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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Tensor density</span></span>

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<p>In <a href="Differential_geometry" title="Differential geometry">differential geometry</a>, a <b>tensor density</b> or <b>relative tensor</b> is a generalization of the <a href="Tensor_field" title="Tensor field">tensor field</a> concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see <a href="Tensor_field" title="Tensor field">tensor field</a>), except that it is additionally multiplied or <i>weighted</i> by a power <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> of the <a href="Jacobian_determinant" class="mw-redirect" title="Jacobian determinant">Jacobian determinant</a> of the coordinate transition function or its absolute value. A tensor density with a single index is called a <b>vector density</b>. A distinction is made among (authentic) tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. Sometimes tensor densities with a negative weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> are called <b>tensor capacity.</b><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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> A tensor density can also be regarded as a <a href="Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> of the <a href="Tensor_product" title="Tensor product">tensor product</a> of a <a href="Tensor_bundle" title="Tensor bundle">tensor bundle</a> with a <a href="Density_bundle" class="mw-redirect" title="Density bundle">density bundle</a>.

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<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

<p>In physics and related fields, it is often useful to work with the components of an algebraic object rather than the object itself. An example would be decomposing a vector into a sum of <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> vectors weighted by some coefficients such as

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=c_{1}{\vec {e}}_{1}+c_{2}{\vec {e}}_{2}+c_{3}{\vec {e}}_{3}}</annotation>

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where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">

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</math></span><img src="./85820588abd7333ef4d0c56539cb31c20e730753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" loading="lazy"></span> is a vector in 3-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, <span class="mwe-math-element mwe-math-element-inline"><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_{i}\in \mathbb {R} ^{1}{\text{ and }}{\vec {e}}_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{i}\in \mathbb {R} ^{1}{\text{ and }}{\vec {e}}_{i}}</annotation>

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</math></span><img src="./bb7fe2fae76df15a35aab12a30dd8080e68ee62a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.311ex; height:3.009ex;" alt="{\displaystyle c_{i}\in \mathbb {R} ^{1}{\text{ and }}{\vec {e}}_{i}}" loading="lazy"></span> are the usual standard basis vectors in Euclidean space. This is usually necessary for computational purposes, and can often be insightful when algebraic objects represent complex abstractions but their components have concrete interpretations. However, with this identification, one has to be careful to track changes of the underlying basis in which the quantity is expanded; it may in the course of a computation become expedient to <a href="Change_of_basis" title="Change of basis">change the basis</a> while the vector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">

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</math></span><img src="./85820588abd7333ef4d0c56539cb31c20e730753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" loading="lazy"></span> remains fixed in physical space. More generally, if an algebraic object represents a geometric object, but is expressed in terms of a particular basis, then it is necessary to, when the basis is changed, also change the representation. Physicists will often call this representation of a geometric object a <a href="Tensor" title="Tensor">tensor</a> if it transforms under a sequence of <a href="Linear_map" title="Linear map">linear maps</a> given a linear change of basis (although confusingly others call the underlying geometric object which hasn't changed under the coordinate transformation a "tensor", a convention this article strictly avoids). In general there are representations which transform in arbitrary ways depending on how the geometric invariant is reconstructed from the representation. In certain special cases it is convenient to use representations which transform almost like tensors, but with an additional, nonlinear factor in the transformation. A prototypical example is a matrix representing the cross product (area of spanned parallelogram) on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}.}</annotation>

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</math></span><img src="./066b155c535a38739cc0c4b288324cbb7a4a227a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}.}" loading="lazy"></span> The representation is given by in the standard basis by

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<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}\times {\vec {v}}={\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}}{\begin{bmatrix}0&1\\-1&0\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}=u_{1}v_{2}-u_{2}v_{1}}</annotation>

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</p><p>If we now try to express this same expression in a basis other than the standard basis, then the components of the vectors will change, say according to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\begin{bmatrix}u'_{1}&u'_{2}\end{bmatrix}}^{\textsf {T}}=A{\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}}^{\textsf {T}}}">

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<annotation encoding="application/x-tex">{\textstyle {\begin{bmatrix}u'_{1}&u'_{2}\end{bmatrix}}^{\textsf {T}}=A{\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}}^{\textsf {T}}}</annotation>

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</math></span><img src="./d1c2c690bc924994a0f2e5305d6c80c96963adff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.706ex; margin-bottom: -0.299ex; width:25.816ex; height:3.676ex;" alt="{\textstyle {\begin{bmatrix}u'_{1}&u'_{2}\end{bmatrix}}^{\textsf {T}}=A{\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}}^{\textsf {T}}}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> is some 2 by 2 matrix of real numbers. Given that the area of the spanned parallelogram is a geometric invariant, it cannot have changed under the change of basis, and so the new representation of this matrix must be:

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<annotation encoding="application/x-tex">{\displaystyle \left(A^{-1}\right)^{\textsf {T}}{\begin{bmatrix}0&1\\-1&0\end{bmatrix}}A^{-1}}</annotation>

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which, when expanded is just the original expression but multiplied by the determinant of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{-1},}">

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</math></span><img src="./cc5a7c51eddc55dce006ed7b1368fef589afb93c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.723ex; height:3.009ex;" alt="{\displaystyle A^{-1},}" loading="lazy"></span> which is also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\det A}}.}">

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Some authors classify tensor densities into the two types called (authentic) tensor densities and pseudotensor densities in this article. Other authors classify them differently, into the types called even tensor densities and odd tensor densities. When a tensor density weight is an integer there is an equivalence between these approaches that depends upon whether the integer is even or odd.

</p><p>Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-<i>reversing</i> coordinate transformations. Regardless of their classifications into these types, there is only one way that tensor densities transform under orientation-<i>preserving</i> coordinate transformations.

</p><p>In this article we have chosen the convention that assigns a weight of +2 to <span class="mwe-math-element mwe-math-element-inline"><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=\det \left(g_{\rho \sigma }\right)}">

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<annotation encoding="application/x-tex">{\displaystyle g=\det \left(g_{\rho \sigma }\right)}</annotation>

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</p><p>In contrast to the meaning used in this article, in general relativity "<a href="Pseudotensor" title="Pseudotensor">pseudotensor</a>" sometimes means an object that does not transform like a tensor or relative tensor of any weight.

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<div class="mw-heading mw-heading3"><h3 id="Tensor_and_pseudotensor_densities">Tensor and pseudotensor densities</h3></div>

<p>For example, a mixed rank-two (authentic) tensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,,}</annotation>

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</math></span><img src="./b95cb37968c46082f22ab0dd2a7f6c8aa605bb48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.181ex; height:6.676ex;" alt="{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,,}" loading="lazy"></span>     <b>((authentic) tensor density of (integer) weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span>)</b></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\mathfrak {T}}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\mathfrak {T}}}}</annotation>

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</math></span><img src="./19daf12851a53a01c15fae38df62c204802ae483.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.571ex; height:2.509ex;" alt="{\displaystyle {\bar {\mathfrak {T}}}}" loading="lazy"></span> is the rank-two tensor density in the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}">

<mover>

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</math></span><img src="./466e03e1c9533b4dab1b9949dad393883f385d80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}" loading="lazy"></span> coordinate system, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {T}}}">

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</math></span><img src="./1ea1f4cdfbe22a1612f083b2cfd87af036f141af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.571ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {T}}}" loading="lazy"></span> is the transformed tensor density in the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x}}">

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<annotation encoding="application/x-tex">{\displaystyle {x}}</annotation>

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</math></span><img src="./5cb8cd0cfa94e69432c076ca30c3bd6facaabb93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}" loading="lazy"></span> coordinate system; and we use the <a href="Jacobian_determinant" class="mw-redirect" title="Jacobian determinant">Jacobian determinant</a>. Because the determinant can be negative, which it is for an orientation-reversing coordinate transformation, this formula is applicable only when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is an integer. (However, see even and odd tensor densities below.)

</p><p>We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation-reversing coordinate transformation. A mixed rank-two pseudotensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> transforms as

</p>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\operatorname {sgn} \left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)\left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,,}</annotation>

</semantics>

</math></span><img src="./f6e13de04ac0e883a1432e057c1fd3ce3df05f6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.654ex; height:6.676ex;" alt="{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\operatorname {sgn} \left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)\left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,,}" loading="lazy"></span>     <b>(pseudotensor density of (integer) weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span>)</b></dd></dl>

<p>where <span class="texhtml"><a href="Sign_function" title="Sign function">sgn</a>(<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation>

</semantics>

</math></span><img src="./ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"></span>)</span> is a function that returns +1 when its argument is positive or −1 when its argument is negative.

</p>

<div class="mw-heading mw-heading3"><h3 id="Even_and_odd_tensor_densities">Even and odd tensor densities</h3></div>

<p>The transformations for even and odd tensor densities have the benefit of being well defined even when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is not an integer. Thus one can speak of, say, an odd tensor density of weight +2 or an even tensor density of weight −1/2.

</p><p>An even tensor density transforms as follows. Although the formula works for any real valued weight <span class="texhtml mvar" style="font-style:italic;">W</span>, the name arises because the transformation is equivalent to the transformation of an (authentic) tensor density transform when its weight is even.

</p>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,.}</annotation>

</semantics>

</math></span><img src="./60a339c0d6b7f088479e216575bf39990552ddf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.053ex; height:6.676ex;" alt="{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,.}" loading="lazy"></span>     <b>(even tensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span>)</b></dd></dl>

<p>Similarly, an odd tensor density transforms as follows. Although the formula works for any real valued weight <span class="texhtml mvar" style="font-style:italic;">W</span>, the name arises because the transformation is equivalent to the transformation of an (authentic) tensor density transform when its weight is odd.

</p>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\operatorname {sgn} \left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,.}</annotation>

</semantics>

</math></span><img src="./ee47f5278c84205123eded6c3045125cb238866c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.526ex; height:6.676ex;" alt="{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\operatorname {sgn} \left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,.}" loading="lazy"></span>     <b>(odd tensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span>)</b></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Weights_of_zero_and_one">Weights of zero and one</h3></div>

<p>A tensor density of any type that has weight zero is also called an <b>absolute tensor</b>. An authentic tensor density of weight zero, which is also an even tensor density of weight zero, is also called an <b>ordinary tensor</b>.

</p><p>If a weight is not specified but the word "relative" or "density" is used in a context where a specific weight is needed, it is usually assumed that the weight is <span class="texhtml">+1</span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_properties">Algebraic properties</h3></div>

<ol><li>A <a href="Linear_combination" title="Linear combination">linear combination</a> (also known as a <a href="Weighted_sum" class="mw-redirect" title="Weighted sum">weighted sum</a>) of tensor densities of the same type and weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is again a tensor density of that type and weight.</li>

<li>A product of two tensor densities of any types, and with weights <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}}">

<msub>

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</math></span><img src="./74ab879909bd9762251f679bbb2fa738100baa45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle W_{2}}</annotation>

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</math></span><img src="./421467fcab2af38ddf3977b9adf66de8ef3abd57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{2}}" loading="lazy"></span>, is a tensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}+W_{2}.}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle W_{1}+W_{2}.}</annotation>

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</math></span><img src="./8fa59c463c663891059bd28aa36f1ce709bcbe8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.983ex; height:2.509ex;" alt="{\displaystyle W_{1}+W_{2}.}" loading="lazy"></span> Furthermore, a product of authentic tensor densities and pseudotensor densities will be an authentic tensor density when an even number of the factors are pseudotensor densities; it will be a pseudotensor density when an odd number of the factors are pseudotensor densities. Similarly, a product of even tensor densities and odd tensor densities will be an even tensor density when an even number of the factors are odd tensor densities; it will be an odd tensor density when an odd number of the factors are odd tensor densities.</li>

<li>The contraction of indices on a tensor density with weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> again yields a tensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W.}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle W.}</annotation>

</semantics>

</math></span><img src="./035a080f11445ba1e7745704a6031989a311a7d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.082ex; height:2.176ex;" alt="{\displaystyle W.}" loading="lazy"></span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>Raising and lowering indices using the metric tensor (which is authentic, even, and of weight 0) leaves the weight unchanged,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> as can be proved by combining (2) and (3).</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Matrix_inversion_and_matrix_determinant_of_tensor_densities">Matrix inversion and matrix determinant of tensor densities</h3></div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\alpha \beta }}</annotation>

</semantics>

</math></span><img src="./22a036c5eebac6449dcba3c4fe38b3c792c8d8b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.781ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {T}}_{\alpha \beta }}" loading="lazy"></span> is a non-singular matrix and a rank-two tensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> with covariant indices then its matrix inverse will be a rank-two tensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -W}</annotation>

</semantics>

</math></span><img src="./422ddec39f03f391cdfa75545fce3ac4f0a0522a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.243ex; height:2.343ex;" alt="{\displaystyle -W}" loading="lazy"></span> with contravariant indices. Similar statements apply when the two indices are contravariant or are mixed covariant and contravariant.

</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {T}}_{\alpha \beta }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\alpha \beta }}</annotation>

</semantics>

</math></span><img src="./22a036c5eebac6449dcba3c4fe38b3c792c8d8b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.781ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {T}}_{\alpha \beta }}" loading="lazy"></span> is a rank-two tensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> with covariant indices then the matrix determinant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\mathfrak {T}}_{\alpha \beta }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det {\mathfrak {T}}_{\alpha \beta }}</annotation>

</semantics>

</math></span><img src="./11576658a0ca3292000d711b1b79bf17ba2b7297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.398ex; height:2.843ex;" alt="{\displaystyle \det {\mathfrak {T}}_{\alpha \beta }}" loading="lazy"></span> will have weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NW+2,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle NW+2,}</annotation>

</semantics>

</math></span><img src="./5ec90c7e453e055f4354ce388f389ef3508ecc23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.149ex; height:2.509ex;" alt="{\displaystyle NW+2,}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> is the number of space-time dimensions. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {T}}^{\alpha \beta }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}^{\alpha \beta }}</annotation>

</semantics>

</math></span><img src="./0fff7601278464f0ac61ba57985b489cf1ec3803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.802ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {T}}^{\alpha \beta }}" loading="lazy"></span> is a rank-two tensor density of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> with contravariant indices then the matrix determinant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\mathfrak {T}}^{\alpha \beta }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det {\mathfrak {T}}^{\alpha \beta }}</annotation>

</semantics>

</math></span><img src="./89536145699670e220104f5cddff8c68a67f9e40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.419ex; height:2.676ex;" alt="{\displaystyle \det {\mathfrak {T}}^{\alpha \beta }}" loading="lazy"></span> will have weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NW-2.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle NW-2.}</annotation>

</semantics>

</math></span><img src="./46a2bb134682c3e650bf24f991b1a87b3296ebd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.149ex; height:2.343ex;" alt="{\displaystyle NW-2.}" loading="lazy"></span> The matrix determinant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\mathfrak {T}}_{~\beta }^{\alpha }}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det {\mathfrak {T}}_{~\beta }^{\alpha }}</annotation>

</semantics>

</math></span><img src="./73417ca25458d1a495a2db51fdf734c1599eec4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.926ex; height:3.176ex;" alt="{\displaystyle \det {\mathfrak {T}}_{~\beta }^{\alpha }}" loading="lazy"></span> will have weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NW.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle NW.}</annotation>

</semantics>

</math></span><img src="./6d9a740b7eb571b2b997bae6882c3ff465bdc366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.146ex; height:2.176ex;" alt="{\displaystyle NW.}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="General_relativity">General relativity</h2></div>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="General_relativity" title="General relativity">General relativity</a></th></tr><tr><td class="sidebar-image"><span class="notpageimage" typeof="mw:File"></span><div class="sidebar-caption" style="padding:0.5em 0.2em 0.6em;border-bottom:1px solid #aaa; display:block;margin-bottom:0.1em;"><span class="mwe-math-element mwe-math-element-inline"><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 }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}</annotation>

</semantics>

</math></span><img src="./124ab80fcb17e2733cc17ff6f93da5e52f355c77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}" loading="lazy"></span></div></td></tr><tr><td class="sidebar-content" style="padding-bottom:0.75em;">

<ul><li><a href="Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li>

<li><div class="hlist"><ul><li><a href="History_of_general_relativity" title="History of general relativity">History</a></li><li><a href="Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">Timeline</a></li><li><a href="Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li></ul></div></li>

<li><a href="Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Fundamental concepts</div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li>

<li><a href="Special_relativity" title="Special relativity">Special relativity</a></li>

<li><a href="World_line" title="World line">World line</a></li>

<li><a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Phenomena</div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content">

<ul><li><a href="Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Kepler problem</a></li>

<li><a href="Gravitational_lens" title="Gravitational lens">Gravitational lensing</a></li>

<li><a href="Gravitational_redshift" title="Gravitational redshift">Gravitational redshift</a></li>

<li><a href="Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li>

<li><a href="Gravitational_wave" title="Gravitational wave">Gravitational waves</a></li>

<li><a href="Frame-dragging" title="Frame-dragging">Frame-dragging</a></li>

<li><a href="Geodetic_effect" title="Geodetic effect">Geodetic effect</a></li>

<li><a href="Event_horizon" title="Event horizon">Event horizon</a></li>

<li><a href="Gravitational_singularity" title="Gravitational singularity">Singularity</a></li>

<li><a href="Black_hole" title="Black hole">Black hole</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:#ececff; font-style:italic;font-weight:normal;">

<a href="Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li>

<li><a href="Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li>

<li><a href="Wormhole" title="Wormhole">Einstein–Rosen bridge</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;">

Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;">

<ul><li><a href="Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li>

<li><a href="Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li>

<li><a href="Friedmann_equations" title="Friedmann equations">Friedmann</a></li>

<li><a href="Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li>

<li><a href="Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon</a></li>

<li><a href="Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;">

Formalisms</th></tr><tr><td class="sidebar-content" style="padding-top:0;">

<li><a href="Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;">

Advanced theory</th></tr><tr><td class="sidebar-content" style="padding-top:0;">

<ul><li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li>

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<ul><li><a href="Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a>)</li>

<li><a href="Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li>

<li><a href="Einstein%E2%80%93Rosen_metric" title="Einstein–Rosen metric">Einstein–Rosen waves</a></li>

<li><a href="Wormhole" title="Wormhole">Wormhole</a></li>

<li><a href="G%C3%B6del_metric" title="Gödel metric">Gödel</a></li>

<li><a href="Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li>

<li><a href="Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de Sitter</a></li>

<li><a href="Kasner_metric" title="Kasner metric">Kasner</a></li>

<li><a href="Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li>

<li><a href="Milne_model" title="Milne model">Milne</a></li>

<li><a href="Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Robertson–Walker</a></li>

<li><a href="Oppenheimer%E2%80%93Snyder_model" title="Oppenheimer–Snyder model">Oppenheimer–Snyder</a></li>

<li><a href="Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li>

<li><a href="Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne</a></li>

<li><a href="Vaidya_metric" title="Vaidya metric">Vaidya</a></li>

<li><a href="Peres_metric" title="Peres metric">Peres</a></li>

<li><a href="De_Sitter%E2%80%93Schwarzschild_metric" title="De Sitter–Schwarzschild metric">De Sitter-Schwarzschild</a></li>

<li><a href="McVittie_metric" title="McVittie metric">McVittie</a></li>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>

<li><a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li>

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<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>

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<li><a href="Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li>

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<li><a href="Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Nordström</a></li>

<li><a href="Arthur_Eddington" title="Arthur Eddington">Eddington</a></li>

<li><a href="Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li>

<li><a href="Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li>

<li><a href="Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li>

<li><a href="Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li>

<li><a href="J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li>

<li><a href="Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li>

<li><a href="John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li>

<li><a href="Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li>

<li><a href="James_M._Bardeen" title="James M. Bardeen">Bardeen</a></li>

<li><a href="Arthur_Geoffrey_Walker" title="Arthur Geoffrey Walker">Walker</a></li>

<li><a href="Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li>

<li><a href="J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li>

<li><a href="Roger_Penrose" title="Roger Penrose">Penrose</a></li>

<li><a href="Stephen_Hawking" title="Stephen Hawking">Hawking</a></li>

<li><a href="Amal_Kumar_Raychaudhuri" title="Amal Kumar Raychaudhuri">Raychaudhuri</a></li>

<li><a href="Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li>

<li><a href="Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li>

<li><a href="Willem_Jacob_van_Stockum" title="Willem Jacob van Stockum">van Stockum</a></li>

<li><a href="Ezra_T._Newman" title="Ezra T. Newman">Newman</a></li>

<li><a href="Kip_Thorne" title="Kip Thorne">Thorne</a></li>

<li><a href="List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul></div></div></td>

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<div class="mw-heading mw-heading3"><h3 id="Relation_of_Jacobian_determinant_and_metric_tensor">Relation of Jacobian determinant and metric tensor</h3></div>

<p>Any non-singular ordinary tensor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mu \nu }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{\mu \nu }}</annotation>

</semantics>

</math></span><img src="./463ab8cef859ece28e33b8460ebd4a6699834dd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.452ex; height:2.843ex;" alt="{\displaystyle T_{\mu \nu }}" loading="lazy"></span> transforms as

<msub>

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</msub>

<mfrac>

<mrow>

<msup>

<mover>

</mover>

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</msup>

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<msup>

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</msup>

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</mfrac>

</mrow>

<msub>

<mover>

</mover>

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</msub>

<mfrac>

<mrow>

<msup>

<mover>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{\mu \nu }={\frac {\partial {\bar {x}}^{\kappa }}{\partial {x}^{\mu }}}{\bar {T}}_{\kappa \lambda }{\frac {\partial {\bar {x}}^{\lambda }}{\partial {x}^{\nu }}}\,,}</annotation>

</semantics>

</math></span></span>

</p><p>where the right-hand side can be viewed as the product of three matrices. Taking the determinant of both sides of the equation (using that the determinant of a matrix product is the product of the determinants), dividing both sides by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left({\bar {T}}_{\kappa \lambda }\right),}">

<mrow>

<msub>

<mover>

</mover>

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</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det \left({\bar {T}}_{\kappa \lambda }\right),}</annotation>

</semantics>

</math></span><img src="./74671c31780291ec829987456829ef0ede9e7446.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.692ex; height:3.176ex;" alt="{\displaystyle \det \left({\bar {T}}_{\kappa \lambda }\right),}" loading="lazy"></span> and taking their square root gives

<mrow>

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<msqrt>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ={\sqrt {\frac {\det({T}_{\mu \nu })}{\det \left({\bar {T}}_{\kappa \lambda }\right)}}}\,.}</annotation>

</semantics>

</math></span></span>

</p><p>When the tensor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> is the <a href="Metric_tensor" title="Metric tensor">metric tensor</a>, <span class="mwe-math-element mwe-math-element-inline"><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}_{\kappa \lambda },}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {g}_{\kappa \lambda },}</annotation>

</semantics>

</math></span><img src="./edccb6df2a2d7d3c1b783b06c28073f48cc3b121.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.893ex; height:2.176ex;" alt="{\displaystyle {g}_{\kappa \lambda },}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}^{\iota }}">

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}^{\iota }}</annotation>

</semantics>

</math></span><img src="./df458b51fd76ce6d2415afb87f1de3f04fbd9e3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.144ex; height:2.343ex;" alt="{\displaystyle {\bar {x}}^{\iota }}" loading="lazy"></span> is a locally inertial coordinate system where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {g}}_{\kappa \lambda }=\eta _{\kappa \lambda }=}">

<msub>

<mover>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {g}}_{\kappa \lambda }=\eta _{\kappa \lambda }=}</annotation>

</semantics>

</math></span><img src="./7e4addc4d74e9827106a4413eb9ee4ff95a9a3fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.214ex; height:2.509ex;" alt="{\displaystyle {\bar {g}}_{\kappa \lambda }=\eta _{\kappa \lambda }=}" loading="lazy"></span><span class="nowrap"> </span>diag(−1,+1,+1,+1), the <a href="Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left({\bar {g}}_{\kappa \lambda }\right)=\det(\eta _{\kappa \lambda })=}">

<mrow>

<msub>

<mover>

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</msub>

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<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \det \left({\bar {g}}_{\kappa \lambda }\right)=\det(\eta _{\kappa \lambda })=}</annotation>

</semantics>

</math></span><img src="./01198298d88235024201470b12f2906d49e2d303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.678ex; height:2.843ex;" alt="{\displaystyle \det \left({\bar {g}}_{\kappa \lambda }\right)=\det(\eta _{\kappa \lambda })=}" loading="lazy"></span><span class="nowrap"> </span>−1 and so

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ={\sqrt {-{g}}}\,,}</annotation>

</semantics>

</math></span></span>

</p><p>where <span class="mwe-math-element mwe-math-element-inline"><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}=\det \left({g}_{\mu \nu }\right)}">

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<mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {g}=\det \left({g}_{\mu \nu }\right)}</annotation>

</semantics>

</math></span><img src="./0da1ac6f0d6e7e7ae3dc0ee64b3f733ac3467eff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.165ex; height:3.343ex;" alt="{\displaystyle {g}=\det \left({g}_{\mu \nu }\right)}" loading="lazy"></span> is the determinant of the metric tensor <span class="mwe-math-element mwe-math-element-inline"><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 }.}">

<msub>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {g}_{\mu \nu }.}</annotation>

</semantics>

</math></span><img src="./7f420b099c3f9b4a47f6059e04b11e4a6ba128b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.851ex; height:2.509ex;" alt="{\displaystyle {g}_{\mu \nu }.}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Use_of_metric_tensor_to_manipulate_tensor_densities">Use of metric tensor to manipulate tensor densities</h3></div>

<p>Consequently, an even tensor density, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots },}">

</mrow>

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</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots },}</annotation>

</semantics>

</math></span><img src="./0a439bd3c3250dcbefc32ac5a5faeb1ba1532f43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots },}" loading="lazy"></span> of weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span>, can be written in the form

</mrow>

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</msubsup>

<msqrt>

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<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots }={\sqrt {-g}}\;^{W}T_{\nu \dots }^{\mu \dots }\,,}</annotation>

</semantics>

</math></span></span>

</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\nu \dots }^{\mu \dots }\,}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{\nu \dots }^{\mu \dots }\,}</annotation>

</semantics>

</math></span><img src="./887e3f7ed3687e9b7aac025a079cf7702fae280c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.256ex; height:2.843ex;" alt="{\displaystyle T_{\nu \dots }^{\mu \dots }\,}" loading="lazy"></span> is an ordinary tensor. In a locally inertial coordinate system, where <span class="mwe-math-element mwe-math-element-inline"><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_{\kappa \lambda }=\eta _{\kappa \lambda },}">

<msub>

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<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{\kappa \lambda }=\eta _{\kappa \lambda },}</annotation>

</semantics>

</math></span><img src="./f9e51e17fed1e498b67def6231fe8f970ad9a6fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.285ex; height:2.176ex;" alt="{\displaystyle g_{\kappa \lambda }=\eta _{\kappa \lambda },}" loading="lazy"></span> it will be the case that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots }}">

</mrow>

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</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots }}</annotation>

</semantics>

</math></span><img src="./165c1e1f641e0cdfafa9fd8d609f52edac6af9b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.725ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots }}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\nu \dots }^{\mu \dots }\,}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{\nu \dots }^{\mu \dots }\,}</annotation>

</semantics>

</math></span><img src="./887e3f7ed3687e9b7aac025a079cf7702fae280c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.256ex; height:2.843ex;" alt="{\displaystyle T_{\nu \dots }^{\mu \dots }\,}" loading="lazy"></span> will be represented with the same numbers.

</p><p>When using the metric connection (<a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a>), the <a href="Covariant_derivative" title="Covariant derivative">covariant derivative</a> of an even tensor density is defined as

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {T}}_{\nu \dots ;\alpha }^{\mu \dots }={\sqrt {-g}}\;^{W}T_{\nu \dots ;\alpha }^{\mu \dots }={\sqrt {-g}}\;^{W}\left({\sqrt {-g}}\;^{-W}{\mathfrak {T}}_{\nu \dots }^{\mu \dots }\right)_{;\alpha }\,.}</annotation>

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</math></span></span>

</p><p>For an arbitrary connection, the covariant derivative is defined by adding an extra term, namely

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<annotation encoding="application/x-tex">{\displaystyle -W\,\Gamma _{~\delta \alpha }^{\delta }\,{\mathfrak {T}}_{\nu \dots }^{\mu \dots }}</annotation>

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</math></span></span>

to the expression that would be appropriate for the covariant derivative of an ordinary tensor.

</p><p>Equivalently, the product rule is obeyed

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<annotation encoding="application/x-tex">{\displaystyle \left({\mathfrak {T}}_{\nu \dots }^{\mu \dots }{\mathfrak {S}}_{\tau \dots }^{\sigma \dots }\right)_{;\alpha }=\left({\mathfrak {T}}_{\nu \dots ;\alpha }^{\mu \dots }\right){\mathfrak {S}}_{\tau \dots }^{\sigma \dots }+{\mathfrak {T}}_{\nu \dots }^{\mu \dots }\left({\mathfrak {S}}_{\tau \dots ;\alpha }^{\sigma \dots }\right)\,,}</annotation>

</semantics>

</math></span></span>

</p><p>where, for the metric connection, the covariant derivative of any function of <span class="mwe-math-element mwe-math-element-inline"><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_{\kappa \lambda }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle g_{\kappa \lambda }}</annotation>

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</math></span><img src="./29ca9aec92f4ab0e3c148c08aedc81bc5c8d6195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.246ex; height:2.009ex;" alt="{\displaystyle g_{\kappa \lambda }}" loading="lazy"></span> is always zero,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{\kappa \lambda ;\alpha }&=0\\\left({\sqrt {-g}}\;^{W}\right)_{;\alpha }&=\left({\sqrt {-g}}\;^{W}\right)_{,\alpha }-W\Gamma _{~\delta \alpha }^{\delta }{\sqrt {-g}}\;^{W}={\frac {W}{2}}g^{\kappa \lambda }g_{\kappa \lambda ,\alpha }{\sqrt {-g}}\;^{W}-W\Gamma _{~\delta \alpha }^{\delta }{\sqrt {-g}}\;^{W}=0\,.\end{aligned}}}</annotation>

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</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Maxwell's_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell's equations in curved spacetime</a></div>

<p>The expression <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-g}}}">

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {-g}}}</annotation>

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</math></span><img src="./8d5737dc40c4e185538cc69efb1487f5b33e6aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.248ex; height:3.509ex;" alt="{\displaystyle {\sqrt {-g}}}" loading="lazy"></span> is a scalar density. By the convention of this article it has a weight of +1.

</p><p>The density of electric current <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {J}}^{\mu }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {J}}^{\mu }}</annotation>

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</math></span><img src="./8ca8078e727ffc93496301cc8dc39b031b8fc94b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-left: -0.022ex; width:2.529ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {J}}^{\mu }}" loading="lazy"></span> (for example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {J}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {J}}^{2}}</annotation>

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</math></span><img src="./5523151173340c02d7271661c4c8cdeffe2ae3bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-left: -0.022ex; width:2.36ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {J}}^{2}}" loading="lazy"></span> is the amount of electric charge crossing the 3-volume element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{3}\,dx^{4}\,dx^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle dx^{3}\,dx^{4}\,dx^{1}}</annotation>

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</math></span><img src="./e3f6eb54a746c37bab8156c15c272fee9e25754d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.574ex; height:2.676ex;" alt="{\displaystyle dx^{3}\,dx^{4}\,dx^{1}}" loading="lazy"></span> divided by that element — do not use the metric in this calculation) is a contravariant vector density of weight +1. It is often written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {J}}^{\mu }=J^{\mu }{\sqrt {-g}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {J}}^{\mu }=J^{\mu }{\sqrt {-g}}}</annotation>

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</math></span><img src="./32bf76ad95e44508eeade9c741db3bffb871f44c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.022ex; width:13.624ex; height:3.509ex;" alt="{\displaystyle {\mathfrak {J}}^{\mu }=J^{\mu }{\sqrt {-g}}}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {J}}^{\mu }=\varepsilon ^{\mu \alpha \beta \gamma }{\mathcal {J}}_{\alpha \beta \gamma }/3!,}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {J}}^{\mu }=\varepsilon ^{\mu \alpha \beta \gamma }{\mathcal {J}}_{\alpha \beta \gamma }/3!,}</annotation>

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</math></span><img src="./941a564b2dfba02944d18a760351b4769e1a6238.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.022ex; width:19.134ex; height:3.343ex;" alt="{\displaystyle {\mathfrak {J}}^{\mu }=\varepsilon ^{\mu \alpha \beta \gamma }{\mathcal {J}}_{\alpha \beta \gamma }/3!,}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{\mu }\,}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle J^{\mu }\,}</annotation>

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</math></span><img src="./26dc65d141aa11a8b3b0051f00f8d67a8e22b10c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.136ex; height:2.343ex;" alt="{\displaystyle J^{\mu }\,}" loading="lazy"></span> and the <a href="Differential_form" title="Differential form">differential form</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}_{\alpha \beta \gamma }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}_{\alpha \beta \gamma }}</annotation>

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</math></span><img src="./97c4711c8db25393a4ab833cfb1b74c6dfebf888.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.695ex; height:2.843ex;" alt="{\displaystyle {\mathcal {J}}_{\alpha \beta \gamma }}" loading="lazy"></span> are absolute tensors, and where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ^{\mu \alpha \beta \gamma }}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon ^{\mu \alpha \beta \gamma }}</annotation>

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</math></span><img src="./8a262ff6f35fe034ead21dc8ab829f6caa000d6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.193ex; height:2.676ex;" alt="{\displaystyle \varepsilon ^{\mu \alpha \beta \gamma }}" loading="lazy"></span> is the <a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a>; see below.

</p><p>The density of <a href="Lorentz_force" title="Lorentz force">Lorentz force</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {f}}_{\mu }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}_{\mu }}</annotation>

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</math></span><img src="./fa8e4788e4953fc2f36791dc0ba18a8896410ef0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.982ex; height:3.009ex;" alt="{\displaystyle {\mathfrak {f}}_{\mu }}" loading="lazy"></span> (that is, the linear momentum transferred from the electromagnetic field to matter within a 4-volume element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{1}\,dx^{2}\,dx^{3}\,dx^{4}}">

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<annotation encoding="application/x-tex">{\displaystyle dx^{1}\,dx^{2}\,dx^{3}\,dx^{4}}</annotation>

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</math></span><img src="./46f6f8c3f36b6254c7b250ea9aff682a3970cac4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.56ex; height:2.676ex;" alt="{\displaystyle dx^{1}\,dx^{2}\,dx^{3}\,dx^{4}}" loading="lazy"></span> divided by that element — do not use the metric in this calculation) is a covariant vector density of weight +1.

</p><p>In <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span>-dimensional space-time, the <a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a> may be regarded as either a rank-<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> contravariant (odd) authentic tensor density of weight +1 <span class="nowrap">(<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon ^{\alpha _{1}\cdots \epsilon _{\alpha _{N}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon ^{\alpha _{1}\cdots \epsilon _{\alpha _{N}}}}</annotation>

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</math></span><img src="./483fdf6fed943d146e77c87cbcfb70024b10977a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.989ex; height:2.343ex;" alt="{\displaystyle \epsilon ^{\alpha _{1}\cdots \epsilon _{\alpha _{N}}}}" loading="lazy"></span>)</span> or a rank-<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> covariant (odd) authentic tensor density of weight −1 <span class="nowrap">(<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{\alpha _{1}\cdots \epsilon _{\alpha _{N}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon _{\alpha _{1}\cdots \epsilon _{\alpha _{N}}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \epsilon ^{\alpha _{1}\cdots \epsilon _{\alpha _{N}}}={\bar {\epsilon }}^{\beta _{1}\cdots \epsilon _{\beta _{N}}}{\frac {\partial x^{\alpha _{1}}}{\partial {\bar {x}}^{\beta _{1}}}}\cdots {\frac {\partial x^{\alpha _{N}}}{\partial {\bar {x}}^{\beta _{N}}}}\left(\det \left[{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\alpha }}}\right]\right)^{+1}}</annotation>

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</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon _{\alpha _{1}\cdots \epsilon _{\alpha _{N}}}={\bar {\epsilon }}_{\beta _{1}\cdots \epsilon _{\beta _{N}}}{\frac {\partial {\bar {x}}^{\beta _{1}}}{\partial x^{\alpha _{1}}}}\cdots {\frac {\partial {\bar {x}}^{\beta _{N}}}{\partial x^{\alpha _{N}}}}\left(\det \left[{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\alpha }}}\right]\right)^{-1}\,.}</annotation>

</semantics>

</math></span></span>

Notice that the Levi-Civita symbol (so regarded) does <em>not</em> obey the usual convention for raising or lowering of indices with the metric tensor. That is, it is true that

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon ^{\alpha \beta \gamma \delta }\,g_{\alpha \kappa }\,g_{\beta \lambda }\,g_{\gamma \mu }g_{\delta \nu }\,=\,\varepsilon _{\kappa \lambda \mu \nu }\,g\,,}</annotation>

</semantics>

</math></span></span>

but in general relativity, where <span class="mwe-math-element mwe-math-element-inline"><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=\det \left(g_{\rho \sigma }\right)}">

<mrow>

<msub>

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</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g=\det \left(g_{\rho \sigma }\right)}</annotation>

</semantics>

</math></span><img src="./82a0f5e256c2dc4119a1afe9455aba950b90211a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.772ex; height:3.009ex;" alt="{\displaystyle g=\det \left(g_{\rho \sigma }\right)}" loading="lazy"></span> is always negative, this is never equal to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\kappa \lambda \mu \nu }.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\kappa \lambda \mu \nu }.}</annotation>

</semantics>

</math></span><img src="./c60ffa9aa496aa7722168b2f4098ad73208fe05f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.73ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{\kappa \lambda \mu \nu }.}" loading="lazy"></span>

</p><p>The <a href="Determinant" title="Determinant">determinant</a> of the metric tensor,

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<annotation encoding="application/x-tex">{\displaystyle g=\det \left(g_{\rho \sigma }\right)={\frac {1}{4!}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }\,,}</annotation>

</semantics>

</math></span></span>

is an (even) authentic scalar density of weight +2, being the contraction of the product of 2 (odd) authentic tensor densities of weight +1 and four (even) authentic tensor densities of weight 0.

</p>

/* start https://en.wikipedia.org/ */

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<ul><li><a href="Action_(physics)" title="Action (physics)">Action (physics)</a> – Physical quantity of dimension energy × time</li>

<li><a href="Conservation_law" title="Conservation law">Conservation law</a> – Scientific law regarding conservation of a physical property</li>

<li><a href="Noether's_theorem" title="Noether's theorem">Noether's theorem</a> – Statement relating differentiable symmetries to conserved quantities</li>

<li><a href="Pseudotensor" title="Pseudotensor">Pseudotensor</a> – Type of physical quantity</li>

<li><a href="Relative_scalar" title="Relative scalar">Relative scalar</a></li>

<li><a href="Variational_principle" title="Variational principle">Variational principle</a> – Scientific principles enabling the use of the calculus of variations</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFWeinreich1998" class="citation book cs1">Weinreich, Gabriel (July 6, 1998). <i>Geometrical Vectors</i>. University of Chicago Press. pp. 112, 115. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0226890487</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFPapastavridis1998" class="citation book cs1">Papastavridis, John G. (Dec 18, 1998). <i>Tensor Calculus and Analytical Dynamics</i>. <a href="CRC_Press" title="CRC Press">CRC Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0849385148</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFRuiz-Tolosa,_Castillo2006" class="citation book cs1">Ruiz-Tolosa, Castillo, Juan R., Enrique (30 Mar 2006). <i>From Vectors to Tensors</i>. Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3540228875</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)</span></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">E.g. <a href="#CITEREFWeinberg1972">Weinberg 1972</a> pp 98. The chosen convention involves in the formulae below the <a href="Jacobian_determinant" class="mw-redirect" title="Jacobian determinant">Jacobian determinant</a> of the inverse transition <span class="texhtml"><i>x</i> → <span style="text-decoration:overline;"><i>x</i></span></span>, while the opposite convention considers the forward transition <span class="texhtml"><span style="text-decoration:overline;"><i>x</i></span> → <i>x</i></span> resulting in a flip of sign of the weight.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFM.R._SpiegelS._LipcshutzD._Spellman2009" class="citation book cs1">M.R. Spiegel; S. Lipcshutz; D. Spellman (2009). <i>Vector Analysis</i> (2nd ed.). New York: Schaum's Outline Series. p. 198. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-161545-7</bdi>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFC.B._Parker1994" class="citation book cs1">C.B. Parker (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/mcgrawhillencycl1993park/page/1417"><i>McGraw Hill Encyclopaedia of Physics</i></a> (2nd ed.). McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mcgrawhillencycl1993park/page/1417">1417</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-051400-3</bdi>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeinberg1972">Weinberg 1972</a> p 100.</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeinberg1972">Weinberg 1972</a> p 100.</span>

</li>

</ol></div></div>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFSpivak1999" class="citation cs2"><a href="Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (1999), <i>A Comprehensive Introduction to Differential Geometry, Vol I</i> (3rd ed.), p. 134</cite>.</li>

<li><cite id="CITEREFKuptsov2001" class="citation cs2">Kuptsov, L.P. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Tensor_Density">"Tensor Density"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite>.</li>

<li><cite id="CITEREFCharles_Misner;_Kip_S_Thorne_&_John_Archibald_Wheeler1973" class="citation book cs1"><a href="Charles_Misner" class="mw-redirect" title="Charles Misner">Charles Misner</a>; <a href="Kip_S_Thorne" class="mw-redirect" title="Kip S Thorne">Kip S Thorne</a> & <a href="John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a> (1973). <i><a href="Gravitation_(book)" title="Gravitation (book)">Gravitation</a></i>. <a href="W._H._Freeman" class="mw-redirect" title="W. H. Freeman">W. H. Freeman</a>. p. 501ff. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7167-0344-0</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)</span></li>

<li><cite id="CITEREFWeinberg1972" class="citation cs2"><a href="Steven_Weinberg" title="Steven Weinberg">Weinberg, Steven</a> (1972), <a rel="nofollow" class="external text" href="https://archive.org/details/gravitationcosmo00stev_0"><i>Gravitation and Cosmology</i></a>, John Wiley & sons, Inc, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-92567-5</bdi></cite></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Tensors176" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Tensors176" style="font-size:114%;margin:0 4em"><a href="Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i><a href="Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</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;"><a href="Mathematics" title="Mathematics">Mathematics</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="Coordinate_system" title="Coordinate system">Coordinate system</a></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential geometry</a></li>

<li><a href="Dyadics" title="Dyadics">Dyadic algebra</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li>

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<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li>

<li><a href="Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="Physics" title="Physics">Physics</a></li><li><a href="Engineering" title="Engineering">Engineering</a></li></ul></div></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="Computer_vision" title="Computer vision">Computer vision</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li>

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<li><a href="General_relativity" title="General relativity">General relativity</a></li>

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</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</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="Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li>

<li><a href="Einstein_notation" title="Einstein notation">Einstein notation</a></li>

<li><a href="Index_notation" title="Index notation">Index notation</a></li>

<li><a href="Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li>

<li><a href="Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li>

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li>

<li><a href="Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li>

<li><a href="Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li>

<li><a href="Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor<br>definitions</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="Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a></li>

<li><a href="Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li>

<li><a href="Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li>

<li><a href="Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li>

<li><a href="Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li>

<li><a href="Tensor_operator" title="Tensor operator">Tensor operator</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li>

<li><a href="Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Operation_(mathematics)" 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="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li>

<li><a href="Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">Raising and lowering indices</a></li>

<li><a href="Symmetrization" title="Symmetrization">Symmetrization</a></li>

<li><a href="Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li>

<li><a href="Tensor_product" title="Tensor product">Tensor product</a></li>

<li><a href="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="Affine_connection" title="Affine connection">Affine connection</a></li>

<li><a href="Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li>

<li><a href="Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li>

<li><a href="Connection_form" title="Connection form">Connection form</a></li>

<li><a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="Dimension" title="Dimension">Dimension</a></li>

<li><a href="Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li>

<li><a href="Geodesic" title="Geodesic">Geodesic</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li>

<li><a href="Linear_map" title="Linear map">Linear map</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li>

<li><a href="Multivector" title="Multivector">Multivector</a></li>

<li><a href="Pseudotensor" title="Pseudotensor">Pseudotensor</a></li>

<li><a href="Spinor" title="Spinor">Spinor</a></li>

<li><a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li>

<li><a href="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="Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li>

<li><a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li>

<li><a href="Metric_tensor" title="Metric tensor">Metric tensor</a></li>

<li><a href="Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li>

<li><a href="Weyl_tensor" title="Weyl tensor">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="Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li>

<li><a href="Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li>

<li><a href="Spin_tensor" title="Spin tensor">Spin tensor</a></li>

<li><a href="Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li>

<li><a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li>

<li><a href="Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li>

<li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li>

<li><a href="Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li>

<li><a href="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="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="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li>

<li><a href="Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li>

<li><a href="Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li>

<li><a href="Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li>

<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li>

<li><a href="Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li>

<li><a href="Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li>

<li><a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Manifolds_(Glossary,_List,_Category)274" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Manifolds_(Glossary,_List,_Category)274" style="font-size:114%;margin:0 4em"><a href="Manifold" title="Manifold">Manifolds</a> (<a href="Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>, <a href="List_of_manifolds" title="List of manifolds">List</a>, Category)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</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="Topological_manifold" title="Topological manifold">Topological manifold</a>

<ul><li><a href="Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li>

<li><a href="Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a>

<ul><li><a href="Differential_structure" title="Differential structure">Differential structure</a></li>

<li><a href="Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li>

<li><a href="Submanifold" title="Submanifold">Submanifold</a></li>

<li><a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li>

<li><a href="Smoothness" title="Smoothness">Smooth map</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="Vector_field" title="Vector field">Vector field</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size: 85%;">(list)</span></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="Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li>

<li><a href="Darboux's_theorem" title="Darboux's theorem">Darboux's</a></li>

<li><a href="Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li>

<li><a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li>

<li><a href="Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li>

<li><a href="Noether's_theorem" title="Noether's theorem">Noether's</a></li>

<li><a href="Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Smoothness" title="Smoothness">Maps</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="Differentiable_curve" title="Differentiable curve">Curve</a></li>

<li><a href="Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a>

<ul><li><a href="Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li>

<li><a href="Geodesic" title="Geodesic">Geodesic</a></li>

<li><a href="Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a>

<ul><li><a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li>

<li><a href="Foliation" title="Foliation">Foliation</a></li>

<li><a href="Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li>

<li><a href="Integral_curve" title="Integral curve">Integral curve</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br>manifolds</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="Closed_manifold" title="Closed manifold">Closed</a></li>

<li><a href="Collapsing_manifold" title="Collapsing manifold">Collapsing</a></li>

<li><a href="Complete_manifold" title="Complete manifold">Complete</a></li>

<li>(<a href="Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="Complex_manifold" title="Complex manifold">Complex</a></li>

<li>(<a href="Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li>

<li><a href="Fibered_manifold" title="Fibered manifold">Fibered</a></li>

<li><a href="Finsler_manifold" title="Finsler manifold">Finsler</a></li>

<li>(<a href="Almost_flat_manifold" title="Almost flat manifold">Almost</a>) <a href="Flat_manifold" title="Flat manifold">Flat</a></li>

<li><a href="G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li>

<li><a href="Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li>

<li><a href="Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li>

<li><a href="Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li>

<li><a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li>

<li><a href="Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li>

<li><a href="Lie_group" title="Lie group">Lie group</a>

<ul><li><a href="Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li>

<li><a href="Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li>

<li><a href="Nilmanifold" title="Nilmanifold">Nilmanifold</a></li>

<li><a href="Orientability" title="Orientability">Oriented</a></li>

<li><a href="Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li>

<li><a href="Poisson_manifold" title="Poisson manifold">Poisson</a></li>

<li><a href="Prime_manifold" title="Prime manifold">Prime</a></li>

<li><a href="Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li>

<li><a href="Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li>

<li>(<a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-</a>, <a href="Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub-</a>) <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li>

<li><a href="Rizza_manifold" title="Rizza manifold">Rizza</a></li>

<li><a href="Stein_manifold" title="Stein manifold">Stein</a></li>

<li>(<a href="Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Tensor" title="Tensor">Tensors</a></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%">Vectors</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="Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li>

<li><a href="Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a>

<ul><li><a href="Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion</a></li>

<li><a href="Vector_field" title="Vector field">Vector field</a></li>

<li><a href="Vector_flow" title="Vector flow">Vector flow</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</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="Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li>

<li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Cotangent_space" title="Cotangent space">Cotangent space</a>

<ul><li><a href="Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li>

<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a>

<ul><li><a href="Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Interior_product" title="Interior product">Interior product</a></li>

<li><a href="Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a>

<li><a href="Volume_form" title="Volume form">Volume form</a></li>

<li><a href="Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Fiber_bundle" title="Fiber bundle">Bundles</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="Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li>

<li><a href="Affine_bundle" title="Affine bundle">Affine</a></li>

<li><a href="Associated_bundle" title="Associated bundle">Associated</a></li>

<li><a href="Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber</a></li>

<li>(<a href="Cofibration" title="Cofibration">Co-</a>) <a href="Fibration" title="Fibration">Fibration</a></li>

<li><a href="Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li>

<li>(<a href="Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="Normal_bundle" title="Normal bundle">Normal</a></li>

<li><a href="Principal_bundle" title="Principal bundle">Principal</a></li>

<li><a href="Spinor_bundle" title="Spinor bundle">Spinor</a></li>

<li><a href="Subbundle" title="Subbundle">Subbundle</a></li>

<li><a href="Tangent_bundle" title="Tangent bundle">Tangent</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor</a></li>

<li><a href="Vector_bundle" title="Vector bundle">Vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Connection_(mathematics)" title="Connection (mathematics)">Connections</a></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="Affine_connection" title="Affine connection">Affine</a></li>

<li><a href="Cartan_connection" title="Cartan connection">Cartan</a></li>

<li><a href="Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li>

<li><a href="Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li>

<li><a href="Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li>

<li><a href="Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li>

<li><a href="Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li>

<li><a href="Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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="Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li>

<li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li>

<li><a href="History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li>

<li><a href="Morse_theory" title="Morse theory">Morse theory</a></li>

<li><a href="Moving_frame" title="Moving frame">Moving frame</a></li>

<li><a href="Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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="Banach_manifold" title="Banach manifold">Banach manifold</a></li>

<li><a href="Diffeology" title="Diffeology">Diffeology</a></li>

<li><a href="Diffiety" title="Diffiety">Diffiety</a></li>

<li><a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li>

<li><a href="K-theory" title="K-theory">K-theory</a></li>

<li><a href="Orbifold" title="Orbifold">Orbifold</a></li>

<li><a href="Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a>

<ul><li><a href="Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li>

<li><a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li>

<li><a href="Stratifold" title="Stratifold">Stratifold</a></li>

<li><a href="Supermanifold" title="Supermanifold">Supermanifold</a></li>

<li><a href="Stratified_space" title="Stratified space">Stratified space</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-30" href="https://en.wikipedia.org/wiki/?title=Tensor_density&oldid=1303395118">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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