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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Retardiertes Potential</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>retardierte Potential</b> (deutsch: <i>verzögertes Potential</i>, manchmal auch <i>retardierendes Potential</i><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> genannt) ist die Bezeichnung für die mathematische Form des <a href="Potential_(Physik)" title="Potential (Physik)">Potentials</a> in der <a href="Elektromagnetismus" class="mw-redirect" title="Elektromagnetismus">elektromagnetischen Feldtheorie</a> oder anderen <a href="Feldtheorie_(Physik)" title="Feldtheorie (Physik)">Feldtheorien</a>, in denen sich Änderungen des <a href="Feld_(Physik)" title="Feld (Physik)">Feldes</a> mit endlicher <a href="Geschwindigkeit" title="Geschwindigkeit">Geschwindigkeit</a> (<a href="Lichtgeschwindigkeit" title="Lichtgeschwindigkeit">Lichtgeschwindigkeit</a>) und <i>nicht</i> <a href="Instantan" class="mw-redirect" title="Instantan">instantan</a> ausbreiten. Es tritt bei der Untersuchung <a href="Dynamik_(Physik)" title="Dynamik (Physik)">zeitabhängiger Probleme</a> auf, zum Beispiel bei der Abstrahlung <a href="Elektromagnetische_Welle" title="Elektromagnetische Welle">elektromagnetischer Wellen</a>.
</p><p>Dagegen werden in der <a href="Elektrostatik" title="Elektrostatik">Elektrostatik</a>, der <a href="Magnetostatik" title="Magnetostatik">Magnetostatik</a> und der klassischen <a href="Newtonsche_Gravitationstheorie" class="mw-redirect" title="Newtonsche Gravitationstheorie">Newtonschen Gravitationstheorie</a> Zeitabhängigkeiten vernachlässigt.
</p>

<div class="mw-heading mw-heading2"><h2 id="Mathematische_Formulierung">Mathematische Formulierung</h2></div>
<p>Mathematisch ist das Potential <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 \,u(t,\mathbf {x} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,u(t,\mathbf {x} )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03c138c068f6c27d4993f3e4d14a15babb19ecdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.811ex; height:2.843ex;" alt="{\displaystyle \,u(t,\mathbf {x} )}" loading="lazy"></span> die Lösung der (aus den <a href="Maxwellgleichungen" class="mw-redirect" title="Maxwellgleichungen">Maxwellgleichungen</a> folgenden) inhomogenen <a href="Wellengleichung" title="Wellengleichung">Wellengleichung</a> in drei Raumdimensionen
</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 {\frac {1}{c^{2}}}\,{\frac {\partial ^{2}u}{\partial t^{2}}}-\sum _{i=1}^{3}\left({\frac {\partial ^{2}u}{\partial x_{i}^{2}}}\right)={\frac {1}{c^{2}}}\,{\frac {\partial ^{2}u}{\partial t^{2}}}-\Delta u=\Box u(t,\mathbf {x} )=v(t,x_{1},x_{2},x_{3})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>u</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</munderover>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>u</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>u</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>u</mi>
<mo>=</mo>
<mi>◻<!-- ◻ --></mi>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{c^{2}}}\,{\frac {\partial ^{2}u}{\partial t^{2}}}-\sum _{i=1}^{3}\left({\frac {\partial ^{2}u}{\partial x_{i}^{2}}}\right)={\frac {1}{c^{2}}}\,{\frac {\partial ^{2}u}{\partial t^{2}}}-\Delta u=\Box u(t,\mathbf {x} )=v(t,x_{1},x_{2},x_{3})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccf370c189518f608b2fafc3dd4d22a290c5b333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.185ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{c^{2}}}\,{\frac {\partial ^{2}u}{\partial t^{2}}}-\sum _{i=1}^{3}\left({\frac {\partial ^{2}u}{\partial x_{i}^{2}}}\right)={\frac {1}{c^{2}}}\,{\frac {\partial ^{2}u}{\partial t^{2}}}-\Delta u=\Box u(t,\mathbf {x} )=v(t,x_{1},x_{2},x_{3})}" loading="lazy"></span>,</dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =\nabla ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo>=</mo>
<msup>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta =\nabla ^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30822e8684827364a4f67d1125a277cd3328ba42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.024ex; height:2.676ex;" alt="{\displaystyle \Delta =\nabla ^{2}}" loading="lazy"></span> für den <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</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 \Box }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>◻<!-- ◻ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Box }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/029b77f09ebeaf7528fc831fe57848be51f2240b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }" loading="lazy"></span> für den <a href="D%E2%80%99Alembert-Operator" title="D’Alembert-Operator">D’Alembert-Operator</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 c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> für die <a href="Wellengeschwindigkeit" title="Wellengeschwindigkeit">Wellengeschwindigkeit</a> und auf der rechten Seite <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 \,v(t,\mathbf {x} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,v(t,\mathbf {x} )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb868a1081cc1531dbd272d8d344e403f923835f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.609ex; height:2.843ex;" alt="{\displaystyle \,v(t,\mathbf {x} )}" loading="lazy"></span> für einen Quellenterm stehen.
</p><p>Die Lösung
</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 u_{\text{retardiert}}(t,\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}\!\,{\frac {v\left(t-{\frac {|\mathbf {x} -\mathbf {y} |}{c}},\mathbf {y} \right)}{|\mathbf {x} -\mathbf {y} |}}\,\mathrm {d} ^{3}\mathbf {y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>retardiert</mtext>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>4</mn>
<mi>π<!-- π --></mi>
</mrow>
</mfrac>
</mrow>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mrow>
</msub>
<mspace width="negativethinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
<mi>c</mi>
</mfrac>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u_{\text{retardiert}}(t,\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}\!\,{\frac {v\left(t-{\frac {|\mathbf {x} -\mathbf {y} |}{c}},\mathbf {y} \right)}{|\mathbf {x} -\mathbf {y} |}}\,\mathrm {d} ^{3}\mathbf {y} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb96d87b5d153d281144672a4975bb3b05371503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.689ex; height:8.343ex;" alt="{\displaystyle u_{\text{retardiert}}(t,\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}\!\,{\frac {v\left(t-{\frac {|\mathbf {x} -\mathbf {y} |}{c}},\mathbf {y} \right)}{|\mathbf {x} -\mathbf {y} |}}\,\mathrm {d} ^{3}\mathbf {y} }" loading="lazy"></span></dd></dl>
<p>heißt <b>retardiertes Potential</b>, weil die Funktion <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 v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> nicht zum Zeitpunkt <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><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"></span> genommen wird, sondern zu einem früheren Zeitpunkt. Nimmt man für die Wellengeschwindigkeit <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 c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> die <a href="Lichtgeschwindigkeit" title="Lichtgeschwindigkeit">Lichtgeschwindigkeit</a> an, so hängt diese Lösung am Ort <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> zur Zeit <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}">
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<mi>t</mi>
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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
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</math></span><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"></span> nur von der Inhomogenität <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 v}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>v</mi>
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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> <i>auf</i> dem <a href="R%C3%BCckw%C3%A4rtslichtkegel" class="mw-redirect" title="Rückwärtslichtkegel">Rückwärtslichtkegel</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">
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<mi mathvariant="bold">x</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> ab. Die Inhomogenität wirkt sich auf die Lösung verspätet mit Lichtgeschwindigkeit aus.
</p><p>Die Lösung
</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 u_{\text{avanciert}}(t,\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}\!\,{\frac {v\left(t+{\frac {|\mathbf {x} -\mathbf {y} |}{c}},\mathbf {y} \right)}{|\mathbf {x} -\mathbf {y} |}}\,\mathrm {d} ^{3}\mathbf {y} }">
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<mi>u</mi>
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<mtext>avanciert</mtext>
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<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
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<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle u_{\text{avanciert}}(t,\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}\!\,{\frac {v\left(t+{\frac {|\mathbf {x} -\mathbf {y} |}{c}},\mathbf {y} \right)}{|\mathbf {x} -\mathbf {y} |}}\,\mathrm {d} ^{3}\mathbf {y} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0daabed340477c1e743b14c3fc78e5978d5e0e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.45ex; height:8.343ex;" alt="{\displaystyle u_{\text{avanciert}}(t,\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}\!\,{\frac {v\left(t+{\frac {|\mathbf {x} -\mathbf {y} |}{c}},\mathbf {y} \right)}{|\mathbf {x} -\mathbf {y} |}}\,\mathrm {d} ^{3}\mathbf {y} }" loading="lazy"></span></dd></dl>
<p>heißt entsprechend avanciertes Potential, weil die Funktion <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 v}">
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<mi>v</mi>
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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> nicht zum Zeitpunkt <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}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>t</mi>
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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><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"></span> genommen wird, sondern zu einem späteren Zeitpunkt. Dies beschreibt zum Beispiel eine Senke, die ein bestehendes Feld absorbiert.
</p><p>Mit retardiertem und avanciertem Potential lassen sich somit Emission und Absorption von Feldern beschreiben.
</p>
<div class="mw-heading mw-heading2"><h2 id="Anwendungsbeispiele">Anwendungsbeispiele</h2></div>
<p>In der <a href="Elektrodynamik" title="Elektrodynamik">Elektrodynamik</a> müssen retardierte Potentiale zum Beispiel in Form der <a href="Li%C3%A9nard-Wiechert-Potential" title="Liénard-Wiechert-Potential">Liénard-Wiechert-Potentiale</a> bei der Erzeugung von <a href="Synchrotronstrahlung" title="Synchrotronstrahlung">Synchrotronstrahlung</a> berücksichtigt werden.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>In der <a href="Gravitation" title="Gravitation">Gravitation</a> gibt es Anwendungsbeispiele zur Berechnung von Abweichungen bei <a href="Umlaufbahn" title="Umlaufbahn">Umlaufbahnen</a> von <a href="Satellitenorbit" title="Satellitenorbit">Satelliten</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>, <a href="Mondbahn" title="Mondbahn">Monden</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> oder <a href="Planetenbahn" title="Planetenbahn">Planeten</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p><p>Laut neueren Arbeiten<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> liefert die Berücksichtigung der retardierten Gravitationspotenziale der sich ändernden Masseverteilungen in <a href="Galaxie" title="Galaxie">Galaxientypen</a> aller Art eine gute Übereinstimmung mit den beobachteten <a href="Rotationskurve" title="Rotationskurve">Rotationskurven</a>. Dadurch könnte das Verhalten von Galaxien erklärt werden, ohne hypothetische <a href="Dunkle_Materie" title="Dunkle Materie">Dunkle Materie</a> berücksichtigen oder eine <a href="Modifizierte_Newtonsche_Dynamik" title="Modifizierte Newtonsche Dynamik">modifizierte Newtonsche Dynamik</a> annehmen zu müssen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Richard_Courant" title="Richard Courant">Richard Courant</a> und <a href="David_Hilbert" title="David Hilbert">David Hilbert</a>: <i>Methoden der mathematischen Physik Band 2.</i> zweite Auflage, Springer Verlag, 1968</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li>Norbert Dragon, <a rel="nofollow" class="external text" href="https://www.itp.uni-hannover.de/fileadmin/arbeitsgruppen/dragon/rech.pdf"><i>Stichworte und Ergänzungen zu Rechenmethoden der Physik</i></a> (PDF; 1,9&nbsp;MB), Kapitel 18</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="Ilja_Nikolajewitsch_Bronschtein" title="Ilja Nikolajewitsch Bronschtein">I. N. Bronstein</a>, <a href="Konstantin_Adolfowitsch_Semendjajew" title="Konstantin Adolfowitsch Semendjajew">K. A. Semendjajew</a>: <a href="Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Taschenbuch der Mathematik</a> 20. Auflage, Verlag Harri Deutsch, Thun und Frankfurt(Main), 1981, Kap. 3.3.2.3.3, S. 545</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Giovanni Perosa, Simone Di Mitri, William A. Barletta, Fulvio Parmigiani: <cite style="font-style:italic">Doppler signature in electrodynamic retarded potentials</cite>. In: <cite style="font-style:italic">Physics Open</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em">&nbsp;</span>14</span>, 1.&nbsp;Februar 2023, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a>&nbsp;<span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%222666-0326%22&amp;key=cql">2666-0326</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>100136</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1016/j.physo.2023.100136">10.1016/j.physo.2023.100136</a></span> (<a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S2666032623000017">sciencedirect.com</a> [abgerufen am 25.&nbsp;November 2024]).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rfr_id=info:sid/de.wikipedia.org:Retardiertes+Potential&amp;rft.atitle=Doppler+signature+in+electrodynamic+retarded+potentials&amp;rft.au=Giovanni+Perosa%2C+Simone+Di+Mitri%2C+William+A.+Barletta%2C+...&amp;rft.date=2023-02-01&amp;rft.doi=10.1016%2Fj.physo.2023.100136&amp;rft.genre=journal&amp;rft.issn=2666-0326&amp;rft.jtitle=Physics+Open&amp;rft.pages=100136&amp;rft.volume=14" style="display:none">&nbsp;</span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">C. K. Raju: <cite style="font-style:italic">Retarded gravitation theory</cite>. In: <cite style="font-style:italic">AIP Conference Proceedings</cite>. AIP, 2012, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>260–276</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1063/1.4756973">10.1063/1.4756973</a></span> (<a rel="nofollow" class="external text" href="https://pubs.aip.org/aip/acp/article-abstract/1483/1/260/727544/Retarded-gravitation-theory?redirectedFrom=fulltext">aip.org</a> [abgerufen am 25.&nbsp;November 2024]).<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:Retardiertes+Potential&amp;rft.atitle=Retarded+gravitation+theory&amp;rft.au=C.+K.+Raju&amp;rft.btitle=AIP+Conference+Proceedings&amp;rft.date=2012&amp;rft.doi=10.1063%2F1.4756973&amp;rft.genre=book&amp;rft.pages=260-276&amp;rft.pub=AIP" style="display:none">&nbsp;</span></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Yin Zhu: <cite style="font-style:italic">The speed of gravity: An observation on galaxy motions</cite>. 2016, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.13140/RG.2.2.30917.45287">10.13140/RG.2.2.30917.45287</a></span> (<a rel="nofollow" class="external text" href="https://rgdoi.net/10.13140/RG.2.2.30917.45287">rgdoi.net</a> [abgerufen am 25.&nbsp;November 2024]).<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:Retardiertes+Potential&amp;rft.au=Yin+Zhu&amp;rft.btitle=The+speed+of+gravity%3A+An+observation+on+galaxy+motions&amp;rft.date=2016&amp;rft.doi=10.13140%2FRG.2.2.30917.45287&amp;rft.genre=book" style="display:none">&nbsp;</span></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Roy J. Kennedy: <cite style="font-style:italic">PLANETARY MOTION IN A RETARDED NEWTONIAN POTENTIAL FIELD</cite>. In: <cite style="font-style:italic">Proceedings of the National Academy of Sciences</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em">&nbsp;</span>15</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em">&nbsp;</span>9</span>, 15.&nbsp;September 1929, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a>&nbsp;<span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220027-8424%22&amp;key=cql">0027-8424</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>744–753</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1073/pnas.15.9.744">10.1073/pnas.15.9.744</a></span>, <a class="external mw-magiclink-pmid" rel="nofollow" href="https://www.ncbi.nlm.nih.gov/pubmed/16577233?dopt=Abstract">PMID 16577233</a> (<a rel="nofollow" class="external text" href="https://pnas.org/doi/full/10.1073/pnas.15.9.744">pnas.org</a> [abgerufen am 25.&nbsp;November 2024]).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rfr_id=info:sid/de.wikipedia.org:Retardiertes+Potential&amp;rft.atitle=PLANETARY+MOTION+IN+A+RETARDED+NEWTONIAN+POTENTIAL+FIELD&amp;rft.au=Roy+J.+Kennedy&amp;rft.date=1929-09-15&amp;rft.doi=10.1073%2Fpnas.15.9.744&amp;rft.genre=journal&amp;rft.issn=0027-8424&amp;rft.issue=9&amp;rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&amp;rft.pages=744-753&amp;rft.pmid=16577233&amp;rft.volume=15" style="display:none">&nbsp;</span></span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Yuval Glass, Tomer Zimmerman, Asher <a href="Asher_Yahalom" title="Asher Yahalom">Yahalom</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Retarded Gravity in Disk Galaxies</cite>. In: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Symmetry</cite>. 16. Jahrgang, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em">&nbsp;</span>4</span>, 26.&nbsp;März 2024, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a>&nbsp;<span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%222073-8994%22&amp;key=cql">2073-8994</a></span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.3390/sym16040387">10.3390/sym16040387</a></span>, <a href="Bibcode" title="Bibcode">bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2024Symm...16..387G">2024Symm...16..387G</a> (englisch, <a rel="nofollow" class="external text" href="https://www.mdpi.com/2073-8994/16/4/387">mdpi.com</a> [abgerufen am 5.&nbsp;Februar 2025]).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rfr_id=info:sid/de.wikipedia.org:Retardiertes+Potential&amp;rft.atitle=Retarded+Gravity+in+Disk+Galaxies&amp;rft.au=Yuval%26%2332%3BGlass%2C%26%2332%3BTomer%26%2332%3BZimmerman%2C%26%2332%3BAsher%26%2332%3BYahalom&amp;rft.date=2024-03-26&amp;rft.doi=10.3390%2Fsym16040387&amp;rft.genre=journal&amp;rft.issn=2073-8994&amp;rft.issue=4&amp;rft.jtitle=Symmetry&amp;rft.volume=16.+Jahrgang" style="display:none">&nbsp;</span></span>
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