Periods of non-circular Schwarzschild orbits - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-06-26T08:18:57Z https://physics.stackexchange.com/feeds/question/446084 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://physics.stackexchange.com/q/446084 2 Periods of non-circular Schwarzschild orbits Gordon https://physics.stackexchange.com/users/195362 2018-12-09T00:11:19Z 2018-12-09T19:53:31Z <p>I have been thinking about non-circular orbits in the Schwarzschild spacetime. How would you define a period of one orbit? I was thinking, in terms of proper time, for <span class="math-container">$r$</span>, how long it takes to go from one apogee to another. For <span class="math-container">$\phi$</span>, again in terms of <span class="math-container">$\tau$</span>, how long it takes to cover <span class="math-container">$2\pi$</span>. What about <span class="math-container">$t$</span>, though? Is my reasoning wrong?</p> https://physics.stackexchange.com/questions/446084/-/446087#446087 3 Answer by Chiral Anomaly for Periods of non-circular Schwarzschild orbits Chiral Anomaly https://physics.stackexchange.com/users/206691 2018-12-09T00:38:32Z 2018-12-09T00:38:32Z <p>You actually described two inequivalent definitions of "period," both legitimate. The paper </p> <ul> <li>Geisler and McVittie (1965), "Orbital periods in the Schwarzschild space-time" (<a href="http://adsabs.harvard.edu/full/1965AJ.....70...14G" rel="nofollow noreferrer">http://adsabs.harvard.edu/full/1965AJ.....70...14G</a>)</li> </ul> <p>considers essentially the same two definitions of "period" that you described. One is the time from one perihelion to the next; they call this the <strong>anomalistic</strong> period. The other is the lapse between two successive passages across <span class="math-container">$\phi=0$</span>; they call this the <strong>sidereal</strong> period. The two periods are not the same. Both periods may be expressed either in terms of the object's own proper time <span class="math-container">$\tau$</span>, or in terms of the coordinate time <span class="math-container">$t$</span>. These again are not the same.</p> <p>The key is to specify the worldline by expressing all of the coordinates as functions of a shared parameter, which we can take to be the object's proper time. Then we have functions <span class="math-container">$r(\tau)$</span>, <span class="math-container">$\phi(\tau)$</span>, and <span class="math-container">$t(\tau)$</span>. We need all of these functions anyway to solve the free-fall equations that define what "orbit" means. Given those functions, we can use <span class="math-container">$r(\tau)$</span> to compute what those authors called the anomalistic period, or we can use <span class="math-container">$\phi(\tau)$</span> to compute what those authors called the sidereal period, both in terms of the object's proper time. To relate those proper-time periods to coordinate-time periods, we can use the function <span class="math-container">$t(\tau)$</span>.</p> https://physics.stackexchange.com/questions/446084/-/446190#446190 2 Answer by Elio Fabri for Periods of non-circular Schwarzschild orbits Elio Fabri https://physics.stackexchange.com/users/204634 2018-12-09T17:21:31Z 2018-12-09T19:53:31Z <p><span class="math-container">$\let\a=\alpha \let\b=\beta \let\phi=\varphi \let\De=\Delta \def\D#1#2{{d#1\over d#2}} \def\dr{\dot r} \def\dt{\dot t} \def\dx{\dot x} \def\dphi{\dot\phi} \def\half{{\textstyle {1 \over 2}}}$</span> If I could assume you can read Italian I would have an easy life - had only to give a link. But since I find it unlikely, I'll write a synthesis of essential points.</p> <p>Consider a general metric <span class="math-container">$$d\tau^2 = g_{\a\b}\,dx^\a dx^\b.\tag1$$</span> We are interested in timelike geodesics, which can be parametrized with proper time <span class="math-container">$\tau$</span>. Then coordinates are functions of <span class="math-container">$\tau$</span>: <span class="math-container">$$x^\a = x^\a(\tau) \qquad dx^\a = \D {x^\a}\tau\,d\tau$$</span> and from (1) we have <span class="math-container">$$g_{\a\b}\,\dx^\a \dx^\b = 1.\tag2$$</span></p> <p>It can be shown that geodesics obey a variational principle with lagrangian <span class="math-container">$$W = \half\,g_{\a\b}\,\dx^\a \dx^\b.$$</span> Eq. (2) shows that <span class="math-container">$W$</span> is a constant of the motion, with <span class="math-container">$2W=1$</span> on a timelike geodesics.</p> <p>Schwarzschild's metric, restricted to the plane <span class="math-container">$\theta=\pi/2$</span>, is <span class="math-container">$$d\tau^2 = \left(\!1 - {1\over r}\!\right) dt^2 - {dr^2\over 1 - 1/r} - r^2 d\phi^2$$</span> where units were so chosen <span class="math-container">$G=1$</span>, <span class="math-container">$c=1$</span>, <span class="math-container">$2M=1$</span> (<span class="math-container">$M$</span> Sun's mass). Then <span class="math-container">$$2W = {r - 1 \over r}\,\dt^2 - {r \over r - 1}\,\dr^2 - r^2 \dphi^2 = 1$$</span> Since <span class="math-container">$W$</span> doesn't depend on <span class="math-container">$t$</span> and on <span class="math-container">$\phi$</span>, we have the constants of the motion <span class="math-container">$${r - 1 \over r}\,\dt = E \qquad r^2 \dphi = J.\tag3$$</span></p> <p>Substituting (3) into (2) we have <span class="math-container">$$\dr^2 = E^2 - \left(\!1 - {1 \over r}\!\right)\! \left(\!1 + {J^2 \over r^2}\right)\!.\tag4$$</span> Integrating eq. (4) by separation of variables we get <span class="math-container">$\tau(r)$</span> and the radial period. Unfortunately an elliptic integral is involved.</p> <p>As to <span class="math-container">$\phi$</span>, from the second of (3) and (4) we have <span class="math-container">$${J^2 \over r^4} \left(\!\D r\phi\!\right)^{\!\!2} = E^2 - \left(\!1 - {1 \over r}\!\right)\! \left(\!1 + {J^2 \over r^2}\right)$$</span> which gives <span class="math-container">$\phi(r)$</span>, again as an elliptic integral. An approximation is possible to deduce perihelion precession (I'll not show how to do it).</p> <p>Instead I find a problem if the azimuthal period is of interest. The reason is the following. Let's start form perihelion: when <span class="math-container">$\phi$</span> increases by <span class="math-container">$2\pi$</span> we are not yet arrived at another perihelion, because of precession. This shows qualitatively that azimuthal period is less than radial one. But a further increment by <span class="math-container">$2\pi$</span> brings us still farther from perihelion, and I expect that the latter <span class="math-container">$2\pi$</span> variation of <span class="math-container">$\phi$</span> takes a different time from the former (larger or smaller?) So it seems that a well definite azimuthal period doesn't exist.</p> <p><strong>Edit</strong>. But an average period can be defined. let <span class="math-container">$T_r$</span> be the radial period, <span class="math-container">$\De\phi$</span> the perihelion advance in time <span class="math-container">$T_r$</span>. Then in the average <span class="math-container">$\phi$</span> advances by <span class="math-container">$2\pi$</span> in time <span class="math-container">$$T_\phi = {2 \pi\,T_r \over 2\pi + \De\phi}.$$</span></p>