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<title>Metrischer Tensor</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Metrischer Tensor</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Der metrische <a href="Tensor" title="Tensor">Tensor</a> (auch Metriktensor oder Maßtensor) dient dazu, mathematische <a href="Raum_(Mathematik)" title="Raum (Mathematik)">Räume</a>, insbesondere differenzierbare <a href="Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeiten</a>, mit einem Maß für Abstände und Winkel auszustatten.

Dieses Maß muss nicht notwendig alle Bedingungen erfüllen, die in der Definition eines <a href="Metrischer_Raum" title="Metrischer Raum">metrischen Raums</a> an eine Metrik gestellt werden: im <a href="Minkowski-Raum" title="Minkowski-Raum">Minkowski-Raum</a> der <a href="Spezielle_Relativit%C3%A4tstheorie" title="Spezielle Relativitätstheorie">Speziellen Relativitätstheorie</a> gelten diese Bedingungen nur für Abstände, die entweder einheitlich <a href="Raumartig" class="mw-redirect" title="Raumartig">raumartig</a> oder einheitlich <a href="Zeitartig" class="mw-redirect" title="Zeitartig">zeitartig</a> sind.

Für die <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a> und die <a href="Allgemeine_Relativit%C3%A4tstheorie" title="Allgemeine Relativitätstheorie">Allgemeine Relativitätstheorie</a> bedeutsam ist, dass der metrische Tensor, anders als eine über <a href="Innenproduktraum" class="mw-redirect" title="Innenproduktraum">inneres Produkt</a> und <a href="Norm_(Mathematik)" title="Norm (Mathematik)">Norm</a> definierte Metrik, vom Ort abhängen kann.

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

Der metrische Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> über einem <a href="Affiner_Raum" title="Affiner Raum">affinen Punktraum</a> <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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"> mit reellem <a href="Vektorraum" title="Vektorraum">Verschiebungsvektorraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> ist eine Abbildung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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"> in den Raum der Skalarprodukte auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. Das heißt, für jeden Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\in A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1502766dc03ce4fa6667d8a850531b607d270335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.329ex; height:2.176ex;" alt="{\displaystyle P\in A}" loading="lazy"> ist

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<annotation encoding="application/x-tex">{\displaystyle g(P)\colon V\times V\to \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c2e12caaa633201a13b69abe0c0134b469f641d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.412ex; height:2.843ex;" alt="{\displaystyle g(P)\colon V\times V\to \mathbb {R} }" loading="lazy"></dd></dl>

eine <a href="Positiv_definit" class="mw-redirect" title="Positiv definit">positiv definite</a>, <a href="Symmetrische_Bilinearform" class="mw-redirect" title="Symmetrische Bilinearform">symmetrische</a> <a href="Bilinearform" title="Bilinearform">Bilinearform</a>.

In Anlehnung an die Unterscheidung zwischen <a href="Metrischer_Raum" title="Metrischer Raum">Metrik und Pseudometrik</a> wird manchmal auch der Fall betrachtet, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(P)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/759388cc2be36e8c039cf605642bd3ea0c43fc67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.671ex; height:2.843ex;" alt="{\displaystyle g(P)}" loading="lazy"> für einige oder alle Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

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<mrow>

<mrow>

<mover>

</mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle g(P)\left({\vec {x}},\,{\vec {x}}\right)>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed3cf635bc9809d97ea2c7d6f990c4bc8408cef1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.208ex; height:2.843ex;" alt="{\displaystyle g(P)\left({\vec {x}},\,{\vec {x}}\right)>0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq {\vec {x}}\in V}">

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle 0\neq {\vec {x}}\in V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6117054e3e62b85c60b9f0fdfb40e85acf22c59b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.218ex; height:2.843ex;" alt="{\displaystyle 0\neq {\vec {x}}\in V}" loading="lazy"></dd></dl>

wird abgeschwächt zu

<mrow>

<mrow>

<mover>

</mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle g(P)\left({\vec {x}},\,{\vec {x}}\right)\geq 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/441352f8762a4ea781f464fa9f95a4026f094349.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.208ex; height:2.843ex;" alt="{\displaystyle g(P)\left({\vec {x}},\,{\vec {x}}\right)\geq 0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}\in V}">

<mover>

</mover>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a8c20331b5ded219ce471a0a8ca64a25ae89a16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}\in V}" loading="lazy">.</dd></dl>

Ein solcher Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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

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Ein metrischer Tensor definiert eine (vom Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> abhängige) Länge (<a href="Norm_(Mathematik)" title="Norm (Mathematik)">Norm</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|{\vec {x}}\|_{P}}">

<mover>

</mover>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \|{\vec {x}}\|_{P}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d43f688813d0899afdb19bac5a478284b440f96c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.121ex; height:2.843ex;" alt="{\displaystyle \|{\vec {x}}\|_{P}}" loading="lazy"> auf dem Vektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</mrow>

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

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<mover>

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</mrow>

<msub>

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<msqrt>

<mrow>

<mrow>

<mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle \|{\vec {x}}\|_{P}={\sqrt {g(P)\left({\vec {x}},\,{\vec {x}}\right)}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12e016cd3aca54ba068ece930a28345d736b86a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.491ex; height:4.843ex;" alt="{\displaystyle \|{\vec {x}}\|_{P}={\sqrt {g(P)\left({\vec {x}},\,{\vec {x}}\right)}}}" loading="lazy"></dd></dl>

Analog zum <a href="Standardskalarprodukt" title="Standardskalarprodukt">Standardskalarprodukt</a> ist der Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \in [0,\pi ]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta \in [0,\pi ]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c833964ea08aa30df8b6f56664461a5499b38144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.753ex; height:2.843ex;" alt="{\displaystyle \theta \in [0,\pi ]}" loading="lazy"> im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> zwischen zwei Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}},{\vec {y}}\in V}">

<mover>

</mover>

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<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}},{\vec {y}}\in V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a520cbc346b98e92f4414ad770a3dd038aef36b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.256ex; height:2.676ex;" alt="{\displaystyle {\vec {x}},{\vec {y}}\in V}" loading="lazy"> definiert durch:

<mfrac>

<mrow>

<mover>

</mover>

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<mover>

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<mrow>

<msqrt>

<mover>

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</mrow>

<mover>

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</mrow>

</msqrt>

</mrow>

<msqrt>

<mover>

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</mrow>

<mover>

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</mrow>

</mfrac>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {g(P)({\vec {x}},{\vec {y}})}{{\sqrt {g(P)({\vec {x}},{\vec {x}})}}\,{\sqrt {g(P)({\vec {y}},{\vec {y}})}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccbb04ef2023a8fc82e8c7542a19cc1d5360292e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.774ex; height:7.009ex;" alt="{\displaystyle \cos \theta ={\frac {g(P)({\vec {x}},{\vec {y}})}{{\sqrt {g(P)({\vec {x}},{\vec {x}})}}\,{\sqrt {g(P)({\vec {y}},{\vec {y}})}}}}}" loading="lazy"></dd></dl>

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

Wenn ein lokales Koordinatensystem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{i})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x^{i})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/929592bc2ec0e7d26941ecbb51787ef3ef0d6a94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.939ex; height:3.176ex;" alt="{\displaystyle (x^{i})}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> mit Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{i})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (e_{i})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/056380dd5cc97207d5798e36224e0a4ba7103f5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.692ex; height:2.843ex;" alt="{\displaystyle (e_{i})}" loading="lazy"> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> gewählt wird, schreibt man die Komponenten von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}(P)=g(P)(e_{i},e_{j})}">

<msub>

</mrow>

</msub>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle g_{ij}(P)=g(P)(e_{i},e_{j})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ec2b195ea473bce9129bae1f855dd332dc8c303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.63ex; height:3.009ex;" alt="{\displaystyle g_{ij}(P)=g(P)(e_{i},e_{j})}" loading="lazy">. Unter Verwendung der <a href="Einsteinsche_Summenkonvention" title="Einsteinsche Summenkonvention">einsteinschen Summenkonvention</a> ist dann für die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}=x^{i}{\vec {e}}_{i}}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}=x^{i}{\vec {e}}_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f65290835b15c2225df680638fd9f39f2c7f251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.58ex; height:3.009ex;" alt="{\displaystyle {\vec {x}}=x^{i}{\vec {e}}_{i}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {y}}=y^{i}{\vec {e}}_{i}}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msub>

<mover>

</mover>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {y}}=y^{i}{\vec {e}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c758430f04c8f55ed78210ccc25a634c8091343c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.346ex; height:3.009ex;" alt="{\displaystyle {\vec {y}}=y^{i}{\vec {e}}_{i}}" loading="lazy">

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msup>

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</msup>

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle g(P)\left({\vec {x}},\,{\vec {y}}\right)=g_{ij}(P)\,x^{i}\,y^{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/381d598d50d6716d5383dc99e669ca71d50f5c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.095ex; height:3.343ex;" alt="{\displaystyle g(P)\left({\vec {x}},\,{\vec {y}}\right)=g_{ij}(P)\,x^{i}\,y^{j}}" loading="lazy">.</dd></dl>

Im Sinne der <a href="Kategorientheorie" title="Kategorientheorie">Kategorientheorie</a> ist der metrische Tensor kontravariant, da unter (affin) linearen injektiven Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon (A,V)\to (B,W)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon (A,V)\to (B,W)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95fcad581799b3a83e81c837b4d65f2cd7b93937.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.584ex; height:2.843ex;" alt="{\displaystyle \varphi \colon (A,V)\to (B,W)}" loading="lazy"> natürlicherweise aus einem metrischen Tensor auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B,W)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229ed7cfb6894c52404f22a2436e5a2f3b20a5e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.042ex; height:2.843ex;" alt="{\displaystyle (B,W)}" loading="lazy"> ein metrischer Tensor auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,V)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef5a5b8b647073cfa96816af85bf3d2402c0b795.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.374ex; height:2.843ex;" alt="{\displaystyle (A,V)}" loading="lazy"> <a href="R%C3%BCcktransport" title="Rücktransport">konstruiert</a> werden kann,

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<annotation encoding="application/x-tex">{\displaystyle (\varphi ^{*}g)(P)({\vec {x}},{\vec {y}})=g{\bigl (}\varphi (P){\bigr )}{\Bigl (}\varphi _{*}({\vec {x}}),\varphi _{*}({\vec {y}}){\Bigr )}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/698b0ae6dfadc9ba551d71281a823c57c3c6002b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.082ex; height:4.843ex;" alt="{\displaystyle (\varphi ^{*}g)(P)({\vec {x}},{\vec {y}})=g{\bigl (}\varphi (P){\bigr )}{\Bigl (}\varphi _{*}({\vec {x}}),\varphi _{*}({\vec {y}}){\Bigr )}}" loading="lazy">.</dd></dl>

In der <a href="Indexnotation_von_Tensoren" title="Indexnotation von Tensoren">Physik</a> wird der metrische Tensor, oder besser seine Koordinatendarstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7c1130c3dec178129b287a3672c72f88e773832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.586ex; height:2.343ex;" alt="{\displaystyle g_{ij}}" loading="lazy"> als <a href="Kovarianz_(Physik)" title="Kovarianz (Physik)">kovariant</a> bezeichnet, da sich seine Komponenten unter einem Koordinatenwechsel in jedem Index wie die Basis transformieren. Ist ein Koordinatenwechsel als

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<annotation encoding="application/x-tex">{\displaystyle x^{k}=A^{k}{}_{i}\;{\tilde {x}}^{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a52d75285bc1aee2ab6660bcfab6269cda915bad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.923ex; height:3.009ex;" alt="{\displaystyle x^{k}=A^{k}{}_{i}\;{\tilde {x}}^{i}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}^{i}=(A^{-1})^{i}{}_{k}\;x^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}^{i}=(A^{-1})^{i}{}_{k}\;x^{k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa75cd950703a650cef281b4a059fe3c0b188245.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.065ex; height:3.176ex;" alt="{\displaystyle {\tilde {x}}^{i}=(A^{-1})^{i}{}_{k}\;x^{k}}" loading="lazy"></dd></dl>

gegeben, so transformieren sich Basisvektoren als

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {e}}_{i}=A^{k}{}_{i}\;e_{k}=(A^{T})_{i}{}^{k}\;e_{k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf83fa4405d24d4ce2988f37c3b2f04d2201f174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.385ex; height:3.176ex;" alt="{\displaystyle {\tilde {e}}_{i}=A^{k}{}_{i}\;e_{k}=(A^{T})_{i}{}^{k}\;e_{k}}" loading="lazy"></dd></dl>

und es gilt für den metrischen Tensor

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {g}}_{ij}=g(P)({\tilde {e}}_{i},\,{\tilde {e}}_{j})=(A^{T})_{i}{}^{k}\,(A^{T})_{j}{}^{l}\;g_{kl}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0efc270a11a7f6369aaa564a4f0b4272b40fe6c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.87ex; height:3.509ex;" alt="{\displaystyle {\tilde {g}}_{ij}=g(P)({\tilde {e}}_{i},\,{\tilde {e}}_{j})=(A^{T})_{i}{}^{k}\,(A^{T})_{j}{}^{l}\;g_{kl}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Länge_von_Kurven">Länge von Kurven</h2></div>

Ist eine differenzierbare Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \colon [a,b]\to A}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma \colon [a,b]\to A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1621bf7c390f455391250ba10583f0a7ccb0f4f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.208ex; height:2.843ex;" alt="{\displaystyle \gamma \colon [a,b]\to A}" loading="lazy"> im affinen Punktraum gegeben, so hat diese in jedem Zeitpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> einen Tangentialvektor

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(t)={\dot {\gamma }}(t)={\frac {\mathrm {d} }{\mathrm {d} t}}\gamma (t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20c42143f1736fd341cb10d1e7183a6cb570b9cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.966ex; height:5.509ex;" alt="{\displaystyle {\vec {x}}(t)={\dot {\gamma }}(t)={\frac {\mathrm {d} }{\mathrm {d} t}}\gamma (t)}" loading="lazy">.</dd></dl>

Der gesamten Kurve oder einem Segment davon kann man nun mit Hilfe des metrischen Tensors eine Länge

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<annotation encoding="application/x-tex">{\displaystyle L_{[a,b]}(\gamma )=\int _{a}^{b}{\sqrt {g{\bigl (}\gamma (t){\bigr )}{\Bigl (}\,{\vec {x}}(t),\,{\vec {x}}(t)\,{\Bigr )}}}\,\mathrm {d} t=\int _{a}^{b}\|{\dot {\gamma }}(t)\|_{\gamma (t)}\,\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ebd854b6e8e55e39b5d214747221638cfdad91c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.29ex; height:6.509ex;" alt="{\displaystyle L_{[a,b]}(\gamma )=\int _{a}^{b}{\sqrt {g{\bigl (}\gamma (t){\bigr )}{\Bigl (}\,{\vec {x}}(t),\,{\vec {x}}(t)\,{\Bigr )}}}\,\mathrm {d} t=\int _{a}^{b}\|{\dot {\gamma }}(t)\|_{\gamma (t)}\,\mathrm {d} t}" loading="lazy"></dd></dl>

zuordnen.

<div class="mw-heading mw-heading3"><h3 id="Linienelement">Linienelement</h3></div>

Der Ausdruck

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=g_{ij}\mathrm {d} x^{i}\mathrm {d} x^{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efea23d453fd97097120e5988bb96f25fe7fd17f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.076ex; height:3.343ex;" alt="{\displaystyle \mathrm {d} s^{2}=g_{ij}\mathrm {d} x^{i}\mathrm {d} x^{j}}" loading="lazy">,</dd></dl>

wieder unter der Verwendung der Summenkonvention, heißt Linienelement. Substituiert man gemäß der Kettenregel

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x^{i}={\frac {\mathrm {d} x^{i}}{\mathrm {d} t}}\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e70674c1b2aabd6d8e5256314ba1c797f82b0877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.91ex; height:5.843ex;" alt="{\displaystyle \mathrm {d} x^{i}={\frac {\mathrm {d} x^{i}}{\mathrm {d} t}}\mathrm {d} t}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x^{j}={\frac {\mathrm {d} x^{j}}{\mathrm {d} t}}\mathrm {d} t}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x^{j}={\frac {\mathrm {d} x^{j}}{\mathrm {d} t}}\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a914113d115e2ad8c03989068d21ddd2f8adce8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.131ex; height:5.843ex;" alt="{\displaystyle \mathrm {d} x^{j}={\frac {\mathrm {d} x^{j}}{\mathrm {d} t}}\mathrm {d} t}" loading="lazy">,</dd></dl>

so ergibt sich

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=g_{ij}{\frac {\mathrm {d} x^{i}}{\mathrm {d} t}}{\frac {\mathrm {d} x^{j}}{\mathrm {d} t}}\mathrm {d} t^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1407900c3a727ab3d4722223982b73fcd8e450dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.934ex; height:5.843ex;" alt="{\displaystyle \mathrm {d} s^{2}=g_{ij}{\frac {\mathrm {d} x^{i}}{\mathrm {d} t}}{\frac {\mathrm {d} x^{j}}{\mathrm {d} t}}\mathrm {d} t^{2}}" loading="lazy">.</dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5496d5dfa7c75b160324a51bc702b3c7abfdb717.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.383ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} s}" loading="lazy"> ist daher der Integrand des obigen Integrals zur Bestimmung einer <a href="L%C3%A4nge_(Mathematik)#Längen_von_Kurven" title="Länge (Mathematik)">Kurvenlänge</a>.

<div class="mw-heading mw-heading2"><h2 id="Induzierter_Metriktensor">Induzierter Metriktensor</h2></div>

Hat man eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-dimensionale <a href="Untermannigfaltigkeit" title="Untermannigfaltigkeit">Untermannigfaltigkeit</a> eines <a href="Riemannscher_Raum" class="mw-redirect" title="Riemannscher Raum">riemannschen Raumes</a> mit der Metrik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g_{ij})}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa25b58cc122d18cf3013026c56dc45af0f47e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.395ex; height:3.009ex;" alt="{\displaystyle (g_{ij})}" loading="lazy">, die mittels der Parameterdarstellung

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<annotation encoding="application/x-tex">{\displaystyle q^{i}=q^{i}(t^{1},t^{2},\dots ,t^{p}),\qquad i=1,\dots ,n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60c394d19a41c9bea7b6be60bf5591fb009c9231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.88ex; height:3.176ex;" alt="{\displaystyle q^{i}=q^{i}(t^{1},t^{2},\dots ,t^{p}),\qquad i=1,\dots ,n}" loading="lazy"></dd></dl>

gegeben ist, wird eine Metrik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{\alpha \beta })}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b56d36b8d8070935d42c28b94b99eaf291a31b6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.265ex; height:3.009ex;" alt="{\displaystyle (a_{\alpha \beta })}" loading="lazy"> induziert. Die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{\alpha }}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f204e6a80be02eeb577bbe32ba1683cd95b84be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.124ex; height:2.343ex;" alt="{\displaystyle t^{\alpha }}" loading="lazy"> nennt man induzierte Koordinaten. Betrachtet man eine Kurve

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<annotation encoding="application/x-tex">{\displaystyle t^{\alpha }=t^{\alpha }(t),\qquad a\leq t\leq b,\qquad \alpha =1,\dots ,p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/302fb5d84d0dbbf5ebe291f48d146b02aae95b44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.713ex; height:2.843ex;" alt="{\displaystyle t^{\alpha }=t^{\alpha }(t),\qquad a\leq t\leq b,\qquad \alpha =1,\dots ,p}" loading="lazy"></dd></dl>

auf dieser Teilmannigfaltigkeit, so erhält man für die Bogenlänge gemäß der <a href="Kettenregel" title="Kettenregel">Kettenregel</a>

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<annotation encoding="application/x-tex">{\displaystyle s=\int _{a}^{b}{\sqrt {g_{ij}{\frac {\mathrm {d} q^{i}}{\mathrm {d} t}}{\frac {\mathrm {d} q^{j}}{\mathrm {d} t}}}}\mathrm {d} t=\int _{a}^{b}{\sqrt {g_{ij}{\frac {\partial q^{i}}{\partial t^{\alpha }}}{\frac {\mathrm {d} t^{\alpha }}{\mathrm {d} t}}{\frac {\partial q^{j}}{\partial t^{\beta }}}{\frac {\mathrm {d} t^{\beta }}{\mathrm {d} t}}}}\mathrm {d} t=\int _{a}^{b}{\sqrt {g_{ij}{\frac {\partial q^{i}}{\partial t^{\alpha }}}{\frac {\partial q^{j}}{\partial t^{\beta }}}{\frac {\mathrm {d} t^{\alpha }}{\mathrm {d} t}}{\frac {\mathrm {d} t^{\beta }}{\mathrm {d} t}}}}\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdc76c25a12c7d225fad05595330e3a25b8a1156.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:84.686ex; height:7.676ex;" alt="{\displaystyle s=\int _{a}^{b}{\sqrt {g_{ij}{\frac {\mathrm {d} q^{i}}{\mathrm {d} t}}{\frac {\mathrm {d} q^{j}}{\mathrm {d} t}}}}\mathrm {d} t=\int _{a}^{b}{\sqrt {g_{ij}{\frac {\partial q^{i}}{\partial t^{\alpha }}}{\frac {\mathrm {d} t^{\alpha }}{\mathrm {d} t}}{\frac {\partial q^{j}}{\partial t^{\beta }}}{\frac {\mathrm {d} t^{\beta }}{\mathrm {d} t}}}}\mathrm {d} t=\int _{a}^{b}{\sqrt {g_{ij}{\frac {\partial q^{i}}{\partial t^{\alpha }}}{\frac {\partial q^{j}}{\partial t^{\beta }}}{\frac {\mathrm {d} t^{\alpha }}{\mathrm {d} t}}{\frac {\mathrm {d} t^{\beta }}{\mathrm {d} t}}}}\mathrm {d} t}" loading="lazy">.</dd></dl>

Die Größe

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<annotation encoding="application/x-tex">{\displaystyle a_{\alpha \beta }:=g_{ij}{\frac {\partial q^{i}}{\partial t^{\alpha }}}{\frac {\partial q^{j}}{\partial t^{\beta }}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ee8ab953acdafbafc2647cb1ab61b30da083dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.233ex; height:6.009ex;" alt="{\displaystyle a_{\alpha \beta }:=g_{ij}{\frac {\partial q^{i}}{\partial t^{\alpha }}}{\frac {\partial q^{j}}{\partial t^{\beta }}}}" loading="lazy"></dd></dl>

ist der induzierte Metriktensor. Mit diesem ergibt sich die Kurvenlänge schließlich als

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<annotation encoding="application/x-tex">{\displaystyle s=\int _{a}^{b}{\sqrt {a_{\alpha \beta }{\frac {\mathrm {d} t^{\alpha }}{\mathrm {d} t}}{\frac {\mathrm {d} t^{\beta }}{\mathrm {d} t}}}}\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dfefa6cd49dc081153608318945ab164dda80d29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.284ex; height:7.676ex;" alt="{\displaystyle s=\int _{a}^{b}{\sqrt {a_{\alpha \beta }{\frac {\mathrm {d} t^{\alpha }}{\mathrm {d} t}}{\frac {\mathrm {d} t^{\beta }}{\mathrm {d} t}}}}\mathrm {d} t}" loading="lazy">.</dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Euklidischer_Raum">Euklidischer Raum</h3></div>

In einem <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</a> mit <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinaten</a> ist der metrische Tensor durch die <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3704c69da4961f99cd8d8e9abaf9c9f5f607bb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.194ex; height:3.009ex;" alt="{\displaystyle g_{ij}=\delta _{ij}}" loading="lazy"></dd></dl>

gegeben. Im euklidischen Raum ist nämlich das Skalarprodukt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \langle x,y\rangle =\sum _{i=1}^{n}x^{i}y^{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \langle x,y\rangle =\sum _{i=1}^{n}x^{i}y^{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ec2a347fb73a945e14b17a12dab1057b9915e21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.258ex; height:3.176ex;" alt="{\displaystyle \textstyle \langle x,y\rangle =\sum _{i=1}^{n}x^{i}y^{i}}" loading="lazy"> gegeben und nach Voraussetzung soll der metrische Tensor diesem Skalarprodukt entsprechen.

Also gilt für diesen in lokalen Koordinaten

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<annotation encoding="application/x-tex">{\displaystyle g_{ij}=\langle e_{i},e_{j}\rangle =\delta _{ij},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cbf600e956c52adeb38dd151ccabfccd57fd3ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.659ex; height:3.009ex;" alt="{\displaystyle g_{ij}=\langle e_{i},e_{j}\rangle =\delta _{ij},}" loading="lazy">

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},\dots ,e_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{1},\dots ,e_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fe2bc876c45938194bb68c2cf977d4948087cd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\dots ,e_{n}}" loading="lazy"> die Vektoren der <a href="Standardbasis" title="Standardbasis">Standardbasis</a> sind.

Für beliebige Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x^{i}e_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cb17eda0e997b5763ca9e7a109dd984e6c9d067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.441ex; height:3.009ex;" alt="{\displaystyle x=x^{i}e_{i}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=y^{j}e_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle y=y^{j}e_{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72d36fba3247237c4a25a3c34fa6b8b0a24ad585.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.317ex; height:3.343ex;" alt="{\displaystyle y=y^{j}e_{j}}" loading="lazy"> des euklidischen Raums gilt

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<annotation encoding="application/x-tex">{\displaystyle g_{ij}\,x^{i}y^{j}=\delta _{ij}x^{i}y^{j}=\sum _{i=1}^{n}x^{i}y^{i}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e17722c9b024d50c713b95c51333feb3de11d1d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.557ex; height:6.843ex;" alt="{\displaystyle g_{ij}\,x^{i}y^{j}=\delta _{ij}x^{i}y^{j}=\sum _{i=1}^{n}x^{i}y^{i}.}" loading="lazy"></dd></dl>

Hier wird die <a href="Einsteinsche_Summenkonvention" title="Einsteinsche Summenkonvention">einsteinsche Summenkonvention</a> verwendet.

Für die Kurvenlänge

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<annotation encoding="application/x-tex">{\displaystyle L=\int _{a}^{b}{\sqrt {\left(\mathrm {d} x\right)^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31cb5d314982bdd0816a58490cb67366290a385f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.28ex; height:6.343ex;" alt="{\displaystyle L=\int _{a}^{b}{\sqrt {\left(\mathrm {d} x\right)^{2}}}}" loading="lazy"></dd></dl>

und den Winkel

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<annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {\mathbf {u} \,\mathbf {v} }{|\mathbf {u} |\cdot |\mathbf {v} |}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ba665aa5cdd1944334a1ee5fd65509049ee9105.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.686ex; height:5.509ex;" alt="{\displaystyle \cos \theta ={\frac {\mathbf {u} \,\mathbf {v} }{|\mathbf {u} |\cdot |\mathbf {v} |}}}" loading="lazy"></dd></dl>

erhält man die üblichen Formeln der <a href="Vektoranalysis" title="Vektoranalysis">Vektoranalysis</a>.

Wenn eine Mannigfaltigkeit in einen euklidischen Raum mit kartesischen Koordinaten eingebettet ist, dann ergibt sich ihr metrischer Tensor aus der <a href="Jacobi-Matrix" title="Jacobi-Matrix">Jacobi-Matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> der Einbettung als

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e98e1b2de32005e35c2c06601a287d9ce8cd127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.248ex; height:3.009ex;" alt="{\displaystyle g=J^{T}J.}" loading="lazy"></dd></dl>

In einigen anderen Koordinatensystemen lautet der metrische Tensor und das Linienelement des Euklidischen Raums wie folgt:

<ul><li>In <a href="Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{1},x^{2})=(r,\theta )}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dc43ef84c19fc21138cf165a0faf41a96a4567c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.692ex; height:3.176ex;" alt="{\displaystyle (x^{1},x^{2})=(r,\theta )}" loading="lazy">:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle g={\begin{bmatrix}1&0\\0&r^{2}\end{bmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc31c35f72f86660dad2c481cc18cd047ab60a90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.009ex; height:6.176ex;" alt="{\displaystyle g={\begin{bmatrix}1&0\\0&r^{2}\end{bmatrix}}}" loading="lazy">, bzw.</dd>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80b99d5e470c5835ebd73c8adbcfb2115ed4fce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.312ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}}" loading="lazy"></dd></dl></dd></dl>

<ul><li>In <a href="Zylinderkoordinaten" class="mw-redirect" title="Zylinderkoordinaten">Zylinderkoordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{1},x^{2},x^{3})=(r,\varphi ,z)}">

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<annotation encoding="application/x-tex">{\displaystyle (x^{1},x^{2},x^{3})=(r,\varphi ,z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19f813373c4dd9c23fad7cd709445c467b2485b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.661ex; height:3.176ex;" alt="{\displaystyle (x^{1},x^{2},x^{3})=(r,\varphi ,z)}" loading="lazy">:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle g={\begin{bmatrix}1&0&0\\0&r^{2}&0\\0&0&1\end{bmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77c96e7de6b429d5e1bb5348d46d0bb0fc62e6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:17.14ex; height:9.509ex;" alt="{\displaystyle g={\begin{bmatrix}1&0&0\\0&r^{2}&0\\0&0&1\end{bmatrix}}}" loading="lazy">, bzw.</dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\mathrm {d} \varphi ^{2}+\mathrm {d} z^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f18b7b3e9de2847b55e145810acfac7848fe25d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.019ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\mathrm {d} \varphi ^{2}+\mathrm {d} z^{2}}" loading="lazy"></dd></dl></dd></dl>

<ul><li>In <a href="Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{1},x^{2},x^{3})=(r,\theta ,\varphi )}">

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<annotation encoding="application/x-tex">{\displaystyle (x^{1},x^{2},x^{3})=(r,\theta ,\varphi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca98a6dc3b77dfe347349d330ef582b67f7f884e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.664ex; height:3.176ex;" alt="{\displaystyle (x^{1},x^{2},x^{3})=(r,\theta ,\varphi )}" loading="lazy">:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle g={\begin{bmatrix}1&0&0\\0&r^{2}&0\\0&0&(r\sin \theta )^{2}\end{bmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a839aa7585739332eab8b9364cd92d6c6db7090e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:24.61ex; height:9.509ex;" alt="{\displaystyle g={\begin{bmatrix}1&0&0\\0&r^{2}&0\\0&0&(r\sin \theta )^{2}\end{bmatrix}}}" loading="lazy">, bzw.</dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}\theta \;\mathrm {d} \varphi ^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34d2812bdbf4944303753369c9ca1c04869c1fb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.929ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}\theta \;\mathrm {d} \varphi ^{2}}" loading="lazy"></dd></dl></dd></dl>

<div style="clear:both; padding-left:0em; padding-right:0" class="mw-collapsible mw-collapsed hintergrundfarbe5">Herleitung für Kugelkoordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \longrightarrow }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c000c43c7ea8069dae7f09e8ded6fc66c58d42a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.128ex; height:1.843ex;" alt="{\displaystyle \quad \longrightarrow }" loading="lazy">

Die <a href="Koordinatentransformation" title="Koordinatentransformation">Koordinatentransformation</a> für die Kugelkoordinaten lautet als Vektorgleichung:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}r\sin \theta \cos \varphi \\r\sin \theta \sin \varphi \\r\cos \theta \end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/495694c012df142c187ec567ea13d5a8aa3b9212.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:29.561ex; height:9.509ex;" alt="{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}r\sin \theta \cos \varphi \\r\sin \theta \sin \varphi \\r\cos \theta \end{pmatrix}}}" loading="lazy">.</dd></dl>

Die lokalen <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basisvektoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}_{1},{\vec {b}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {b}}_{1},{\vec {b}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04f3d39306deb3aaff5c1bf025b21d51f240227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:3.176ex;" alt="{\displaystyle {\vec {b}}_{1},{\vec {b}}_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}_{3}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43761463e071612dba27d83f72d1f26824ce0214.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.148ex; height:3.176ex;" alt="{\displaystyle {\vec {b}}_{3}}" loading="lazy"> verlaufen tangential zu den <a href="Koordinatenlinie" title="Koordinatenlinie">Koordinatenlinien</a> und ergeben sich somit aus der Koordinatentransformation durch partielle Ableitung nach den Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,\theta }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07b5f2263d4272798d6ff1f9ddea1e0cae379323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.173ex; height:2.509ex;" alt="{\displaystyle r,\theta }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy">. Also gilt:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {b}}_{1}={\frac {\partial {\vec {r}}}{\partial r}}={\begin{pmatrix}\sin \theta \cos \varphi \\\sin \theta \sin \varphi \\\cos \theta \end{pmatrix}},\quad {\vec {b}}_{2}={\frac {\partial {\vec {r}}}{\partial \theta }}={\begin{pmatrix}r\cos \theta \cos \varphi \\r\cos \theta \sin \varphi \\-r\sin \theta \end{pmatrix}},\quad {\vec {b}}_{3}={\frac {\partial {\vec {r}}}{\partial \varphi }}={\begin{pmatrix}-r\sin \theta \sin \varphi \\r\sin \theta \cos \varphi \\0\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/348c105ab91afe4df5741ab4bc807b729c8198c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:90.528ex; height:9.509ex;" alt="{\displaystyle {\vec {b}}_{1}={\frac {\partial {\vec {r}}}{\partial r}}={\begin{pmatrix}\sin \theta \cos \varphi \\\sin \theta \sin \varphi \\\cos \theta \end{pmatrix}},\quad {\vec {b}}_{2}={\frac {\partial {\vec {r}}}{\partial \theta }}={\begin{pmatrix}r\cos \theta \cos \varphi \\r\cos \theta \sin \varphi \\-r\sin \theta \end{pmatrix}},\quad {\vec {b}}_{3}={\frac {\partial {\vec {r}}}{\partial \varphi }}={\begin{pmatrix}-r\sin \theta \sin \varphi \\r\sin \theta \cos \varphi \\0\end{pmatrix}}}" loading="lazy">.</dd></dl>

Die Komponenten des metrischen Tensors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=(g_{ij})}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e36129d1e65f26b765daad53fc02c24f1a275aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.61ex; height:3.009ex;" alt="{\displaystyle g=(g_{ij})}" loading="lazy"> sind die Skalarprodukte dieser Basisvektoren:

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<annotation encoding="application/x-tex">{\displaystyle g_{ij}={\vec {b}}_{i}{\vec {b}}_{j}\quad (i,j\in \{1,2,3\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0bf3a6ddfe2f4d3271dedcfe471eba8537177a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.324ex; height:3.509ex;" alt="{\displaystyle g_{ij}={\vec {b}}_{i}{\vec {b}}_{j}\quad (i,j\in \{1,2,3\}}" loading="lazy">.</dd></dl>

Die Rechnung ergibt:

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<annotation encoding="application/x-tex">{\displaystyle g_{11}=1,\quad g_{22}=r^{2},\quad und\quad g_{33}=r^{2}\sin ^{2}\theta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eff21bc3e3dddfd84c3ac1e3484bdb28e815cd64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:42.37ex; height:3.009ex;" alt="{\displaystyle g_{11}=1,\quad g_{22}=r^{2},\quad und\quad g_{33}=r^{2}\sin ^{2}\theta }" loading="lazy">.</dd></dl>

Die übrigen Skalarprodukte sind null. Dies bedeutet, dass die Basisvektoren paarweise aufeinander senkrecht stehen: die Kugelkoordinaten bilden ein orthogonales Koordinatensystem.

Für das Linienelement ergibt sich somit

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}\theta \;\mathrm {d} \varphi ^{2}}</annotation>

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Die Herleitungen für die anderen Koordinatensysteme verlaufen entsprechend.</div></div>

<div class="mw-heading mw-heading3"><h3 id="Minkowski-Raum_(spezielle_Relativitätstheorie)">Minkowski-Raum (spezielle Relativitätstheorie)</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Minkowski-Raum" title="Minkowski-Raum">Minkowski-Raum</a></div>

Der flache Minkowski-Raum der <a href="Spezielle_Relativit%C3%A4tstheorie" title="Spezielle Relativitätstheorie">speziellen Relativitätstheorie</a> beschreibt eine vierdimensionale <a href="Raumzeit" title="Raumzeit">Raum-Zeit</a> ohne <a href="Gravitation" title="Gravitation">Gravitation</a>. Räumliche Abstände und Zeitspannen hängen in diesem Raum von der Wahl eines <a href="Inertialsystem" title="Inertialsystem">Inertialsystems</a> ab; wenn man einen physikalischen Vorgang in zwei verschiedenen, gleichförmig gegeneinander bewegten Inertialsystemen beschreibt, können sie verschiedene Werte annehmen.

Invariant unter <a href="Lorentztransformation" class="mw-redirect" title="Lorentztransformation">Lorentztransformationen</a> ist hingegen der sogenannte Viererabstand, der räumliche und zeitliche Abstände zusammenfasst. Unter Verwendung der <a href="Lichtgeschwindigkeit" title="Lichtgeschwindigkeit">Lichtgeschwindigkeit</a> c berechnet sich dieser Viererabstand aus räumlichem Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \mathbf {r} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f09df207c8e619140a90e8781a22d6f141211341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.395ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \mathbf {r} }" loading="lazy"> und Zeitspanne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\mathrm {d} t}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22822fa24c79dba519e103c80df9d952f9172bef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.519ex; height:2.176ex;" alt="{\displaystyle \,\mathrm {d} t}" loading="lazy"> als

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=c^{2}\,\left(\mathrm {d} t\right)^{2}\,-\left(\mathrm {d} \mathbf {r} \right)^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a9516a9cf80cf29d91ea8969fdbefd7d7f66dcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.465ex; height:3.343ex;" alt="{\displaystyle \mathrm {d} s^{2}=c^{2}\,\left(\mathrm {d} t\right)^{2}\,-\left(\mathrm {d} \mathbf {r} \right)^{2}}" loading="lazy"></dd></dl>

Im Minkowski-Raum wird der kontravariante Orts-Vierervektor definiert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,x^{\mu }=(x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)}">

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<annotation encoding="application/x-tex">{\displaystyle \,x^{\mu }=(x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f9a5c05903c47e94cdd2a36d7dbc44d370289f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.915ex; height:3.176ex;" alt="{\displaystyle \,x^{\mu }=(x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)}" loading="lazy">.

Der metrische Tensor lautet in einer Konvention, die vor allem in der <a href="Quantenfeldtheorie" title="Quantenfeldtheorie">Quantenfeldtheorie</a> verwendet wird (<a href="Signatur_(lineare_Algebra)" class="mw-redirect" title="Signatur (lineare Algebra)">Signatur</a> −2, also +,−,−,−)

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<annotation encoding="application/x-tex">{\displaystyle \eta _{\mu \nu }={\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}}\equiv \operatorname {diag} (1,-1,-1,-1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06d55ad84f4d9758f493b266af6aeac29f4695d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:49.591ex; height:12.509ex;" alt="{\displaystyle \eta _{\mu \nu }={\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}}\equiv \operatorname {diag} (1,-1,-1,-1)}" loading="lazy">.</dd></dl>

In einer Konvention, die hauptsächlich in der <a href="Allgemeine_Relativit%C3%A4tstheorie" title="Allgemeine Relativitätstheorie">Allgemeinen Relativitätstheorie</a> benutzt wird (Signatur +2, also −,+,+,+), schreibt man

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<annotation encoding="application/x-tex">{\displaystyle \eta _{\mu \nu }={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\equiv \operatorname {diag} (-1,1,1,1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f398af7d7c271de8e43207d6bdec548b20375572.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:42.358ex; height:12.509ex;" alt="{\displaystyle \eta _{\mu \nu }={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\equiv \operatorname {diag} (-1,1,1,1)}" loading="lazy">.</dd></dl>

Dabei ist allerdings zu beachten, dass es sich hierbei trotz der allgemein verwendeten Bezeichnung weder um einen metrischen noch um einen pseudometrischen Tensor handelt, weil er nicht positiv (semi-) definit ist, was sofort aus der Signatur hervorgeht. Das heißt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\nu \mu }}">

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</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fa51e6b83add4b6c2ae644e26b3306a5fa014b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.25ex; height:2.343ex;" alt="{\displaystyle \eta _{\nu \mu }}" loading="lazy"> stellt lediglich eine symmetrische Bilinearform bezüglich einer bestimmten Basis dar, keine positiv (semi-)definite symmetrische Bilinearform.

In der Allgemeinen Relativitätstheorie wird der metrische Tensor durch Lösen der <a href="Einsteinsche_Feldgleichungen" title="Einsteinsche Feldgleichungen">Einsteinschen Feldgleichungen</a> gefunden. Er ist ortsabhängig und bildet daher ein <a href="Tensorfeld" title="Tensorfeld">Tensorfeld</a>, da die Krümmung der <a href="Raumzeit" title="Raumzeit">Raumzeit</a> an verschiedenen Punkten meist verschieden ist.

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

<ul><li>Rainer Oloff: <cite style="font-style:italic">Geometrie der Raumzeit: Eine mathematische Einführung in die Relativitätstheorie</cite>. Springer-Verlag, 2013, ISBN 3-322-94260-0.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Metrischer+Tensor&rft.au=Rainer+Oloff&rft.btitle=Geometrie+der+Raumzeit%3A+Eine+mathematische+Einf%C3%BChrung+in+die+Relativit%C3%A4tstheorie&rft.date=2013&rft.genre=book&rft.isbn=3322942600&rft.pub=Springer-Verlag" style="display:none"> </li>

<li>Chris Isham: <cite style="font-style:italic">Modern Differential Geometry for Physicists</cite>. Allied Publishers, 2002, ISBN 81-7764-316-9.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Metrischer+Tensor&rft.au=Chris+Isham&rft.btitle=Modern+Differential+Geometry+for+Physicists&rft.date=2002&rft.genre=book&rft.isbn=8177643169&rft.pub=Allied+Publishers" style="display:none"> </li>

<li>W. Werner: <cite style="font-style:italic">Vektoren und Tensoren als universelle Sprache in Physik und Technik</cite>. Band 1. Springer Vieweg, ISBN 978-3-658-25271-7.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Metrischer+Tensor&rft.au=W.+Werner&rft.btitle=Vektoren+und+Tensoren+als+universelle+Sprache+in+Physik+und+Technik&rft.genre=book&rft.isbn=9783658252717&rft.pub=Springer+Vieweg&rft.volume=1" style="display:none"> </li></ul></div><div><div class="zim-footer">

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