Editing Weyl tensor (section)

{{Short description|Measure of the curvature of a pseudo-Riemannian manifold}}
In [[differential geometry]], the '''Weyl curvature tensor''', named after [[Hermann Weyl]],<ref>{{Cite journal|last=Weyl|first=Hermann|date=1918-09-01|title=Reine Infinitesimalgeometrie|url=https://doi.org/10.1007/BF01199420|journal=Mathematische Zeitschrift|language=de|volume=2|issue=3|pages=384–411|doi=10.1007/BF01199420|bibcode=1918MatZ....2..384W |s2cid=186232500 |issn=1432-1823}}</ref> 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 (linear algebra)|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: [[Tensor contraction#Metric contraction|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&mdash;a solution of the [[Einstein field equation|vacuum Einstein equation]]&mdash;and it governs the propagation of [[gravitational waves]] through regions of space devoid of matter.<ref name="Danehkar2009">{{cite journal | last1=Danehkar | first1=A. | date=2009 | title=On the Significance of the Weyl Curvature in a Relativistic Cosmological Model | journal=Mod. Phys. Lett. A | volume=24 | issue=38 | pages=3113–3127 | doi=10.1142/S0217732309032046 | bibcode=2009MPLA...24.3113D| arxiv=0707.2987 | s2cid=15949217 }}</ref>  More generally, the Weyl curvature is the only component of curvature for [[Ricci-flat manifold]]s and always governs the [[method of characteristics|characteristics]] of the field equations of an [[Einstein manifold]].<ref name="Danehkar2009"/>

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 locally [[conformally flat]]: there exists a [[local coordinate system]] in which the metric tensor is proportional to a constant tensor.  This fact was a key component of [[Nordström's theory of gravitation]], which was a precursor of [[general relativity]].