Micron Document
<!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>ADM-Masse</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/ADM-Masse"> <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-ADM-Masse rootpage-ADM-Masse 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">ADM-Masse</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>Die <b>ADM-Masse</b> (nach <a href="Richard_Arnowitt" title="Richard Arnowitt">Richard <b>A</b>rnowitt</a>, <a href="Stanley_Deser" title="Stanley Deser">Stanley <b>D</b>eser</a> und <a href="Charles_W._Misner" class="mw-redirect" title="Charles W. Misner">Charles W. <b>M</b>isner</a> 1961) ordnet Lösungen der Feldgleichungen der <a href="Allgemeine_Relativit%C3%A4tstheorie" title="Allgemeine Relativitätstheorie">Allgemeinen Relativitätstheorie</a> eine <a href="Masse_(Physik)" title="Masse (Physik)">Masse</a> zu, die an ihrer gravitativen Auswirkung in großem Abstand abgelesen werden kann. Die ADM-Masse ist für <a href="Asymptotisch" class="mw-redirect" title="Asymptotisch">asymptotisch</a> flache <a href="Raumzeit" title="Raumzeit">Raumzeiten</a> definiert.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Sei <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> eine <a href="Asymptotisch_flache_Mannigfaltigkeit" title="Asymptotisch flache Mannigfaltigkeit">asymptotisch flache</a> <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">Riemannsche Mannigfaltigkeit</a> (also ein Raum, dessen <a href="Kr%C3%BCmmung" title="Krümmung">Krümmungstensor</a> im Unendlichen verschwindet) mit <a href="Metrischer_Tensor" title="Metrischer Tensor">Metrik</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><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"></span>. Dann ist die ADM-Masse gegeben durch
</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 m_{ADM}(M,g):=\lim _{R\to \infty }\,{\frac {1}{16\,\pi }}\sum _{\mu ,\nu =1,2,3}\,\,\int _{\partial K_{R}}\left({\frac {\partial }{\partial x_{\mu }}}\,g_{\nu \nu }-{\frac {\partial }{\partial x_{\nu }}}\,g_{\nu \mu }\right)\,\mathrm {d} n^{\mu }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
<mi>D</mi>
<mi>M</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>R</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munder>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>16</mn>
<mspace width="thinmathspace"></mspace>
<mi>π<!-- π --></mi>
</mrow>
</mfrac>
</mrow>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
</mrow>
</munder>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>R</mi>
</mrow>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mi>μ<!-- μ --></mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<msup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m_{ADM}(M,g):=\lim _{R\to \infty }\,{\frac {1}{16\,\pi }}\sum _{\mu ,\nu =1,2,3}\,\,\int _{\partial K_{R}}\left({\frac {\partial }{\partial x_{\mu }}}\,g_{\nu \nu }-{\frac {\partial }{\partial x_{\nu }}}\,g_{\nu \mu }\right)\,\mathrm {d} n^{\mu }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/353a9287645978a4690dcd5b062fab40db97f6d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:68.174ex; height:7.009ex;" alt="{\displaystyle m_{ADM}(M,g):=\lim _{R\to \infty }\,{\frac {1}{16\,\pi }}\sum _{\mu ,\nu =1,2,3}\,\,\int _{\partial K_{R}}\left({\frac {\partial }{\partial x_{\mu }}}\,g_{\nu \nu }-{\frac {\partial }{\partial x_{\nu }}}\,g_{\nu \mu }\right)\,\mathrm {d} n^{\mu }}" loading="lazy"></span>,</dd></dl>
<p>Dabei ist <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_{R}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>R</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K_{R}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e36eac01f2a535e2559c9b18f3401355bc6232bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.453ex; height:2.509ex;" alt="{\displaystyle K_{R}}" loading="lazy"></span> eine Kugel mit Radius <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 R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> und Oberfläche <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 \partial K_{R}\,,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>R</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial K_{R}\,,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45fa96d590d44ce415a9f0c1453837205469361e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.805ex; height:2.509ex;" alt="{\displaystyle \partial K_{R}\,,}" loading="lazy"></span> <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> ist die nach außen zeigende Oberflächennormale.
</p><p>Die ADM-Masse kann also aus metrischen Größen in großer Entfernung von der Materie bestimmt werden. Nach dem <a href="Positive-Masse-Theorem" title="Positive-Masse-Theorem">Positive-Masse-Theorem</a> ist die ADM-Masse positiv, <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_{ADM}>0\,,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
<mi>D</mi>
<mi>M</mi>
</mrow>
</msub>
<mo>&gt;</mo>
<mn>0</mn>
<mspace width="thinmathspace"></mspace>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m_{ADM}&gt;0\,,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/040c956bccd6c494396fa61add71f4c5fadb2e4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.888ex; height:2.509ex;" alt="{\displaystyle m_{ADM}>0\,,}" loading="lazy"></span> wenn die schwache Energiebedingung erfüllt ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Für die <a href="Schwarzschild-Metrik" title="Schwarzschild-Metrik">Schwarzschild-Metrik</a> ist die ADM-Masse <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_{ADM}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
<mi>D</mi>
<mi>M</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m_{ADM}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ef000081aa02de57fb481f36c77ddb72dc0470e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.593ex; height:2.009ex;" alt="{\displaystyle m_{ADM}}" loading="lazy"></span> gleich der Masse des <a href="Schwarzes_Loch" title="Schwarzes Loch">schwarzen Lochs</a>, die man am Schwarzschildradius abliest. Dabei ist überall, außer im Ursprung, Vakuum, das heißt dort verschwindet der <a href="Energie-Impuls-Tensor" title="Energie-Impuls-Tensor">Energie-Impuls-Tensor</a>, also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mu \nu }=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T_{\mu \nu }=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57e631ab50204e2e947f23341ee10e13beca1ac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.713ex; height:2.843ex;" alt="{\displaystyle T_{\mu \nu }=0}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>R. Arnowitt, S. Deser, C. Misner: <i>Coordinate Invariance and Energy Expressions in General Relativity</i>, Phys. Rev. <b>122</b> (1961) 997–1006.</li>
<li><span class="book">Norbert Straumann: <cite class="lang" lang="en" dir="auto" style="font-style:italic">General Relativity</cite>. 2. Auflage. Springer Netherlands, Dordrecht 2013, ISBN 978-94-007-5409-6 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:ADM-Masse&amp;rft.au=Norbert%26%2332%3BStraumann&amp;rft.btitle=General+Relativity&amp;rft.date=2013&amp;rft.edition=2.&amp;rft.genre=book&amp;rft.isbn=9789400754096&amp;rft.place=Dordrecht&amp;rft.pub=Springer+Netherlands" style="display:none">&nbsp;</span></span></li>
<li><span class="book">Robert M. Wald: <cite class="lang" lang="en" dir="auto" style="font-style:italic">General Relativity</cite>. University of Chicago Press, Chicago 1984, ISBN 978-0-226-87033-5 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:ADM-Masse&amp;rft.au=Robert+M.%26%2332%3BWald&amp;rft.btitle=General+Relativity&amp;rft.date=1984&amp;rft.genre=book&amp;rft.isbn=9780226870335&amp;rft.place=Chicago&amp;rft.pub=University+of+Chicago+Press" style="display:none">&nbsp;</span></span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-04-27" href="https://de.wikipedia.org/wiki/?title=ADM-Masse&amp;oldid=255509870">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>