Weyl tensor - Wikipedia

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href="https://de.wikipedia.org/wiki/Weyl-Tensor" title="Weyl-Tensor – German" lang="de" hreflang="de" data-title="Weyl-Tensor" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tenseur_de_Weyl" title="Tenseur de Weyl – French" lang="fr" hreflang="fr" data-title="Tenseur de Weyl" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%94%EC%9D%BC_%EA%B3%A1%EB%A5%A0_%ED%85%90%EC%84%9C" title="바일 곡률 텐서 – Korean" lang="ko" hreflang="ko" data-title="바일 곡률 텐서" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tensore_di_Weyl" title="Tensore di Weyl – Italian" lang="it" hreflang="it" data-title="Tensore di Weyl" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tensor_de_Weyl" title="Tensor de Weyl – Portuguese" lang="pt" hreflang="pt" data-title="Tensor de Weyl" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80_%D0%92%D0%B5%D0%B9%D0%BB%D1%8F" title="Тензор Вейля – Russian" lang="ru" hreflang="ru" data-title="Тензор Вейля" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Measure of the curvature of a pseudo-Riemannian manifold</div> <p>In <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, the <b>Weyl curvature tensor</b>, named after <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a measure of the <a href="/wiki/Curvature" title="Curvature">curvature</a> of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> or, more generally, a <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a>. Like the <a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a>, the Weyl tensor expresses the <a href="/wiki/Tidal_force" title="Tidal force">tidal force</a> that a body feels when moving along a <a href="/wiki/Geodesic" title="Geodesic">geodesic</a>. 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 <a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a>, or <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> 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 <a href="/wiki/Traceless" class="mw-redirect" title="Traceless">traceless</a> component of the Riemann tensor. This <a href="/wiki/Tensor" title="Tensor">tensor</a> has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: <a href="/wiki/Tensor_contraction#Metric_contraction" title="Tensor contraction">metric contraction</a> 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. </p><p>In <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, the Weyl curvature is the only part of the curvature that exists in free space—a solution of the <a href="/wiki/Einstein_field_equation" class="mw-redirect" title="Einstein field equation">vacuum Einstein equation</a>—and it governs the propagation of <a href="/wiki/Gravitational_waves" class="mw-redirect" title="Gravitational waves">gravitational waves</a> through regions of space devoid of matter.<sup id="cite_ref-Danehkar2009_2-0" class="reference"><a href="#cite_note-Danehkar2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> More generally, the Weyl curvature is the only component of curvature for <a href="/wiki/Ricci-flat_manifold" title="Ricci-flat manifold">Ricci-flat manifolds</a> and always governs the <a href="/wiki/Method_of_characteristics" title="Method of characteristics">characteristics</a> of the field equations of an <a href="/wiki/Einstein_manifold" title="Einstein manifold">Einstein manifold</a>.<sup id="cite_ref-Danehkar2009_2-1" class="reference"><a href="#cite_note-Danehkar2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>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 <a href="/wiki/Conformally_flat" class="mw-redirect" title="Conformally flat">conformally flat</a>: there exists a <a href="/wiki/Local_coordinate_system" class="mw-redirect" title="Local coordinate system">local coordinate system</a> in which the metric tensor is proportional to a constant tensor. This fact was a key component of <a href="/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation" title="Nordström's theory of gravitation">Nordström's theory of gravitation</a>, which was a precursor of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weyl_tensor&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 (<a href="#CITEREFPetersen2006">Petersen 2006</a>, p. 92) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>R</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>n</mi> </mfrac> </mrow> <mi>g</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mo>∧</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="thickmathspace" /> <mo>◯</mo> <mtext> </mtext> </mrow> <mi>g</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mo>∧</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="thickmathspace" /> <mo>◯</mo> <mtext> </mtext> </mrow> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d878459824eea7f331c8d47577ee00ce162f9e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.85ex; height:6.009ex;" alt="{\displaystyle 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}"></span></dd></dl> <p>where <i>n</i> is the dimension of the manifold, <i>g</i> is the metric, <i>R</i> is the Riemann tensor, <i>Ric</i> is the <a href="/wiki/Ricci_tensor" class="mw-redirect" title="Ricci tensor">Ricci tensor</a>, <i>s</i> is the <a href="/wiki/Scalar_curvature" title="Scalar curvature">scalar curvature</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mo>∧</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="thickmathspace" /> <mo>◯</mo> <mtext> </mtext> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3971e29210af939122f390c97753e1f7f8e1ea8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.166ex; height:2.676ex;" alt="{\displaystyle h{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}k}"></span> denotes the <a href="/wiki/Kulkarni%E2%80%93Nomizu_product" title="Kulkarni–Nomizu product">Kulkarni–Nomizu product</a> of two symmetric (0,2) tensors: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(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{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mo>∧</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="thickmathspace" /> <mo>◯</mo> <mtext> </mtext> </mrow> <mi>k</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mspace width="1em" /> </mtd> <mtd> <mi>h</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi>h</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>h</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b578a16e98ba36a9b7668e8c828c149be3f8d728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.62ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(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{aligned}}}"></span></dd></dl> <p>In tensor component notation, this can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}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{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>ℓ</mi> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>ℓ</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ℓ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ℓ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ℓ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mtext> </mtext> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd46717d72d85b6d34a3a2b3b2cdb9fd2a1dbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:62.196ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}"></span></dd></dl> <p>The ordinary (1,3) valent Weyl tensor is then given by contracting the above with the inverse of the metric. </p><p>The decomposition (<b><a href="#math_1">1</a></b>) expresses the Riemann tensor as an <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> <a href="/wiki/Direct_sum_of_vector_bundles" class="mw-redirect" title="Direct sum of vector bundles">direct sum</a>, in the sense that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |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}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>R</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>n</mi> </mfrac> </mrow> <mi>g</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mo>∧</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="thickmathspace" /> <mo>◯</mo> <mtext> </mtext> </mrow> <mi>g</mi> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mo>∧</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="thickmathspace" /> <mo>◯</mo> <mtext> </mtext> </mrow> <mi>g</mi> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |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}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad2034fb8a4f47953723891f68f22bdea3bc8ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:62.889ex; height:6.843ex;" alt="{\displaystyle |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}.}"></span></dd></dl> <p>This decomposition, known as the <a href="/wiki/Ricci_decomposition" title="Ricci decomposition">Ricci decomposition</a>, expresses the Riemann curvature tensor into its <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible</a> components under the action of the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>.<sup id="cite_ref-FOOTNOTESingerThorpe1969_3-0" class="reference"><a href="#cite_note-FOOTNOTESingerThorpe1969-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In dimension 4, the Weyl tensor further decomposes into invariant factors for the action of the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a>, the self-dual and antiself-dual parts <i>C</i><sup>+</sup> and <i>C</i><sup>−</sup>. </p><p>The Weyl tensor can also be expressed using the <a href="/wiki/Schouten_tensor" title="Schouten tensor">Schouten tensor</a>, which is a trace-adjusted multiple of the Ricci tensor, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\frac {1}{n-2}}\left(\mathrm {Ric} -{\frac {s}{2(n-1)}}g\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>g</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {1}{n-2}}\left(\mathrm {Ric} -{\frac {s}{2(n-1)}}g\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a81653c7d509b8157c31dc93e6ea3dfed2b7de3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.471ex; height:6.343ex;" alt="{\displaystyle P={\frac {1}{n-2}}\left(\mathrm {Ric} -{\frac {s}{2(n-1)}}g\right).}"></span></dd></dl> <p>Then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=R-P{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}g.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>R</mi> <mo>−</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mo>∧</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="thickmathspace" /> <mo>◯</mo> <mtext> </mtext> </mrow> <mi>g</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=R-P{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}g.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae0d6e51303421ceccae74c21076b76c9713f696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.593ex; height:2.676ex;" alt="{\displaystyle C=R-P{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}g.}"></span></dd></dl> <p>In indices,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo stretchy="false">[</mo> <mi>c</mi> </mrow> </msub> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo stretchy="false">]</mo> <mi>b</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo stretchy="false">[</mo> <mi>c</mi> </mrow> </msub> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo stretchy="false">]</mo> <mi>a</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>R</mi> <mtext> </mtext> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo stretchy="false">[</mo> <mi>c</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo stretchy="false">]</mo> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1fe6a1fc5ad6e2ad77b9fc603fdaae8c3bba1f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:69.691ex; height:6.009ex;" alt="{\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}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{abcd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{abcd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30d0d3245d30b6553b1e3fd857d6c4f473501807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.143ex; height:2.509ex;" alt="{\displaystyle R_{abcd}}"></span> is the Riemann tensor, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{ab}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{ab}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e2940a775e31cc7bb12718daeac703f3f74367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.571ex; height:2.509ex;" alt="{\displaystyle R_{ab}}"></span> is the Ricci tensor, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the Ricci scalar (the scalar curvature) and brackets around indices refers to the <a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">antisymmetric part</a>. Equivalently, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {C_{ab}}^{cd}={R_{ab}}^{cd}-4S_{[a}^{[c}\delta _{b]}^{d]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>c</mi> </mrow> </msubsup> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo stretchy="false">]</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {C_{ab}}^{cd}={R_{ab}}^{cd}-4S_{[a}^{[c}\delta _{b]}^{d]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/179a28f7ac753ff166f6bb694dd0fbde056ebf4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:23.337ex; height:4.176ex;" alt="{\displaystyle {C_{ab}}^{cd}={R_{ab}}^{cd}-4S_{[a}^{[c}\delta _{b]}^{d]}}"></span></dd></dl> <p>where <i>S</i> denotes the <a href="/wiki/Schouten_tensor" title="Schouten tensor">Schouten tensor</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weyl_tensor&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Conformal_rescaling">Conformal rescaling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weyl_tensor&action=edit&section=3" title="Edit section: Conformal rescaling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Weyl tensor has the special property that it is invariant under <a href="/wiki/Conformal_map" title="Conformal map">conformal</a> changes to the <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a>. That is, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\mu \nu }\mapsto g'_{\mu \nu }=fg_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi>f</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }\mapsto g'_{\mu \nu }=fg_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/784ce7689ed8096ab7f1e29435259b3b075c2ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.602ex; height:2.843ex;" alt="{\displaystyle g_{\mu \nu }\mapsto g'_{\mu \nu }=fg_{\mu \nu }}"></span> for some positive scalar function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> then the (1,3) valent Weyl tensor satisfies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {C'}_{\ \ bcd}^{a}=C_{\ \ bcd}^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mtext> </mtext> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mtext> </mtext> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {C'}_{\ \ bcd}^{a}=C_{\ \ bcd}^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/307dc13a68bcbb0125a2b7a5c277ab616a2933ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.584ex; height:3.176ex;" alt="{\displaystyle {C'}_{\ \ bcd}^{a}=C_{\ \ bcd}^{a}}"></span>. For this reason the Weyl tensor is also called the <b>conformal tensor</b>. It follows that a <a href="/wiki/Necessary_condition" class="mw-redirect" title="Necessary condition">necessary condition</a> for a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> to be <a href="/wiki/Conformally_flat" class="mw-redirect" title="Conformally flat">conformally flat</a> is that the Weyl tensor vanish. In dimensions ≥ 4 this condition is <a href="/wiki/Sufficient_condition" class="mw-redirect" title="Sufficient condition">sufficient</a> as well. In dimension 3 the vanishing of the <a href="/wiki/Cotton_tensor" title="Cotton tensor">Cotton tensor</a> 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 <a href="/wiki/Isothermal_coordinates" title="Isothermal coordinates">isothermal coordinates</a>. </p><p>Indeed, the existence of a conformally flat scale amounts to solving the overdetermined partial differential equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ddf-df\otimes df+\left(|df|^{2}+{\frac {\Delta f}{n-2}}\right)g=\operatorname {Ric} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>d</mi> <mi>f</mi> <mo>−</mo> <mi>d</mi> <mi>f</mi> <mo>⊗</mo> <mi>d</mi> <mi>f</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>f</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>g</mi> <mo>=</mo> <mi>Ric</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ddf-df\otimes df+\left(|df|^{2}+{\frac {\Delta f}{n-2}}\right)g=\operatorname {Ric} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/872eb01576329713d0b24cbd349bc931ffd2a0f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.905ex; height:6.176ex;" alt="{\displaystyle Ddf-df\otimes df+\left(|df|^{2}+{\frac {\Delta f}{n-2}}\right)g=\operatorname {Ric} .}"></span></dd></dl> <p>In dimension ≥ 4, the vanishing of the Weyl tensor is the only <a href="/wiki/Integrability_condition" class="mw-redirect" title="Integrability condition">integrability condition</a> for this equation; in dimension 3, it is the <a href="/wiki/Cotton_tensor" title="Cotton tensor">Cotton tensor</a> instead. </p> <div class="mw-heading mw-heading3"><h3 id="Symmetries">Symmetries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weyl_tensor&action=edit&section=4" title="Edit section: Symmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Weyl tensor has the same symmetries as the Riemann tensor. This includes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}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{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>C</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>w</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f608eda2cab70fd00723268c7168242cefe711a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:50.194ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}"></span></dd></dl> <p>In addition, of course, the Weyl tensor is trace free: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} C(u,\cdot )v=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} C(u,\cdot )v=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e75729e51f30561a05331a8252cc1e181c06dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.178ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} C(u,\cdot )v=0}"></span></dd></dl> <p>for all <i>u</i>, <i>v</i>. In indices these four conditions are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}C_{abcd}=-C_{bacd}&=-C_{abdc}\\C_{abcd}+C_{acdb}+C_{adbc}&=0\\{C^{a}}_{bac}&=0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>d</mi> <mi>c</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> <mi>d</mi> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>d</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> <mi>c</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}C_{abcd}=-C_{bacd}&=-C_{abdc}\\C_{abcd}+C_{acdb}+C_{adbc}&=0\\{C^{a}}_{bac}&=0.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c09a672226b54ca8e579b9f2ccb1742cd12013b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:31.502ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}C_{abcd}=-C_{bacd}&=-C_{abdc}\\C_{abcd}+C_{acdb}+C_{adbc}&=0\\{C^{a}}_{bac}&=0.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Bianchi_identity">Bianchi identity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weyl_tensor&action=edit&section=5" title="Edit section: Bianchi identity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Taking traces of the usual second Bianchi identity of the Riemann tensor eventually shows that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{a}{C^{a}}_{bcd}=2(n-3)\nabla _{[c}S_{d]b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>c</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo stretchy="false">]</mo> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{a}{C^{a}}_{bcd}=2(n-3)\nabla _{[c}S_{d]b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95bc5946c57649b107d431fc644496b3176823cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.932ex; height:3.176ex;" alt="{\displaystyle \nabla _{a}{C^{a}}_{bcd}=2(n-3)\nabla _{[c}S_{d]b}}"></span></dd></dl> <p>where <i>S</i> is the <a href="/wiki/Schouten_tensor" title="Schouten tensor">Schouten tensor</a>. The valence (0,3) tensor on the right-hand side is the <a href="/wiki/Cotton_tensor" title="Cotton tensor">Cotton tensor</a>, apart from the initial factor. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weyl_tensor&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Curvature_of_Riemannian_manifolds" title="Curvature of Riemannian manifolds">Curvature of Riemannian manifolds</a></li> <li><a href="/wiki/Christoffel_symbols" title="Christoffel symbols">Christoffel symbols</a> provides a coordinate expression for the Weyl tensor.</li> <li><a href="/wiki/Lanczos_tensor" title="Lanczos tensor">Lanczos tensor</a></li> <li><a href="/wiki/Peeling_theorem" title="Peeling theorem">Peeling theorem</a></li> <li><a href="/wiki/Petrov_classification" title="Petrov classification">Petrov classification</a></li> <li><a href="/wiki/Plebanski_tensor" title="Plebanski tensor">Plebanski tensor</a></li> <li><a href="/wiki/Weyl_curvature_hypothesis" title="Weyl curvature hypothesis">Weyl curvature hypothesis</a></li> <li><a href="/wiki/Weyl_scalar" title="Weyl scalar">Weyl scalar</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weyl_tensor&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeyl1918" 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(identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186232500">186232500</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=Reine+Infinitesimalgeometrie&rft.volume=2&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E384-%3C%2Fspan%3E411&rft.date=1918-09-01&rft_id=info%3Adoi%2F10.1007%2FBF01199420&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186232500%23id-name%3DS2CID&rft.issn=1432-1823&rft_id=info%3Abibcode%2F1918MatZ....2..384W&rft.aulast=Weyl&rft.aufirst=Hermann&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2FBF01199420&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeyl+tensor" class="Z3988"></span></span> </li> <li id="cite_note-Danehkar2009-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Danehkar2009_2-0"><sup><i><b>a</b></i></sup></a> <a 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(2009). "On the Significance of the Weyl Curvature in a Relativistic Cosmological Model". <i>Mod. Phys. Lett. A</i>. <b>24</b> (38): <span class="nowrap">3113–</span>3127. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0707.2987">0707.2987</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009MPLA...24.3113D">2009MPLA...24.3113D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217732309032046">10.1142/S0217732309032046</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15949217">15949217</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mod.+Phys.+Lett.+A&rft.atitle=On+the+Significance+of+the+Weyl+Curvature+in+a+Relativistic+Cosmological+Model&rft.volume=24&rft.issue=38&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3113-%3C%2Fspan%3E3127&rft.date=2009&rft_id=info%3Aarxiv%2F0707.2987&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15949217%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS0217732309032046&rft_id=info%3Abibcode%2F2009MPLA...24.3113D&rft.aulast=Danehkar&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeyl+tensor" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTESingerThorpe1969-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESingerThorpe1969_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSingerThorpe1969">Singer & Thorpe 1969</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrønHervik2007">Grøn & Hervik 2007</a>, p. 490</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weyl_tensor&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHawkingEllis1973" class="citation cs2"><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking, Stephen W.</a>; <a href="/wiki/George_Francis_Rayner_Ellis" class="mw-redirect" title="George Francis Rayner Ellis">Ellis, George F. R.</a> (1973), <i><a href="/wiki/The_Large_Scale_Structure_of_Space-Time" class="mw-redirect" title="The Large Scale Structure of Space-Time">The Large Scale Structure of Space-Time</a></i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-09906-4" title="Special:BookSources/0-521-09906-4"><bdi>0-521-09906-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Large+Scale+Structure+of+Space-Time&rft.pub=Cambridge+University+Press&rft.date=1973&rft.isbn=0-521-09906-4&rft.aulast=Hawking&rft.aufirst=Stephen+W.&rft.au=Ellis%2C+George+F.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeyl+tensor" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetersen2006" class="citation cs2">Petersen, Peter (2006), <i>Riemannian geometry</i>, Graduate Texts in Mathematics, vol. 171 (2nd ed.), Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0387292462" title="Special:BookSources/0387292462"><bdi>0387292462</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2243772">2243772</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemannian+geometry&rft.place=Berlin%2C+New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=2006&rft.isbn=0387292462&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2243772%23id-name%3DMR&rft.aulast=Petersen&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeyl+tensor" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSharpe1997" class="citation cs2">Sharpe, R.W. (1997), <i>Differential Geometry: Cartan's Generalization of Klein's Erlangen Program</i>, Springer-Verlag, New York, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94732-9" title="Special:BookSources/0-387-94732-9"><bdi>0-387-94732-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Geometry%3A+Cartan%27s+Generalization+of+Klein%27s+Erlangen+Program&rft.pub=Springer-Verlag%2C+New+York&rft.date=1997&rft.isbn=0-387-94732-9&rft.aulast=Sharpe&rft.aufirst=R.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeyl+tensor" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSingerThorpe1969" class="citation cs2"><a href="/wiki/Isadore_Singer" title="Isadore Singer">Singer, I.M.</a>; Thorpe, J.A. (1969), "The curvature of 4-dimensional Einstein spaces", <i>Global Analysis (Papers in Honor of K. Kodaira)</i>, Univ. Tokyo Press, pp. <span class="nowrap">355–</span>365</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+curvature+of+4-dimensional+Einstein+spaces&rft.btitle=Global+Analysis+%28Papers+in+Honor+of+K.+Kodaira%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E355-%3C%2Fspan%3E365&rft.pub=Univ.+Tokyo+Press&rft.date=1969&rft.aulast=Singer&rft.aufirst=I.M.&rft.au=Thorpe%2C+J.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeyl+tensor" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Weyl_tensor">"Weyl tensor"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Weyl+tensor&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DWeyl_tensor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeyl+tensor" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrønHervik2007" class="citation cs2"><a href="/wiki/%C3%98yvind_Gr%C3%B8n" title="Øyvind Grøn">Grøn, Øyvind</a>; Hervik, Sigbjørn (2007), <i>Einstein's General Theory of Relativity</i>, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-69199-2" title="Special:BookSources/978-0-387-69199-2"><bdi>978-0-387-69199-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein%27s+General+Theory+of+Relativity&rft.place=New+York&rft.pub=Springer&rft.date=2007&rft.isbn=978-0-387-69199-2&rft.aulast=Gr%C3%B8n&rft.aufirst=%C3%98yvind&rft.au=Hervik%2C+Sigbj%C3%B8rn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeyl+tensor" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a class="mw-selflink selflink">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li> <li><a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li> <li><a href="/wiki/Woldemar_Voigt" 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Weyl tensor - Wikipedia