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av-blog-meta-html-info-disabled av-main-single' role="main" itemscope="itemscope" itemtype="https://schema.org/Blog" ><article class="post-entry post-entry-type-standard post-entry-45831 post-loop-1 post-parity-odd post-entry-last single-big with-slider post-45831 post type-post status-publish format-standard has-post-thumbnail hentry category-physics-articles tag-special-relativity" itemscope="itemscope" itemtype="https://schema.org/BlogPosting" itemprop="blogPost" ><div class="big-preview single-big" itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" ><a href="https://www.physicsforums.com/insights/wp-content/uploads/2024/05/addition_of_velocities.png" title="addition of velocities" ><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/05/addition_of_velocities.png" class="wp-image-45978 avia-img-lazy-loading-45978 attachment-entry_with_sidebar size-entry_with_sidebar wp-post-image" alt="" decoding="async" /><noscript><img loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/05/addition_of_velocities.png" class="wp-image-45978 avia-img-lazy-loading-45978 attachment-entry_with_sidebar size-entry_with_sidebar wp-post-image" alt="" decoding="async" /></noscript></a></div><div class="blog-meta"></div><div class='entry-content-wrapper clearfix standard-content'><header class="entry-content-header" aria-label="Post: Addition of Velocities (Velocity Composition) in Special Relativity"><h1 class='post-title entry-title ' itemprop="headline" >Addition of Velocities (Velocity Composition) in Special Relativity<span class="post-format-icon minor-meta"></span></h1><span class="post-meta-infos"><time class="date-container minor-meta updated" itemprop="datePublished" datetime="2024-05-20T06:37:49-05:00" >May 20, 2024</time><span class="text-sep">/</span><span class="comment-container minor-meta"><a href="https://www.physicsforums.com/insights/addition-of-velocities-velocity-composition-in-special-relativity/#comments" class="comments-link" >1 Comment</a></span><span class="text-sep">/</span><span class="blog-categories minor-meta">in <a href="https://www.physicsforums.com/insights/science-math-articles/physics-articles/" rel="tag">Physics Articles</a></span><span class="text-sep">/</span><span class="blog-author minor-meta">by <span class="entry-author-link" itemprop="author" ><span class="author"><span class="fn"><a href="https://www.physicsforums.com/insights/author/robphy/" title="Posts by robphy" rel="author">robphy</a></span></span></span></span></span></header><div class="entry-content" itemprop="text" ><div><strong>Estimated Read Time:</strong> 5 minute(s)</div><div><strong>Common Topics:</strong> os, wrt, galilean, respect, minkowski</div><div><p>The “Addition of Velocities” formula (more correctly, the “Composition of Velocities” formula) in Special Relativity</p><p>[tex]\frac{v_{AC}}{c}=\frac{ \frac{v_{AB}}{c}+\frac{v_{BC}}{c} }{1 + \frac{v_{AB}}{c} \frac{v_{BC}}{c}}[/tex]</p><p>is a non-intuitive result that arises from a “hyperbolic-tangent of a sum”-identity in Minkowski spacetime geometry, with its use of hyperbolic trigonometry.</p><p>However, I claim <strong>it is difficult to obtain this by looking at the Galilean version of this formula and then motivating the special-relativistic version.</strong></p><p>Instead, one should start with the Euclidean analogue (in what could be mistakenly called the “addition of slopes” formula… “composition of slopes” is better),<br /> then do the special-relativistic analogue, then do the Galilean analogue (to obtain the familiar but unfortunately-“our common sense” formula).</p><p>In response to <a href="https://www.physicsforums.com/threads/construct-a-diagram-that-illustrates-the-galilean-law-of-addition-of-velocities.1062466/post-7085225">“Construct a Diagram that Illustrates The Galilean Law of Addition of Velocities”</a><br /> I posed a sequence of trigonometry questions (which hints at the above motivation).</p><p>Here I provide the answers to those questions.</p><div id="ez-toc-container" class="ez-toc-v2_0_69_1 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction"><p class="ez-toc-title" style="cursor:inherit">Table of Contents</p> <label for="ez-toc-cssicon-toggle-item-6743b1f2a6024" class="ez-toc-cssicon-toggle-label"><span class="ez-toc-cssicon"><span class="eztoc-hide" style="display:none;">Toggle</span><span class="ez-toc-icon-toggle-span"><svg style="fill: #4c4c4c;color:#4c4c4c" xmlns="http://www.w3.org/2000/svg" class="list-377408" width="20px" height="20px" viewBox="0 0 24 24" fill="none"><path d="M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z" fill="currentColor"></path></svg><svg style="fill: #4c4c4c;color:#4c4c4c" class="arrow-unsorted-368013" xmlns="http://www.w3.org/2000/svg" width="10px" height="10px" viewBox="0 0 24 24" version="1.2" baseProfile="tiny"><path d="M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z"/></svg></span></span></label><input type="checkbox" id="ez-toc-cssicon-toggle-item-6743b1f2a6024" aria-label="Toggle" /><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class="ez-toc-link ez-toc-heading-1" href="#Euclidean-geometry" title="Euclidean geometry">Euclidean geometry</a></li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class="ez-toc-link ez-toc-heading-2" href="#Minkowski-%E2%80%93-Einstein-Special-Relativity" title="Minkowski – Einstein Special Relativity">Minkowski – Einstein Special Relativity</a></li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class="ez-toc-link ez-toc-heading-3" href="#Galilean-relativity" title="Galilean relativity">Galilean relativity</a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class="ez-toc-link ez-toc-heading-4" href="#More-Related-Articles" title="More Related Articles">More Related Articles</a></li></ul></li></ul></li></ul></nav></div><h1 style="margin-top: 1.5em; letter-spacing: normal;"><span class="ez-toc-section" id="Euclidean-geometry"></span>Euclidean geometry<span class="ez-toc-section-end"></span></h1><p>In the figure below [<strong>where O is the center</strong>],<br /> express<br /> <b>the “slope of OS [with respect to OP]” (PS/OP)</b><br /> in terms of<br /> <b>the “slope of ON [with respect to OP or OM]” (MN/OM)</b><br /> and<br /> <b>the “slope of OS [with respect to ON]” NS/ON).</b><br /> Everything can be done using ratios of segments<br /> (and one can use some trigonometric intuition to guide you). Note: in this Euclidean geometry, <b>NS is Euclidean-perpendicular to ON</b><br /> since, for radius vector ON, the segment NS is tangent to the Euclidean-circle.</p><p><img data-lazyloaded="1" src="data:image/gif;base64,R0lGODdhAQABAPAAAMPDwwAAACwAAAAAAQABAAACAkQBADs=" decoding="async" data-src="https://www.physicsforums.com/attachments/1715198868081-png.344790/" alt="1715198868081.png" /><noscript><img decoding="async" src="https://www.physicsforums.com/attachments/1715198868081-png.344790/" alt="1715198868081.png" /></noscript></p><p><strong>Answer:</strong> (note the minus sign in the second line)[tex]\begin{align*}<br /> (\mbox{slope of OS wrt OP})<br /> &=\frac{PS}{OP}\\<br /> &=\frac{PR+RS}{OM➖MP}\\<br /> &=\frac{MN+RS}{OM-NR}\\<br /> &=\frac{\frac{MN}{OM}+\frac{RS}{OM}}{1-\frac{NR}{OM}}\\<br /> &=\frac{\frac{MN}{OM}+\frac{RS}{OM} }{1-\color{green}{\frac{MN}{OM}\frac{NR}{MN}}}\\<br /> &=\frac{\frac{MN}{OM}+\color{red}{\left(\frac{-SR}{OM}\right) } }{1-\color{green}{\frac{MN}{OM}\color{blue}{\left( \frac{-RN}{MN} \right) } }}\\<br /> &=\frac{\frac{MN}{OM}+\color{red}{\left(\frac{-SN}{ON}\right) } }{1-\frac{MN}{OM}\color{blue}{\left( \frac{-SN}{ON} \right) } }\\<br /> &=\frac{\frac{MN}{OM}+\color{red}{\left(\frac{NS}{ON}\right) } }{1-\frac{MN}{OM} \color{blue}{\left( \frac{NS}{ON} \right) }}\\<br /> &=\frac{(\mbox{slope of ON wrt OM})+(\mbox{slope of OS wrt ON}) }{1- (\mbox{slope of ON wrt OM})(\mbox{slope of OS wrt ON}) }<br /> \end{align*}<br /> [/tex]<br /> where we used the similarity of triangles OMN and SRN.<br /> The “slopes” don’t add…because the spatial displacements are not parallel in this plane. However, the associated radial angles<br /> [interpreted as intercepted Euclidean-arc-lengths on a unit Euclidean-circle<br /> or as intercepted sector-areas in the unit Euclidean-disk]<br /> do add: [itex]\phi_{POS}=\phi_{MON}+\phi_{NOS}[/itex]<br /> with slopes [itex]\frac{PS}{OP}=\tan\phi_{POS}[/itex], [itex]\frac{MN}{OM}=\tan\phi_{MON}[/itex], [itex]\frac{NS}{ON}=\tan\phi_{NOS}[/itex].<br /> Thus,<br /> [tex]\begin{align*}<br /> \tan\phi_{POS}<br /> &=\tan\left( \phi_{MON}+\phi_{NOS} \right) \\&=\frac{ \tan \phi_{MON}+\tan\phi_{NOS} }{ 1 – \tan\phi_{MON}\tan\phi_{NOS} } \\<br /> \end{align*}<br /> [/tex]</p><h1><span class="ez-toc-section" id="Minkowski-%E2%80%93-Einstein-Special-Relativity"></span>Minkowski – Einstein Special Relativity<span class="ez-toc-section-end"></span></h1><p>Then repeat for this figure in special relativity<strong> [where O is the center]</strong>,.<br /> That is,<br /> express<br /> <b>the “velocity of OS [with respect to OP]” (PS/OP)</b><br /> in terms of<br /> <b>the “velocity of ON [with respect to OP or OM]” (MN/OM)</b><br /> and<br /> <b>the “velocity of OS [with respect to ON]” NS/ON).</b><br /> (The method is almost the same…<br /> You might wish to follow your earlier steps and see what has become of them in this case.<br /> However, you’ll have to accept that<br /> in this [Minkowski spacetime] geometry, <b>NS is Minkowski-perpendicular to ON</b><br /> since, for radius vector ON, the segment NS is tangent to the Minkowski-circle.)<img data-lazyloaded="1" src="data:image/gif;base64,R0lGODdhAQABAPAAAMPDwwAAACwAAAAAAQABAAACAkQBADs=" decoding="async" data-src="https://www.physicsforums.com/attachments/1715199384160-png.344791/" alt="1715199384160.png" /><noscript><img decoding="async" src="https://www.physicsforums.com/attachments/1715199384160-png.344791/" alt="1715199384160.png" /></noscript></p><p><strong>Answer:</strong> (note the plus sign in the second line)[tex]\begin{align*}<br /> (\mbox{velocity of OS wrt OP})<br /> &=\frac{PS}{OP}\\<br /> &=\frac{PR+RS}{OM ➕ MP}\\<br /> &=\frac{MN+RS}{OM+NR}\\<br /> &=\frac{\frac{MN}{OM}+\frac{RS}{OM}}{1+\frac{NR}{OM}}\\<br /> &=\frac{\frac{MN}{OM}+\frac{RS}{OM} }{1+\color{green}{\frac{MN}{OM}\frac{NR}{MN}}}\\<br /> &=\frac{\frac{MN}{OM}+\color{red}{\left(\frac{-SR}{OM}\right) } }{1+\color{green}{\frac{MN}{OM}\color{blue}{\left( \frac{-RN}{MN} \right) } }}\\<br /> &=\frac{\frac{MN}{OM}+\color{red}{\left(\frac{-SN}{ON}\right) } }{1+\frac{MN}{OM}\color{blue}{\left( \frac{-SN}{ON} \right) } }\\<br /> &=\frac{\frac{MN}{OM}+\color{red}{\left(\frac{NS}{ON}\right) } }{1+\frac{MN}{OM} \color{blue}{\left( \frac{NS}{ON} \right) }}\\<br /> &=\frac{(\mbox{velocity of ON wrt OM})+(\mbox{velocity of OS wrt ON}) }{1+ (\mbox{velocity of ON wrt OM})(\mbox{velocity of OS wrt ON}) }<br /> \end{align*}<br /> [/tex]<br /> where we used the similarity of triangles OMN and SRN.<br /> The “velocities” don’t add…because the spatial displacements are not coplanar (collinear) in spacetime.<br /> Physically, the corresponding observers do not mutually share the same notion of which events are simultaneous. However, the associated radial Minkowski-angles<br /> [interpreted as intercepted Minkowski-arc-lengths on a unit Minkowski-circle<br /> or as intercepted sector-areas in the unit Minkowski -disk]<br /> do add: [itex]\theta_{POS}=\theta_{MON}+\theta_{NOS}[/itex]<br /> with velocities [itex]\frac{PS}{OP}=\theta\phi_{POS}[/itex], [itex]\frac{MN}{OM}=\tan\theta_{MON}[/itex], [itex]\frac{NS}{ON}=\tan\theta_{NOS}[/itex].<br /> Thus,<br /> [tex]\begin{align*}<br /> \tan\theta_{POS}<br /> &=\tan\left( \theta_{MON}+\theta_{NOS} \right) \\&=\frac{ \tan \theta_{MON}+\tan\theta_{NOS} }{ 1 + \tan\theta_{MON}\tan\theta_{NOS} } \\<br /> \end{align*}<br /> [/tex]</p><h1><span class="ez-toc-section" id="Galilean-relativity"></span>Galilean relativity<span class="ez-toc-section-end"></span></h1><p>And, now finally,<br /> repeat for this figure in <a href="https://www.physicsforums.com/insights/speed-light-galilean-relativity/" data-wpil-monitor-id="3">Galilean relativity</a>.<br /> That is,<br /> express<br /> <b>the “velocity of OS [with respect to OP]” (PS/OP)</b><br /> in terms of<br /> <b>the “velocity of ON [with respect to OP or OM]” (MN/OM)</b><br /> and<br /> <b>the “velocity of OS [with respect to ON]” NS/ON).</b><br /> (The method is almost the same…<br /> You might wish to follow your earlier steps and see what has become of them in this case.<br /> However, you’ll have to accept that<br /> in this [Galilean spacetime] geometry, <b>NS is Galilean-perpendicular to ON</b><br /> since, for radius vector ON, the segment NS is tangent to the Galilean-circle.<br /> )<img data-lazyloaded="1" src="data:image/gif;base64,R0lGODdhAQABAPAAAMPDwwAAACwAAAAAAQABAAACAkQBADs=" decoding="async" data-src="https://www.physicsforums.com/attachments/1715199747887-png.344792/" alt="1715199747887.png" /><noscript><img decoding="async" src="https://www.physicsforums.com/attachments/1715199747887-png.344792/" alt="1715199747887.png" /></noscript></p><p><strong>Answer:</strong> (note the second line, as if there was a zero instead of a sign seen above)[tex]\begin{align*}<br /> (\mbox{velocity of OS wrt OP})<br /> &=\frac{PS}{OP}\\<br /> &=\frac{PR+RS}{OM }\\<br /> &=\frac{MN+RS}{OM}\\<br /> &=\frac{\frac{MN}{OM}+\frac{RS}{OM}}{1}\\<br /> &=\frac{\frac{MN}{OM}+\frac{RS}{OM} }{1}\\<br /> &=\frac{\frac{MN}{OM}+\color{red}{\left(\frac{-SR}{OM}\right) } }{1 }\\<br /> &=\frac{\frac{MN}{OM}+\color{red}{\left(\frac{-SN}{ON}\right) } }{1 }\\<br /> &=\frac{\frac{MN}{OM}+\color{red}{\left(\frac{NS}{ON}\right) } }{1}\\<br /> &=\frac{(\mbox{velocity of ON wrt OM})+(\mbox{velocity of OS wrt ON}) }{1}<br /> \end{align*}<br /> [/tex]<br /> where we used the similarity of triangles OMN and SRN.</p><p>So, in this [degenerate] case, the “slopes” (the “velocities”) do add…because the spatial displacements are coplanar (collinear) in spacetime.<br /> However, <strong>this “additivity of velocities” is actually the exception, rather than the rule. </strong>(There is an analogous notion of Galilean angles and Galilean trigonometry [due to I.M. Yaglom]… but I won’t discuss this here.]</p></div><div class="saboxplugin-wrap" itemtype="http://schema.org/Person" itemscope itemprop="author"><div class="saboxplugin-tab"><div class="saboxplugin-gravatar"></div><div class="saboxplugin-authorname"><a href="https://www.physicsforums.com/insights/author/robphy/" class="vcard author" rel="author"><span class="fn">robphy</span></a></div><div class="saboxplugin-desc"><div itemprop="description"><p>Professor of Physics (BS,MS,PhD), Math (BS). 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data-av_iconfont='entypo-fontello'></span></span></span><strong class="av-related-title">Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details</strong></a></div></div></div><div class='comment-entry post-entry'><div class='comment_meta_container'><div class='side-container-comment'><div class='side-container-comment-inner'> <span class='comment-count'>1</span> <span class='comment-text'>reply</span> <span class='center-border center-border-left'></span> <span class='center-border center-border-right'></span></div></div></div><div class='comment_container'><ol class="commentlist" id="comments"><li class="comment byuser comment-author-mister-t even thread-even depth-1" id="li-comment-13327"><div id="comment-13327"><article><div class="gravatar"></div><div class='comment_content'><header class="comment-header" aria-label="Comment Info" > <cite class="author_name heading"><cite class="comment_author_name">Mister T</cite></cite> <span class="says">says:</span><div class="comment-meta commentmetadata"> <a href="https://www.physicsforums.com/insights/addition-of-velocities-velocity-composition-in-special-relativity/#comment-13327"> <time > May 21, 2024 at 8:38 am </time> </a></div></header><div class='comment_text entry-content-wrapper clearfix' ><div class="bbWrapper">Composition of slopes is also called the stacked wedge analogy.</div> <a rel="nofollow" class="comment-reply-login" href="https://www.physicsforums.com/insights/wp-login.php?redirect_to=https%3A%2F%2Fwww.physicsforums.com%2Finsights%2Faddition-of-velocities-velocity-composition-in-special-relativity%2F">Log in to Reply</a></div></div></article></div></li></ol></div><div class="comment_container"><h3 class='miniheading '>Leave a Reply</h3><span class="minitext">Want to join the discussion?<br />Feel free to contribute!</span><div id="respond" class="comment-respond"><h3 id="reply-title" class="comment-reply-title">Leave a Reply <small><a rel="nofollow" id="cancel-comment-reply-link" 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