<p>An alternative expression for the extra, (supra-Newtonian, relativistic) acceleration (brought to my attention by user /u/uhoh) is presented by Shahid-Saless (Colorado) and Yeomans (JPL) in their 1994 Astronomical Journal paper: <a href="https://www.google.co.uk/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=6&amp;cad=rja&amp;uact=8&amp;ved=0ahUKEwic2LfchcXZAhXL-qQKHeJxAusQFghZMAU&amp;url=http%3A%2F%2Fadsabs.harvard.edu%2Ffull%2F1994AJ....107.1885S&amp;usg=AOvVaw1sOMUCKcBLnUaI427q9N7S" rel="nofollow noreferrer">Relativistic Effects on the Motion of Asteroids and Comets</a>.</p>
<p>Their equation 3.11 for the Newtonian + Relativistic acceleration of a single target body orbiting the Sun is as follows:-
<span class="math-container">$$
\frac{d^2 \textbf{r}}{c^2dt^2} 
= \frac{-\mu}{r^3}\textbf{r} 
+ \frac{\mu}{r^3} \left[ \left(4\frac{\mu}{r}-\frac{v^2}{c^2}\right)\textbf{r} 
+ 4 \frac{(\textbf{r} \cdot \textbf{v})\textbf{v}}{c^2}\right]
$$</span>
where</p>
<p><span class="math-container">$\textbf{r}$</span>  is the instantaneous position vector of the target body relative to the Sun,</p>
<p><span class="math-container">$\textbf{v}$</span>  is the instantaneous velocity vector of the target body relative to the Sun,</p>
<p><span class="math-container">$c$</span>  is the speed of light,</p>
<p><span class="math-container">$\mu = GM/c^2$</span> is the Schwartzschild Gravitational Radius of the Sun,</p>
<p><span class="math-container">$G$</span> is the universal gravitational constant,</p>
<p><span class="math-container">$M$</span> is the (post-Newtonian) mass of the Sun,</p>
<p><span class="math-container">$\frac{-\mu}{r^3} $</span> the first term on the RHS, is the Newtonian radial acceleration with the negative sign indicating acceleration towards the source.</p>
<p>The authors present a derivation of the equation (which is beyond my expertise). They also use it to derive an expression for <span class="math-container">$\delta \omega$</span> the amount (in radians) of rotation of the line of apsides per complete (<span class="math-container">$2\pi$</span> radians) orbital revolution:-</p>
<p><span class="math-container">$$
\delta\omega = \frac{6\pi\mu}{a(1-e^2)} 
\equiv \frac{6\pi GM}{c^2 a(1-e^2)} 
\equiv \frac{24\pi^3 a^2}{T^2c^2 (1-e^2)}
$$</span></p>
<p>where <span class="math-container">$a$</span> is semi-major axis of orbit, <span class="math-container">$e$</span> is eccentricity of orbit, and <span class="math-container">$T$</span> is the orbit period.</p>
<p>The right-most version is identical to the <a href="http://www.gsjournal.net/old/eeuro/vankov.pdf" rel="nofollow noreferrer">Einstein 1915</a> equation presented in the question.</p>
<p>It is noteworthy that the Shahid-Saless &amp; Yeomans equation 3.11 indicates that when <span class="math-container">$\textbf{v}$</span> is not perpendicular to <span class="math-container">$\textbf{r}$</span> a part of the non-Newtonian acceleration will be directed in the direction transverse to the radial direction.</p>
<p>Note that caveats apply when moving between a General Relativistic model of Space-Time and a Euclidean-Galilean model -see Stan Liou's answer and the Shahid-Saless &amp; Yeomans paper itself.</p>
<hr />
<p><strong>Update</strong></p>
<p>I am not sure that the Shahid-Saless (Colorado) and Yeomans equation makes sense. Consider a circular orbit of a test particle around the Sun (where, given the absence of antipodes, orbital precession would be observed as a decrease in the period of revolution relative to the Newtonian period, and a corresponding increase in the circular velocity <span class="math-container">$v_c$</span>, with no change in orbit radius <span class="math-container">$r$</span>.) Then the vectors <span class="math-container">$\textbf{r}$</span> and <span class="math-container">$\textbf{v}$</span> will be perpendicular, and so the dot product <span class="math-container">$\textbf{r} \cdot \textbf{v}$</span> will be zero and so we will obtain (replacing <span class="math-container">$\mu$</span> by <span class="math-container">$GMc^2$</span>):-</p>
<p><span class="math-container">$$
\frac{d^2 \textbf{r}}{ dt^2} 
= \frac{-GM}{r^3}\textbf{r} 
+ \frac{GM}{r^3} \left[ \left(4\frac{GM}{r}-\frac{v^2}{c^2}\right)\textbf{r} 
\right]
$$</span></p>
<p>Also, in a circular orbit we have <span class="math-container">$\frac{GM}{r}=\frac{v^2}{c^2}$</span> and so the radial &quot;acceleration&quot; (specific force) can be written as:-
<span class="math-container">$$
\frac{d^2 \textbf{r}}{dt^2} 
= \frac{-GM}{r^3}\textbf{r} 
+ \frac{GM}{r^3}  \left(3\frac{GM}{r} \textbf{r} 
\right)
$$</span>
and so
<span class="math-container">$$
\frac{d^2 \textbf{r}}{dt^2} 
= \frac{-GM}{r^3}\textbf{r}   \left(1 -\frac{3GM}{r} \right)
= \frac{-GM}{r^3}\textbf{r}   \left(1 -\frac{3v^2}{c^2} \right)
= \frac{-GM}{r^2}             \left(1 -\frac{3v^2}{c^2} \right) \hat{r}
$$</span>
Now this final expression is in similar form to that obtained in my <a href="https://astronomy.stackexchange.com/questions/7700/what-is-the-correct-ratio-of-newtonian-to-general-relativistic-gravitational-eff?rq=1">earlier SE Astronomy post</a>. However the sign of the term <span class="math-container">$\frac{3v^2}{c^2}$</span> is positive in the other post and that formula does produce positive orbital precession whereas the formula above produces &quot;negative precession&quot;.  It seems to me that either the Shahid-Saless &amp; Yeomans formula has the wrong sign or there is something wrong in my analysis.</p>
<p>Update: I have posted a follow-up question about formulae similar to that of Shahid-Saless &amp; Yeomans here:-<a href="https://astronomy.stackexchange.com/questions/51649/problem-with-the-orbital-precession-predicted-by-the-relativistic-acceleration-e?noredirect=1#comment116077_51649">Problem with the Orbital Precession predicted by the Relativistic Acceleration equation used in the JPL D440-D441 Ephemerides</a></p>

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2018-03-08T01:14:40.910Z
2023-01-23T22:30:16.277Z
2023-01-23T22:30:16.277Z
steveOw
<p>Since I don't have Walter's book, I'm uncertain as the context of the derivation of the equation you quote. Therefore, I've simply re-derived it here; apologies if there's some repetition of things you already know, but perhaps it'll be useful for anyone else reading this regardless.</p>

<h2>Constants of Motion</h2>

<p>The Schwarzschild solution is the unique nontrivial spherically symmetric vacuum solution of general relativity. In the Schwarzschild coordinate chart and units of $G = c = 1$, the metric takes the form
$$\mathrm{d}s^2 = -\left(1-\frac{2M}{r}\right)\mathrm{d}t^2 + \left(1-\frac{2M}{r}\right)\mathrm{d}r^2 + r^2\left(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2\right)\text{,}$$
and one can immediately note that that the metric coefficients are completely independent of $t$ and $\phi$, which implies that $\partial_t$ and $\partial_\phi$ are <em><a href="https://en.wikipedia.org/wiki/Killing_vector_field" rel="noreferrer">Killing vector fields</a></em>. They are important here because along with generating symmetries of the geometry, they also produce conserved orbital quantities in the following way: given an orbit with four-velocity $u^\mu = (\dot{t},\dot{r},\dot{\theta},\dot{\phi})$, the inner product with a Killing vector field is conserved:
$$\epsilon = -\langle\partial_t,u\rangle = \left(1-\frac{2M}{r}\right)\frac{\mathrm{d}t}{\mathrm{d}\tau}\text{,}$$
$$h = \langle\partial_\phi,u\rangle = r^2\sin^2\theta\,\frac{\mathrm{d}\phi}{\mathrm{d}\tau}\text{.}$$
The overdot indicates differentiation with respect to any affine parameter of the orbit, which for timelike geodesics appropriate for massive particles we can take without loss of generality to be the proper time $\tau$. An alternative way to find these constants of motion is to integrate the $t$ and $\phi$ components of the geodesic equation, but in this way they can read off immediately from the metric. These are the specific energy and specific angular momentum of the orbit, respectively. Also note that the coordinates are analogues of the spherical coordinates for Euclidean space, where $\theta$ is the zenith angle while $\phi$ is the azimuth; if we take the orbital plane to be the equatorial plane ($\theta = \pi/2$), then $\phi$ would represent the true anomaly.</p>

<h2>Effective Potential</h2>

<p>Substituting the above constants of motion into the timelike worldline condition $\langle u,u\rangle \equiv g_{\mu\nu} u^\mu u^\nu = -1$, i.e.,
$$-\left(1-\frac{2M}{r}\right)\dot{t}^2 + \left(1-\frac{2M}{r}\right)^{-1}\dot{r}^2 + r^2\dot{\phi}^2 = -1\text{,}$$
one can immediately derive the effective gravitational potential:
$$\frac{1}{2}(\epsilon^2-1) = \frac{1}{2}\dot{r}^2 + \underbrace{\left[-\frac{M}{r}+\frac{h^2}{2r^2} - \frac{Mh^2}{r^3}\right]}_{V_\text{eff}}\text{,}$$
or if one insists on a <em>formal</em> comparison with the <a href="https://en.wikipedia.org/wiki/Effective_potential" rel="noreferrer">Newtonian effective potential</a> ($L\equiv mh$),
$$E = \underbrace{\frac{1}{2}m\dot{r}^2 + \frac{L^2}{2mr^2} - \frac{GMm}{r}}_{\text{Newtonian form}} - \frac{GML^2}{mr^3c^2}\text{.}$$</p>

<h2>Orbit Equation</h2>

<p>Differentiation of the above effective potential gives
$$\ddot{r} + \frac{M}{r^2} - \frac{h^2}{r^3} + 3\frac{Mh^2}{r^4} = 0\text{.}$$
In terms of $u \equiv 1/r$ with prime denoting differentiation with respect to $\phi$,
$$u''= \frac{\mathrm{d}\tau}{\mathrm{d}\phi}\frac{\mathrm{d}}{\mathrm{d}\tau}\left(\frac{\mathrm{d}\tau}{\mathrm{d}\phi}\dot{u}\right) = \frac{r^2}{h}\frac{\mathrm{d}}{\mathrm{d}\tau}\left(\frac{r^2}{h}\left(-r^{-2}\dot{r}\right)\right) = -\frac{\ddot{r}r^2}{h^2}\text{,}$$
this gives, after multiplication through by $-r^2/h^2$,
$$u'' + u = \frac{M}{h^2} + 3Mu^2\text{.}$$
However, there is really no need to consider a second-order at any point; there's a simpler one in terms of $V \equiv V_\text{eff} - h^2/2r^2$, the effective potential sans the centrifugal potential term:
$$\begin{eqnarray*}
\frac{2}{h^2}\left[\frac{E}{m}-V\right] &amp;=&amp; \frac{\dot{r}^2}{h^2} + \frac{1}{r^2}
\\&amp;=&amp; \frac{1}{r^4}\left[\frac{\mathrm{d}r}{\mathrm{d}\phi}\right]^2 + u^2
\\&amp;=&amp; (u')^2 + u^2\text{.}
\end{eqnarray*}$$</p>

<blockquote>
  <p>Walter derived an approximate relationship assuming a circular orbit. Goldstein focused on deriving an orbit-average expression for perihelion precession. On re-examining these texts it seems to me that GR provides more than just an orbit-averaged approximation.
  ...
  Walter presents the following equation for a GR orbit (Schwarzschild model)
  $$u''_\theta + u_\theta =\frac{\mu}{h^2} + \frac{3\mu}{c^2}\,u_\theta^2$$ </p>
</blockquote>

<p>One can immediately see that Walter's equation is the above second-order equation, just in normal units rather than $G = c = 1$. I don't know what Walter's argument is (I'm willing to bet the approximation is because Walter substituted a circular-orbit case for $L^2$ or $h^2$ somewhere, though), but that particular relationship holds <em>exactly</em> for massive test particles in Schwarzschild spacetime. It does not even have to be a bound orbit, although of course if one is interested in precession specifically, it would have to be at least bound for precession to make sense. Lightlike geodesics are described by nearly the same equation, just without the $M/h^2$ term.</p>

<p>Furthermore, we can also restate it as
$$u'' + u = \frac{M}{h^2}\left[1 + 3\frac{h^2}{r^2}\right] \leadsto \frac{\mu}{h^2}\left[1+3\frac{h^2}{r^2c^2}\right]\text{,}$$
which after substitution of $V_t\equiv r\dot{\phi} = h/r$ is what you have.</p>

<h2>Conclusion</h2>

<blockquote>
  <p>... so an alternative, more palatable, distance-dependent form of the ratio of accelerations ... would be:-
  $$1\;\text{to}\;\frac {3 h^2}{c^2 \, r_\theta^2}\text{.}$$
  The GR/Schwarzschild equations relate to proper time and Schwarzschild radial distance, not their Newtonian equivalents so strictly the ratio of accelerations is still an approximation.</p>
  
  <p>Is this analysis valid, or have I missed something?</p>
</blockquote>

<p>It is mostly valid, but I would like you to caution you on several points regarding the way you frame the problem and interpret the result, although you are likely already aware of some of them:</p>

<ol>
<li>The Schwarzschild time coordinate $t$ is quite different from the proper time $\tau$. The former is a special coordinate in which the Schwarzschild geometry is time-independent. It defines the worldlines of a family of observers that are stationary with respect to the geometry, and its scaling matches a stationary observer at infinity. On the other hand, proper time is simply the time measured along some particular worldline; in this context, by the orbiting test particle.</li>
<li><p>The Schwarzschild radial coordinate $r$ is not a radial distance. It could be called an areal radius in the sense that it is chosen to make a sphere of constant $r$ have area of exactly $4\pi r^2$, but usually it is simply called the <em>Schwarzschild radial coordinate</em>. In the Schwarzschild coordinate chart, the radial distance between Schwarzschild radial coordinates $r = r_0$ and $r = r_1$ would be given by
$$D = \int_{r_0}^{r_1}\frac{\mathrm{d}r}{\sqrt{1-\frac{2GM}{rc^2}}}\text{,}$$
and would be the distance one would measure if one slowly crawled along the radial direction from $r = r_0$ to $r = r_1$ with some ideal meter-stick, in the limit of zero speed. Of course, $r$ could serve as an approximation to the radial distance in appropriate contexts, but the point is that not only does $r$ fail to be the Newtonian radial distance, it's not actually the 'Schwarzschild radial <em>distance</em>' either.</p></li>
<li><p><em>Acceleration</em> is a bit of a loaded word here. If we mean the second derivative of our radial coordinate with respect to proper time, then no, $\ddot{r}_\text{GTR}/\ddot{r}_\text{Newtonian}$ does not simplify quite that nicely, but you calculate it from the above anyway. On the other hand, if we mean the second derivative of the inverse radial coordinate with respect to the azimuthal angle, then yes, the above correct.</p></li>
</ol>

<p>But then, it doesn't really make sense to actually call it 'acceleration', does it? This explains (if your previous question was accurate in this phrasing) why Walter uses a more vague term of 'effects' when talking about the above ratio.</p>

<p>Instead (once again using the intentional conflation between $r$,$\tau$ and their Newtonian counterparts as an approximation or analogy), it would probably be better to simply think of the Schwarzschild geometry as introducing a new term in the potential that is analogous to a quadrupole moment, which would also put a $\propto 1/r^3$ term into the potential, with the corresponding Newtonian equation being
$$(u')^2 + u^2 = \frac{2}{h^2}\left(\frac{E}{m} - \Phi(u)\right)\text{.}$$
Both the effective potential and the first-order equation in $u$ provide a much more straightforward analogy between the Newtonian and Schwarzschild cases.</p>

<p>This is actually pretty interesting: if one assumes that the Sun does indeed have a quadrupole moment, e.g., caused by solar oblateness, then one can easily account for the perihelion advance of Mercury. However, because this is simply an analogy, blaming Mercury's behavior on this would simultaneously mess up the behavior of other planets (since the new term depends on orbital angular momentum) and be even more inconsistent for orbits outside the equatorial plane (since actual oblateness should have the quadrupole term dependent on zenith angle, whereas GTR's is not).</p>

<p>It is also possible to think of the Schwarzschild geometry itself as a scalar field, which we can similarly decompose into spherical harmonic components. Naturally, like most of the above, this peculiarity is specific to the niceness of the spherically symmetric vacuum.</p>

CC BY-SA 3.0
2014-11-09T21:47:06.100Z
.Many thanks for this rich and helpful answer. I see I need to choose words carefully, like Goldstein "GTR predicts a correction to Newtonian motion that **can be construed as** an $r^-3$ potential"...similar to your "analogous to...a quadrupole moment". I shall ponder some more.  I note that GTR was "first proved" with Mercury's perihelion precession anomaly determined using a Newtonian model and the perturbation analyses which endorse Einstein's precession formula (e.g. Walter & Goldstein) are also Newtonian. So the "fence" between GTR & Newton models must be jumped somewhere!.
steveOw
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2014-11-09T22:31:36.537Z
@steveOw You're welcome. Side note: the relationship between a solar quadrupole and Mercury is interesting and was at least somewhat controversial. In the 60's, Dicke et al found a value for the quadrupole moment $J_2$ that would account for nearly a tenth of the value of anomalous perihelion advance of Mercury, which would be an obvious problem since GTR is supposed to account for virtually all of it. But through the years, advances in helioseismology have pushed their value of quadrupole moment down by two orders of magnitude. The Sun's field turned out to be very nearly spherical after all.
Stan Liou
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2014-11-10T12:12:15.840Z
Yes I had read about the Dicke et al Oblate Sun hypothesis and its rejection.
steveOw
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2014-11-10T13:23:27.553Z
I have added an update to my question reflecting your answer.
steveOw
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2014-11-10T20:08:24.120Z
@steveOw You're touching upon a very large topic if you're going towards something like "why is GTR's anomalous precession impressive when things like Gerber's theory predicted it before?" There is a good answer to that, and it relates to what (in modern terms) would be the parametrized post-Newtonian formalism, which also shows that the most natural correspondence between Schwarzschild geometry and Newton is actually its form in isotropic coordinates rather than Schwarzschild coordinates. If you like, I could elaborate here, but I think that's a large enough issue for a separate question.
Stan Liou
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2014-11-10T21:23:17.697Z
I am not a fan of Gerber's theory but I am interested in the (electrodynamic gravity) ideas of people like Weber, Riemann and Ritz who I first encountered in the entertaining book by [Roseveare](http://www.jstor.org/discover/10.2307/686971?uid=3738032&uid=2129&uid=2&uid=70&uid=4&sid=21104501457261).  I find PPNF, GTR and STR difficult to accept on the basis of their axioms.  I would greatly enjoy a dialogue/debate about these contentious topics but I suspect this forum may not be the place to host debate on such non-mainstream (heretical?) topics.
steveOw
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2014-11-11T01:43:09.670Z
@steveOw STR? What about do you find difficult to accept? On an intuitive level, it's just as easy as Euclidean geometry, and geometrically the Euclidean/Galilean/Lorentzian cases form a natural trifecta. Recognizing that space is different form time only narrows it down to to possibilities consistent with the principle of relativity. The only physical question is whether the Galilean or Lorentzian is more effective at describing the world, and all of fundamental physics is a testament to the latter.
Stan Liou
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2014-11-11T13:28:18.910Z
For me physics (+science + knowledge) is essentially about Useful Rules (not Platonic Universal Laws).  Useful is subjective.  Arguments invoking different rulesets (axiom sets) are generally irreconcilable.  Galilean mechanics have IMO proved more useful for humans than Lorentzian.  STR has IMO been useful to fundamental physicists. IMO this is because (pending a Better Model) it acts as a temporary placeholder to paper over the deficiencies of aether-based Maxwellian electrodynamics, itself very useful in electronics etc. My "project" seeks that Better Model (ruleset).  GTR IMO ?
steveOw
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2014-11-11T17:41:35.787Z
@steveOw No one claimed Platonic truth; I think most physicists would simply say STR is far, far more useful. You're correct that this is a poor forum for this, but I will say this: I think by characterizing STR as having an intrinsic connection to or basis in any sort of EM, you're placing too much value in the historical and accidental circumstances of its initial formulation.
Stan Liou
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2014-11-11T21:28:00.750Z
I consider any statement about "Universal Laws of Physics" or "Beautiful Theories" to be suspiciously Platonic, such talk is often used wrt STR & GTR. To graduate in Physics these days I suspect it is necessary to believe firmly in STR (at least in the West!). Would you like to continue in chat?
steveOw
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2014-11-11T22:28:52.010Z
@steveOw Everyone working on gravity beyond the classical believes GTR to be an [effective field theory](https://en.wikipedia.org/wiki/Effective_field_theory), which is a rather anti-Platonic notion that permeated physics in 1940s on the heels of QED. When physicists say that STR is true, what it means is that working physics theories are Lorentz-invariant, and this is *very useful* in guidance to new working theories because it is a much more restrictive condition. Sorry, but it's the denial of STR that throws out usefulness to pander to philosophical preconceptions. As for chat, go ahead.
Stan Liou
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2014-11-11T22:48:36.693Z
Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/18574/discussion-between-steveow-and-stan-liou).
steveOw
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2014-11-08T04:42:48.733Z
2014-11-08T18:36:31.063Z
2014-11-08T18:36:31.063Z
Stan Liou
<p>In a previous question about <a href="https://astronomy.stackexchange.com/questions/632/determining-effect-of-small-variable-force-on-planetary-perihelion-precession">differences in Newtonian and GTR gravitional force</a> for the case of star-planet gravitational interactions an <em>approximate</em> relationship was noted between the expressions for the gravitational force exerted by a star on an orbiting planet (the solution is valid situations with low gravity, slow orbital speed, and spherical source).  This was based on texts in <a href="http://books.google.co.uk/books?id=V4_s91-Vun4C&amp;pg=PA482&amp;lpg=PA482&amp;dq=we+recall+from+section+7.6.1&amp;source=bl&amp;ots=RyU1Adp8h7&amp;sig=qQ5_KyjoDhaSaOadgP8zyy8ubIc&amp;hl=en&amp;sa=X&amp;ei=Hlc-VJDJK9TzaoHlgKgL&amp;ved=0CCEQ6AEwAA#v=onepage&amp;q=we%20recall%20from%20section%207.6.1&amp;f=false" rel="nofollow noreferrer">Walter 2008</a> and <a href="http://rads.stackoverflow.com/amzn/click/0201657023" rel="nofollow noreferrer">Goldstein et al 2001</a>.</p>

<p>Walter derived an approximate relationship assuming a circular orbit.  Goldstein focussed on deriving an orbit-average expression for perihelion precession.</p>

<p>On re-examining these texts it seems to me that GTR (General Theory of Relativity) provides more than just an orbit-averaged approximation.  Rather it provides a phase-specific formula for total acceleration (in Scwharzschild space-time).</p>

<p>The equation of motion for a Newtonian orbit is
$$u''_\theta + u_\theta =\frac{\mu}{h^2}$$
where $u_\theta = 1/r_\theta$ and $u'_\theta =$ d$(u_\theta)/$d$\theta$ and $h$ is specific angular momentum which is constant, $\mu = GM$ the gravitational parameter, $G$ is Newton's Universal gravitational constant, $M$ is mass of the star, $r$ is distance from star to planet, $\theta$ is the True anomaly.</p>

<p>Using $h^2 = Vt_\theta^2 \,r_\theta^2$ where $Vt$ is the instantaneous transverse component of velocity (in vector terms = full velocity minus radial velocity) we can  obtain
$$u''_\theta + u_\theta = \frac{1}{Vt_\theta^2} \left( \frac{\mu}{r_\theta^2} \right)$$ where the term in brackets is the Newtonian acceleration.</p>

<p>Walter presents the following equation for a GTR orbit (Schwarzschild model)</p>

<p>$$u''_\theta + u_\theta =\frac{\mu}{h^2} + \frac{3\mu}{c^2}\,u_\theta^2$$ </p>

<p>Now, using $h^2 = Vt_\theta^2 \,r_\theta^2$ and $u_\theta^2 = 1/r_\theta^2$ we get</p>

<p>$$u''_\theta + u_\theta 
=\frac{\mu}{Vt_\theta^2 \,r_\theta^2} + \frac{3\mu}{c^2 \,r_\theta^2}
=\frac{1}{Vt_\theta^2 } 
\left( \frac{\mu}{r_\theta^2} \,+ \frac{\mu}{r_\theta^2} \,\frac{3\,Vt_\theta^2}{c^2 } \right)$$ 
where the terms in brackets are the Newtonian acceleration and the extra acceleration according to GTR.  And so the ratio of Newtonian acceleration to GTR-specific acceleration, at any angle $\theta$ is 
$$1 \, \mathrm{to} \, \frac {3 Vt_\theta^2}{c^2}$$.
Goldstein emphasises that the GR acceleration does not indicate a velocity-dependence, so an alternative, more palatable, distance-dependent form of the ratio of accelerations, at any angle $\theta$, would be:-
$$1 \, \mathrm{to} \, \frac {3 h^2}{c^2 \, r_\theta^2} \equiv 
1 \, \mathrm{to} \, \frac {3 GM.P}{c^2 \, r_\theta^2}
$$
where $h$ (specific angular momentum) and $P$ (semi-latus rectum) are the values for the orbit of the particular subject planet. </p>

<p>And so (ignoring other massive perturbing bodies) the magnitude of the total instantaneous radial force on the planet (of mass $m$) towards the Sun is given by:-
$$ \mathrm{F/m} \,=  \, \frac {GM}{r_\theta^2} + \frac {3 GM.GM.P}{c^2 \, r_\theta^4}
$$</p>

<p>N.B. The GTR/Schwarzschild equations relate to proper time and Schwarzschild radial distance, not their Newtonian equivalents so strictly the ratio of accelerations is still an approximation.</p>

<p>Is this analysis valid, or have I missed something?</p>

<h2>Update</h2>

<p>I have accepted Stan Liou's answer as being very helpful in (a) providing a derivation from GTR/Schwarzschild of the formulae presented by Walter and Goldstein et al. and (b) indicating the imperfect correspondence between GTR terms/concepts and Newtonian terms/concepts.</p>

<p>My understanding is as follows. In a Newtonian central-force elliptical-orbit model the addition of an extra centre-directed acceleration ($dV/dt$), which varies through the orbit as $(+3 V_t^2/c^2\,\equiv \, +3 h^2/r^2c^2)$ times the co-temporal standard Newtonian centre-directed gravitational acceleration, can be shown (by first order perturbation analysis or numerical modelling) to produce apsidal precession at a rate (radians per orbit) defined by a Newtonian formula $$\epsilon = 24 \pi^3 \frac{a^2}{T^2 c^2(1-e^2)}$$ .  This formula is well-known (see for example <a href="http://en.wikipedia.org/wiki/Apsidal_precession" rel="nofollow noreferrer">Wikipedia Apsidal precession</a>). According to, but not clearly referenced by, <a href="http://www.mathpages.com/home/kmath527/kmath527.htm" rel="nofollow noreferrer">this article</a>, the formula (or an algebraic equivalent using other terms) was well known c. 1895 (i.e. <em>before</em> the publications of Gerber 1898 and Einstein 1915).  The formula predicts very well the long-term values of apsidal precession determined using Newtonian models from observations of solar planets. </p>

<p>Various writers (<a href="http://www.gsjournal.net/old/eeuro/vankov.pdf" rel="nofollow noreferrer">Einstein</a>, Goldstein, Walter, presumably many others) present mathematical arguments indicating how an identical formula can be derived from Einstein's GTR.  The presented arguments may involve approximations (e.g. Walter's use of near-circular orbits, Goldstein's use of orbit-averaged precession) and non-mathematical "correspondences" between the concepts/terms of the GTR model and the concepts/terms of the Newtonian model.</p>

<orbit><general-relativity>
Can General Relativity indicate phase-dependent variations in planetary orbital acceleration?
steveOw
# Can General Relativity indicate phase-dependent variations in planetary orbital acceleration?

(Asked by: username_0 on 2014-11-07)

In a previous question about [differences in Newtonian and GTR gravitional force](https://astronomy.stackexchange.com/questions/632/determining-effect-of-small-variable-force-on-planetary-perihelion-precession) for the case of star-planet gravitational interactions an *approximate* relationship was noted between the expressions for the gravitational force exerted by a star on an orbiting planet (the solution is valid situations with low gravity, slow orbital speed, and spherical source). This was based on texts in [Walter 2008](http://books.google.co.uk/books?id=V4_s91-Vun4C&pg=PA482&lpg=PA482&dq=we+recall+from+section+7.6.1&source=bl&ots=RyU1Adp8h7&sig=qQ5_KyjoDhaSaOadgP8zyy8ubIc&hl=en&sa=X&ei=Hlc-VJDJK9TzaoHlgKgL&ved=0CCEQ6AEwAA#v=onepage&q=we%20recall%20from%20section%207.6.1&f=false) and [Goldstein et al 2001](http://rads.stackoverflow.com/amzn/click/0201657023).

Walter derived an approximate relationship assuming a circular orbit. Goldstein focussed on deriving an orbit-average expression for perihelion precession.

On re-examining these texts it seems to me that GTR (General Theory of Relativity) provides more than just an orbit-averaged approximation. Rather it provides a phase-specific formula for total acceleration (in Scwharzschild space-time).

The equation of motion for a Newtonian orbit is
$$u''\_\theta + u\_\theta =\frac{\mu}{h^2}$$
where $u\_\theta = 1/r\_\theta$ and $u'\_\theta =$ d$(u\_\theta)/$d$\theta$ and $h$ is specific angular momentum which is constant, $\mu = GM$ the gravitational parameter, $G$ is Newton's Universal gravitational constant, $M$ is mass of the star, $r$ is distance from star to planet, $\theta$ is the True anomaly.

Using $h^2 = Vt\_\theta^2 \,r\_\theta^2$ where $Vt$ is the instantaneous transverse component of velocity (in vector terms = full velocity minus radial velocity) we can obtain
$$u''\_\theta + u\_\theta = \frac{1}{Vt\_\theta^2} \left( \frac{\mu}{r\_\theta^2} \right)$$ where the term in brackets is the Newtonian acceleration.

Walter presents the following equation for a GTR orbit (Schwarzschild model)

$$u''\_\theta + u\_\theta =\frac{\mu}{h^2} + \frac{3\mu}{c^2}\,u\_\theta^2$$

Now, using $h^2 = Vt\_\theta^2 \,r\_\theta^2$ and $u\_\theta^2 = 1/r\_\theta^2$ we get

$$u''\_\theta + u\_\theta
=\frac{\mu}{Vt\_\theta^2 \,r\_\theta^2} + \frac{3\mu}{c^2 \,r\_\theta^2}
=\frac{1}{Vt\_\theta^2 }
\left( \frac{\mu}{r\_\theta^2} \,+ \frac{\mu}{r\_\theta^2} \,\frac{3\,Vt\_\theta^2}{c^2 } \right)$$
where the terms in brackets are the Newtonian acceleration and the extra acceleration according to GTR. And so the ratio of Newtonian acceleration to GTR-specific acceleration, at any angle $\theta$ is
$$1 \, \mathrm{to} \, \frac {3 Vt\_\theta^2}{c^2}$$.
Goldstein emphasises that the GR acceleration does not indicate a velocity-dependence, so an alternative, more palatable, distance-dependent form of the ratio of accelerations, at any angle $\theta$, would be:-
$$1 \, \mathrm{to} \, \frac {3 h^2}{c^2 \, r\_\theta^2} \equiv
1 \, \mathrm{to} \, \frac {3 GM.P}{c^2 \, r\_\theta^2}
$$
where $h$ (specific angular momentum) and $P$ (semi-latus rectum) are the values for the orbit of the particular subject planet.

And so (ignoring other massive perturbing bodies) the magnitude of the total instantaneous radial force on the planet (of mass $m$) towards the Sun is given by:-
$$ \mathrm{F/m} \,= \, \frac {GM}{r\_\theta^2} + \frac {3 GM.GM.P}{c^2 \, r\_\theta^4}
$$

N.B. The GTR/Schwarzschild equations relate to proper time and Schwarzschild radial distance, not their Newtonian equivalents so strictly the ratio of accelerations is still an approximation.

Is this analysis valid, or have I missed something?

## Update

I have accepted Stan Liou's answer as being very helpful in (a) providing a derivation from GTR/Schwarzschild of the formulae presented by Walter and Goldstein et al. and (b) indicating the imperfect correspondence between GTR terms/concepts and Newtonian terms/concepts.

My understanding is as follows. In a Newtonian central-force elliptical-orbit model the addition of an extra centre-directed acceleration ($dV/dt$), which varies through the orbit as $(+3 V\_t^2/c^2\,\equiv \, +3 h^2/r^2c^2)$ times the co-temporal standard Newtonian centre-directed gravitational acceleration, can be shown (by first order perturbation analysis or numerical modelling) to produce apsidal precession at a rate (radians per orbit) defined by a Newtonian formula $$\epsilon = 24 \pi^3 \frac{a^2}{T^2 c^2(1-e^2)}$$ . This formula is well-known (see for example [Wikipedia Apsidal precession](http://en.wikipedia.org/wiki/Apsidal_precession)). According to, but not clearly referenced by, [this article](http://www.mathpages.com/home/kmath527/kmath527.htm), the formula (or an algebraic equivalent using other terms) was well known c. 1895 (i.e. *before* the publications of Gerber 1898 and Einstein 1915). The formula predicts very well the long-term values of apsidal precession determined using Newtonian models from observations of solar planets.

Various writers ([Einstein](http://www.gsjournal.net/old/eeuro/vankov.pdf), Goldstein, Walter, presumably many others) present mathematical arguments indicating how an identical formula can be derived from Einstein's GTR. The presented arguments may involve approximations (e.g. Walter's use of near-circular orbits, Goldstein's use of orbit-averaged precession) and non-mathematical "correspondences" between the concepts/terms of the GTR model and the concepts/terms of the Newtonian model.

**Tags:** orbit, general-relativity

---

## 2 answers

### Answer by username_1 on 2014-11-08. Score: 7 | Accepted answer

Since I don't have Walter's book, I'm uncertain as the context of the derivation of the equation you quote. Therefore, I've simply re-derived it here; apologies if there's some repetition of things you already know, but perhaps it'll be useful for anyone else reading this regardless.

## Constants of Motion

The Schwarzschild solution is the unique nontrivial spherically symmetric vacuum solution of general relativity. In the Schwarzschild coordinate chart and units of $G = c = 1$, the metric takes the form
$$\mathrm{d}s^2 = -\left(1-\frac{2M}{r}\right)\mathrm{d}t^2 + \left(1-\frac{2M}{r}\right)\mathrm{d}r^2 + r^2\left(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2\right)\text{,}$$
and one can immediately note that that the metric coefficients are completely independent of $t$ and $\phi$, which implies that $\partial\_t$ and $\partial\_\phi$ are *[Killing vector fields](https://en.wikipedia.org/wiki/Killing_vector_field)*. They are important here because along with generating symmetries of the geometry, they also produce conserved orbital quantities in the following way: given an orbit with four-velocity $u^\mu = (\dot{t},\dot{r},\dot{\theta},\dot{\phi})$, the inner product with a Killing vector field is conserved:
$$\epsilon = -\langle\partial\_t,u\rangle = \left(1-\frac{2M}{r}\right)\frac{\mathrm{d}t}{\mathrm{d}\tau}\text{,}$$
$$h = \langle\partial\_\phi,u\rangle = r^2\sin^2\theta\,\frac{\mathrm{d}\phi}{\mathrm{d}\tau}\text{.}$$
The overdot indicates differentiation with respect to any affine parameter of the orbit, which for timelike geodesics appropriate for massive particles we can take without loss of generality to be the proper time $\tau$. An alternative way to find these constants of motion is to integrate the $t$ and $\phi$ components of the geodesic equation, but in this way they can read off immediately from the metric. These are the specific energy and specific angular momentum of the orbit, respectively. Also note that the coordinates are analogues of the spherical coordinates for Euclidean space, where $\theta$ is the zenith angle while $\phi$ is the azimuth; if we take the orbital plane to be the equatorial plane ($\theta = \pi/2$), then $\phi$ would represent the true anomaly.

## Effective Potential

Substituting the above constants of motion into the timelike worldline condition $\langle u,u\rangle \equiv g\_{\mu\nu} u^\mu u^\nu = -1$, i.e.,
$$-\left(1-\frac{2M}{r}\right)\dot{t}^2 + \left(1-\frac{2M}{r}\right)^{-1}\dot{r}^2 + r^2\dot{\phi}^2 = -1\text{,}$$
one can immediately derive the effective gravitational potential:
$$\frac{1}{2}(\epsilon^2-1) = \frac{1}{2}\dot{r}^2 + \underbrace{\left[-\frac{M}{r}+\frac{h^2}{2r^2} - \frac{Mh^2}{r^3}\right]}\_{V\_\text{eff}}\text{,}$$
or if one insists on a *formal* comparison with the [Newtonian effective potential](https://en.wikipedia.org/wiki/Effective_potential) ($L\equiv mh$),
$$E = \underbrace{\frac{1}{2}m\dot{r}^2 + \frac{L^2}{2mr^2} - \frac{GMm}{r}}\_{\text{Newtonian form}} - \frac{GML^2}{mr^3c^2}\text{.}$$

## Orbit Equation

Differentiation of the above effective potential gives
$$\ddot{r} + \frac{M}{r^2} - \frac{h^2}{r^3} + 3\frac{Mh^2}{r^4} = 0\text{.}$$
In terms of $u \equiv 1/r$ with prime denoting differentiation with respect to $\phi$,
$$u''= \frac{\mathrm{d}\tau}{\mathrm{d}\phi}\frac{\mathrm{d}}{\mathrm{d}\tau}\left(\frac{\mathrm{d}\tau}{\mathrm{d}\phi}\dot{u}\right) = \frac{r^2}{h}\frac{\mathrm{d}}{\mathrm{d}\tau}\left(\frac{r^2}{h}\left(-r^{-2}\dot{r}\right)\right) = -\frac{\ddot{r}r^2}{h^2}\text{,}$$
this gives, after multiplication through by $-r^2/h^2$,
$$u'' + u = \frac{M}{h^2} + 3Mu^2\text{.}$$
However, there is really no need to consider a second-order at any point; there's a simpler one in terms of $V \equiv V\_\text{eff} - h^2/2r^2$, the effective potential sans the centrifugal potential term:
$$\begin{eqnarray\*}
\frac{2}{h^2}\left[\frac{E}{m}-V\right] &=& \frac{\dot{r}^2}{h^2} + \frac{1}{r^2}
\\&=& \frac{1}{r^4}\left[\frac{\mathrm{d}r}{\mathrm{d}\phi}\right]^2 + u^2
\\&=& (u')^2 + u^2\text{.}
\end{eqnarray\*}$$

> Walter derived an approximate relationship assuming a circular orbit. Goldstein focused on deriving an orbit-average expression for perihelion precession. On re-examining these texts it seems to me that GR provides more than just an orbit-averaged approximation.
> ...
> Walter presents the following equation for a GR orbit (Schwarzschild model)
> $$u''\_\theta + u\_\theta =\frac{\mu}{h^2} + \frac{3\mu}{c^2}\,u\_\theta^2$$

One can immediately see that Walter's equation is the above second-order equation, just in normal units rather than $G = c = 1$. I don't know what Walter's argument is (I'm willing to bet the approximation is because Walter substituted a circular-orbit case for $L^2$ or $h^2$ somewhere, though), but that particular relationship holds *exactly* for massive test particles in Schwarzschild spacetime. It does not even have to be a bound orbit, although of course if one is interested in precession specifically, it would have to be at least bound for precession to make sense. Lightlike geodesics are described by nearly the same equation, just without the $M/h^2$ term.

Furthermore, we can also restate it as
$$u'' + u = \frac{M}{h^2}\left[1 + 3\frac{h^2}{r^2}\right] \leadsto \frac{\mu}{h^2}\left[1+3\frac{h^2}{r^2c^2}\right]\text{,}$$
which after substitution of $V\_t\equiv r\dot{\phi} = h/r$ is what you have.

## Conclusion

> ... so an alternative, more palatable, distance-dependent form of the ratio of accelerations ... would be:-
> $$1\;\text{to}\;\frac {3 h^2}{c^2 \, r\_\theta^2}\text{.}$$
> The GR/Schwarzschild equations relate to proper time and Schwarzschild radial distance, not their Newtonian equivalents so strictly the ratio of accelerations is still an approximation.
>
> Is this analysis valid, or have I missed something?

It is mostly valid, but I would like you to caution you on several points regarding the way you frame the problem and interpret the result, although you are likely already aware of some of them:

1. The Schwarzschild time coordinate $t$ is quite different from the proper time $\tau$. The former is a special coordinate in which the Schwarzschild geometry is time-independent. It defines the worldlines of a family of observers that are stationary with respect to the geometry, and its scaling matches a stationary observer at infinity. On the other hand, proper time is simply the time measured along some particular worldline; in this context, by the orbiting test particle.
2. The Schwarzschild radial coordinate $r$ is not a radial distance. It could be called an areal radius in the sense that it is chosen to make a sphere of constant $r$ have area of exactly $4\pi r^2$, but usually it is simply called the *Schwarzschild radial coordinate*. In the Schwarzschild coordinate chart, the radial distance between Schwarzschild radial coordinates $r = r\_0$ and $r = r\_1$ would be given by
   $$D = \int\_{r\_0}^{r\_1}\frac{\mathrm{d}r}{\sqrt{1-\frac{2GM}{rc^2}}}\text{,}$$
   and would be the distance one would measure if one slowly crawled along the radial direction from $r = r\_0$ to $r = r\_1$ with some ideal meter-stick, in the limit of zero speed. Of course, $r$ could serve as an approximation to the radial distance in appropriate contexts, but the point is that not only does $r$ fail to be the Newtonian radial distance, it's not actually the 'Schwarzschild radial *distance*' either.
3. *Acceleration* is a bit of a loaded word here. If we mean the second derivative of our radial coordinate with respect to proper time, then no, $\ddot{r}\_\text{GTR}/\ddot{r}\_\text{Newtonian}$ does not simplify quite that nicely, but you calculate it from the above anyway. On the other hand, if we mean the second derivative of the inverse radial coordinate with respect to the azimuthal angle, then yes, the above correct.

But then, it doesn't really make sense to actually call it 'acceleration', does it? This explains (if your previous question was accurate in this phrasing) why Walter uses a more vague term of 'effects' when talking about the above ratio.

Instead (once again using the intentional conflation between $r$,$\tau$ and their Newtonian counterparts as an approximation or analogy), it would probably be better to simply think of the Schwarzschild geometry as introducing a new term in the potential that is analogous to a quadrupole moment, which would also put a $\propto 1/r^3$ term into the potential, with the corresponding Newtonian equation being
$$(u')^2 + u^2 = \frac{2}{h^2}\left(\frac{E}{m} - \Phi(u)\right)\text{.}$$
Both the effective potential and the first-order equation in $u$ provide a much more straightforward analogy between the Newtonian and Schwarzschild cases.

This is actually pretty interesting: if one assumes that the Sun does indeed have a quadrupole moment, e.g., caused by solar oblateness, then one can easily account for the perihelion advance of Mercury. However, because this is simply an analogy, blaming Mercury's behavior on this would simultaneously mess up the behavior of other planets (since the new term depends on orbital angular momentum) and be even more inconsistent for orbits outside the equatorial plane (since actual oblateness should have the quadrupole term dependent on zenith angle, whereas GTR's is not).

It is also possible to think of the Schwarzschild geometry itself as a scalar field, which we can similarly decompose into spherical harmonic components. Naturally, like most of the above, this peculiarity is specific to the niceness of the spherically symmetric vacuum.

> **Comments:**
> **username_0**: .Many thanks for this rich and helpful answer. I see I need to choose words carefully, like Goldstein "GTR predicts a correction to Newtonian motion that **can be construed as** an $r^-3$ potential"...similar to your "analogous to...a quadrupole moment". I shall ponder some more.  I note that GTR was "first proved" with Mercury's perihelion precession anomaly determined using a Newtonian model and the perturbation analyses which endorse Einstein's precession formula (e.g. Walter & Goldstein) are also Newtonian. So the "fence" between GTR & Newton models must be jumped somewhere!.  
> **username_1**: @steveOw You're welcome. Side note: the relationship between a solar quadrupole and Mercury is interesting and was at least somewhat controversial. In the 60's, Dicke et al found a value for the quadrupole moment $J_2$ that would account for nearly a tenth of the value of anomalous perihelion advance of Mercury, which would be an obvious problem since GTR is supposed to account for virtually all of it. But through the years, advances in helioseismology have pushed their value of quadrupole moment down by two orders of magnitude. The Sun's field turned out to be very nearly spherical after all.  
> **username_0**: Yes I had read about the Dicke et al Oblate Sun hypothesis and its rejection.  
> **username_0**: I have added an update to my question reflecting your answer.  
> **username_1**: @steveOw You're touching upon a very large topic if you're going towards something like "why is GTR's anomalous precession impressive when things like Gerber's theory predicted it before?" There is a good answer to that, and it relates to what (in modern terms) would be the parametrized post-Newtonian formalism, which also shows that the most natural correspondence between Schwarzschild geometry and Newton is actually its form in isotropic coordinates rather than Schwarzschild coordinates. If you like, I could elaborate here, but I think that's a large enough issue for a separate question.  
> **username_0**: I am not a fan of Gerber's theory but I am interested in the (electrodynamic gravity) ideas of people like Weber, Riemann and Ritz who I first encountered in the entertaining book by [Roseveare](http://www.jstor.org/discover/10.2307/686971?uid=3738032&uid=2129&uid=2&uid=70&uid=4&sid=21104501457261).  I find PPNF, GTR and STR difficult to accept on the basis of their axioms.  I would greatly enjoy a dialogue/debate about these contentious topics but I suspect this forum may not be the place to host debate on such non-mainstream (heretical?) topics.  
> **username_1**: @steveOw STR? What about do you find difficult to accept? On an intuitive level, it's just as easy as Euclidean geometry, and geometrically the Euclidean/Galilean/Lorentzian cases form a natural trifecta. Recognizing that space is different form time only narrows it down to to possibilities consistent with the principle of relativity. The only physical question is whether the Galilean or Lorentzian is more effective at describing the world, and all of fundamental physics is a testament to the latter.  
> **username_0**: For me physics (+science + knowledge) is essentially about Useful Rules (not Platonic Universal Laws).  Useful is subjective.  Arguments invoking different rulesets (axiom sets) are generally irreconcilable.  Galilean mechanics have IMO proved more useful for humans than Lorentzian.  STR has IMO been useful to fundamental physicists. IMO this is because (pending a Better Model) it acts as a temporary placeholder to paper over the deficiencies of aether-based Maxwellian electrodynamics, itself very useful in electronics etc. My "project" seeks that Better Model (ruleset).  GTR IMO ?  
> **username_1**: @steveOw No one claimed Platonic truth; I think most physicists would simply say STR is far, far more useful. You're correct that this is a poor forum for this, but I will say this: I think by characterizing STR as having an intrinsic connection to or basis in any sort of EM, you're placing too much value in the historical and accidental circumstances of its initial formulation.  
> **username_0**: I consider any statement about "Universal Laws of Physics" or "Beautiful Theories" to be suspiciously Platonic, such talk is often used wrt STR & GTR. To graduate in Physics these days I suspect it is necessary to believe firmly in STR (at least in the West!). Would you like to continue in chat?  
> **username_1**: @steveOw Everyone working on gravity beyond the classical believes GTR to be an [effective field theory](https://en.wikipedia.org/wiki/Effective_field_theory), which is a rather anti-Platonic notion that permeated physics in 1940s on the heels of QED. When physicists say that STR is true, what it means is that working physics theories are Lorentz-invariant, and this is *very useful* in guidance to new working theories because it is a much more restrictive condition. Sorry, but it's the denial of STR that throws out usefulness to pander to philosophical preconceptions. As for chat, go ahead.  
> **username_0**: Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/18574/discussion-between-steveow-and-stan-liou).  

---

### Answer by username_0 on 2018-03-08. Score: 1

An alternative expression for the extra, (supra-Newtonian, relativistic) acceleration (brought to my attention by user /u/uhoh) is presented by Shahid-Saless (Colorado) and Yeomans (JPL) in their 1994 Astronomical Journal paper: [Relativistic Effects on the Motion of Asteroids and Comets](https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&cad=rja&uact=8&ved=0ahUKEwic2LfchcXZAhXL-qQKHeJxAusQFghZMAU&url=http%3A%2F%2Fadsabs.harvard.edu%2Ffull%2F1994AJ....107.1885S&usg=AOvVaw1sOMUCKcBLnUaI427q9N7S).

Their equation 3.11 for the Newtonian + Relativistic acceleration of a single target body orbiting the Sun is as follows:-
$$
\frac{d^2 \textbf{r}}{c^2dt^2}
= \frac{-\mu}{r^3}\textbf{r}
+ \frac{\mu}{r^3} \left[ \left(4\frac{\mu}{r}-\frac{v^2}{c^2}\right)\textbf{r}
+ 4 \frac{(\textbf{r} \cdot \textbf{v})\textbf{v}}{c^2}\right]
$$
where

$\textbf{r}$ is the instantaneous position vector of the target body relative to the Sun,

$\textbf{v}$ is the instantaneous velocity vector of the target body relative to the Sun,

$c$ is the speed of light,

$\mu = GM/c^2$ is the Schwartzschild Gravitational Radius of the Sun,

$G$ is the universal gravitational constant,

$M$ is the (post-Newtonian) mass of the Sun,

$\frac{-\mu}{r^3} $ the first term on the RHS, is the Newtonian radial acceleration with the negative sign indicating acceleration towards the source.

The authors present a derivation of the equation (which is beyond my expertise). They also use it to derive an expression for $\delta \omega$ the amount (in radians) of rotation of the line of apsides per complete ($2\pi$ radians) orbital revolution:-

$$
\delta\omega = \frac{6\pi\mu}{a(1-e^2)}
\equiv \frac{6\pi GM}{c^2 a(1-e^2)}
\equiv \frac{24\pi^3 a^2}{T^2c^2 (1-e^2)}
$$

where $a$ is semi-major axis of orbit, $e$ is eccentricity of orbit, and $T$ is the orbit period.

The right-most version is identical to the [Einstein 1915](http://www.gsjournal.net/old/eeuro/vankov.pdf) equation presented in the question.

It is noteworthy that the Shahid-Saless & Yeomans equation 3.11 indicates that when $\textbf{v}$ is not perpendicular to $\textbf{r}$ a part of the non-Newtonian acceleration will be directed in the direction transverse to the radial direction.

Note that caveats apply when moving between a General Relativistic model of Space-Time and a Euclidean-Galilean model -see Stan Liou's answer and the Shahid-Saless & Yeomans paper itself.

---

**Update**

I am not sure that the Shahid-Saless (Colorado) and Yeomans equation makes sense. Consider a circular orbit of a test particle around the Sun (where, given the absence of antipodes, orbital precession would be observed as a decrease in the period of revolution relative to the Newtonian period, and a corresponding increase in the circular velocity $v\_c$, with no change in orbit radius $r$.) Then the vectors $\textbf{r}$ and $\textbf{v}$ will be perpendicular, and so the dot product $\textbf{r} \cdot \textbf{v}$ will be zero and so we will obtain (replacing $\mu$ by $GMc^2$):-

$$
\frac{d^2 \textbf{r}}{ dt^2}
= \frac{-GM}{r^3}\textbf{r}
+ \frac{GM}{r^3} \left[ \left(4\frac{GM}{r}-\frac{v^2}{c^2}\right)\textbf{r}
\right]
$$

Also, in a circular orbit we have $\frac{GM}{r}=\frac{v^2}{c^2}$ and so the radial "acceleration" (specific force) can be written as:-
$$
\frac{d^2 \textbf{r}}{dt^2}
= \frac{-GM}{r^3}\textbf{r}
+ \frac{GM}{r^3} \left(3\frac{GM}{r} \textbf{r}
\right)
$$
and so
$$
\frac{d^2 \textbf{r}}{dt^2}
= \frac{-GM}{r^3}\textbf{r} \left(1 -\frac{3GM}{r} \right)
= \frac{-GM}{r^3}\textbf{r} \left(1 -\frac{3v^2}{c^2} \right)
= \frac{-GM}{r^2} \left(1 -\frac{3v^2}{c^2} \right) \hat{r}
$$
Now this final expression is in similar form to that obtained in my [earlier SE Astronomy post](https://astronomy.stackexchange.com/questions/7700/what-is-the-correct-ratio-of-newtonian-to-general-relativistic-gravitational-eff?rq=1). However the sign of the term $\frac{3v^2}{c^2}$ is positive in the other post and that formula does produce positive orbital precession whereas the formula above produces "negative precession". It seems to me that either the Shahid-Saless & Yeomans formula has the wrong sign or there is something wrong in my analysis.

Update: I have posted a follow-up question about formulae similar to that of Shahid-Saless & Yeomans here:-[Problem with the Orbital Precession predicted by the Relativistic Acceleration equation used in the JPL D440-D441 Ephemerides](https://astronomy.stackexchange.com/questions/51649/problem-with-the-orbital-precession-predicted-by-the-relativistic-acceleration-e?noredirect=1#comment116077_51649)


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