Properties of Born rigid congruence

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This concept emphasizes the preservation of distances and the uniform motion of rigid bodies in a curved spacetime, highlighting the implications for both classical mechanics and general relativity. Key properties include the limitations on acceleration, the effects of gravitational fields, and the mathematical formulation of the congruence, which provides insights into the behavior of objects moving through spacetime without deformation. </div> <script src="https://www.physicsforums.com/check-for-tex.js" defer></script> <div class="block block--messages" data-xf-init="" data-type="post" data-href="/inline-mod/" data-search-target="*"> <span class="u-anchorTarget" id="posts"></span> <div class="block-outer"><div class="block-outer-main"> <nav class="pageNavWrapper pageNavWrapper--logarithmic pageNavWrapper--full " data-block-size="1" data-last-page="5"> <div class="pageNav"> <ul class="pageNav-main"> <li class="pageNav-page pageNav-page--current "><a href="/threads/properties-of-born-rigid-congruence.1064188/">1</a></li> <li class="pageNav-page "><a href="/threads/properties-of-born-rigid-congruence.1064188/page-2">2</a></li> <li class="pageNav-page "><a href="/threads/properties-of-born-rigid-congruence.1064188/page-3">3</a></li> <li class="pageNav-page "><a href="/threads/properties-of-born-rigid-congruence.1064188/page-4">4</a></li> <li class="pageNav-page "><a href="/threads/properties-of-born-rigid-congruence.1064188/page-5">5</a></li>  </ul>  </div> </nav></div></div> <div class="block-outer js-threadStatusField"></div> <div class="block-container lbContainer" data-xf-init="lightbox select-to-quote old-new-message-indicator" data-sv-live-content-time-element-selector=".message-attribution-main time.u-dt" data-message-selector=".js-post" data-lb-id="thread-1064188" data-lb-universal="0"> <div class="block-body js-replyNewMessageContainer">  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102080" id="js-post-7102080" > <span class="u-anchorTarget" id="post-7102080"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T01:43:26-0500" data-time="1720161806" data-date-string="Jul 5, 2024" data-time-string="1:43 AM" title="Jul 5, 2024 at 1:43 AM">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102080/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #1 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" > <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697">cianfa72</span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102080" data-lb-caption-desc="cianfa72 · Jul 5, 2024 at 1:43 AM"> <article class="message-body js-selectToQuote"> <div class="googleads_fixed"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-6580726045122001" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-6580726045122001" data-ad-slot="3049969336" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> <div > <div class="bbWrapper">Consider a Born rigid timelike congruence. Starting from any point P on each congruence's worldline take a spacelike direction at P and build the spacelike geodesic from P pointing in that direction. It will eventually intersect other congruence's worldlines.<br /> <br /> Does the Born rigid condition imply that the spacetime lenght of such spacelike geodesic "segments" won't change regardless the point P chosen along a given congruence's member ?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <div class="message-cell message-cell--main" id="post-physorg"> <a href="https://phys.org/physics-news/" target="_blank">Physics news</a> on Phys.org <ul class="phys_items"> <li><a href="https://phys.org/news/2024-11-classical-black-hole-inequalities-quantum.html" target="_blank">Extending classical black hole inequalities into the quantum realm</a></li> <li><a href="https://phys.org/news/2024-11-accessing-lesser-nucleon-neutron-physicists.html" target="_blank">Accessing the lesser known nucleon: New neutron measurement can help physicists learn about nucleon structure and spin</a></li> <li><a href="https://phys.org/news/2024-11-prototype-lenses-block-epileptic-seizure.html" target="_blank">Prototype lenses can block epileptic-seizure causing wavelengths</a></li> </ul> </div> <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102228" id="js-post-7102228" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102228/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102228"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T13:06:28-0500" data-time="1720202788" data-date-string="Jul 5, 2024" data-time-string="1:06 PM" title="Jul 5, 2024 at 1:06 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102228/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #2 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102228" data-lb-caption-desc="PeterDonis · Jul 5, 2024 at 1:06 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102080" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Consider a Born rigid timelike congruence. Starting from any point P on each congruence's worldline take a spacelike direction at P and build the spacelike geodesic from P pointing in that direction. It will eventually intersect other congruence's worldlines.<br /> <br /> Does the Born rigid condition imply that the spacetime lenght of such spacelike geodesic "segments" won't change regardless the point P chosen along a given congruence's member ? </div> </div> </blockquote>Not as you state it, because you haven't specified how the spacelike direction is to be taken.<br /> <br /> I would suggest looking at some specific examples, such as the Rindler congruence and the Langevin congruence in flat spacetime, and seeing how you might try to do what you describe. For extra credit, you could then try some congruences in curved spacetimes.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102237" id="js-post-7102237" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102237/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102237"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T14:21:24-0500" data-time="1720207284" data-date-string="Jul 5, 2024" data-time-string="2:21 PM" title="Jul 5, 2024 at 2:21 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102237/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #3 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102237" data-lb-caption-desc="cianfa72 · Jul 5, 2024 at 2:21 PM"> <article class="message-body js-selectToQuote"> <div class="googleads_fixed"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-6580726045122001" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-6580726045122001" data-ad-slot="5961342082" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102228" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Not as you state it, because you haven't specified how the spacelike direction is to be taken. </div> </div> </blockquote>Yes, the spacelike direction I was talking about belongs to the orthogonal complement at each point P along the timelike congruence.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102238" id="js-post-7102238" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102238/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102238"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T14:22:13-0500" data-time="1720207333" data-date-string="Jul 5, 2024" data-time-string="2:22 PM" title="Jul 5, 2024 at 2:22 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102238/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #4 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102238" data-lb-caption-desc="PeterDonis · Jul 5, 2024 at 2:22 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102237" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> the spacelike direction I was talking about belongs to the orthogonal complement at each point along the timelike congruence. </div> </div> </blockquote>Have you tried the specific examples I gave using this specification?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102253" id="js-post-7102253" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102253/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102253"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T15:30:06-0500" data-time="1720211406" data-date-string="Jul 5, 2024" data-time-string="3:30 PM" title="Jul 5, 2024 at 3:30 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102253/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #5 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102253" data-lb-caption-desc="cianfa72 · Jul 5, 2024 at 3:30 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102238" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Have you tried the specific examples I gave using this specification? </div> </div> </blockquote>For Rindler congruence in 2D flat spacetime, the spacelike direction orthogonal to the congruence's member tangent vector at each point P is unique. If we draw its integral geodesic line in Minkowski standard inertial coordinate chart, the lenght of such spacelike geodesic "segment" between two given congruence's worldlines doesn't change regardless the point P picked.<br /> <br /> I believe the same argument applies to Langevin congruence as well.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102257" id="js-post-7102257" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102257/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102257"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T15:46:13-0500" data-time="1720212373" data-date-string="Jul 5, 2024" data-time-string="3:46 PM" title="Jul 5, 2024 at 3:46 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102257/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #6 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102257" data-lb-caption-desc="PeterDonis · Jul 5, 2024 at 3:46 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102253" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> For Rindler congruence in 2D flat spacetime, the spacelike direction orthogonal to the congruence's member tangent vector at each point P is unique. If we draw its integral geodesic line in Minkowski standard inertial coordinate chart, the lenght of such spacelike geodesic "segment" between two given congruence's worldlines doesn't change regardless the point P picked. </div> </div> </blockquote>Yes.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102253" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I believe the same argument applies to Langevin congruence as well. </div> </div> </blockquote>The Langevin congruence, unlike the Rindler congruence, is not hypersurface orthogonal. So you need to be more careful in that case.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="DrGreg" data-content="post-7102271" id="js-post-7102271" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102271/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102271"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T16:06:31-0500" data-time="1720213591" data-date-string="Jul 5, 2024" data-time-string="4:06 PM" title="Jul 5, 2024 at 4:06 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102271/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #7 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/drgreg.35885/"> <meta itemprop="url" content="https://www.physicsforums.com/members/drgreg.35885/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/drgreg.35885/" class="username " dir="auto" data-user-id="35885" data-xf-init="member-tooltip"><span itemprop="name">DrGreg</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,474</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>2,082</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102271" data-lb-caption-desc="DrGreg · Jul 5, 2024 at 4:06 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102253" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> in 2D flat spacetime, the spacelike direction orthogonal to the congruence's member tangent vector at each point P is unique </div> </div> </blockquote>But in 4D spacetime, there are an infinite number of such directions to choose from. How do you relate the choices at two different Ps?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102276" id="js-post-7102276" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102276/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102276"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T16:20:19-0500" data-time="1720214419" data-date-string="Jul 5, 2024" data-time-string="4:20 PM" title="Jul 5, 2024 at 4:20 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102276/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #8 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102276" data-lb-caption-desc="PeterDonis · Jul 5, 2024 at 4:20 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102253" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> in 2D flat spacetime </div> </div> </blockquote>Note that you can't describe the Langevin congruence in 2D flat spacetime. So <a href="https://www.physicsforums.com/members/35885/" class="username" data-xf-init="member-tooltip" data-user-id="35885" data-username="@DrGreg">@DrGreg</a>'s question is already relevant even for that (still relatively easy) case.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102280" id="js-post-7102280" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102280/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102280"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T16:38:33-0500" data-time="1720215513" data-date-string="Jul 5, 2024" data-time-string="4:38 PM" title="Jul 5, 2024 at 4:38 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102280/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #9 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102280" data-lb-caption-desc="cianfa72 · Jul 5, 2024 at 4:38 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102257" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> The Langevin congruence, unlike the Rindler congruence, is not hypersurface orthogonal. So you need to be more careful in that case. </div> </div> </blockquote>Pick a point P along a Langevin congruence's worldline in 4D flat spacetime and consider the orthogonal complement to the worldline's tangent vector at that point. Pick a spacelike direction in it and build the integral geodesic emanating from P. It is a straight line in Minkowski standard inertial coordinates that will intersect other congruence's members. Calculate the spacetime lenght of a such spacelike geodesic segment starting from P between two given congruence's worldlines/members.<br /> <br /> Now pick a different point Q on the chosen congruence's worldline and repeat the above construction starting from it.<br /> <br /> How the fact that Langevin congruence is not hypersurface orthogonal related to the result that the above calculated lenghts are not the same (note that Langevin congruence is Born rigid) ?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102281" id="js-post-7102281" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102281/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102281"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T16:40:46-0500" data-time="1720215646" data-date-string="Jul 5, 2024" data-time-string="4:40 PM" title="Jul 5, 2024 at 4:40 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102281/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #10 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102281" data-lb-caption-desc="PeterDonis · Jul 5, 2024 at 4:40 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102280" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> repeat the above construction </div> </div> </blockquote>You can't because there are an infinite number of possible results of "pick a spacelike direction in the orthogonal complement to the worldline's tangent vector", and you have given no way to pick a unique one.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102280" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> How the fact that Langevin congruence is not hypersurface orthogonal related to the result that the above calculated lenght are not the same </div> </div> </blockquote>That's not what I said. I said you need to first come up with a unique specification for <i>how to pick the spacelike geodesics</i> before we can even evaluate the question of whether the calculated lengths change or not.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102283" id="js-post-7102283" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102283/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102283"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T16:45:41-0500" data-time="1720215941" data-date-string="Jul 5, 2024" data-time-string="4:45 PM" title="Jul 5, 2024 at 4:45 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102283/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #11 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102283" data-lb-caption-desc="cianfa72 · Jul 5, 2024 at 4:45 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102281" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> That's not what I said. I said you need to first come up with a unique specification for <i>how to pick the spacelike geodesics</i> before we can even evaluate the question of whether the calculated lengths change or not. </div> </div> </blockquote>Ah ok, we can do that by parallel transporting the spacelike direction picked at point P from P to Q along the given congruence's worldline.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102284" id="js-post-7102284" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102284/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102284"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T16:47:07-0500" data-time="1720216027" data-date-string="Jul 5, 2024" data-time-string="4:47 PM" title="Jul 5, 2024 at 4:47 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102284/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #12 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102284" data-lb-caption-desc="PeterDonis · Jul 5, 2024 at 4:47 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102283" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> we can do that by parallel transporting the spacelike direction picked at point P from P to Q along the given congruence's worldline. </div> </div> </blockquote>That would be one way, but not the only way. Will doing it this way make the spacelike length to a given other member of the congruence constant, or not? What do you think?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102285" id="js-post-7102285" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102285/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102285"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T16:58:24-0500" data-time="1720216704" data-date-string="Jul 5, 2024" data-time-string="4:58 PM" title="Jul 5, 2024 at 4:58 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102285/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #13 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102285" data-lb-caption-desc="cianfa72 · Jul 5, 2024 at 4:58 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102284" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Will doing it this way make the spacelike length to a given other member of the congruence constant, or not? What do you think? </div> </div> </blockquote>No, I don't think so. Which other way could there be to uniquely specify how to pick a spacelike direction in the 3D orthogonal complement at each point P along a worldline?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> <div class="message-lastEdit"> Last edited: <time class="u-dt" dir="auto" datetime="2024-07-05T17:25:55-0500" data-time="1720218355" data-date-string="Jul 5, 2024" data-time-string="5:25 PM" title="Jul 5, 2024 at 5:25 PM" itemprop="dateModified">Jul 5, 2024</time> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102289" id="js-post-7102289" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102289/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102289"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-05T17:35:41-0500" data-time="1720218941" data-date-string="Jul 5, 2024" data-time-string="5:35 PM" title="Jul 5, 2024 at 5:35 PM" itemprop="datePublished">Jul 5, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102289/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #14 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102289" data-lb-caption-desc="PeterDonis · Jul 5, 2024 at 5:35 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102285" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I don't think so. </div> </div> </blockquote>You're correct.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102285" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Which other way could there be to uniquely specify how to pick a spacelike direction in the 3D orthogonal complement at each point P along a worldline? </div> </div> </blockquote>What other transport laws are there? What other properties of the worldline could you use?<br /> <br /> (You can get one hint from considering what property of the Rindler congruence picks out a 2-D surface in spacetime. Note that even in 4-D spacetime there is <i>still</i> a Rindler congruence with a property that picks out 2-D surfaces.)</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102389" id="js-post-7102389" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102389/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102389"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-06T06:58:39-0500" data-time="1720267119" data-date-string="Jul 6, 2024" data-time-string="6:58 AM" title="Jul 6, 2024 at 6:58 AM" itemprop="datePublished">Jul 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102389/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #15 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102389" data-lb-caption-desc="cianfa72 · Jul 6, 2024 at 6:58 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102289" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> What other transport laws are there? What other properties of the worldline could you use? </div> </div> </blockquote>Other transport laws I'm aware of are Fermi-Walker transport and Lie dragging/transport.<br /> <br /> The Rindler congruence's members are integral orbits of a KVF and have zero expansion scalar. Perhaps these are useful properties to uniquely pick out a spacelike direction at each point along a congruence's member/worldline.<br /> <br /> <blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102289" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> (You can get one hint from considering what property of the Rindler congruence picks out a 2-D surface in spacetime. Note that even in 4-D spacetime there is <i>still</i> a Rindler congruence with a property that picks out 2-D surfaces.) </div> </div> </blockquote>Sorry, in 2D flat spacetime how can the Rindler congruence picks out a 2D surface when the spacetime itself has dimension 2 ?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102484" id="js-post-7102484" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102484/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102484"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-06T13:57:00-0500" data-time="1720292220" data-date-string="Jul 6, 2024" data-time-string="1:57 PM" title="Jul 6, 2024 at 1:57 PM" itemprop="datePublished">Jul 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102484/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #16 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102484" data-lb-caption-desc="PeterDonis · Jul 6, 2024 at 1:57 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102389" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Other transport laws I'm aware of are Fermi-Walker transport and Lie dragging/transport. </div> </div> </blockquote>Have you tried those to see what they say?<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102389" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> The Rindler congruence's members are integral orbits of a KVF and have zero expansion scalar. </div> </div> </blockquote>So are the members of the Langevin congruence.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102389" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Perhaps these are useful properties to uniquely pick out a spacelike direction at each point along a congruence's member/worldline. </div> </div> </blockquote>Since these properties are shared by both the Rindler and Langevin congruences, as above, it would not seem so.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102389" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> in 2D flat spacetime how can the Rindler congruence picks out a 2D surface when the spacetime itself has dimension 2 ? </div> </div> </blockquote>I didn't say in 2D flat spacetime, I said in 4D flat spacetime.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102522" id="js-post-7102522" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102522/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102522"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-06T15:49:49-0500" data-time="1720298989" data-date-string="Jul 6, 2024" data-time-string="3:49 PM" title="Jul 6, 2024 at 3:49 PM" itemprop="datePublished">Jul 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102522/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #17 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102522" data-lb-caption-desc="cianfa72 · Jul 6, 2024 at 3:49 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102484" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Have you tried those to see what they say? </div> </div> </blockquote>What about Fermi-Walker transport? As far as I know, if one picks a spacelike direction ##e_1## at a point P along a worldline (a spacelike direction ##e_1## in the orthogonal complement w.r.t. the timelike tangent vector ##e_0## at that point P) then, using Fermi-Walker transport along the worldline, one gets a "non-rotated" version of ##e_1## at point Q.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102525" id="js-post-7102525" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102525/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102525"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-06T16:01:43-0500" data-time="1720299703" data-date-string="Jul 6, 2024" data-time-string="4:01 PM" title="Jul 6, 2024 at 4:01 PM" itemprop="datePublished">Jul 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102525/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #18 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102525" data-lb-caption-desc="PeterDonis · Jul 6, 2024 at 4:01 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102522" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> What about Fermi-Walker transport? </div> </div> </blockquote>Again, have you tried it? Do the math and see.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102522" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> using Fermi-Walker transport along the worldline, one gets a "non-rotated" version of ##e_1## at point Q. </div> </div> </blockquote>For a particular sense of "non-rotated", yes. But is that sense of "non-rotated" the right one to make the spacelike distance you have defined constant for the Langevin congruence?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102531" id="js-post-7102531" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102531/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102531"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-06T16:16:31-0500" data-time="1720300591" data-date-string="Jul 6, 2024" data-time-string="4:16 PM" title="Jul 6, 2024 at 4:16 PM" itemprop="datePublished">Jul 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102531/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #19 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102531" data-lb-caption-desc="cianfa72 · Jul 6, 2024 at 4:16 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102525" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> For a particular sense of "non-rotated", yes. But is that sense of "non-rotated" the right one to make the spacelike distance you have defined constant for the Langevin congruence? </div> </div> </blockquote>Sorry, I really don't know. Perhaps it isn't the right one since the Langevin congruence is not "irrotational" -- i.e. it isn't hypersurface orthogonal.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102536" id="js-post-7102536" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102536/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102536"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-06T16:28:52-0500" data-time="1720301332" data-date-string="Jul 6, 2024" data-time-string="4:28 PM" title="Jul 6, 2024 at 4:28 PM" itemprop="datePublished">Jul 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102536/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #20 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102536" data-lb-caption-desc="PeterDonis · Jul 6, 2024 at 4:28 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102531" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I really don't know. </div> </div> </blockquote>That's why I suggested doing the math to see.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102531" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> the Langevin congruence is not "irrotational" -- i.e. it isn't hypersurface orthogonal. </div> </div> </blockquote>That does make a difference, yes.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102655" id="js-post-7102655" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102655/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102655"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-07T06:21:19-0500" data-time="1720351279" data-date-string="Jul 7, 2024" data-time-string="6:21 AM" title="Jul 7, 2024 at 6:21 AM" itemprop="datePublished">Jul 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102655/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #21 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102655" data-lb-caption-desc="cianfa72 · Jul 7, 2024 at 6:21 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102536" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> That's why I suggested doing the math to see. </div> </div> </blockquote>Langevin congruence's worldlines in flat spacetime are represented by <a href="https://en.m.wikipedia.org/wiki/Born_coordinates#Langevin_observers_in_the_cylindrical_chart" target="_blank" class="link link--external" rel="nofollow ugc noopener">helices</a> in Minkowski standard inertial coordinates.<br /> <br /> Now the point is: at a point P along a given congruence's worldline/member (say ##a##) pick a spacelike direction orthogonal to the (timelike) tangent vector at P. The spacelike geodesic built in that direction will intersect another member of the Langevin congruence (say ##b##). Now take a different point Q along the same member ##a##. In the orthogonal complement at Q I think there will be a unique (spacelike) direction such that the spacelike geodesic built in that direction will intersect the member ##b##. Therefore the goal is to single out such a spacelike direction at Q starting from the spacelike direction picked at point P.<br /> <br /> As you pointed out, Langevin congruence is not hypersurface orthogonal, therefore neighboring congruence's worldlines of member ##a## "twist/rotate" w.r.t. it.<br /> <br /> If the above is correct then, from an intuitive point of view, Fermi-Walker transport might not be the right answer to get that unique spacelike direction at Q.<br /> <br /> Edit: I know the definition of <a href="https://en.m.wikipedia.org/wiki/Fermi%E2%80%93Walker_transport#Fermi%E2%80%93Walker_differentiation" target="_blank" class="link link--external" rel="nofollow ugc noopener">Fermi-Walker transport</a>, namely $$ \frac {D_F X} {ds} = 0$$ where ##X=e_1(\tau)##, however I'm in trouble to work out the math for this specific case.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> <div class="message-lastEdit"> Last edited: <time class="u-dt" dir="auto" datetime="2024-07-07T10:09:00-0500" data-time="1720364940" data-date-string="Jul 7, 2024" data-time-string="10:09 AM" title="Jul 7, 2024 at 10:09 AM" itemprop="dateModified">Jul 7, 2024</time> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102752" id="js-post-7102752" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102752/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102752"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-07T13:32:35-0500" data-time="1720377155" data-date-string="Jul 7, 2024" data-time-string="1:32 PM" title="Jul 7, 2024 at 1:32 PM" itemprop="datePublished">Jul 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102752/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #22 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102752" data-lb-caption-desc="PeterDonis · Jul 7, 2024 at 1:32 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102655" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> from an intuitive point of view, Fermi-Walker transport might not be the right answer to get that unique spacelike direction at Q. </div> </div> </blockquote>Your intuition is correct; it isn't. But you should still take the time to learn how to work out the math. You might find this Insights article helpful:<br /> <br /> <a href="https://www.physicsforums.com/insights/how-to-study-fermi-walker-transport-in-minkowski-spacetime/" class="link link--internal">https://www.physicsforums.com/insights/how-to-study-fermi-walker-transport-in-minkowski-spacetime/</a></div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102857" id="js-post-7102857" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102857/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102857"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T01:49:28-0500" data-time="1720421368" data-date-string="Jul 8, 2024" data-time-string="1:49 AM" title="Jul 8, 2024 at 1:49 AM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102857/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #23 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102857" data-lb-caption-desc="cianfa72 · Jul 8, 2024 at 1:49 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102752" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Your intuition is correct; it isn't. But you should still take the time to learn how to work out the math. You might find this Insights article helpful:<br /> <br /> <a href="https://www.physicsforums.com/insights/how-to-study-fermi-walker-transport-in-minkowski-spacetime/" class="link link--internal">https://www.physicsforums.com/insights/how-to-study-fermi-walker-transport-in-minkowski-spacetime/</a> </div> </div> </blockquote>Thanks, I read the insight and I have some questions on it. Is it fine to discuss it here?<br /> <br /> For example how the vorticity vector $$<br /> \vec{\Omega} = \frac{\gamma^2 v}{R} \hat{p}_1 = \gamma^2 \omega \hat{p}_1$$ is related to the frame base vectors spinning about Z axis relative to gyro-stabilized vectors.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> <div class="message-lastEdit"> Last edited: <time class="u-dt" dir="auto" datetime="2024-07-08T02:05:15-0500" data-time="1720422315" data-date-string="Jul 8, 2024" data-time-string="2:05 AM" title="Jul 8, 2024 at 2:05 AM" itemprop="dateModified">Jul 8, 2024</time> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102956" id="js-post-7102956" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102956/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102956"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T09:14:24-0500" data-time="1720448064" data-date-string="Jul 8, 2024" data-time-string="9:14 AM" title="Jul 8, 2024 at 9:14 AM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102956/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #24 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102956" data-lb-caption-desc="PeterDonis · Jul 8, 2024 at 9:14 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102857" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Is it fine to discuss it here? </div> </div> </blockquote>It is now that I've moved this discussion to a new thread. <img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite2" alt=":wink:" title="Wink :wink:" loading="lazy" data-shortname=":wink:" /><br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102857" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> how the vorticity vector $$<br /> \vec{\Omega} = \frac{\gamma^2 v}{R} \hat{p}_1 = \gamma^2 \omega \hat{p}_1$$ is related to the frame base vectors spinning about Z axis relative to gyro-stabilized vectors. </div> </div> </blockquote>Have you actually read the Insights article? It addresses this exact point towards the end.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102974" id="js-post-7102974" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102974/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102974"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T11:09:14-0500" data-time="1720454954" data-date-string="Jul 8, 2024" data-time-string="11:09 AM" title="Jul 8, 2024 at 11:09 AM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102974/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #25 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102974" data-lb-caption-desc="cianfa72 · Jul 8, 2024 at 11:09 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102956" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Have you actually read the Insights article? It addresses this exact point towards the end. </div> </div> </blockquote>Yes, I read it. There you claim that<br /> $$D_F \hat{p}_2 = \frac{\gamma^2 v}{R} \hat{p}_3$$ and $$D_F \hat{p}_3 = – \frac{\gamma^2 v}{R} \hat{p}_2$$ can be written using vorticity vector ##\vec {\Omega}## $$<br /> \vec{\Omega} = \frac{\gamma^2 v}{R} \hat{p}_1 = \gamma^2 \omega \hat{p}_1$$ Is there a typo ? The latter includes only the vector ##\hat{p}_1##.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> In other words, we have a combination of the hyperbolic rotation in spacetime of ##\hat{p}_0## and ##\hat{p}_2## due to Fermi-Walker transport, and the extra rotation of ##\hat{p}_2## and ##\hat{p}_3## due to the nonzero vorticity. </div> </div> </blockquote>So, the terms that include the "vorticity" scalar ##\Omega## represent the extra rotation due non-zero vorticity. So far so good.<br /> <br /> Coming back to my original question how can I use the above to uniquely define a spacelike vector with the properties as in post #21 ?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102976" id="js-post-7102976" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102976/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102976"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T11:13:29-0500" data-time="1720455209" data-date-string="Jul 8, 2024" data-time-string="11:13 AM" title="Jul 8, 2024 at 11:13 AM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102976/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #26 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102976" data-lb-caption-desc="PeterDonis · Jul 8, 2024 at 11:13 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102974" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Is there a typo ? </div> </div> </blockquote>No. ##\hat{p}_1## is the direction of the rotation axis, or, to put it another way that might help, it is the spatial direction that is orthogonal to the plane of rotation (which in this case is the ##\hat{p}_2##, ##\hat{p}_3## plane). When you describe rotation using a vector, that is what the vector means.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102978" id="js-post-7102978" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102978/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102978"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T11:15:57-0500" data-time="1720455357" data-date-string="Jul 8, 2024" data-time-string="11:15 AM" title="Jul 8, 2024 at 11:15 AM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102978/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #27 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102978" data-lb-caption-desc="PeterDonis · Jul 8, 2024 at 11:15 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102974" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> how can I use the above to uniquely define a spacelike vector with the properties as in post #21 ? </div> </div> </blockquote>I didn't say you could. I just said that the Insights article might help you to work out the math for Fermi-Walker transport, which would in turn help you to see why your intuition is correct that Fermi-Walker transport is <i>not</i> the right transport law to define the spacelike vector you are trying to define.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102981" id="js-post-7102981" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102981/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102981"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T11:22:58-0500" data-time="1720455778" data-date-string="Jul 8, 2024" data-time-string="11:22 AM" title="Jul 8, 2024 at 11:22 AM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102981/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #28 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102981" data-lb-caption-desc="cianfa72 · Jul 8, 2024 at 11:22 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102976" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> No. ##\hat{p}_1## is the direction of the rotation axis, or, to put it another way that might help, it is the spatial direction that is orthogonal to the plane of rotation (which in this case is the ##\hat{p}_2##, ##\hat{p}_3## plane). When you describe rotation using a vector, that is what the vector means. </div> </div> </blockquote>Sorry, coming back to your Insight, I'm confused about the following: what can be written as "vorticity vector" ##\vec{\Omega}## ? $$<br /> \vec{\Omega} = \frac{\gamma^2 v}{R} \hat{p}_1 = \gamma^2 \omega \hat{p}_1$$</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102989" id="js-post-7102989" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102989/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102989"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T11:53:19-0500" data-time="1720457599" data-date-string="Jul 8, 2024" data-time-string="11:53 AM" title="Jul 8, 2024 at 11:53 AM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102989/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #29 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102989" data-lb-caption-desc="PeterDonis · Jul 8, 2024 at 11:53 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102981" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> what can be written as "vorticity vector" ##\vec{\Omega}## ? $$<br /> \vec{\Omega} = \frac{\gamma^2 v}{R} \hat{p}_1 = \gamma^2 \omega \hat{p}_1$$ </div> </div> </blockquote>I'm not sure what you're asking. The vorticity vector is right there in what you posted.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7102990" id="js-post-7102990" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102990/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102990"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T11:58:13-0500" data-time="1720457893" data-date-string="Jul 8, 2024" data-time-string="11:58 AM" title="Jul 8, 2024 at 11:58 AM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102990/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #30 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102990" data-lb-caption-desc="cianfa72 · Jul 8, 2024 at 11:58 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">In the Insight, where the vorticity vector ##\vec {\Omega}## is actually used ?<br /> <br /> In particular what is the meaning of the claim<br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> This can be written as a “vorticity vector”: $$\vec{\Omega} = \frac{\gamma^2 v}{R} \hat{p}_1 = \gamma^2 \omega \hat{p}_1$$ which captures the failure of the basis vectors to be Fermi-Walker transported; in other words, relative to gyro-stabilized vectors, these basis vectors (I believe ##\hat{p}_2## and ##\hat{p}_3##) are spinning about the Z axis </div> </div> </blockquote></div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> <div class="message-lastEdit"> Last edited: <time class="u-dt" dir="auto" datetime="2024-07-08T12:18:56-0500" data-time="1720459136" data-date-string="Jul 8, 2024" data-time-string="12:18 PM" title="Jul 8, 2024 at 12:18 PM" itemprop="dateModified">Jul 8, 2024</time> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7102994" id="js-post-7102994" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7102994/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7102994"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T12:26:19-0500" data-time="1720459579" data-date-string="Jul 8, 2024" data-time-string="12:26 PM" title="Jul 8, 2024 at 12:26 PM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7102994/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #31 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7102994" data-lb-caption-desc="PeterDonis · Jul 8, 2024 at 12:26 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102990" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> where the vorticity vector ##\vec {\Omega}## is actually used ? </div> </div> </blockquote>I'm not sure what you mean by "used". The point of the article is, among other things, to show how the vorticity vector is derived.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7102990" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> In particular what is the meaning of the claim </div> </div> </blockquote>What is unclear about it? You do understand that "Fermi-Walker transported vectors" are the same thing as "gyro-stabilized vectors", right?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7103015" id="js-post-7103015" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7103015/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7103015"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T13:55:07-0500" data-time="1720464907" data-date-string="Jul 8, 2024" data-time-string="1:55 PM" title="Jul 8, 2024 at 1:55 PM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7103015/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #32 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7103015" data-lb-caption-desc="cianfa72 · Jul 8, 2024 at 1:55 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7102994" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> PeterDonis said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> What is unclear about it? You do understand that "Fermi-Walker transported vectors" are the same thing as "gyro-stabilized vectors", right? </div> </div> </blockquote>Yes, I do. Since ##D_F \hat{p}_2 \neq 0## and ##D_F \hat{p}_3 \neq 0## that means ##\hat{p}_2## and ##\hat{p}_3## vector in the frame field are not Fermi-Walker (FW) transported along the congruence.<br /> <br /> I don't understand why you claim that, relative to gyro-stabilized vectors, these two basis vectors are spinning about Z axis.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7103016" id="js-post-7103016" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7103016/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7103016"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T13:57:10-0500" data-time="1720465030" data-date-string="Jul 8, 2024" data-time-string="1:57 PM" title="Jul 8, 2024 at 1:57 PM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7103016/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #33 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7103016" data-lb-caption-desc="PeterDonis · Jul 8, 2024 at 1:57 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7103015" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Since ##D_F \hat{p}_2 \neq 0## and ##D_F \hat{p}_3 \neq 0## that means ##\hat{p}_2## and ##\hat{p}_3## vector in the frame field are not Fermi-Walker (FW) transported along the congruence. </div> </div> </blockquote>That's correct.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7103015" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I don't understand why you claim that, relative to gyro-stabilized vectors, these two basis vectors are spinning about Z axis. </div> </div> </blockquote>The vorticity vector points along the Z axis; that is the axis about which the ##\hat{p}_2## and ##\hat{p}_3## vectors are spinning relative to Fermi-Walker transported, i.e., gyro-stabilized, vectors.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="1" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList is-active"> <ul class="reactionSummary"> <li><span class="reaction reaction--small reaction--1" data-reaction-id="1"><i aria-hidden="true"></i><img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="reaction-sprite js-reaction" alt="Like" title="Like" /></span></li> </ul> <span class="u-srOnly">Likes</span> <bdi>cianfa72</bdi> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="cianfa72" data-content="post-7103024" id="js-post-7103024" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7103024/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7103024"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T14:31:20-0500" data-time="1720467080" data-date-string="Jul 8, 2024" data-time-string="2:31 PM" title="Jul 8, 2024 at 2:31 PM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7103024/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #34 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/cianfa72.651697/"> <meta itemprop="url" content="https://www.physicsforums.com/members/cianfa72.651697/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="651697"><span itemprop="name">cianfa72</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>2,454</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>255</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7103024" data-lb-caption-desc="cianfa72 · Jul 8, 2024 at 2:31 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">Coming back to your post #27, the FW transport of spacelike basis vectors ##\hat{p}_2## and ##\hat{p}_3## along a Langevin congruence's member (say ##a##) from point P to point Q <i>doesn't</i> give those frame field's basis vectors at Q.<br /> <br /> I believe the right spacelike vector I'm interested in actually lies in the 2D plane spanned by ##\hat{p}_2## and ##\hat{p}_3## of the frame field at any point along the congruence's member ##a##.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="PeterDonis" data-content="post-7103026" id="js-post-7103026" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7103026/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/" /> <span class="u-anchorTarget" id="post-7103026"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-07-08T14:39:02-0500" data-time="1720467542" data-date-string="Jul 8, 2024" data-time-string="2:39 PM" title="Jul 8, 2024 at 2:39 PM" itemprop="datePublished">Jul 8, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7103026/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #35 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/peterdonis.197831/"> <meta itemprop="url" content="https://www.physicsforums.com/members/peterdonis.197831/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/peterdonis.197831/" class="username " dir="auto" data-user-id="197831" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">PeterDonis</span></a></span>    <div class="post-badges"> <div class="post-badge badge-mentor" data-xf-init="tooltip" title="PF Community Staff"> <i class="fa--xf fal fa-user-graduate" aria-hidden="true"></i><span> Mentor</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/PeterDonis/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>47,494</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>23,767</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7103026" data-lb-caption-desc="PeterDonis · Jul 8, 2024 at 2:39 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7103024" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> the FW transport of spacelike basis vectors ##\hat{p}_2## and ##\hat{p}_3## along a Langevin congruence's member (say ##a##) from point P to point Q <i>doesn't</i> give those frame field's basis vectors at Q. </div> </div> </blockquote>Yes. You are restating what you said in the first paragraph of your post #32.<br /> <br /> <blockquote data-attributes="member: 651697" data-quote="cianfa72" data-source="post: 7103024" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> cianfa72 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I believe the right spacelike vector I'm interested in actually lies in the 2D plane spanned by ##\hat{p}_2## and ##\hat{p}_3## of the frame field at any point along the congruence's member ##a##. </div> </div> </blockquote>This is correct. However, that still leaves an infinite number of such vectors.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article> </div> </div> <div class="block-outer block-outer--after"> <div class="block-outer-main"> <nav class="pageNavWrapper pageNavWrapper--logarithmic pageNavWrapper--full " data-block-size="1" data-last-page="5"> <div class="pageNav"> <ul class="pageNav-main"> <li class="pageNav-page pageNav-page--current "><a 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Starting from any point P on each congruence's worldline take a spacelike direction at P and build the spacelike geodesic from P pointing in that direction. It will eventually intersect other congruence's worldlines.\n\nDoes the Born rigid condition imply that the spacetime lenght of such spacelike geodesic \"segments\" won't change regardless the point P chosen along a given congruence's member ?" }, "publisher": { "@type": "Organization", "name": "Physics Forums: Science Discussion, Homework Help, Articles", "alternateName": "Physics Forums", "description": "Join Physics Forums, where students, scientists, and enthusiasts come together to explore and discuss the current understanding and practice of various scientific fields.", "url": "https://www.physicsforums.com", "logo": "https://www.physicsforums.com/styles/physicsforums/images/physics-forums-logo-og.png" } } </script> <script type="text/javascript" src="https://www.physicsforums.com/copyright.js" defer></script> <script> function parseNewPost(postid) { var newpost = document.getElementById("js-post-" + postid), npa = [newpost]; MathJax.typeset(npa); } function parsePreview() { var preview = document.getElementsByClassName("xfPreview"); MathJax.typeset(preview); } function parseNewConversationPost(postid) { var newpost = document.getElementsByClassName("convMessage-" + postid); MathJax.typeset(newpost); } function parseNewProfilePost(postid) { var newpost = document.getElementById("js-profilePost-" + postid), npa = [newpost]; MathJax.typeset(npa); } function parseTooltip() { MathJax.typeset(); } </script> </body> </html>

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Properties of Born rigid congruence