<!DOCTYPE html>
<html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-0 vector-toc-not-available vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-0 skin-theme-clientpref-day vector-sticky-header-enabled" lang="de" dir="ltr"><head>
<meta charset="UTF-8">
<title>Vierertensor</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="icon" type="image/png" href="./_res_/favicon.png">
<link rel="canonical" href="https://de.wikipedia.org/wiki/Vierertensor"> <link href="./_mw_/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.wikimediamessages.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./_mw_/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/skins.vector.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link href="./_mw_/ext.gadget.citeRef.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.defaultPlainlinks.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiCommonHide.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiCommonLayout.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiCommonStyle.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiDarkmode.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiResponsive.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.specialSearch.css" rel="stylesheet" type="text/css">
<link rel="stylesheet" type="text/css" href="./_mw_/site.styles.css">
<link rel="stylesheet" type="text/css" href="./_mw_/noscript.css">
<link rel="stylesheet" type="text/css" href="./_res_/footer.css">
<link rel="stylesheet" type="text/css" href="./_res_/vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Vierertensor rootpage-Vierertensor skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Vierertensor</span></h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="contentSub">
<div id="mw-content-subtitle"></div>
</div>
<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"><p><b>Vierertensor</b> ist ein Begriff aus der <a href="Relativit%C3%A4tstheorie" title="Relativitätstheorie">Relativitätstheorie</a>. Ein Vierertensor ist ein <a href="Tensor" title="Tensor">Tensor</a> über dem <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">4-dimensionalen</a> <a href="Vektorraum" title="Vektorraum">Vektorraum</a> der <a href="Minkowskiraum" class="mw-redirect" title="Minkowskiraum">Minkowski-Raum-Zeit</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> und seinem <a href="Dualraum" title="Dualraum">Dualraum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{*}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M^{*}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6d760e72a9f5f578f8ba166127f0713d56dc589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.553ex; height:2.343ex;" alt="{\displaystyle M^{*}}" loading="lazy"></span>, oder in der <a href="Allgemeine_Relativit%C3%A4tstheorie" title="Allgemeine Relativitätstheorie">Allgemeinen Relativitätstheorie</a> über dem <a href="Tangentialraum" title="Tangentialraum">Tangentialraum</a> an die <a href="Raumzeit" title="Raumzeit">Raumzeit</a>, eine vierdimensionale <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">Riemannsche Mannigfaltigkeit</a>.
</p><p>Ein Vierertensor der Stufe <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 (k,l)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>l</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (k,l)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6596d38bbba96f6eda9c984af791cdd7790a802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.748ex; height:2.843ex;" alt="{\displaystyle (k,l)}" loading="lazy"></span> ist ein Element des <a href="Tensorprodukt" title="Tensorprodukt">Tensorprodukts</a>
</p>
<dl><dd><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 \underbrace {M\otimes M\otimes \dotsb \otimes M} _{k{\text{ mal}}}\otimes \underbrace {M^{*}\otimes M^{*}\otimes \dotsb \otimes M^{*}} _{l{\text{ mal}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mrow>
<mi>M</mi>
<mo>⊗<!-- ⊗ --></mo>
<mi>M</mi>
<mo>⊗<!-- ⊗ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>⊗<!-- ⊗ --></mo>
<mi>M</mi>
</mrow>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext> mal</mtext>
</mrow>
</mrow>
</munder>
<mo>⊗<!-- ⊗ --></mo>
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mrow>
<msup>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
<mo>⊗<!-- ⊗ --></mo>
<msup>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
<mo>⊗<!-- ⊗ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>⊗<!-- ⊗ --></mo>
<msup>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
</mrow>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext> mal</mtext>
</mrow>
</mrow>
</munder>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \underbrace {M\otimes M\otimes \dotsb \otimes M} _{k{\text{ mal}}}\otimes \underbrace {M^{*}\otimes M^{*}\otimes \dotsb \otimes M^{*}} _{l{\text{ mal}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1dcf5f0966a5f053305fe0583516ca35f5589a5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.094ex; margin-right: -0.028ex; width:43.437ex; height:6.009ex;" alt="{\displaystyle \underbrace {M\otimes M\otimes \dotsb \otimes M} _{k{\text{ mal}}}\otimes \underbrace {M^{*}\otimes M^{*}\otimes \dotsb \otimes M^{*}} _{l{\text{ mal}}}}" loading="lazy"></span></dd></dl>
<p>Ein solcher Tensor der Stufe <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 (k,l)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>l</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (k,l)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6596d38bbba96f6eda9c984af791cdd7790a802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.748ex; height:2.843ex;" alt="{\displaystyle (k,l)}" loading="lazy"></span> heißt <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 k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-fach kontravariant und <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 l}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"></span>-fach kovariant. Vierertensoren der Stufe <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 (1,0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (1,0)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b53cc1773694affcc1d4d6c2c778d43156a1206.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,0)}" loading="lazy"></span> bzw. <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 (0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}" loading="lazy"></span> heißen auch kontravariante bzw. kovariante <a href="Vierervektor" title="Vierervektor">Vierervektoren</a>.
</p><p>Vierertensoren erster Stufe lassen sich durch einen Vektor mit vier Einträgen darstellen. Beispiele:
</p>
<ul><li><a href="Vierergeschwindigkeit" class="mw-redirect" title="Vierergeschwindigkeit">Vierergeschwindigkeit</a></li>
<li><a href="Viererimpuls" title="Viererimpuls">Viererimpuls</a></li></ul>
<p>Vierertensoren zweiter Stufe lassen sich durch eine <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 4\times 4}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>4</mn>
<mo>×<!-- × --></mo>
<mn>4</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 4\times 4}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89eb2e0f4ddfe5f30c8016a0f2aa1fb5ecedfe20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 4\times 4}" loading="lazy"></span> <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> darstellen. Beispiele:
</p>
<ul><li><a href="Metrischer_Tensor" title="Metrischer Tensor">metrischer Tensor</a></li>
<li><a href="Elektromagnetischer_Feldst%C3%A4rketensor" title="Elektromagnetischer Feldstärketensor">elektromagnetischer Feldstärketensor</a></li>
<li><a href="Energie-Impuls-Tensor" title="Energie-Impuls-Tensor">Energie-Impuls-Tensor</a></li></ul>
<p>Ein Vierertensor vierter Stufe lässt sich durch <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 4^{4}=256}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>4</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>256</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 4^{4}=256}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/452a3f9eeda2b4556dbf80a2bac1bdd9e716eb8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.803ex; height:2.676ex;" alt="{\displaystyle 4^{4}=256}" loading="lazy"></span> Einträge darstellen. Beispiel:
</p>
<ul><li><a href="Riemannscher_Kr%C3%BCmmungstensor" title="Riemannscher Krümmungstensor">Riemannscher Krümmungstensor</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://www.spektrum.de/lexikon/physik/vierertensor/15254">Vierertensor</a> im Lexikon der Physik</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2021-07-04" href="https://de.wikipedia.org/wiki/?title=Vierertensor&oldid=213541869">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
<script src="./_webp_/webpHandler.js"></script>
</body></html>