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<title>Retarded potential</title>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks em-sidebar"><tbody><tr><th class="sidebar-title"><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content hlist">

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<li><a href="Magnetic_moment" title="Magnetic moment">Magnetic dipole</a></li>

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<li><a href="Magnetic_flux" title="Magnetic flux">Magnetic flux</a></li>

<li><a href="Magnetic_scalar_potential" title="Magnetic scalar potential">Magnetic scalar potential</a></li>

<li><a href="Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li>

<li><a href="Magnetization" title="Magnetization">Magnetization</a></li>

<li><a href="Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></li>

<li><a href="Right-hand_rule#Electromagnetism" title="Right-hand rule">Right-hand rule</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Classical_electromagnetism" title="Classical electromagnetism">Electrodynamics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

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<li><a href="Electromagnetic_field" title="Electromagnetic field">Electromagnetic field</a></li>

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<li><a href="Electromagnetic_pulse" title="Electromagnetic pulse">Electromagnetic pulse</a></li>

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<li><a href="Faraday's_law_of_induction" title="Faraday's law of induction">Faraday's law</a></li>

<li><a href="Jefimenko's_equations" title="Jefimenko's equations">Jefimenko equations</a></li>

<li><a href="Larmor_formula" title="Larmor formula">Larmor formula</a></li>

<li><a href="Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li>

<li><a href="London_equations" title="London equations">London equations</a></li>

<li><a href="Lorentz_force" title="Lorentz force">Lorentz force</a></li>

<li><a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a></li>

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<ul><li><a href="AC_motor" title="AC motor">AC motor</a></li>

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<li><a href="Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">Gyrator–capacitor</a></li>

<li><a href="Induction_motor" title="Induction motor">Induction motor</a></li>

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<ul><li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li>

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<li><a href="Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li>

<li><a href="Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li>

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<li><a href="Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li>

<li><a href="Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li>

<li><a href="Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>

<li><a href="Michael_Faraday" title="Michael Faraday">Faraday</a></li>

<li><a href="Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>

<li><a href="Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li>

<li><a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li>

<li><a href="Joseph_Henry" title="Joseph Henry">Henry</a></li>

<li><a href="Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li>

<li><a href="John_Hopkinson" title="John Hopkinson">Hopkinson</a></li>

<li><a href="Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li>

<li><a href="James_Prescott_Joule" title="James Prescott Joule">Joule</a></li>

<li><a href="Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li>

<li><a href="Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li>

<li><a href="Joseph_Larmor" title="Joseph Larmor">Larmor</a></li>

<li><a href="Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li>

<li><a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li>

<li><a href="James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li>

<li><a href="Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li>

<li><a href="Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li>

<li><a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li>

<li><a href="John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li>

<li><a href="William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li>

<li><a href="F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li>

<li><a href="George_Singer" title="George Singer">Singer</a></li>

<li><a href="Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li>

<li><a href="Nikola_Tesla" title="Nikola Tesla">Tesla</a></li>

<li><a href="J._J._Thomson" title="J. J. Thomson">Thomson</a></li>

<li><a href="Alessandro_Volta" title="Alessandro Volta">Volta</a></li>

<li><a href="Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li>

<li><a href="Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td>

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<p>In <a href="Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>, the <b>retarded potentials</b> are the <a href="Electromagnetic_potential" class="mw-redirect" title="Electromagnetic potential">electromagnetic potentials</a> for the <a href="Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> generated by <a href="Time-variant_system" title="Time-variant system">time-varying</a> <a href="Electric_current" title="Electric current">electric current</a> or <a href="Charge_distribution" class="mw-redirect" title="Charge distribution">charge distributions</a> in the past. The fields propagate at the <a href="Speed_of_light" title="Speed of light">speed of light</a> <i>c</i>, so the delay of the fields connecting <a href="Causality_(physics)" title="Causality (physics)">cause and effect</a> at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution (the point of cause) to another point in space (where the effect is measured), see figure below.<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>

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<div class="mw-heading mw-heading2"><h2 id="In_the_Lorenz_gauge">In the Lorenz gauge</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a> and <a href="Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions of the electromagnetic field</a></div>

<p>The starting point is <a href="Maths_of_EM_field" class="mw-redirect" title="Maths of EM field">Maxwell's equations in the potential formulation</a> using the <a href="Lorenz_gauge" class="mw-redirect" title="Lorenz gauge">Lorenz gauge</a>:

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<annotation encoding="application/x-tex">{\displaystyle \Box \varphi ={\dfrac {\rho }{\epsilon _{0}}}\,,\quad \Box \mathbf {A} =\mu _{0}\mathbf {J} }</annotation>

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</math></span><img src="./255417e081105d3c8590433efe219bd34201edf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.768ex; height:5.176ex;" alt="{\displaystyle \Box \varphi ={\dfrac {\rho }{\epsilon _{0}}}\,,\quad \Box \mathbf {A} =\mu _{0}\mathbf {J} }" loading="lazy"></span></dd></dl>

<p>where φ(<b>r</b>, <i>t</i>) is the <a href="Electric_potential" title="Electric potential">electric potential</a> and <b>A</b>(<b>r</b>, <i>t</i>) is the <a href="Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a>, for an arbitrary source of <a href="Charge_density" title="Charge density">charge density</a> ρ(<b>r</b>, <i>t</i>) and <a href="Current_density" title="Current density">current density</a> <b>J</b>(<b>r</b>, <i>t</i>), and <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 }">

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

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</math></span><img src="./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> is the <a href="D'Alembert_operator" title="D'Alembert operator">D'Alembert operator</a>.<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> Solving these gives the retarded potentials below (all in <a href="SI_units" class="mw-redirect" title="SI units">SI units</a>).

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<div class="mw-heading mw-heading3"><h3 id="For_time-dependent_fields">For time-dependent fields</h3></div>

<p>For time-dependent fields, the retarded potentials are:<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><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>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {\varphi } (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '}</annotation>

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</math></span><img src="./b276a4436fe844512ba74bc290d9be7780cae102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.702ex; height:6.509ex;" alt="{\displaystyle \mathrm {\varphi } (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int {\frac {\mathbf {J} (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,.}</annotation>

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</math></span><img src="./326ecbe15cb1de08fd1ecd1e991e7fdbe406e825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.416ex; height:6.509ex;" alt="{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int {\frac {\mathbf {J} (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,.}" loading="lazy"></span></dd></dl>

<p>where <b>r</b> is a <a href="Position_vector" class="mw-redirect" title="Position vector">point</a> in space, <i>t</i> is time,

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<annotation encoding="application/x-tex">{\displaystyle t_{r}=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}</annotation>

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</math></span><img src="./2ef685a7f72fd0a61c7d6755ee442a23e29f22f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.451ex; height:5.676ex;" alt="{\displaystyle t_{r}=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}" loading="lazy"></span></dd></dl>

<p>is the <a href="Retarded_time" title="Retarded time">retarded time</a>, and d<sup>3</sup><b>r'</b> is the <a href="List_of_integration_and_measure_theory_topics" title="List of integration and measure theory topics">integration measure</a> using <b>r'</b>.

</p><p>From φ(<b>r</b>, t) and <b>A</b>(<b>r</b>, <i>t</i>), the fields <b>E</b>(<b>r</b>, <i>t</i>) and <b>B</b>(<b>r</b>, <i>t</i>) can be calculated using the definitions of the potentials:

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<annotation encoding="application/x-tex">{\displaystyle -\mathbf {E} =\nabla \varphi +{\frac {\partial \mathbf {A} }{\partial t}}\,,\quad \mathbf {B} =\nabla \times \mathbf {A} \,.}</annotation>

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</math></span><img src="./b0993578b67378225b0e4767dae99f5ac4d331e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.707ex; height:5.509ex;" alt="{\displaystyle -\mathbf {E} =\nabla \varphi +{\frac {\partial \mathbf {A} }{\partial t}}\,,\quad \mathbf {B} =\nabla \times \mathbf {A} \,.}" loading="lazy"></span></dd></dl>

<p>and this leads to <a href="Jefimenko's_equations" title="Jefimenko's equations">Jefimenko's equations</a>. The corresponding advanced potentials have an identical form, except the advanced time

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<annotation encoding="application/x-tex">{\displaystyle t_{a}=t+{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}</annotation>

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<p>replaces the retarded time.

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<div class="mw-heading mw-heading3"><h3 id="In_comparison_with_static_potentials_for_time-independent_fields">In comparison with static potentials for time-independent fields</h3></div>

<p>In the case the fields are time-independent (<a href="Electrostatic" class="mw-redirect" title="Electrostatic">electrostatic</a> and <a href="Magnetostatic" class="mw-redirect" title="Magnetostatic">magnetostatic</a> fields), the time derivatives in the <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 }">

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\varphi =-{\dfrac {\rho }{\epsilon _{0}}}\,,\quad \nabla ^{2}\mathbf {A} =-\mu _{0}\mathbf {J} \,,}</annotation>

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</math></span><img src="./37e66c2cc6856484b9fd7ec8f80ef96b0cfa7e84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.782ex; height:5.176ex;" alt="{\displaystyle \nabla ^{2}\varphi =-{\dfrac {\rho }{\epsilon _{0}}}\,,\quad \nabla ^{2}\mathbf {A} =-\mu _{0}\mathbf {J} \,,}" loading="lazy"></span></dd></dl>

<p>where ∇<sup>2</sup> is the <a href="Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a>, which take the form of <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a> in four components (one for φ and three for <b>A</b>), and the solutions are:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {\varphi } (\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int {\frac {\mathbf {J} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,.}</annotation>

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<p>These also follow directly from the retarded potentials.

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<div class="mw-heading mw-heading2"><h2 id="In_the_Coulomb_gauge">In the Coulomb gauge</h2></div>

<p>In the <a href="Coulomb_gauge" class="mw-redirect" title="Coulomb gauge">Coulomb gauge</a>, Maxwell's equations are<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>

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\varphi =-{\dfrac {\rho }{\epsilon _{0}}}}</annotation>

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</math></span><img src="./9566fefbf818819daad7207046007ef6c8f75ce8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.251ex; height:5.176ex;" alt="{\displaystyle \nabla ^{2}\varphi =-{\dfrac {\rho }{\epsilon _{0}}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\mathbf {A} -{\dfrac {1}{c^{2}}}{\dfrac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}=-\mu _{0}\mathbf {J} +{\dfrac {1}{c^{2}}}\nabla \left({\dfrac {\partial \varphi }{\partial t}}\right)\,,}</annotation>

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<p>although the solutions contrast the above, since <b>A</b> is a retarded potential yet φ changes <i>instantly</i>, given by:

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<annotation encoding="application/x-tex">{\displaystyle \varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi \epsilon _{0}}}\int {\dfrac {\rho (\mathbf {r} ',t)}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '}</annotation>

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</math></span><img src="./e351ad4aa6fea93e6922ffc25bd1e3da90989fed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.693ex; height:6.509ex;" alt="{\displaystyle \varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi \epsilon _{0}}}\int {\dfrac {\rho (\mathbf {r} ',t)}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\dfrac {1}{4\pi \varepsilon _{0}}}\nabla \times \int \mathrm {d} ^{3}\mathbf {r'} \int _{0}^{|\mathbf {r} -\mathbf {r} '|/c}\mathrm {d} t_{r}{\dfrac {t_{r}\mathbf {J} (\mathbf {r'} ,t-t_{r})}{|\mathbf {r} -\mathbf {r} '|^{3}}}\times (\mathbf {r} -\mathbf {r} ')\,.}</annotation>

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</math></span><img src="./45a53e5e21f74924f269b204fd7ef7d0c19acf70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.822ex; height:7.176ex;" alt="{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\dfrac {1}{4\pi \varepsilon _{0}}}\nabla \times \int \mathrm {d} ^{3}\mathbf {r'} \int _{0}^{|\mathbf {r} -\mathbf {r} '|/c}\mathrm {d} t_{r}{\dfrac {t_{r}\mathbf {J} (\mathbf {r'} ,t-t_{r})}{|\mathbf {r} -\mathbf {r} '|^{3}}}\times (\mathbf {r} -\mathbf {r} ')\,.}" loading="lazy"></span></dd></dl>

<p>This presents an advantage and a disadvantage of the Coulomb gauge - φ is easily calculable from the charge distribution ρ but <b>A</b> is not so easily calculable from the current distribution <b>j</b>. However, provided we require that the potentials vanish at infinity, they can be expressed neatly in terms of fields:

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<annotation encoding="application/x-tex">{\displaystyle \varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi }}\int {\dfrac {\nabla \cdot \mathbf {E} (\mathbf {r} ',t)}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '}</annotation>

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</math></span><img src="./552d09089dea3b721a0f3ce7ae1ace69e22eff0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.513ex; height:6.509ex;" alt="{\displaystyle \varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi }}\int {\dfrac {\nabla \cdot \mathbf {E} (\mathbf {r} ',t)}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\dfrac {1}{4\pi }}\int {\dfrac {\nabla \times \mathbf {B} (\mathbf {r} ',t)}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '}</annotation>

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</math></span><img src="./c97651b2b47029cb7fd513990c6667d08fc74d97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.318ex; height:6.509ex;" alt="{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\dfrac {1}{4\pi }}\int {\dfrac {\nabla \times \mathbf {B} (\mathbf {r} ',t)}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="In_linearized_gravity">In linearized gravity</h2></div>

<p>The retarded potential in <a href="Linearized_gravity" title="Linearized gravity">linearized general relativity</a> is closely analogous to the electromagnetic case. The trace-reversed tensor <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="{\textstyle {\tilde {h}}_{\mu \nu }=h_{\mu \nu }-{\frac {1}{2}}\eta _{\mu \nu }h}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\tilde {h}}_{\mu \nu }=h_{\mu \nu }-{\frac {1}{2}}\eta _{\mu \nu }h}</annotation>

</semantics>

</math></span><img src="./8e6fa872f3596bc8a5199fe4a1d520c5440367d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.053ex; height:3.509ex;" alt="{\textstyle {\tilde {h}}_{\mu \nu }=h_{\mu \nu }-{\frac {1}{2}}\eta _{\mu \nu }h}" loading="lazy"></span> plays the role of the four-vector potential, the <a href="Harmonic_coordinate_condition" title="Harmonic coordinate condition">harmonic gauge</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 {\tilde {h}}^{\mu \nu }{}_{,\mu }=0}">

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {h}}^{\mu \nu }{}_{,\mu }=0}</annotation>

</semantics>

</math></span><img src="./2e86bd1fe56a12eedfc9cc67af9ba49658f02796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.375ex; height:3.509ex;" alt="{\displaystyle {\tilde {h}}^{\mu \nu }{}_{,\mu }=0}" loading="lazy"></span> replaces the electromagnetic Lorenz gauge, the field equations are <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 {\tilde {h}}_{\mu \nu }=-16\pi GT_{\mu \nu }}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Box {\tilde {h}}_{\mu \nu }=-16\pi GT_{\mu \nu }}</annotation>

</semantics>

</math></span><img src="./6af2396642f9db22c36106c5c9fe93fc645a3890.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.084ex; height:3.343ex;" alt="{\displaystyle \Box {\tilde {h}}_{\mu \nu }=-16\pi GT_{\mu \nu }}" loading="lazy"></span>, and the retarded-wave solution is<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>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {h}}_{\mu \nu }(\mathbf {r} ,t)=4G\int {\frac {T_{\mu \nu }(\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '.}</annotation>

</semantics>

</math></span></span>

Using SI units, the expression must be divided by <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^{4}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c^{4}}</annotation>

</semantics>

</math></span><img src="./7b2f32d6df5fdcb43c0206db7fe322a4c5d29ba0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.061ex; height:2.676ex;" alt="{\displaystyle c^{4}}" loading="lazy"></span>, as can be confirmed by dimensional analysis.

</p>

<div class="mw-heading mw-heading2"><h2 id="Occurrence_and_application">Occurrence and application</h2></div>

<p>A many-body theory which includes an average of retarded and <i>advanced</i> <a href="Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potentials</a> is the <a href="Wheeler%E2%80%93Feynman_absorber_theory" title="Wheeler–Feynman absorber theory">Wheeler–Feynman absorber theory</a> also known as the Wheeler–Feynman time-symmetric theory.

</p><p>In <a href="Gravitation" class="mw-redirect" title="Gravitation">gravitation</a>, there are application examples for calculating deviations in <a href="Orbit" title="Orbit">orbits</a> of <a href="Satellite" title="Satellite">satellites</a>,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="Orbit_of_the_Moon" title="Orbit of the Moon">moons</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> or <a href="Orbits_of_planets" class="mw-redirect" title="Orbits of planets">planets</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The anomalies in the <a href="Rotation_curve" class="mw-redirect" title="Rotation curve">rotation curves</a> of more than one hundred <a href="Spiral_galaxy" title="Spiral galaxy">spiral galaxys</a> of different <a href="Galaxy_morphological_classification" title="Galaxy morphological classification">types</a> could also be explained. The data of the “SPARC (Spitzer Photometry and Accurate Rotation Curves) Galaxy collection”, which were recorded with the <a href="Spitzer_Space_Telescope" title="Spitzer Space Telescope">Spitzer Space Telescope</a>, were used for this purpose. In this way, neither the assumption of <a href="Dark_matter" title="Dark matter">dark matter</a> nor a modification of <a href="General_relativity" title="General relativity">general relativity</a> is required to explain the observations.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> On even larger scales, the retarded gravitational potentials result in effects such as an <a href="Expansion_of_the_universe" title="Expansion of the universe">accelerated expansion</a>, which leads to an <a href="Isotropy" title="Isotropy">isotropic</a>, but not homogeneous <a href="Universe" title="Universe">universe</a> with an outer shell of dark matter with an increased <a href="Density" title="Density">mass density</a> as well as a strong <a href="Gravitational_redshift" title="Gravitational redshift">gravitational redshift</a> of distant <a href="Astronomical_object" title="Astronomical object">astronomical objects</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

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

<p>The potential of charge with uniform speed on a straight line has <a href="Inversion_in_a_point" class="mw-redirect" title="Inversion in a point">inversion in a point</a> that is in the recent position. The potential is not changed in the direction of movement.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a></li>

<li><a href="Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li>

<li><a href="Whitehead's_theory_of_gravitation" title="Whitehead's theory of gravitation">Whitehead's theory of gravitation</a></li></ul>

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

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</style><cite id="CITEREFRohrlich1993" class="citation encyclopaedia cs1">Rohrlich, F (1993). <a rel="nofollow" class="external text" href="https://archive.org/details/mcgrawhillencycl1993park/page/1072/">"Potentials"</a>. In Parker, S.P. (ed.). <i>McGraw Hill Encyclopaedia of Physics</i> (2nd ed.). New York. p. 1072. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-051400-3</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite encyclopedia}}</code>: CS1 maint: location missing publisher (link)</span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Garg, A., <i>Classical Electromagnetism in a Nutshell</i>, 2012, p. 129</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-92712-9</bdi></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>81-7758-293-3</bdi></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>81-7758-293-3</bdi></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Sean M. Carroll, "Lecture Notes on General Relativity" (<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/9712019">arXiv:gr-qc/9712019</a>), equations 6.20, 6.21, 6.22, 6.74</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFC._K._Raju2012" class="citation journal cs1">C. K. Raju (2012). <a rel="nofollow" class="external text" href="https://pubs.aip.org/aip/acp/article-abstract/1483/1/260/727544/Retarded-gravitation-theory?redirectedFrom=fulltext">"Retarded gravitation theory"</a>. <i>AIP Conference Proceedings</i>. <b>1483</b> (1): <span class="nowrap">260–</span>276. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1102.2945">1102.2945</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012AIPC.1483..260R">2012AIPC.1483..260R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.4756973">10.1063/1.4756973</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-25</span></span>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFYin_Zhu2016" class="citation web cs1">Yin Zhu (2016). <a rel="nofollow" class="external text" href="http://rgdoi.net/10.13140/RG.2.2.30917.45287">"The speed of gravity: An observation on galaxy motions"</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.13140%2FRG.2.2.30917.45287">10.13140/RG.2.2.30917.45287</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-25</span></span>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFRoy_J._Kennedy1929" class="citation cs2">Roy J. Kennedy (1929-09-15), <a rel="nofollow" class="external text" href="https://pnas.org/doi/full/10.1073/pnas.15.9.744">"PLANETARY MOTION IN A RETARDED NEWTONIAN POTENTIAL FIELD"</a>, <i>Proceedings of the National Academy of Sciences</i>, vol. 15, no. 9, pp. <span class="nowrap">744–</span>753, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.15.9.744">10.1073/pnas.15.9.744</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>, <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC522551">522551</a></span>, <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16577233">16577233</a><span class="reference-accessdate">, retrieved <span class="nowrap">2024-11-25</span></span></cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFGlassZimmermanYahalom2024" class="citation journal cs1">Glass, Yuval; Zimmerman, Tomer; Yahalom, Asher (2024-02-20). <a rel="nofollow" class="external text" href="https://www.preprints.org/manuscript/202402.1088/v1">"Retarded Gravity in Disk Galaxies"</a>. <i>Symmetry</i>. <b>16</b> (4): 387. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2024Symm...16..387G">2024Symm...16..387G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsym16040387">10.3390/sym16040387</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2073-8994">2073-8994</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-25</span></span>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFMarkus_Bautsch2024" class="citation cs2">Markus Bautsch (2024), <a rel="nofollow" class="external text" href="https://rgdoi.net/10.13140/RG.2.2.27349.23529"><i>Retarded gravitational potentials on the scale of the universe</i></a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.13140%2FRG.2.2.27349.23529">10.13140/RG.2.2.27349.23529</a><span class="reference-accessdate">, retrieved <span class="nowrap">2024-11-25</span></span></cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_26.html">Feynman, Lecture 26, Lorentz Transformations of the Fields</a></span>

</li>

</ol></div></div></div><div><div class="zim-footer">

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