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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Positive-Masse-Theorem</span></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"><p>Das <b>Positive-Masse-Theorem</b> aus der <a href="Allgemeine_Relativit%C3%A4tstheorie" title="Allgemeine Relativitätstheorie">allgemeinen Relativitätstheorie</a> besagt, dass die Gesamtmasse eines isolierten Gravitationssystems mit nicht-negativer, nicht verschwindender lokaler Massendichte stets positiv ist.
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<div class="mw-heading mw-heading2"><h2 id="Mathematische_Formulierung">Mathematische Formulierung</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}">
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<mi>M</mi>
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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
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</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="Vollst%C3%A4ndige_Riemannsche_Mannigfaltigkeit" class="mw-redirect" title="Vollständige Riemannsche Mannigfaltigkeit">vollständige</a>, <a href="Asymptotisch_flache_Mannigfaltigkeit" title="Asymptotisch flache Mannigfaltigkeit">asymptotisch flache Mannigfaltigkeit</a> mit nicht-negativer <a href="Skalarkr%C3%BCmmung" class="mw-redirect" title="Skalarkrümmung">Skalarkrümmung</a>. Dann ist die <a href="ADM-Masse" title="ADM-Masse">ADM-Masse</a> nicht-negativ.
</p><p>Wenn die ADM-Masse null ist, zum Beispiel wenn <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}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>M</mi>
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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
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</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> außerhalb eines <a href="Kompaktum" class="mw-redirect" title="Kompaktum">Kompaktums</a> <a href="Flache_Mannigfaltigkeit" title="Flache Mannigfaltigkeit">flach</a> ist, dann 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 M}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>M</mi>
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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
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</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> <a href="Isometrie_(Riemannsche_Geometrie)" title="Isometrie (Riemannsche Geometrie)">isometrisch</a> zum <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</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 \mathbb {R} ^{n}}">
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<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span>.
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<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Richard_Schoen" title="Richard Schoen">Richard Schoen</a>, <a href="Shing-Tung_Yau" title="Shing-Tung Yau">Shing-Tung Yau</a>: <i>Incompressible minimal surfaces, three-dimensional manifolds with nonnegative scalar curvature, and the positive mass conjecture in general relativity.</i> Proc. Natl. Acad. Sci. USA 75, 2567 (1978)</li>
<li><a href="Edward_Witten" title="Edward Witten">Edward Witten</a>: <i>A new proof of the positive energy theorem.</i> Comm. Math. Phys. 80, 381-402 (1981)</li>
<li>Joachim Lohkamp: <i>Positive scalar curvature in dim ≥ 8.</i> C. R. Math. Acad. Sci. Paris 343, 585-588 (2006)</li>
<li>Schoen, Yau: <i>Positive scalar curvature and minimal hypersurface singularities.</i> ArXiv 1704.05490</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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