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<title>Line element</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Line element</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about lines in mathematics. For Long Interspersed Nuclear Elements in DNA, see <a href="Retrotransposon#LINEs" title="Retrotransposon">Retrotransposon § LINEs</a>.</div>

<p>In <a href="Geometry" title="Geometry">geometry</a>, the <b>line element</b> or <b>length element</b> can be informally thought of as a line segment associated with an <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> <a href="Displacement_vector" class="mw-redirect" title="Displacement vector">displacement vector</a> in a <a href="Metric_space" title="Metric space">metric space</a>. The length of the line element, which may be thought of as a differential <a href="Arc_length" title="Arc length">arc length</a>, is a function of the <a href="Metric_tensor" title="Metric tensor">metric tensor</a> and is denoted by <i><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 ds}">

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</math></span><img src="./5f0fb36e4308227d3e4a1f809c2571ec02527100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.306ex; height:2.176ex;" alt="{\displaystyle ds}" loading="lazy"></span></i>.

</p><p>Line elements are used in <a href="Physics" title="Physics">physics</a>, especially in theories of <a href="Gravitation" class="mw-redirect" title="Gravitation">gravitation</a> (most notably <a href="General_relativity" title="General relativity">general relativity</a>) where <a href="Spacetime" title="Spacetime">spacetime</a> is modelled as a curved <a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a> with an appropriate <a href="Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric tensor</a>.<sup id="cite_ref-WheelerMisnerThorne_1-0" class="reference"><a href="#cite_note-WheelerMisnerThorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="General_formulation">General formulation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">For notation used, see <a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a> and <a href="Einstein_notation" title="Einstein notation">Einstein notation</a>.</div>

<div class="mw-heading mw-heading3"><h3 id="Definition_of_the_line_element_and_arc_length">Definition of the line element and arc length</h3></div>

<p>The <a href="Coordinate" class="mw-redirect" title="Coordinate">coordinate</a>-independent definition of the square of the line element <i>ds</i> in an <i>n</i>-<a href="Dimension" title="Dimension">dimensional</a> <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a> or <a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a> (in physics usually a <a href="Spacetime_manifold" class="mw-redirect" title="Spacetime manifold">Lorentzian manifold</a>) is the "square of the length" of an infinitesimal displacement <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 d\mathbf {q} }">

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

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</math></span><img src="./5ed55779a508eaf2aaadce7bfa3c185361234763.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.631ex; height:2.509ex;" alt="{\displaystyle d\mathbf {q} }" loading="lazy"></span><sup id="cite_ref-Kay_2-0" class="reference"><a href="#cite_note-Kay-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (in pseudo-Riemannian manifolds possibly negative) whose square root should be used for computing curve length:

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<annotation encoding="application/x-tex">{\displaystyle ds^{2}=d\mathbf {q} \cdot d\mathbf {q} =g(d\mathbf {q} ,d\mathbf {q} )}</annotation>

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where <i>g</i> is the <a href="Metric_tensor" title="Metric tensor">metric tensor</a>, <b>·</b> denotes <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a>, and <i>d</i><b>q</b> an <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> <a href="Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> on the (pseudo) Riemannian manifold. By parametrizing a curve <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 \mathbf {q} (\lambda )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} (\lambda )}</annotation>

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</math></span><img src="./ff0139a22353f3c897a9cbe21d501a93e152f858.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.58ex; height:2.843ex;" alt="{\displaystyle \mathbf {q} (\lambda )}" loading="lazy"></span>, we can define the <a href="Arc_length" title="Arc length">arc length</a> of the curve length of the curve between <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 \mathbf {q} _{1}=\mathbf {q} (\lambda _{1})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} _{1}=\mathbf {q} (\lambda _{1})}</annotation>

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</math></span><img src="./652864596260ed5282f68f7095c772f07fb435a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.198ex; height:2.843ex;" alt="{\displaystyle \mathbf {q} _{1}=\mathbf {q} (\lambda _{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 \mathbf {q} _{2}=\mathbf {q} (\lambda _{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} _{2}=\mathbf {q} (\lambda _{2})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle s=\int _{\mathbf {q} _{1}}^{\mathbf {q} _{2}}{\sqrt {\left|ds^{2}\right|}}=\int _{\lambda _{1}}^{\lambda _{2}}d\lambda {\sqrt {\left|g\left({\frac {d\mathbf {q} }{d\lambda }},{\frac {d\mathbf {q} }{d\lambda }}\right)\right|}}=\int _{\lambda _{1}}^{\lambda _{2}}d\lambda {\sqrt {\left|g_{ij}{\frac {dq^{i}}{d\lambda }}{\frac {dq^{j}}{d\lambda }}\right|}}.}</annotation>

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</p><p>To compute a sensible length of curves in pseudo Riemannian manifolds, it is best to assume that the infinitesimal displacements have the same sign everywhere. E.g. in physics the square of a line element along a timeline curve would (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 -+++}">

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From this point of view, the metric also defines in addition to line element the <a href="Surface_(topology)" title="Surface (topology)">surface</a> and <a href="Volume_element" title="Volume element">volume elements</a> etc.

</p>

<div class="mw-heading mw-heading3"><h3 id="Identification_of_the_square_of_the_line_element_with_the_metric_tensor">Identification of the square of the line element with the metric tensor</h3></div>

<p>Since <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 d\mathbf {q} }">

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</math></span><img src="./5ed55779a508eaf2aaadce7bfa3c185361234763.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.631ex; height:2.509ex;" alt="{\displaystyle d\mathbf {q} }" loading="lazy"></span> is an arbitrary "square of the arc length", <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 ds^{2}}">

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This identification of the square of arc length <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 ds^{2}}">

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</math></span><img src="./dcbeadecfd3d657628269afa2f6a1d20b0bf1cce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.361ex; height:2.676ex;" alt="{\displaystyle ds^{2}}" loading="lazy"></span> with the metric is even more easy to see in <i>n</i>-dimensional general <a href="Curvilinear_coordinates" title="Curvilinear coordinates">curvilinear coordinates</a> <span class="nowrap"><b>q</b> = (<i>q</i><sup>1</sup>, <i>q</i><sup>2</sup>, <i>q</i><sup>3</sup>, ..., <i>q<sup>n</sup></i>)</span>, where it is written as a symmetric rank 2 tensor<sup id="cite_ref-SpiegelLipschutzSpellman_3-1" class="reference"><a href="#cite_note-SpiegelLipschutzSpellman-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> coinciding with the metric tensor:

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<annotation encoding="application/x-tex">{\displaystyle ds^{2}=g_{ij}dq^{i}dq^{j}=g.}</annotation>

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</p><p>Here the <a href="Ricci_calculus" title="Ricci calculus">indices</a> <i>i</i> and <i>j</i> take values 1, 2, 3, ..., <i>n</i> and <a href="Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a> is used. Common examples of (pseudo-) Riemannian spaces include <a href="Three-dimensional" class="mw-redirect" title="Three-dimensional">three-dimensional</a> <a href="Space" title="Space">space</a> (no inclusion of <a href="Time" title="Time">time</a> coordinates), and indeed <a href="Four-dimensional" class="mw-redirect" title="Four-dimensional">four-dimensional</a> <a href="Spacetime" title="Spacetime">spacetime</a>.

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<div class="mw-heading mw-heading2"><h2 id="Line_elements_in_Euclidean_space">Line elements in Euclidean space</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Euclidean_space" title="Euclidean space">Euclidean space</a></div>

<p>Following are examples of how the line elements are found from the metric.

</p>

<div class="mw-heading mw-heading3"><h3 id="Cartesian_coordinates">Cartesian coordinates</h3></div>

<p>The simplest line element is in <a href="Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> - in which case the metric is just the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>:

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<annotation encoding="application/x-tex">{\displaystyle g_{ij}=\delta _{ij}}</annotation>

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(here <i>i, j</i> = 1, 2, 3 for space) or in <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> form (<i>i</i> denotes row, <i>j</i> denotes column):

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<annotation encoding="application/x-tex">{\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}</annotation>

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</p><p>The general curvilinear coordinates reduce to Cartesian coordinates:

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<annotation encoding="application/x-tex">{\displaystyle (q^{1},q^{2},q^{3})=(x,y,z)\,\Rightarrow \,d\mathbf {r} =(dx,dy,dz)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle ds^{2}=g_{ij}dq^{i}dq^{j}=dx^{2}+dy^{2}+dz^{2}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Orthogonal_curvilinear_coordinates">Orthogonal curvilinear coordinates</h3></div>

<p>For all <a href="Orthogonal_coordinates" title="Orthogonal coordinates">orthogonal coordinates</a> the metric is given by:<sup id="cite_ref-SpiegelLipschutzSpellman_3-2" class="reference"><a href="#cite_note-SpiegelLipschutzSpellman-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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</p><p>for <i>i</i> = 1, 2, 3 are <a href="Curvilinear_coordinates#Orthogonal_curvilinear_coordinates_in_3d" title="Curvilinear coordinates">scale factors</a>, so the square of the line element is:

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</p><p>Some examples of line elements in these coordinates are below.<sup id="cite_ref-Kay_2-1" class="reference"><a href="#cite_note-Kay-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<th>Coordinate system

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<th>Metric

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<th>Line element

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<td><a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian</a>

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<annotation encoding="application/x-tex">{\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\\end{pmatrix}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2}}</annotation>

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<td><a href="Polar_coordinate_system" title="Polar coordinate system">Plane polars</a>

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<mtr>

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</mtable>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle [g_{ij}]={\begin{pmatrix}1&0\\0&r^{2}\\\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./aa1f50d848d23df21508fccee27959a791034aa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.739ex; height:6.176ex;" alt="{\displaystyle [g_{ij}]={\begin{pmatrix}1&0\\0&r^{2}\\\end{pmatrix}}}" loading="lazy"></span>

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<msup>

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<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}}</annotation>

</semantics>

</math></span><img src="./c3df0767ce94841d40bd6193c411754cbe1ff59a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.082ex; height:2.843ex;" alt="{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}}" loading="lazy"></span>

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<tr>

<td><a href="Spherical_coordinate_system" title="Spherical coordinate system">Spherical polars</a>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&r^{2}&0\\0&0&r^{2}\sin ^{2}\theta \\\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./24c23e8cb9b860c90c418b1c8745b59b873f85b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:27.585ex; height:9.509ex;" alt="{\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&r^{2}&0\\0&0&r^{2}\sin ^{2}\theta \\\end{pmatrix}}}" loading="lazy"></span>

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<msup>

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<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta \ ^{2}+r^{2}\sin ^{2}\theta d\varphi ^{2}}</annotation>

</semantics>

</math></span><img src="./7d1be5bf0e4e8f58572723519ba0682b15c19a9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.17ex; height:3.176ex;" alt="{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta \ ^{2}+r^{2}\sin ^{2}\theta d\varphi ^{2}}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Cylindrical_polar_coordinates" class="mw-redirect" title="Cylindrical polar coordinates">Cylindrical polars</a>

</td>

</td>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

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<mtr>

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<mtd>

<msup>

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<mtr>

<mtd>

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<mtd>

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<mtd>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&r^{2}&0\\0&0&1\\\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./277a047ee06a5e0ae85c9597e45d451ff1e833dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:20.87ex; height:9.509ex;" alt="{\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&r^{2}&0\\0&0&1\\\end{pmatrix}}}" loading="lazy"></span>

</td>

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ds^{2}=dr^{2}+r^{2}d\varphi ^{2}+dz^{2}}</annotation>

</semantics>

</math></span><img src="./c52eaea9ac7adac4597f1077857b158eec307c19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.712ex; height:3.176ex;" alt="{\displaystyle ds^{2}=dr^{2}+r^{2}d\varphi ^{2}+dz^{2}}" loading="lazy"></span>

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

<div class="mw-heading mw-heading2"><h2 id="General_curvilinear_coordinates">General curvilinear coordinates</h2></div>

<p>Given an arbitrary basis <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 \{{\hat {b}}_{i}\}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\hat {b}}_{i}\}}</annotation>

</semantics>

</math></span><img src="./34e9c46171bb1a97ea11808ea4343dd70666919f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.287ex; height:3.343ex;" alt="{\displaystyle \{{\hat {b}}_{i}\}}" loading="lazy"></span> of a space of dimension <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="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>, the metric is defined as the inner product of the basis vectors.

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

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</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{ij}=\langle {\hat {b}}_{i},{\hat {b}}_{j}\rangle }</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 1\leq i,j\leq n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\leq i,j\leq n}</annotation>

</semantics>

</math></span><img src="./a261efc4a9aa731385da988d0ccb68a202d522ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.549ex; height:2.509ex;" alt="{\displaystyle 1\leq i,j\leq n}" loading="lazy"></span> and the inner product is with respect to the ambient space (usually its <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 \delta _{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math></span><img src="./fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"></span>)

</p><p>In a coordinate basis <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 {\hat {b}}_{i}={\frac {\partial }{\partial x^{i}}}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {b}}_{i}={\frac {\partial }{\partial x^{i}}}}</annotation>

</semantics>

</math></span><img src="./4423fbafaaa3437a9f0d486fa2b077a041267574.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.344ex; height:5.676ex;" alt="{\displaystyle {\hat {b}}_{i}={\frac {\partial }{\partial x^{i}}}}" loading="lazy"></span>

</p><p>The coordinate basis is a special type of basis that is regularly used in differential geometry.

</p>

<div class="mw-heading mw-heading2"><h2 id="Line_elements_in_4d_spacetime">Line elements in 4d spacetime</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Minkowski_spacetime">Minkowski spacetime</h3></div>

<p>The <a href="Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a> is:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-WheelerMisnerThorne_1-1" class="reference"><a href="#cite_note-WheelerMisnerThorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

<msub>

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</msub>

<mrow>

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<mtd>

</mtd>

<mtd>

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<mtd>

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<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [g_{ij}]=\pm {\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\\\end{pmatrix}}}</annotation>

</semantics>

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

where one sign or the other is chosen, both conventions are used. This applies only for <a href="Flat_spacetime" class="mw-redirect" title="Flat spacetime">flat spacetime</a>. The coordinates are given by the <a href="4-position" class="mw-redirect" title="4-position">4-position</a>:

</mrow>

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =(x^{0},x^{1},x^{2},x^{3})=(ct,\mathbf {r} )\,\Rightarrow \,d\mathbf {x} =(cdt,d\mathbf {r} )}</annotation>

</semantics>

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

</p><p>so the line element is:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle ds^{2}=\pm (c^{2}dt^{2}-d\mathbf {r} \cdot d\mathbf {r} ).}</annotation>

</semantics>

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

</p>

<div class="mw-heading mw-heading3"><h3 id="Schwarzschild_coordinates">Schwarzschild coordinates</h3></div>

<p>In <a href="Schwarzschild_coordinates" title="Schwarzschild coordinates">Schwarzschild coordinates</a> coordinates are <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 \left(t,r,\theta ,\phi \right)}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(t,r,\theta ,\phi \right)}</annotation>

</semantics>

</math></span><img src="./883b2d68ef317adb70e76285ffc4edf6f8bc7ea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.275ex; height:2.843ex;" alt="{\displaystyle \left(t,r,\theta ,\phi \right)}" loading="lazy"></span>, being the general metric of the form:

<msub>

</mrow>

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<mrow>

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<msup>

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</mtd>

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<mtd>

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<mtr>

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<mtd>

<msup>

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<msup>

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<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [g_{ij}]={\begin{pmatrix}-a(r)^{2}&0&0&0\\0&b(r)^{2}&0&0\\0&0&r^{2}&0\\0&0&0&r^{2}\sin ^{2}\theta \\\end{pmatrix}}}</annotation>

</semantics>

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

</p><p>(note the similitudes with the metric in 3D spherical polar coordinates).

</p><p>so the line element is:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

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<msup>

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</msup>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ds^{2}=-a(r)^{2}\,dt^{2}+b(r)^{2}\,dr^{2}+r^{2}\,d\theta ^{2}+r^{2}\sin ^{2}\theta \,d\phi ^{2}.}</annotation>

</semantics>

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

</p>

<div class="mw-heading mw-heading3"><h3 id="General_spacetime">General spacetime</h3></div>

<p>The coordinate-independent definition of the square of the line element d<i>s</i> in <a href="Spacetime#Spacetime_intervals" title="Spacetime">spacetime</a> is:<sup id="cite_ref-WheelerMisnerThorne_1-2" class="reference"><a href="#cite_note-WheelerMisnerThorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

<msup>

</mrow>

</msup>

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</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ds^{2}=d\mathbf {x} \cdot d\mathbf {x} =g(d\mathbf {x} ,d\mathbf {x} )}</annotation>

</semantics>

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

</p><p>In terms of coordinates:

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ds^{2}=g_{\alpha \beta }dx^{\alpha }dx^{\beta }}</annotation>

</semantics>

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

where for this case the indices <span class="texhtml"><i>α</i></span> and <span class="texhtml"><i>β</i></span> run over 0, 1, 2, 3 for spacetime.

</p><p>This is the <a href="Spacetime_interval" class="mw-redirect" title="Spacetime interval">spacetime interval</a> - the measure of separation between two arbitrarily close <a href="Event_(relativity)" title="Event (relativity)">events</a> in <a href="Spacetime" title="Spacetime">spacetime</a>. In <a href="Special_relativity" title="Special relativity">special relativity</a> it is invariant under <a href="Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>. In <a href="General_relativity" title="General relativity">general relativity</a> it is invariant under arbitrary <a href="Inverse_function" title="Inverse function">invertible</a> <a href="Differentiable_function" title="Differentiable function">differentiable</a> <a href="Coordinate_transformations" class="mw-redirect" title="Coordinate transformations">coordinate transformations</a>.

</p>

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

<li><a href="First_fundamental_form" title="First fundamental form">First fundamental form</a></li>

<li><a href="List_of_integration_and_measure_theory_topics" title="List of integration and measure theory topics">List of integration and measure theory topics</a></li>

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

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</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="Volume_element" title="Volume element">Volume element</a></li></ul>

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

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

<li id="cite_note-WheelerMisnerThorne-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-WheelerMisnerThorne_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-WheelerMisnerThorne_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-WheelerMisnerThorne_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7167-0344-0</bdi></span>

</li>

<li id="cite_note-Kay-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kay_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kay_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Tensor Calculus, D.C. Kay, Schaum’s Outlines, McGraw Hill (USA), 1988, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-033484-6</bdi></span>

</li>

<li id="cite_note-SpiegelLipschutzSpellman-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-SpiegelLipschutzSpellman_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SpiegelLipschutzSpellman_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-SpiegelLipschutzSpellman_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-161545-7</bdi></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">An introduction to Tensor Analysis: For Engineers and Applied Scientists, J.R. Tyldesley, Longman, 1975, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-582-44355-5</bdi></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">Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-145545-0</bdi></span>

</li>

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