where${\displaystyle R_{abcd}}$ is the Riemann tensor,${\displaystyle R_{ab}}$ is the Ricci tensor,${\displaystyle R}$ is the Ricci scalar (the scalar curvature) and brackets around indices refers to theantisymmetric part. Equivalently,

${\displaystyle {C_{ab}}^{cd}={R_{ab}}^{cd}-4S_{[a}^{[c}{\delta }_{b]}^{d]}}$

whereS denotes theSchouten tensor.

The Weyl tensor has the special property that it is invariant underconformal changes to themetric. That is, if${\displaystyle g_{\mu \nu }\mapsto g_{\mu \nu }^{\prime }=f{g}_{\mu \nu }}$ for some positive scalar function${\displaystyle f}$ then the (1,3) valent Weyl tensor satisfies${\displaystyle {C^{\prime }}_{bcd}^a=C_{bcd}^a}$. For this reason the Weyl tensor is also called theconformal tensor. It follows that anecessary condition for aRiemannian manifold to beconformally flat is that the Weyl tensor vanish. In dimensions ≥ 4 this condition issufficient as well. In dimension 3 the vanishing of theCotton 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 ofisothermal coordinates.

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

${\displaystyle Ddf-df\otimes df+\left(|df{|}^2+\frac{\mathrm{\Delta }f}{n-2}\right)g=\mathrm{Ric}.}$

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

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

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

${\displaystyle \mathrm{tr}C(u,\cdot )v=0}$

for allu,v. In indices these four conditions are

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

${\displaystyle {\mathrm{\nabla }}_a{C^a}_{bcd}=2(n-3){\mathrm{\nabla }}_{[c}{S}_{d]b}}$

whereS is theSchouten tensor. The valence (0,3) tensor on the right-hand side is theCotton tensor, apart from the initial factor.

• 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

• Hawking, Stephen W.;Ellis, George F. R. (1973),The Large Scale Structure of Space-Time, Cambridge University Press,ISBN 0-521-09906-4
• Petersen, Peter (2006),Riemannian geometry, Graduate Texts in Mathematics, vol. 171 (2nd ed.), Berlin, New York:Springer-Verlag,ISBN 0387292462,MR 2243772.
• Sharpe, R.W. (1997),Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York,ISBN 0-387-94732-9.
• Singer, I.M.; Thorpe, J.A. (1969), "The curvature of 4-dimensional Einstein spaces",Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, pp. 355–365
• "Weyl tensor",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Grøn, Øyvind; Hervik, Sigbjørn (2007),Einstein's General Theory of Relativity, New York: Springer,ISBN 978-0-387-69199-2

• Curvature tensors
• Riemannian geometry
• Tensors in general relativity

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