Editing Weyl tensor

{{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]].

==Definition==
{{See also|Ricci decomposition}}
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 {{Harv|Petersen|2006|p=92}}

:<math>C = R - \frac{1}{n-2}\left(\mathrm{Ric} - \frac{s}{n}g\right) {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g - \frac{s}{2n(n - 1)}g {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g</math>

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 <math>h {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} k</math> denotes the [[Kulkarni–Nomizu product]] of two symmetric (0,2) tensors:

:<math>\begin{align} 
  (h {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} k)\left(v_1, v_2, v_3, v_4\right) =\quad
          &h\left(v_1, v_3\right)k\left(v_2, v_4\right) + h\left(v_2, v_4\right)k\left(v_1, v_3\right) \\
    {}-{} &h\left(v_1, v_4\right)k\left(v_2, v_3\right) - h\left(v_2, v_3\right)k\left(v_1, v_4\right)
\end{align}</math>

In tensor component notation, this can be written as 
:<math>\begin{align} 
  C_{ik\ell m} = R_{ik\ell m}
      +{} &\frac{1}{n - 2} \left(R_{im}g_{k\ell} - R_{i\ell}g_{km} + R_{k\ell}g_{im} - R_{km}g_{i\ell} \right) \\
    {}+{} &\frac{1}{(n - 1)(n - 2)} R \left(g_{i\ell}g_{km} - g_{im}g_{k\ell} \right).\ 
\end{align}</math>

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

The decomposition ({{EquationNote|1}}) expresses the Riemann tensor as an [[orthogonal]] [[direct sum of vector bundles|direct sum]], in the sense that
:<math>|R|^2 = |C|^2 + \left|\frac{1}{n - 2}\left(\mathrm{Ric} - \frac{s}{n}g\right) {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g\right|^2 + \left|\frac{s}{2n(n - 1)}g {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g\right|^2.</math>

This decomposition, known as the [[Ricci decomposition]], expresses the Riemann curvature tensor into its [[irreducible representation|irreducible]] components under the action of the [[orthogonal group]].{{sfn|Singer|Thorpe|1969}} 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''<sup>+</sup> and ''C''<sup>−</sup>.

The Weyl tensor can also be expressed using the [[Schouten tensor]], which is a trace-adjusted multiple of the Ricci tensor,
:<math>P = \frac{1}{n - 2}\left(\mathrm{Ric} - \frac{s}{2(n-1)}g\right).</math>

Then
:<math>C = R - P {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g.</math>

In indices,<ref>{{Harvnb|Grøn|Hervik|2007|loc=p. 490}}</ref>
:<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 tensor|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 map|conformal]] changes to the [[metric tensor|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 condition|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 ==
{{reflist}}

==References==
*{{Citation
 | last1=Hawking
 | first1=Stephen W.
 | author1-link=Stephen Hawking
 | last2=Ellis
 | first2=George F. R.
 | author2-link=George Francis Rayner Ellis
 | title=[[The Large Scale Structure of Space-Time]]
 | publisher=Cambridge University Press
 | isbn=0-521-09906-4
 | year=1973
 }}
*{{Citation | last1=Petersen | first1=Peter | title=Riemannian geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=0387292462 |mr=2243772 | year=2006 | volume=171}}.
* {{Citation | first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997 | isbn = 0-387-94732-9}}.
* {{Citation | first1=I.M.|last1=Singer|author-link1=Isadore Singer|first2=J.A.|last2=Thorpe|title=Global Analysis (Papers in Honor of K. Kodaira)|contribution=The curvature of 4-dimensional Einstein spaces|publisher=Univ. Tokyo Press|year=1969|pages=355&ndash;365}}
*{{Springer|id=Weyl_tensor|title=Weyl tensor}}
*{{Citation|last1=Grøn|first1=Øyvind|author-link=Øyvind Grøn|last2=Hervik|first2=Sigbjørn|title=Einstein's General Theory of Relativity|location=New York|publisher=Springer|year=2007|isbn=978-0-387-69199-2}}

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{{DEFAULTSORT:Weyl Tensor}}
[[Category:Curvature tensors]]
[[Category:Riemannian geometry]]
[[Category:Tensors in general relativity]]