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class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%A4%E0%A6%BF%E0%A6%86%E0%A6%B2%E0%A7%8B%E0%A6%95%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%97%E0%A6%A4%E0%A6%BF" title="অতিআলোকীয় গতি – Bangla" lang="bn" hreflang="bn" data-title="অতিআলোকীয় গতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Moviment_superlum%C3%ADnic" title="Moviment superlumínic – Catalan" lang="ca" hreflang="ca" data-title="Moviment superlumínic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A5%CF%80%CE%B5%CF%81%CF%86%CF%89%CF%84%CE%B5%CE%B9%CE%BD%CE%AE_%CE%BA%CE%AF%CE%BD%CE%B7%CF%83%CE%B7" title="Υπερφωτεινή κίνηση – Greek" lang="el" hreflang="el" data-title="Υπερφωτεινή κίνηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Movimiento_superlum%C3%ADnico" title="Movimiento superlumínico – Spanish" lang="es" hreflang="es" data-title="Movimiento superlumínico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%B1%DA%A9%D8%AA_%D9%88%D8%B1%D8%A7%DB%8C_%D8%B3%D8%B1%D8%B9%D8%AA_%D9%86%D9%88%D8%B1" title="حرکت ورای سرعت نور – Persian" lang="fa" hreflang="fa" data-title="حرکت ورای سرعت نور" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mouvement_supraluminique" title="Mouvement supraluminique – French" lang="fr" hreflang="fr" data-title="Mouvement supraluminique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Flugger-as-ljochtreizgjen" title="Flugger-as-ljochtreizgjen – Western Frisian" lang="fy" hreflang="fy" data-title="Flugger-as-ljochtreizgjen" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Taisteal_forsholasach" title="Taisteal forsholasach – Irish" lang="ga" hreflang="ga" data-title="Taisteal forsholasach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B4%88%EA%B4%91%EC%86%8D_%EC%9A%B4%EB%8F%99" title="초광속 운동 – Korean" lang="ko" hreflang="ko" data-title="초광속 운동" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D5%A5%D6%80%D5%AC%D5%B8%D6%82%D5%BD%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B7%D5%A1%D6%80%D5%AA%D5%B8%D6%82%D5%B4" title="Գերլուսային շարժում – Armenian" lang="hy" hreflang="hy" data-title="Գերլուսային շարժում" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gerakan_superluminal" title="Gerakan superluminal – Indonesian" lang="id" hreflang="id" data-title="Gerakan superluminal" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Moto_superluminale" title="Moto superluminale – Italian" lang="it" hreflang="it" data-title="Moto superluminale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%B6%85%E5%85%89%E9%80%9F%E9%81%8B%E5%8B%95" title="超光速運動 – Japanese" lang="ja" hreflang="ja" data-title="超光速運動" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Movimento_superluminal" 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typeof="mw:File/Thumb"><a href="/wiki/File:Superluminalmotion.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Superluminalmotion.gif/300px-Superluminalmotion.gif" decoding="async" width="300" height="312" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/32/Superluminalmotion.gif 1.5x" data-file-width="405" data-file-height="421" /></a><figcaption>Superluminal motion</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Original_research plainlinks metadata ambox ambox-content ambox-Original_research" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>possibly contains <a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research">original research</a></b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Superluminal_motion&action=edit">improve it</a> by <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verifying</a> the claims made and adding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a>. Statements consisting only of original research should be removed.</span> <span class="date-container"><i>(<span class="date">August 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, <b>superluminal motion</b> is the apparently <a href="/wiki/Faster-than-light" title="Faster-than-light">faster-than-light</a> motion seen in some <a href="/wiki/Radio_galaxy" title="Radio galaxy">radio galaxies</a>, <a href="/wiki/BL_Lac_object" class="mw-redirect" title="BL Lac object">BL Lac objects</a>, <a href="/wiki/Quasar" title="Quasar">quasars</a>, <a href="/wiki/Blazars" class="mw-redirect" title="Blazars">blazars</a> and recently also in some galactic sources called <a href="/wiki/Microquasar" title="Microquasar">microquasars</a>. Bursts of energy moving out along the <a href="/wiki/Relativistic_jet" class="mw-redirect" title="Relativistic jet">relativistic jets</a> emitted from these objects can have a <a href="/wiki/Proper_motion" title="Proper motion">proper motion</a> that appears greater than the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. All of these sources are thought to contain a <a href="/wiki/Black_hole" title="Black hole">black hole</a>, responsible for the ejection of mass at high velocities. <a href="/wiki/Light_echo" title="Light echo">Light echoes</a> can also produce apparent superluminal motion.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Explanation">Explanation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superluminal_motion&action=edit&section=1" title="Edit section: Explanation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Superluminal motion occurs as a special case of a more general phenomenon arising from the difference between the apparent speed of distant objects moving across the sky and their actual speed as measured at the source.<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 tracking the movement of such objects across the sky, a naive calculation of their speed can be derived by a simple distance divided by time calculation. If the distance of the object from the Earth is known, the angular speed of the object can be measured, and the speed can be naively calculated via: </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;"><i>apparent speed = distance to object × angular speed.</i></div> <p>This calculation does not yield the actual speed of the object, as it fails to account for the fact that the speed of light is finite. When measuring the movement of distant objects across the sky, there is a large time delay between what has been observed and what has occurred, due to the large distance the light from the distant object has to travel to reach us. The error in the above naive calculation comes from the fact that when an object has a component of velocity directed towards the Earth, as the object moves closer to the Earth that time delay becomes smaller. This means that the apparent speed as calculated above is <em>greater</em> than the actual speed. Correspondingly, if the object is moving away from the Earth, the above calculation underestimates the actual speed. </p><p>This effect in itself does not generally lead to superluminal motion being observed. But when the actual speed of the object is close to the speed of light, the apparent speed can be observed as greater than the speed of light, as a result of the above effect. As the actual speed of the object approaches the speed of light, the effect is most pronounced as the component of the velocity towards the Earth increases. This means that in most cases, 'superluminal' objects are travelling almost directly towards the Earth. However it is not strictly necessary for this to be the case, and superluminal motion can still be observed in objects with appreciable velocities not directed towards the Earth.<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> </p><p>Superluminal motion is most often observed in two opposing jets emanating from the core of a star or black hole. In this case, one jet is moving away from and one towards the Earth. If <a href="/wiki/Doppler_shift" class="mw-redirect" title="Doppler shift">Doppler shifts</a> are observed in both sources, the velocity and the distance can be determined independently of other observations. </p> <div class="mw-heading mw-heading3"><h3 id="Some_contrary_evidence">Some contrary evidence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superluminal_motion&action=edit&section=2" title="Edit section: Some contrary evidence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As early as 1983, at the "superluminal workshop" held at <a href="/wiki/Jodrell_Bank_Observatory" title="Jodrell Bank Observatory">Jodrell Bank Observatory</a>, referring to the seven then-known superluminal jets, </p> <blockquote><p>Schilizzi ... presented maps of arc-second resolution [showing the large-scale outer jets] ... which ... have revealed outer double structure in all but one (<a href="/wiki/3C_273" title="3C 273">3C 273</a>) of the known superluminal sources. An embarrassment is that the average projected size [on the sky] of the outer structure is no smaller than that of the normal radio-source population.<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></p></blockquote> <p>In other words, the jets are evidently not, on average, close to the Earth's line-of-sight. (Their apparent length would appear much shorter if they were.) </p><p>In 1993, Thomson et al. suggested that the (outer) jet of the quasar <a href="/wiki/3C_273" title="3C 273">3C 273</a> is nearly collinear to the Earth's line-of-sight. Superluminal motion of up to ~9.6<i>c</i> has been observed along the (inner) jet of this quasar.<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><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><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> </p><p>Superluminal motion of up to 6<i>c</i> has been observed in the inner parts of the jet of <a href="/wiki/Messier_87" title="Messier 87">M87</a>. To explain this in terms of the "narrow-angle" model, the jet must be no more than 19° from the Earth's line-of-sight.<sup id="cite_ref-nature891_8-0" class="reference"><a href="#cite_note-nature891-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> But evidence suggests that the jet is in fact at about 43° to the Earth's line-of-sight.<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 same group of scientists later revised that finding and argue in favour of a superluminal bulk movement in which the jet is embedded.<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> </p><p>Suggestions of turbulence and/or "wide cones" in the inner parts of the jets have been put forward to try to counter such problems, and there seems to be some evidence for this.<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-heading4"><h4 id="Signal_velocity">Signal velocity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superluminal_motion&action=edit&section=3" title="Edit section: Signal velocity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The model identifies a difference between the information carried by the wave at its signal velocity <i>c</i>, and the information about the wave front's apparent rate of change of position. If a light pulse is envisaged in a wave guide (glass tube) moving across an observer's field of view, the pulse can only move at <i>c</i> through the guide. If that pulse is also directed towards the observer, he will receive that wave information, at <i>c</i>. If the wave guide is moved in the same direction as the pulse, the information on its position, passed to the observer as lateral emissions from the pulse, changes. He may see the rate of change of position as apparently representing motion faster than <i>c</i> when calculated, like the edge of a shadow across a curved surface. This is a different signal, containing different information, to the pulse and does not break the second postulate of special relativity. <i>c</i> is strictly maintained in all local fields. </p> <div class="mw-heading mw-heading2"><h2 id="Derivation_of_the_apparent_velocity">Derivation of the apparent velocity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superluminal_motion&action=edit&section=4" title="Edit section: Derivation of the apparent velocity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Superluminal_motion" title="Special:EditPage/Superluminal motion">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">August 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A <a href="/wiki/Relativistic_jet" class="mw-redirect" title="Relativistic jet">relativistic jet</a> coming out of the center of an <a href="/wiki/Active_galactic_nucleus" title="Active galactic nucleus">active galactic nucleus</a> is moving along AB with a velocity <i>v</i>, and is observed from the point O. At time <span class="mwe-math-element"><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0768c0bd659f2f84fb5ef9f4b74f336123d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}"></span> a light ray leaves the jet from point A and another ray leaves at time <span class="mwe-math-element"><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_{2}=t_{1}+\delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}=t_{1}+\delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf2cc03e1de3aa8bd369da247b20b08ccbc28b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.615ex; height:2.676ex;" alt="{\displaystyle t_{2}=t_{1}+\delta t}"></span> from point B. An observer at O receives the rays at time <span class="mwe-math-element"><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_{1}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b95dfcfdb79a4e47aad87ab7cae014cea2f7445e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.894ex; height:2.843ex;" alt="{\displaystyle t_{1}^{\prime }}"></span> and <span class="mwe-math-element"><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_{2}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b8e6986cc7182d6b2b4b44473055e23c597033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.894ex; height:2.843ex;" alt="{\displaystyle t_{2}^{\prime }}"></span> respectively. The angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is small enough that the two distances marked <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44b39c353d4634b76f550679dcc7aed38a0cd47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.276ex; height:2.509ex;" alt="{\displaystyle D_{L}}"></span> can be considered equal. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Superluminal_motion_in_AGN_jets.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Superluminal_motion_in_AGN_jets.png/400px-Superluminal_motion_in_AGN_jets.png" decoding="async" width="400" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/a8/Superluminal_motion_in_AGN_jets.png 1.5x" data-file-width="540" data-file-height="232" /></a><figcaption></figcaption></figure> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB=v\,\delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB=v\,\delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c363de965d2af78b8a3c4a86a32e1be28f279e29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.009ex; height:2.343ex;" alt="{\displaystyle AB=v\,\delta t}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AC=v\,\delta t\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>C</mi> <mo>=</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>t</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC=v\,\delta t\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d22eb1f18a233994ebc6fda7dcd8477ff670d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.987ex; height:2.343ex;" alt="{\displaystyle AC=v\,\delta t\cos \theta }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BC=v\,\delta t\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> <mo>=</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>t</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BC=v\,\delta t\sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76e8388b7f8a8fd3f04c0d7a6ac66f8ede1c541c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.752ex; height:2.343ex;" alt="{\displaystyle BC=v\,\delta t\sin \theta }"></span></dd> <dd><span class="mwe-math-element"><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_{2}-t_{1}=\delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}-t_{1}=\delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34530abf149daedabd4c8e3e056f3e0a7e92440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.615ex; height:2.676ex;" alt="{\displaystyle t_{2}-t_{1}=\delta t}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><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_{1}'=t_{1}+{\frac {D_{L}+v\,\delta t\cos \theta }{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>+</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>t</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}'=t_{1}+{\frac {D_{L}+v\,\delta t\cos \theta }{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b1392fa70f57593987b0e2614ab821202e3b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.058ex; height:5.343ex;" alt="{\displaystyle t_{1}'=t_{1}+{\frac {D_{L}+v\,\delta t\cos \theta }{c}}}"></span></dd> <dd><span class="mwe-math-element"><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_{2}'=t_{2}+{\frac {D_{L}}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}'=t_{2}+{\frac {D_{L}}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f65b4f228663a265aa97548a3e505f14316915e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.838ex; height:5.176ex;" alt="{\displaystyle t_{2}'=t_{2}+{\frac {D_{L}}{c}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><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 t'=t_{2}'-t_{1}'=t_{2}-t_{1}-{\frac {v\,\delta t\cos \theta }{c}}=\delta t-{\frac {v\,\delta t\cos \theta }{c}}=\delta t(1-\beta \cos \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>−</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>t</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mi>δ</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>t</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mi>δ</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta t'=t_{2}'-t_{1}'=t_{2}-t_{1}-{\frac {v\,\delta t\cos \theta }{c}}=\delta t-{\frac {v\,\delta t\cos \theta }{c}}=\delta t(1-\beta \cos \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0acff1c19ab546289f00a46507d237621d89ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:68.23ex; height:5.343ex;" alt="{\displaystyle \delta t'=t_{2}'-t_{1}'=t_{2}-t_{1}-{\frac {v\,\delta t\cos \theta }{c}}=\delta t-{\frac {v\,\delta t\cos \theta }{c}}=\delta t(1-\beta \cos \theta )}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =v/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =v/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c09197ac61cf6c55baab7eaaf25cbde57010efd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.727ex; height:2.843ex;" alt="{\displaystyle \beta =v/c}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><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 t={\frac {\delta t'}{1-\beta \cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta t={\frac {\delta t'}{1-\beta \cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103ac50158d61d2d9b046152dace8495d7eb86e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.133ex; height:6.009ex;" alt="{\displaystyle \delta t={\frac {\delta t'}{1-\beta \cos \theta }}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BC=D_{L}\sin \phi \approx \phi D_{L}=v\,\delta t\sin \theta \Rightarrow \phi D_{L}=v\sin \theta {\frac {\delta t'}{1-\beta \cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>≈</mo> <mi>ϕ</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>t</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">⇒</mo> <mi>ϕ</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mi>v</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BC=D_{L}\sin \phi \approx \phi D_{L}=v\,\delta t\sin \theta \Rightarrow \phi D_{L}=v\sin \theta {\frac {\delta t'}{1-\beta \cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba4caaf73f36fcbd519c2165c631e159943e61c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:62.27ex; height:6.009ex;" alt="{\displaystyle BC=D_{L}\sin \phi \approx \phi D_{L}=v\,\delta t\sin \theta \Rightarrow \phi D_{L}=v\sin \theta {\frac {\delta t'}{1-\beta \cos \theta }}}"></span></dd></dl> <p>Apparent transverse velocity along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfec1f2fa1995085360fe94a62a49e376ada7d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\displaystyle CB}"></span>, <span class="mwe-math-element"><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_{\text{T}}={\frac {\phi D_{L}}{\delta t'}}={\frac {v\sin \theta }{1-\beta \cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> <mrow> <mi>δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{T}}={\frac {\phi D_{L}}{\delta t'}}={\frac {v\sin \theta }{1-\beta \cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c6aac0e204054161258c4b5309f39559e537a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.387ex; height:5.843ex;" alt="{\displaystyle v_{\text{T}}={\frac {\phi D_{L}}{\delta t'}}={\frac {v\sin \theta }{1-\beta \cos \theta }}}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{\text{T}}={\frac {v_{\text{T}}}{c}}={\frac {\beta \sin \theta }{1-\beta \cos \theta }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{\text{T}}={\frac {v_{\text{T}}}{c}}={\frac {\beta \sin \theta }{1-\beta \cos \theta }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/948ea537ca1f18aca0cab0c3b70f9f1857fbadb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.108ex; height:5.843ex;" alt="{\displaystyle \beta _{\text{T}}={\frac {v_{\text{T}}}{c}}={\frac {\beta \sin \theta }{1-\beta \cos \theta }}.}"></span></dd></dl> <p>The apparent transverse velocity is maximal for angle (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\beta <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>β</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\beta <1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5823463cddbbeea88e7e3a68c879cc6f391f7e4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.854ex; height:2.509ex;" alt="{\displaystyle 0<\beta <1}"></span> is used) </p> <dl><dd><span class="mwe-math-element"><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 {\partial \beta _{\text{T}}}{\partial \theta }}={\frac {\partial }{\partial \theta }}\left[{\frac {\beta \sin \theta }{1-\beta \cos \theta }}\right]={\frac {\beta \cos \theta }{1-\beta \cos \theta }}-{\frac {(\beta \sin \theta )^{2}}{(1-\beta \cos \theta )^{2}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \beta _{\text{T}}}{\partial \theta }}={\frac {\partial }{\partial \theta }}\left[{\frac {\beta \sin \theta }{1-\beta \cos \theta }}\right]={\frac {\beta \cos \theta }{1-\beta \cos \theta }}-{\frac {(\beta \sin \theta )^{2}}{(1-\beta \cos \theta )^{2}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa245a61b6beadbe6e8f355c568aeb75e014d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.578ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial \beta _{\text{T}}}{\partial \theta }}={\frac {\partial }{\partial \theta }}\left[{\frac {\beta \sin \theta }{1-\beta \cos \theta }}\right]={\frac {\beta \cos \theta }{1-\beta \cos \theta }}-{\frac {(\beta \sin \theta )^{2}}{(1-\beta \cos \theta )^{2}}}=0}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow \beta \cos \theta (1-\beta \cos \theta )^{2}=(1-\beta \cos \theta )(\beta \sin \theta )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow \beta \cos \theta (1-\beta \cos \theta )^{2}=(1-\beta \cos \theta )(\beta \sin \theta )^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/365c7b5b179229d3c936dae9b813513d25251038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.585ex; height:3.176ex;" alt="{\displaystyle \Rightarrow \beta \cos \theta (1-\beta \cos \theta )^{2}=(1-\beta \cos \theta )(\beta \sin \theta )^{2}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow \beta \cos \theta (1-\beta \cos \theta )=(\beta \sin \theta )^{2}\Rightarrow \beta \cos \theta -\beta ^{2}\cos ^{2}\theta =\beta ^{2}\sin ^{2}\theta \Rightarrow \cos \theta _{\text{max}}=\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">⇒</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">⇒</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mo>=</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow \beta \cos \theta (1-\beta \cos \theta )=(\beta \sin \theta )^{2}\Rightarrow \beta \cos \theta -\beta ^{2}\cos ^{2}\theta =\beta ^{2}\sin ^{2}\theta \Rightarrow \cos \theta _{\text{max}}=\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbe9bb5351cbd0b2ccc62717b7b36f7cefc5afe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:81.783ex; height:3.176ex;" alt="{\displaystyle \Rightarrow \beta \cos \theta (1-\beta \cos \theta )=(\beta \sin \theta )^{2}\Rightarrow \beta \cos \theta -\beta ^{2}\cos ^{2}\theta =\beta ^{2}\sin ^{2}\theta \Rightarrow \cos \theta _{\text{max}}=\beta }"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow \sin \theta _{\text{max}}={\sqrt {1-\cos ^{2}\theta _{\text{max}}}}={\sqrt {1-\beta ^{2}}}={\frac {1}{\gamma }}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow \sin \theta _{\text{max}}={\sqrt {1-\cos ^{2}\theta _{\text{max}}}}={\sqrt {1-\beta ^{2}}}={\frac {1}{\gamma }}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a143c4d6e60aa3b7edfed1778eb0ca08ce9083e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.353ex; height:5.676ex;" alt="{\displaystyle \Rightarrow \sin \theta _{\text{max}}={\sqrt {1-\cos ^{2}\theta _{\text{max}}}}={\sqrt {1-\beta ^{2}}}={\frac {1}{\gamma }}\,}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb91924eaa6e5a593ba98cc7e2dfda04f764f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.915ex; height:6.509ex;" alt="{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \therefore \beta _{\text{T}}^{\text{max}}={\frac {\beta \sin \theta _{\text{max}}}{1-\beta \cos \theta _{\text{max}}}}={\frac {\beta /\gamma }{1-\beta ^{2}}}=\beta \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∴</mo> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>γ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>β</mi> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \therefore \beta _{\text{T}}^{\text{max}}={\frac {\beta \sin \theta _{\text{max}}}{1-\beta \cos \theta _{\text{max}}}}={\frac {\beta /\gamma }{1-\beta ^{2}}}=\beta \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05444e9fe8f76109cc3e8c3fb1280a6896743ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.381ex; height:6.343ex;" alt="{\displaystyle \therefore \beta _{\text{T}}^{\text{max}}={\frac {\beta \sin \theta _{\text{max}}}{1-\beta \cos \theta _{\text{max}}}}={\frac {\beta /\gamma }{1-\beta ^{2}}}=\beta \gamma }"></span></dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \gg 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>≫</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \gg 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39806de711cf242023193a701fef32f66dd7412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.676ex;" alt="{\displaystyle \gamma \gg 1}"></span> (i.e. when velocity of jet is close to the velocity of light) then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{\text{T}}^{\text{max}}>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msubsup> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{\text{T}}^{\text{max}}>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdcfde6f77c73fa0be43c84b18188e7c7d7e7986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.889ex; height:2.843ex;" alt="{\displaystyle \beta _{\text{T}}^{\text{max}}>1}"></span> despite the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta <1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a25a2be3655a09dd4938ea4991c5749cb46708c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta <1}"></span>. And of course <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{\text{T}}>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{\text{T}}>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33d706a313adf8e67d15d451821626ff4889c804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.995ex; height:2.509ex;" alt="{\displaystyle \beta _{\text{T}}>1}"></span> means that the apparent transverse velocity along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfec1f2fa1995085360fe94a62a49e376ada7d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\displaystyle CB}"></span>, the only velocity on the sky that can be measured, is larger than the velocity of light in vacuum, i.e. the motion is apparently superluminal. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superluminal_motion&action=edit&section=5" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The apparent superluminal motion in the faint nebula surrounding Nova Persei was first observed in 1901 by <a href="/wiki/Charles_Dillon_Perrine" title="Charles Dillon Perrine">Charles Dillon Perrine</a>.<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> “Mr. Perrine’s photograph of November 7th and 8th, 1901, secured with the Crossley Reflector, led to the remarkable discovery that the masses of nebulosity were apparently in motion, with a speed perhaps several hundred times as great as hitherto observed.”<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> “Using the 36-in. telescope (Crossley), he discovered the apparent superluminal motion of the expanding light bubble around Nova Persei (1901). Thought to be a nebula, the visual appearance was actually caused by light from the nova event reflected from the surrounding interstellar medium as the light moved outward from the star. Perrine studied this phenomenon using photographic, spectroscopic, and polarization techniques.”<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Superluminal motion was first observed in 1902 by <a href="/wiki/Jacobus_Kapteyn" title="Jacobus Kapteyn">Jacobus Kapteyn</a> in the ejecta of the <a href="/wiki/Nova" title="Nova">nova</a> <a href="/wiki/GK_Persei" title="GK Persei">GK Persei</a>, which had exploded in 1901.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> His discovery was published in the <a href="/wiki/German_language" title="German language">German</a> journal <i><a href="/wiki/Astronomische_Nachrichten" title="Astronomische Nachrichten">Astronomische Nachrichten</a></i>, and received little attention from English-speaking astronomers until many decades later.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>In 1966, <a href="/wiki/Martin_Rees" title="Martin Rees">Martin Rees</a> pointed out that "an object moving relativistically in suitable directions may appear to a distant observer to have a transverse velocity much greater than the velocity of light".<sup id="cite_ref-Rees_18-0" class="reference"><a href="#cite_note-Rees-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In 1969 and 1970 such sources were found as very distant astronomical radio sources, such as radio galaxies and quasars,<sup id="cite_ref-Gubbay_19-0" class="reference"><a href="#cite_note-Gubbay-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> and were called superluminal sources. The discovery was the result of a new technique called <a href="/wiki/Very_Long_Baseline_Interferometry" class="mw-redirect" title="Very Long Baseline Interferometry">Very Long Baseline Interferometry</a>, which allowed astronomers to set limits to the angular size of components and to determine positions to better than <a href="/wiki/Minute_of_arc" class="mw-redirect" title="Minute of arc">milli-arcseconds</a>, and in particular to determine the change in positions on the sky, called <a href="/wiki/Proper_motion" title="Proper motion">proper motions</a>, in a timespan of typically years. The apparent velocity is obtained by multiplying the observed proper motion by the distance, which could be up to 6 times the speed of light. </p><p>In the introduction to a workshop on superluminal radio sources, Pearson and Zensus reported </p> <blockquote><p>The first indications of changes in the structure of some sources were obtained by an American-Australian team in a series of transpacific VLBI observations between 1968 and 1970 (Gubbay et al. 1969).<sup id="cite_ref-Gubbay_19-1" class="reference"><a href="#cite_note-Gubbay-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Following the early experiments, they had realised the potential of the NASA tracking antennas for VLBI measurements and set up an interferometer operating between California and Australia. The change in the source visibility that they measured for <a href="/wiki/3C_279" title="3C 279">3C 279</a>, combined with changes in total flux density, indicated that a component first seen in 1969 had reached a diameter of about 1 milliarcsecond, implying expansion at an apparent velocity of at least twice the speed of light. Aware of Rees's model,<sup id="cite_ref-Rees_18-1" class="reference"><a href="#cite_note-Rees-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> (Moffet et al. 1972)<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> concluded that their measurement presented evidence for relativistic expansion of this component. This interpretation, although by no means unique, was later confirmed, and in hindsight it seems fair to say that their experiment was the first interferometric measurement of superluminal expansion.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>In 1994, a galactic speed record was obtained with the discovery of a superluminal source in the <a href="/wiki/Milky_Way" title="Milky Way">Milky Way</a>, the <a href="/wiki/Cosmic_x-ray_source" class="mw-redirect" title="Cosmic x-ray source">cosmic x-ray source</a> <a href="/wiki/GRS_1915%2B105" title="GRS 1915+105">GRS 1915+105</a>. The expansion occurred on a much shorter timescale. Several separate blobs were seen to expand in pairs within weeks by typically 0.5 <a href="/wiki/Minute_of_arc#Symbols.2C_abbreviations_and_subdivisions" class="mw-redirect" title="Minute of arc">arcsec</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Because of the analogy with quasars, this source was called a <a href="/wiki/Microquasar" title="Microquasar">microquasar</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superluminal_motion&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Quantum entanglement</a></li> <li><a href="/wiki/Superluminal_communication" title="Superluminal communication">Superluminal communication</a></li> <li><a href="/wiki/Ultra-high-energy_cosmic_ray" title="Ultra-high-energy cosmic ray">Ultra-high-energy cosmic ray</a></li> <li><a href="/wiki/List_of_quasars#List_of_quasars_with_apparent_superluminal_jet_motion" title="List of quasars">List of quasars § List of quasars with apparent superluminal jet motion</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superluminal_motion&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist 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(2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F381529">"Echoes of an Explosive Past: Solving the Mystery of the First Superluminal Source"</a>. <i>The Astrophysical Journal</i>. <b>600</b> (1): L63–L66. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004ApJ...600L..63B">2004ApJ...600L..63B</a>. <a href="/wiki/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/10.1086%2F381529">10.1086/381529</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121645094">121645094</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astrophysical+Journal&rft.atitle=Echoes+of+an+Explosive+Past%3A+Solving+the+Mystery+of+the+First+Superluminal+Source&rft.volume=600&rft.issue=1&rft.pages=L63-L66&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121645094%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1086%2F381529&rft_id=info%3Abibcode%2F2004ApJ...600L..63B&rft.aulast=Bode&rft.aufirst=M.+F.&rft.au=O%27Brien%2C+T.+J.&rft.au=Simpson%2C+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1086%252F381529&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperluminal+motion" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1902AN....157..201K">Kapteyn's paper</a></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/cgi-bin/nph-ref_query?bibcode=1901AN....157..201K&refs=CITATIONS&db_key=AST">Index of citations to Kapteyn's paper</a></span> </li> <li id="cite_note-Rees-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rees_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rees_18-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRees1966" class="citation journal cs1"><a href="/wiki/Martin_Rees,_Baron_Rees_of_Ludlow" class="mw-redirect" title="Martin Rees, Baron Rees of Ludlow">Rees, M. J.</a> (1966). 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Cambridge; New York: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 3. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1987slrs.work....1P">1987slrs.work....1P</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521345606" title="Special:BookSources/9780521345606"><bdi>9780521345606</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Superluminal+Radio+Sources%3A+proceedings+of+a+workshop+in+honor+of+Professor+Marshall+H.+Cohen%2C+held+at+Big+Bear+Solar+Observatory%2C+California%2C+October+28-30%2C+1986&rft.place=Cambridge%3B+New+York&rft.pages=3&rft.pub=Cambridge+University+Press&rft.date=1987&rft_id=info%3Abibcode%2F1987slrs.work....1P&rft.isbn=9780521345606&rft.aulast=Pearson&rft.aufirst=Timothy+J.&rft.au=Zensus%2C+J.+Anton&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperluminal+motion" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|journal=</code> ignored (<a href="/wiki/Help:CS1_errors#periodical_ignored" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMirabelRodriguez1994" class="citation journal cs1">Mirabel, I.F.; Rodriguez, L.F. (1994). "A superluminal source in the Galaxy". <i>Nature</i>. <b>371</b> (6492): 46–48. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994Natur.371...46M">1994Natur.371...46M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F371046a0">10.1038/371046a0</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4347263">4347263</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=A+superluminal+source+in+the+Galaxy&rft.volume=371&rft.issue=6492&rft.pages=46-48&rft.date=1994&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4347263%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F371046a0&rft_id=info%3Abibcode%2F1994Natur.371...46M&rft.aulast=Mirabel&rft.aufirst=I.F.&rft.au=Rodriguez%2C+L.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperluminal+motion" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superluminal_motion&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/Superluminal/superluminal.html">A more detailed explanation.</a></li> <li><a rel="nofollow" class="external text" href="http://aether.lbl.gov/www/classes/p139/homework/superluminal.pdf">A mathematical deduction of superluminal motion.</a></li> <li><a rel="nofollow" class="external text" href="http://www.physics.purdue.edu/MOJAVE/superluminal.html">Superluminal motion Flash Applet.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130705062516/http://www.physics.purdue.edu/MOJAVE/superluminal.html">Archived</a> 2013-07-05 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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