Einstein brother-in-law elevator

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It may not display this or other websites correctly.<br />You should upgrade or use an <a href="https://www.google.com/chrome/" target="_blank" rel="noopener">alternative browser</a>.</div> <div class="p-body-main p-body-main--withSidebar "> <div class="p-body-contentCol"></div> <div class="p-body-sidebarCol"></div> <div class="p-body-content"> <div class="p-body-pageContent"> <div class="p-breadcrumbs--parent "> <ul class="p-breadcrumbs " itemscope itemtype="https://schema.org/BreadcrumbList"> <li itemprop="itemListElement" itemscope itemtype="https://schema.org/ListItem"> <a href="https://www.physicsforums.com/" itemprop="item"> <span itemprop="name">Forums</span> </a> <meta itemprop="position" content="1" /> </li> <li itemprop="itemListElement" itemscope itemtype="https://schema.org/ListItem"> <a href="/#physics.9" itemprop="item"> <span itemprop="name">Physics</span> </a> <meta itemprop="position" content="2" /> </li> <li itemprop="itemListElement" itemscope itemtype="https://schema.org/ListItem"> <a href="/forums/special-and-general-relativity.70/" itemprop="item"> <span itemprop="name">Special and General Relativity</span> </a> <meta itemprop="position" content="3" /> </li> </ul> </div> <div class="p-body-header"> <div class="p-title "> <h1 class="p-title-value"> Einstein brother-in-law elevator </h1> </div> <div class="p-description"> <ul class="listInline listInline--bullet"> <li> <span data-xf-init="tooltip" title="Level: Intermediate (Undergrad)" ><span class="prefix inter" dir="auto">I</span></span> </li> <li> <i class="fa--xf fal fa-user" aria-hidden="true" title="Thread starter"></i> <span class="u-srOnly">Thread starter</span> <span>member 728827</span> </li> <li> <i class="fa--xf fal fa-clock" aria-hidden="true" title="Start date"></i> <span class="u-srOnly">Start date</span> <time class="u-dt" dir="auto" datetime="2024-04-17T15:51:09-0500" data-time="1713387069" data-date-string="Apr 17, 2024" data-time-string="3:51 PM" title="Apr 17, 2024 at 3:51 PM">Apr 17, 2024</time> </li> <li> <dl class="tagList tagList--thread-1061915 "> <dt> <i class="fa--xf fal fa-tags" aria-hidden="true" title="Tags"></i> <span class="u-srOnly">Tags</span> </dt> <dd> <span class="js-tagList"> <a href="/tags/geodesics-general-relativity/" data-xf-init="preview-tooltip" data-preview-url="/tags/geodesics-general-relativity/preview" class="tagItem tagItem--tag_geodesics-general-relativity" dir="auto"> Geodesics general relativity </a> <a href="/tags/schwarzchild-metric/" data-xf-init="preview-tooltip" data-preview-url="/tags/schwarzchild-metric/preview" class="tagItem tagItem--tag_schwarzchild-metric" dir="auto"> Schwarzchild metric </a> </span> </dd> </dl> </li> </ul> </div> </div> <div class="block-container lbContainer" style="padding:5px;"> In summary, the "Einstein brother-in-law elevator" refers to an anecdote involving Albert Einstein's brother-in-law, who humorously recounted a moment in an elevator where he experienced the juxtaposition of his mundane life against Einstein's genius. This story highlights the contrast between ordinary experiences and extraordinary intellect, illustrating how personal relationships can provide a unique perspective on renowned figures. </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> <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-1061915" data-lb-universal="0"> <div class="block-body js-replyNewMessageContainer">  <article class="message message--post js-post js-inlineModContainer " data-author="member 728827" data-content="post-7079035" id="js-post-7079035" > <span class="u-anchorTarget" id="post-7079035"></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="2024-04-17T15:51:09-0500" data-time="1713387069" data-date-string="Apr 17, 2024" data-time-string="3:51 PM" title="Apr 17, 2024 at 3:51 PM">Apr 17, 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/7079035/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="0">member 728827</span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </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-7079035" data-lb-caption-desc="member 728827 · Apr 17, 2024 at 3:51 PM"> <div class="message-fields message-fields--before"> <dl class="pairs pairs--columns pairs--fixedSmall pairs--customField" data-field="summary"> <dt>TL;DR Summary</dt> <dd>As personal curiosity, I want to calculate which is the difference in "travelled height" between a photon that goes across the width of an elevator - which is more or less 2[m] in my country - and a tiny mass particle that free-falls starting at the same "height" as the photon origin, and is released exactly at the same time that the photon is shot from the other side.</dd> </dl> </div> <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">As personal curiosity, I want to calculate which is the difference in "travelled height" between a photon that goes across the width of an elevator - which is more or less 2[m] in my country - and a tiny mass particle that free-falls starting at the same "height" as the photon origin, and is released exactly at the same time that the photon is shot from the other side.<br /> <br /> I'll make some assumptions in order to make this race "fair enough". Hope I don't make many mistakes and correct me if I'm wrong. We'll see...<br /> <br /> First of all, thanks to the sponsors of this "tough" experiment: Coke© light (supplying the laser assembly) and Nanomass© dynamics for the test particle (which they sell by the weight and are dirty cheap).<br /> <br /> Also, due the lack of budget, the test will be conducted in Earth, simplifying and considering the <b>Schwarzschild metric</b> solution to the Einstein Field Equations, so Earth and surrounding spacetime are not flat.<br /> <br /> This defines the "Schwarzschild radius", and will be used in the metric:<br /> $$[1]\quad R_s=\frac{2GM}{c^2}$$<br /> <br /> The Schwarzschild metric itself:<br /> $$[2]\quad ds^2=-c^2(1-\frac{R_s}{r})dt^2+\frac{dr^2}{(1-\frac{R_s}{r})}+r^2d\theta^2+r^2sin^2(\theta)d\phi^2$$<br /> <br /> Both the photon and the test particle will be in "<b>free-fall</b>" once released, and thus will follow a <b>Geodesic</b> trajectory. To obtain the motion equations, first I'll begin with the <b>Euler-Lagrange</b> condition, because such trajectories will maximize some Lagrangian function.<br /> <br /> $$[3]\quad (x^0,x^1,x^2,x^3)=(ct,r,\theta,\phi) \qquad \dot{x}^\mu=\frac{\partial x^\mu}{\partial λ} \qquad \frac{\partial}{\partial λ}(\frac{\partial\mathscr{L}}{\partial\dot{x}^\mu})-\frac{\partial\mathscr{L}}{\partial x^\mu}=0$$<br /> <br /> Defining the Lagrangian as the derivative of the line interval by a suitable parameter λ:<br /> <br /> $$[4]\quad \mathscr{L}=(\frac{ds}{dλ})^2=-c^2(1-\frac{R_s}{r})\dot{t}^2+\frac{\dot{r}^2}{(1-\frac{R_s}{r})}+r^2\dot{\theta}^2+r^2sin^2(\theta)\dot{\phi}^2$$<br /> <br /> Applying the Euler-Lagrange rule to the coordinates, we get the following general conditions that the Geodesic trajectories must follow:<br /> <br /> - We can consider the trajectory contained in a plane. Let's choose for sake of simplicity $\theta=\pi/2$:<br /> $$[5]\quad \mathscr{L}(\theta) \Rightarrow 2r^2\ddot{\theta}-2r^2\dot{\theta}^2sin(\theta)cos(\theta)=0 \Rightarrow<br /> \theta=\pi/2$$<br /> <br /> - There're a two conserved quantities (constants of motion) to be determined for a Geodesic path:<br /> <br /> $$[6]\quad \mathscr{L}(ct) \Rightarrow -2c^2(1-\frac{R_s}{r})\ddot{t}=0 \Rightarrow (1-\frac{R_s}{r})\dot{t}=k$$<br /> $$[7]\quad \mathscr{L}(\phi)\Rightarrow 2r^2sin^2(\theta)\ddot{\phi}=0\Rightarrow r^2sin^2(\theta)\dot{\phi}=l$$<br /> <br /> Dividing [7] into [6], we get rid of the derivatives with respect to parameter λ, and can write the constant of motion directly as a consequence of the coordinate components:<br /> <br /> $$[9]\quad b=\frac{l}{k}=\frac{r^2\dot{\phi}}{(1-\frac{R_s}{r})\dot{t}}=\frac{r^2}{(1-\frac{R_s}{r})}\frac{d\phi}{dt}$$<br /> <br /> Combining all this stuff, we obtain our <b>first general equation of motion</b> for a Geodesic path:<br /> <br /> $$[10]\quad \Rightarrow \frac{d\phi}{dt}=\frac{1}{r^2}(1-\frac{R_s}{r})\cdot b$$<br /> <br /> Now, we have to distinguish between a test particle (with mass), and a (massless) photon.<br /> <br /> For light, and using the original metric equation [2]:<br /> <br /> $$[11]\quad photon: ds^2=0=-c^2(1-\frac{R_s}{r})dt^2+\frac{dr^2}{(1-\frac{R_s}{r})}+r^2d\phi^2$$<br /> <br /> And then, here we have our second equation of motion, related to a constant b to be determined afterwards:<br /> <br /> $$[12]\quad (\frac{dr}{dt})^2=-\frac{b^2}{r^2}(1-\frac{R_s}{r})^3+c^2(1-\frac{R_s}{r})^2$$<br /> <br /> For the constant of motion, if consider the launch angle ##\varphi## - from the horizontal to the initial direction -, and the primed coordinates (r' and t') for a local observer which always measures (as Einstein likes) the speed of light as c, we have:<br /> <br /> $$[13]\quad t=t_0=0,r=r_0 \qquad \frac{dr'}{dt'}|_{t0}=c \cdot sin(\varphi)$$<br /> $$c^2dt'^2=c^2(1-\frac{R_s}{r})dt^2 \qquad dr'^2=\frac{dr^2}{(1-\frac{R_s}{r})}\Rightarrow \frac{dr'}{dt'}=\frac{1}{(1-\frac{R_s}{r})} \cdot \frac{dr}{dt}$$<br /> $$\frac{dr}{dt}|_{t_0}=(1-\frac{R_s}{r_0}) \cdot c \cdot sin(\varphi)$$<br /> <br /> $$[14]\quad (1-\frac{R_s}{r_0})^2\cdot c^2 \cdot sin^2(\varphi)=-\frac{b^2}{r_0^2}(1-\frac{R_s}{r_0})^3+c^2(1-\frac{R_s}{r_0})^2$$<br /> $$\Rightarrow b^2=\frac{c^2r_0^2}{(1-\frac{R_s}{r_0})} \cdot \left\{ 1-sin^2(\varphi) \right\}=\frac{c^2r_0^2}{(1-\frac{R_s}{r_0})} \cdot cos^2(\varphi)$$<br /> <br /> Now for the test particle. We do the same, except that is not a null but a timelike Geodesic:<br /> <br /> $$[15]\quad \mathscr{L}=(\frac{ds}{d\tau})^2=-c^2=-\frac{c^2k^2}{(1-\frac{R_s}{r})}+\frac{(\frac{dr}{d\tau})^2}{(1-\frac{R_s}{r})} \qquad \theta=\pi/2 \quad d\theta=0 \quad d\phi=0$$<br /> <br /> $$[16]\quad -c^2(1-\frac{R_s}{r})=-c^2k^2+(\frac{dr}{d\tau})^2$$<br /> <br /> $$[17]\quad \frac{dr}{d\tau}=\frac{dr}{dt}\frac{dt}{d\tau}=\frac{dr}{dt}\frac{k}{(1-\frac{R_s}{r})}$$<br /> <br /> The second equation of motion for a mass particle:<br /> $$[18]\quad (\frac{dr}{dt})^2=-\frac{c^2}{k^2}(1-\frac{R_s}{r})^3+c^2(1-\frac{R_s}{r})^2$$<br /> <br /> From the initial condition, we can get the constant of motion:<br /> $$[19]\quad t=t_0=0,r=r_0 \qquad \frac{dr}{dt}|_{t_0}=0 \qquad \Rightarrow k=±\sqrt{1-\frac{R_s}{r_0}}$$<br /> <br /> As for the setup of the test, if we shoot the photon with an initial non-zero launch angle ##\varphi## (inwards), then it will have obviously an initial inward geometric radial velocity, and being c a big number seems unfair to compare then the radial travelled distance of the photon with a poor test particle that starts quiet and still.<br /> <br /> So, we'll shot the photon tangentially (##\varphi=0##). This means that if the path of light were unaffected by Gravity, would hit the other side of the elevator at a higher position (i.e., if the starting radius is r_0, would reach the other side at a radius r_1>r_0). In a nutshell, we don't point the laser towards the test particle, but a little above it.<br /> <br /> For the test particle, it's starting position will be r_0, and so both the photon and the test particle begin at the same distance (radius) from the center of the planet.<br /> <br /> That's fair enough for me.<br /> <br /> With a simple Python code (attached), I obtain the following:<br /> <br /> FINAL RESULT:<br /> r1-rend=6.556e-16[m] r0-rend_m=2.185e-16[m] tend=6.671e-09<s><br /> <br /> where r1 is the point-of-impact radius of a straight line with a length of 2[m] (we fix as condition the arc covered to be L/r_0), rend if the final radius of the photon position, rend_m is the final radius of the test particle position, and tend is the total flight time (in the coordinate time). So the radial distance displacement for the photon is 3.0004.. times greater than the radial distance displacement of the test particle.<br /> <br /> <div class="bbCodeBlock bbCodeBlock--screenLimited bbCodeBlock--code language-python " data-xf-init="CMTV-code-block"> <div class="bbCodeBlock-header"> <div class="bbCodeBlock-title"> <span title="Python"> Python code: </span> </div> <div class="bbCodeBlock-actions"> <i class="fa--xf fal fa-expand-alt action action--expand action--hidden" aria-hidden="true" title="Expand" data-xf-init="tooltip" data-xf-click="CMTV-code-block-expand"></i> <i class="fa--xf fal fa-compress-alt action action--collapse action--hidden" aria-hidden="true" title="Collapse" data-xf-init="tooltip" data-xf-click="CMTV-code-block-collapse"></i> <i class="fa--xf fal fa-paste action action--copy" aria-hidden="true" title="Copy" data-xf-init="tooltip"></i> </div> </div> <div class="bbCodeBlock-content" dir="ltr"> <pre class="bbCodeCode line-numbers" dir="ltr" data-line="" data-xf-init="CMTV-code-block-extend" data-lang="python"><code>from mpmath import * import numpy as np import matplotlib.pyplot as plt import math from datetime import datetime mp.dps = 100 mp.pretty = True c=299792458 # speed of light c2=pow(c,2) # square of speed of light G=mpf(6.6743e-11) # Gravitional constant M=mpf(5.9722e+24) # Earth's mass r0=mpf(6371000) # Earth's radius L=mpf(2) # Elevator's width alp=L/r0 Rs=(2*G*M)/c2 Rs2=pow(Rs,2) r0s=1-Rs/r0 b2=((pow(r0,2)*c2)/r0s) b=sqrt(b2) def zeq(r): return c2-(b2/pow(r,2))*(1-Rs/r) def fph_dt_dr(r): zr=zeq(r) if (zr>0): return 1/((1-Rs/r)*sqrt(zr)) else: raise Exception('NEGATIVE SQRT! r=%5.3e[m]'%(r)) def fph_dphi_dt(r): return (b/pow(r,2))*(1-Rs/r) def fph_dphi_dr(r): zr=zeq(r) if (zr>0): return b/sqrt(r*(c2*pow(r,3)-b2*(r-Rs))) else: raise Exception('NEGATIVE SQRT! r=%5.3e[m]'%(r)) def fmp_dt_dr(r): return -1/(c*(1-Rs/r)*sqrt(1-(1-Rs/r)/r0s)) print('') r1=r0/cos(alp) [rend] = findroot([lambda r: quad(fph_dphi_dr,[r0,r])-alp],(r1)) tend = quad(fph_dt_dr,[r0,rend]) d=pow(tend,2)*G*M/(2*pow(r0,2)) # initial guess [rend_m] = findroot([lambda r: quad(fmp_dt_dr,[r0-1e-51,r])-tend],(r0-d)) print('') print('FINAL RESULT:') print('r1-rend=%5.3e[m] r0-rend_m=%5.3e[m] tend=%9.3e[s]'%(r1-rend,r0-rend_m,tend))</code></pre> </div> <div class="bbCodeBlock-grip resizer--hidden" data-xf-init="CMTV-code-block-resizer"> <i class="fa--xf fal fa-grip-horizontal" aria-hidden="true"></i> </div> </div></s></div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <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> 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js-inlineModContainer " data-author="Ibix" data-content="post-7079044" id="js-post-7079044" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7079044/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/einstein-brother-in-law-elevator.1061915/" /> <span class="u-anchorTarget" id="post-7079044"></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="2024-04-17T16:26:03-0500" data-time="1713389163" data-date-string="Apr 17, 2024" data-time-string="4:26 PM" title="Apr 17, 2024 at 4:26 PM" itemprop="datePublished">Apr 17, 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/7079044/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/ibix.365269/"> <meta itemprop="url" content="https://www.physicsforums.com/members/ibix.365269/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/ibix.365269/" class="username " dir="auto" data-user-id="365269" data-xf-init="member-tooltip"><span itemprop="name">Ibix</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-file" data-xf-init="tooltip" title="Click to View My Articles"> <a 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that the equations of motion for this case can be found in textbooks. Carroll's lecture notes equations 7.43, 7.44, 7.47 and 7.48 are there, and you can check your results.<br /> <br /> There is a closed-form expression for ##t(r)## for radial motion. Wolfram Alpha should be able to determine it given ##dr/dt## in that case. Sadly you cannot get ##r(t)##, but it may provide a better way of calculating the fall distance of the mass.<br /> <br /> You can sanity check your results with Newtonian gravity. Light travels ##l=2\mathrm{m}## in ##\delta t=l/c## seconds. It will rise by approximately ##l^2/2R_E## compared to its initial altitude. The mass will fall approximately ##\frac 12g\delta t^2## in that time. You won't be far wrong with this - how much does it disagree with your detailed calculations?</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>member 728827</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="PeterDonis" data-content="post-7079046" id="js-post-7079046" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7079046/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/einstein-brother-in-law-elevator.1061915/" /> <span class="u-anchorTarget" id="post-7079046"></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="2024-04-17T16:30:54-0500" data-time="1713389454" data-date-string="Apr 17, 2024" data-time-string="4:30 PM" title="Apr 17, 2024 at 4:30 PM" itemprop="datePublished">Apr 17, 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/7079046/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/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-7079046" data-lb-caption-desc="PeterDonis · Apr 17, 2024 at 4:30 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: 728827" data-quote="Lluis Olle" data-source="post: 7079035" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Lluis Olle said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> the difference in "travelled height" between a photon that goes across the width of an elevator - which is more or less 2[m] in my country - and a tiny mass particle that free-falls starting at the same "height" as the photon origin, and is released exactly at the same time that the photon is shot from the other side. </div> </div> </blockquote>If I'm understanding you correctly, you want to release a test particle and a light pulse (the word "photon" is not appropriate in a classical context, you're really firing a very short pulse of light) at the same height in an "elevator", with the test particle starting at rest relative to the elevator and the light pulse released pointing horizontally, and watch how far they both "fall" in the time it takes the light to travel 2 meters across the elevator.<br /> <br /> Assuming that my understanding is correct, you are doing an awful lot of work to derive a result that the equivalence principle can tell you immediately: the fall distances will be the same, i.e., the difference in "travelled height" will be <i>zero</i>. Just do the analysis in a local inertial frame centered on the elevator. In that frame, the light pulse and the test object are both in free fall, and the test particle will simply remain at rest in the local inertial frame while the light pulse will travel horizontally across it, with both staying at the same "vertical" coordinate. So in the local inertial frame the "fall distance" is <i>zero</i> for both test particle and light pulse. The elevator itself is accelerating upward relative to the local inertial frame, so the "fall distance" relative to the elevator is just the distance the elevator moves upward in the 6.6 nanonseconds it takes for the light pulse to travel 2 meters.</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>ersmith</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="member 728827" data-content="post-7079166" id="js-post-7079166" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7079166/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/einstein-brother-in-law-elevator.1061915/" /> <span class="u-anchorTarget" id="post-7079166"></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="2024-04-18T03:31:29-0500" data-time="1713429089" data-date-string="Apr 18, 2024" data-time-string="3:31 AM" title="Apr 18, 2024 at 3:31 AM" itemprop="datePublished">Apr 18, 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/7079166/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=""> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="0"><span itemprop="name">member 728827</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </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-7079166" data-lb-caption-desc="member 728827 · Apr 18, 2024 at 3:31 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">I attach an extremely exaggerated figure, to better understand the scenario.<br /> <br /> The laser is fixed at point A, at the left side of the elevator. The test particle is located at point B. Both laser origin and test particle are at the same radial coordinate r0.<br /> <br /> For reasons given in the thread, the laser is pointed towards C.<br /> <br /> At t0, the laser emits a light pulse and the test particle is released. At the time the front of the light pulse covers the arc between A-C (arc which is given as L/r0, being L the width of the elevator), I want the difference of r1 vs the radial coordinate of the light pulse, and the difference of r0 vs the radial coordinate of the test particle. For me, that is the distance radially travelled for each of them.<br /> <br /> This awful lot of work is to test where the "uniform gravitational field" concept is applicable, if you can do the calculations up to n digits of precision.<br /> <br /> <script class="js-extraPhrases" type="application/json"> { "lightbox_close": "Close", "lightbox_next": "Next", "lightbox_previous": "Previous", "lightbox_error": "The requested content cannot be loaded. 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Light travels ##l=2\mathrm{m}## in ##\delta t=l/c## seconds. It will rise by approximately ##l^2/2R_E## compared to its initial altitude. The mass will fall approximately ##\frac 12g\delta t^2## in that time. You won't be far wrong with this - how much does it disagree with your detailed calculations? </div> </div> </blockquote>Sure enough...<br /> <br /> For the test particle, the initial guess for the findroot() function is just the Newtonian approximation.<br /> <br /> <code class="bbCodeInline">d=pow(tend,2)*G*M/(2*pow(r0,2)) # initial guess</code><br /> <code class="bbCodeInline">[rend_m] = findroot([lambda r: quad(fmp_dt_dr,[r0-1e-51,r])-tend],(r0-d))</code><br /> <br /> Newton:<br /> <b>0.0000000000000002185311982</b>482473936267804473389293545547985151184677544202782604335814083432271557989293021512927874<br /> <br /> Schwarzschild:<br /> <b>0.0000000000000002185311979</b>4399465414896083431526218218450276234588121518223277040941841501688951624519599357665033</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>Dale</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="member 728827" data-content="post-7079182" id="js-post-7079182" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7079182/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/einstein-brother-in-law-elevator.1061915/" /> <span class="u-anchorTarget" id="post-7079182"></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="2024-04-18T05:17:46-0500" data-time="1713435466" data-date-string="Apr 18, 2024" data-time-string="5:17 AM" title="Apr 18, 2024 at 5:17 AM" itemprop="datePublished">Apr 18, 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/7079182/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=""> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="0"><span itemprop="name">member 728827</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </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-7079182" data-lb-caption-desc="member 728827 · Apr 18, 2024 at 5:17 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7079046" 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 "> ...with both staying at the same "vertical" coordinate. </div> </div> </blockquote>Don't know what this "vertical coordinate" is ¿?. <b>I'm working with the Schwarzschild coordinates</b>. The light pulse, moving from side to side will not maintain a constant <b>radial coordinate</b>, no matter what the "equivalence principle" says. Not even a mathematical straight line will do so, ...is what happens with polar or spherical coordinate systems.<br /> <br /> <blockquote data-attributes="member: 197831" data-quote="PeterDonis" data-source="post: 7079046" 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 "> so the "fall distance" relative to the elevator is just the distance the elevator moves upward in the 6.6 nanonseconds... </div> </div> </blockquote>I don't care the "fall distance" relative to the elevator, that's why the thread is titled "Einstein brother-in-law elevator"... the only think related to an elevator is the width scale of such a thing. Will not enter in discussions about rigid body movement in a non-uniform Gravitational field and so on.<br /> <br /> And, of course, if the "Equivalence principle" applies in this case, the spacetime is flat, and why use the <b>Schwarzschild</b> metric? But I do use that metric, and I do obtain those results, being pretty careful as to maintain the required precision solving the equations of motion (that can be wrong, because are handcrafted).<br /> <br /> If the difference is in the order of 10<sup>-16</sup> m, of course that the number will show as zero with a casio calculator.</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="Nugatory" 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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/nugatory.382138/"> <meta itemprop="url" content="https://www.physicsforums.com/members/nugatory.382138/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/nugatory.382138/" class="username " dir="auto" data-user-id="382138" data-xf-init="member-tooltip"><span class="username--staff username--moderator" itemprop="name">Nugatory</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>  </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>15,120</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>9,889</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-7079203" data-lb-caption-desc="Nugatory · Apr 18, 2024 at 6:44 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 728827" data-quote="Lluis Olle" data-source="post: 7079182" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Lluis Olle said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> But I do use that metric, </div> </div> </blockquote>If you need to use the Schwarzschild metric the tidal effects must be significant, and then we don’t expect the drop within the elevator to be the same for the falling object and the horizontal beam of light.<br /> <br /> I’m really at a loss to understand what your point is.</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="Ibix" data-content="post-7079205" id="js-post-7079205" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7079205/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/einstein-brother-in-law-elevator.1061915/" /> <span class="u-anchorTarget" id="post-7079205"></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="2024-04-18T06:55:48-0500" data-time="1713441348" data-date-string="Apr 18, 2024" data-time-string="6:55 AM" title="Apr 18, 2024 at 6:55 AM" itemprop="datePublished">Apr 18, 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/7079205/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/ibix.365269/"> <meta itemprop="url" content="https://www.physicsforums.com/members/ibix.365269/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/ibix.365269/" class="username " dir="auto" data-user-id="365269" data-xf-init="member-tooltip"><span itemprop="name">Ibix</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-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/Ibix/"><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>12,496</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>14,560</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-7079205" data-lb-caption-desc="Ibix · Apr 18, 2024 at 6:55 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 382138" data-quote="Nugatory" data-source="post: 7079203" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Nugatory said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> If you need to use the Schwarzschild metric the tidal effects must be significant, </div> </div> </blockquote>I think he's trying to quantify the tidal effects in a small room on Earth. An easier way is just to hang two plumb lines and measure their failure to be parallel.<br /> <br /> The general approach, I think, is to transform the metric to make all first derivatives vanish at the middle of the lift, and then look at the second derivatives times the length scale of the lift.</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="member 728827" data-content="post-7079224" id="js-post-7079224" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7079224/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/einstein-brother-in-law-elevator.1061915/" /> <span class="u-anchorTarget" id="post-7079224"></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="2024-04-18T08:31:16-0500" data-time="1713447076" data-date-string="Apr 18, 2024" data-time-string="8:31 AM" title="Apr 18, 2024 at 8:31 AM" itemprop="datePublished">Apr 18, 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/7079224/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=""> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="0"><span itemprop="name">member 728827</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </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-7079224" data-lb-caption-desc="member 728827 · Apr 18, 2024 at 8:31 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 382138" data-quote="Nugatory" data-source="post: 7079203" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Nugatory said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> If you need to use the Schwarzschild metric the tidal effects must be significant, and then we don’t expect the drop within the elevator to be the same for the falling object and the horizontal beam of light.<br /> <br /> I’m really at a loss to understand what your point is. </div> </div> </blockquote>The point is: none.<br /> <br /> I just set up an scenario, derive the required Geodesic equations using the Schwarzschild metric, stating at the beginning of the thread that "...the test will be conducted in Earth, simplifying and considering the <b>Schwarzschild metric</b> solution to the Einstein Field Equations, so Earth and <b>surrounding spacetime are not flat</b>."<br /> <br /> The Geodesic equations of motion are what they are! Did I got a wrong reasoning in some step, o did some algebraic error, which makes the equations wrong? But if are not wrong, then the final result is what it is, and for the extreme scenario of a 2 meter distance, the light beam will "travel radially" a distance that is 3 times the distance for the test particle. The fact that the scale is 10<sup>-16</sup> is ... irrelevant.<br /> <br /> What you call "tidal effect", is just the application of the metric, is that easy to understand. That makes the Gravity non uniform, of course.</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="member 728827" data-content="post-7079226" id="js-post-7079226" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7079226/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/einstein-brother-in-law-elevator.1061915/" /> <span class="u-anchorTarget" id="post-7079226"></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="2024-04-18T08:38:36-0500" data-time="1713447516" data-date-string="Apr 18, 2024" data-time-string="8:38 AM" title="Apr 18, 2024 at 8:38 AM" itemprop="datePublished">Apr 18, 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/7079226/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=""> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="0"><span itemprop="name">member 728827</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </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-7079226" data-lb-caption-desc="member 728827 · Apr 18, 2024 at 8:38 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 365269" data-quote="Ibix" data-source="post: 7079205" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Ibix said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I think he's trying to quantify the tidal effects in a small room on Earth. An easier way is just to hang two plumb lines and measure their failure to be parallel. </div> </div> </blockquote>That will not give the correct numbers, because in the scenario the test particle do fall radially, but the light beam does not, and is "launched" tangentially.</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-7079254" id="js-post-7079254" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7079254/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/einstein-brother-in-law-elevator.1061915/" /> <span class="u-anchorTarget" id="post-7079254"></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="2024-04-18T10:17:15-0500" data-time="1713453435" data-date-string="Apr 18, 2024" data-time-string="10:17 AM" title="Apr 18, 2024 at 10:17 AM" itemprop="datePublished">Apr 18, 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/7079254/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/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-7079254" data-lb-caption-desc="PeterDonis · Apr 18, 2024 at 10:17 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 728827" data-quote="Lluis Olle" data-source="post: 7079166" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Lluis Olle said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> radial coordinate </div> </div> </blockquote>Why would you even use radial coordinates in this problem? If your "radial coordinate" is measured from the center of the Earth, you are not testing anything related to what you say you're testing:<br /> <br /> <blockquote data-attributes="member: 728827" data-quote="Lluis Olle" data-source="post: 7079166" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Lluis Olle said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> This awful lot of work is to test where the "uniform gravitational field" concept is applicable </div> </div> </blockquote>The Earth's gravitational field is already known <i>not</i> to be "uniform", so it's pointless to use radial coordinates centered on the Earth. You should be using Cartesian coordinates centered on the elevator itself.<br /> <br /> <blockquote data-attributes="member: 728827" data-quote="Lluis Olle" data-source="post: 7079182" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Lluis Olle said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> <b>I'm working with the Schwarzschild coordinates</b>. </div> </div> </blockquote>Then you are not doing what you say you want to do; the "gravitational field" in Schwarzschild coordinates is <i>not</i> "uniform", and your entire analysis is pointless.</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-7079255" id="js-post-7079255" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7079255/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/einstein-brother-in-law-elevator.1061915/" /> <span class="u-anchorTarget" id="post-7079255"></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="2024-04-18T10:18:06-0500" data-time="1713453486" data-date-string="Apr 18, 2024" data-time-string="10:18 AM" title="Apr 18, 2024 at 10:18 AM" itemprop="datePublished">Apr 18, 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/7079255/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-7079255" data-lb-caption-desc="PeterDonis · Apr 18, 2024 at 10:18 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 728827" data-quote="Lluis Olle" data-source="post: 7079224" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Lluis Olle said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> The point is: none. </div> </div> </blockquote>In which case, given the fatal flaws I've already pointed out, you are just wasting everyone's time, so this thread is now closed.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" 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Hope I don't make many mistakes and correct me if I'm wrong. We'll see...\n\nFirst of all, thanks to the sponsors of this \"tough\" experiment: Coke\u00a9 light (supplying the laser assembly) and Nanomass\u00a9 dynamics for the test particle (which they sell by the weight and are dirty cheap).\n\nAlso, due the lack of budget, the test will be conducted in Earth, simplifying and considering the Schwarzschild metric solution to the Einstein Field Equations, so Earth and surrounding spacetime are not flat.\n\nThis defines the \"Schwarzschild radius\", and will be used in the metric:\n$$[1]\\quad R_s=\\frac{2GM}{c^2}$$\n\nThe Schwarzschild metric itself:\n$$[2]\\quad ds^2=-c^2(1-\\frac{R_s}{r})dt^2+\\frac{dr^2}{(1-\\frac{R_s}{r})}+r^2d\\theta^2+r^2sin^2(\\theta)d\\phi^2$$\n\nBoth the photon and the test particle will be in \"free-fall\" once released, and thus will follow a Geodesic trajectory. To obtain the motion equations, first I'll begin with the Euler-Lagrange condition, because such trajectories will maximize some Lagrangian function.\n\n$$[3]\\quad (x^0,x^1,x^2,x^3)=(ct,r,\\theta,\\phi) \\qquad \\dot{x}^\\mu=\\frac{\\partial x^\\mu}{\\partial \u03bb} \\qquad \\frac{\\partial}{\\partial \u03bb}(\\frac{\\partial\\mathscr{L}}{\\partial\\dot{x}^\\mu})-\\frac{\\partial\\mathscr{L}}{\\partial x^\\mu}=0$$\n\nDefining the Lagrangian as the derivative of the line interval by a suitable parameter \u03bb:\n\n$$[4]\\quad \\mathscr{L}=(\\frac{ds}{d\u03bb})^2=-c^2(1-\\frac{R_s}{r})\\dot{t}^2+\\frac{\\dot{r}^2}{(1-\\frac{R_s}{r})}+r^2\\dot{\\theta}^2+r^2sin^2(\\theta)\\dot{\\phi}^2$$\n\nApplying the Euler-Lagrange rule to the coordinates, we get the following general conditions that the Geodesic trajectories must follow:\n\n- We can consider the trajectory contained in a plane. Let's choose for sake of simplicity $\\theta=\\pi/2$:\n$$[5]\\quad \\mathscr{L}(\\theta) \\Rightarrow 2r^2\\ddot{\\theta}-2r^2\\dot{\\theta}^2sin(\\theta)cos(\\theta)=0 \\Rightarrow\n\\theta=\\pi/2$$\n\n- There're a two conserved quantities (constants of motion) to be determined for a Geodesic path:\n\n$$[6]\\quad \\mathscr{L}(ct) \\Rightarrow -2c^2(1-\\frac{R_s}{r})\\ddot{t}=0 \\Rightarrow (1-\\frac{R_s}{r})\\dot{t}=k$$\n$$[7]\\quad \\mathscr{L}(\\phi)\\Rightarrow 2r^2sin^2(\\theta)\\ddot{\\phi}=0\\Rightarrow r^2sin^2(\\theta)\\dot{\\phi}=l$$\n\nDividing [7] into [6], we get rid of the derivatives with respect to parameter \u03bb, and can write the constant of motion directly as a consequence of the coordinate components:\n\n$$[9]\\quad b=\\frac{l}{k}=\\frac{r^2\\dot{\\phi}}{(1-\\frac{R_s}{r})\\dot{t}}=\\frac{r^2}{(1-\\frac{R_s}{r})}\\frac{d\\phi}{dt}$$\n\nCombining all this stuff, we obtain our first general equation of motion for a Geodesic path:\n\n$$[10]\\quad \\Rightarrow \\frac{d\\phi}{dt}=\\frac{1}{r^2}(1-\\frac{R_s}{r})\\cdot b$$\n\nNow, we have to distinguish between a test particle (with mass), and a (massless) photon.\n\nFor light, and using the original metric equation [2]:\n\n$$[11]\\quad photon: ds^2=0=-c^2(1-\\frac{R_s}{r})dt^2+\\frac{dr^2}{(1-\\frac{R_s}{r})}+r^2d\\phi^2$$\n\nAnd then, here we have our second equation of motion, related to a constant b to be determined afterwards:\n\n$$[12]\\quad (\\frac{dr}{dt})^2=-\\frac{b^2}{r^2}(1-\\frac{R_s}{r})^3+c^2(1-\\frac{R_s}{r})^2$$\n\nFor the constant of motion, if consider the launch angle ##\\varphi## - from the horizontal to the initial direction -, and the primed coordinates (r' and t') for a local observer which always measures (as Einstein likes) the speed of light as c, we have:\n\n$$[13]\\quad t=t_0=0,r=r_0 \\qquad \\frac{dr'}{dt'}|_{t0}=c \\cdot sin(\\varphi)$$\n$$c^2dt'^2=c^2(1-\\frac{R_s}{r})dt^2 \\qquad dr'^2=\\frac{dr^2}{(1-\\frac{R_s}{r})}\\Rightarrow \\frac{dr'}{dt'}=\\frac{1}{(1-\\frac{R_s}{r})} \\cdot \\frac{dr}{dt}$$\n$$\\frac{dr}{dt}|_{t_0}=(1-\\frac{R_s}{r_0}) \\cdot c \\cdot sin(\\varphi)$$\n\n$$[14]\\quad (1-\\frac{R_s}{r_0})^2\\cdot c^2 \\cdot sin^2(\\varphi)=-\\frac{b^2}{r_0^2}(1-\\frac{R_s}{r_0})^3+c^2(1-\\frac{R_s}{r_0})^2$$\n$$\\Rightarrow b^2=\\frac{c^2r_0^2}{(1-\\frac{R_s}{r_0})} \\cdot \\left\\{ 1-sin^2(\\varphi) \\right\\}=\\frac{c^2r_0^2}{(1-\\frac{R_s}{r_0})} \\cdot cos^2(\\varphi)$$\n\nNow for the test particle. We do the same, except that is not a null but a timelike Geodesic:\n\n$$[15]\\quad \\mathscr{L}=(\\frac{ds}{d\\tau})^2=-c^2=-\\frac{c^2k^2}{(1-\\frac{R_s}{r})}+\\frac{(\\frac{dr}{d\\tau})^2}{(1-\\frac{R_s}{r})} \\qquad \\theta=\\pi/2 \\quad d\\theta=0 \\quad d\\phi=0$$\n\n$$[16]\\quad -c^2(1-\\frac{R_s}{r})=-c^2k^2+(\\frac{dr}{d\\tau})^2$$\n\n$$[17]\\quad \\frac{dr}{d\\tau}=\\frac{dr}{dt}\\frac{dt}{d\\tau}=\\frac{dr}{dt}\\frac{k}{(1-\\frac{R_s}{r})}$$\n\nThe second equation of motion for a mass particle:\n$$[18]\\quad (\\frac{dr}{dt})^2=-\\frac{c^2}{k^2}(1-\\frac{R_s}{r})^3+c^2(1-\\frac{R_s}{r})^2$$\n\nFrom the initial condition, we can get the constant of motion:\n$$[19]\\quad t=t_0=0,r=r_0 \\qquad \\frac{dr}{dt}|_{t_0}=0 \\qquad \\Rightarrow k=\u00b1\\sqrt{1-\\frac{R_s}{r_0}}$$\n\nAs for the setup of the test, if we shoot the photon with an initial non-zero launch angle ##\\varphi## (inwards), then it will have obviously an initial inward geometric radial velocity, and being c a big number seems unfair to compare then the radial travelled distance of the photon with a poor test particle that starts quiet and still.\n\nSo, we'll shot the photon tangentially (##\\varphi=0##). This means that if the path of light were unaffected by Gravity, would hit the other side of the elevator at a higher position (i.e., if the starting radius is r_0, would reach the other side at a radius r_1>r_0). In a nutshell, we don't point the laser towards the test particle, but a little above it.\n\nFor the test particle, it's starting position will be r_0, and so both the photon and the test particle begin at the same distance (radius) from the center of the planet.\n\nThat's fair enough for me.\n\nWith a simple Python code (attached), I obtain the following:\n\nFINAL RESULT:\nr1-rend=6.556e-16[m] r0-rend_m=2.185e-16[m] tend=6.671e-09\n\nwhere r1 is the point-of-impact radius of a straight line with a length of 2[m] (we fix as condition the arc covered to be L/r_0), rend if the final radius of the photon position, rend_m is the final radius of the test particle position, and tend is the total flight time (in the coordinate time). So the radial distance displacement for the photon is 3.0004.. times greater than the radial distance displacement of the test particle.\n\n\nfrom mpmath import *\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport math\nfrom datetime import datetime\n\nmp.dps = 100\nmp.pretty = True\n\nc=299792458 # speed of light\nc2=pow(c,2) # square of speed of light\nG=mpf(6.6743e-11) # Gravitional constant\nM=mpf(5.9722e+24) # Earth's mass\nr0=mpf(6371000) # Earth's radius\nL=mpf(2) # Elevator's width\n\nalp=L/r0\n\nRs=(2*G*M)/c2\nRs2=pow(Rs,2)\nr0s=1-Rs/r0\n\nb2=((pow(r0,2)*c2)/r0s)\nb=sqrt(b2)\n\ndef zeq(r):\n return c2-(b2/pow(r,2))*(1-Rs/r)\n\ndef fph_dt_dr(r):\n zr=zeq(r)\n\n if (zr>0):\n return 1/((1-Rs/r)*sqrt(zr))\n else:\n raise Exception('NEGATIVE SQRT! r=%5.3e[m]'%(r))\n\ndef fph_dphi_dt(r):\n return (b/pow(r,2))*(1-Rs/r)\n\ndef fph_dphi_dr(r):\n zr=zeq(r)\n\n if (zr>0):\n return b/sqrt(r*(c2*pow(r,3)-b2*(r-Rs)))\n else:\n raise Exception('NEGATIVE SQRT! r=%5.3e[m]'%(r))\n\ndef fmp_dt_dr(r):\n return -1/(c*(1-Rs/r)*sqrt(1-(1-Rs/r)/r0s))\n\n\nprint('')\n\nr1=r0/cos(alp)\n\n[rend] = findroot([lambda r: quad(fph_dphi_dr,[r0,r])-alp],(r1))\ntend = quad(fph_dt_dr,[r0,rend])\n\nd=pow(tend,2)*G*M/(2*pow(r0,2)) # initial guess\n[rend_m] = findroot([lambda r: quad(fmp_dt_dr,[r0-1e-51,r])-tend],(r0-d))\n\nprint('') \nprint('FINAL RESULT:')\nprint('r1-rend=%5.3e[m] r0-rend_m=%5.3e[m] tend=%9.3e[s]'%(r1-rend,r0-rend_m,tend))" }, "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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Einstein brother-in-law elevator