Can General Relativity indicate phase-dependent variations in planetary orbital acceleration?

Question

In a previous question about differences in Newtonian and GTR gravitional force 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 and Goldstein et al 2001.

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). According to, but not clearly referenced by, this article, 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, 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.

Answer 1

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. 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 ($L\equiv mh$), $$E = \underbrace{\frac{1}{2}m\dot{r}^2 + \frac{L^2}{2mr^2} - \frac{GMm}{r}}_{\text{Newtonian form}} - \frac{GML^2}{mr^3c^2}\text{.}$$

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.

Answer 2

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.

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 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. 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