general relativity - 3PN and higher order post-Newtonian Schwarzschild approximation

This expression can be found in documentation from the JPL determining the relativistic acceleration under Schwarzschild conditions in the euclidean approximation they use to calculate the orbits of celestial bodies:

$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\left(1-\frac{4GM}{rc^2}+\frac{v^2}{c^2}\right)\hat{r} +\frac{4GM}{r^2}\left(\hat{r}\cdot \hat{v}\right)\frac{v^2}{c^2}\hat{v}$$

This is expression 4-26 on page 4-19 in Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation, by Theodore Moyer. Most of the terms becomes zero when you have only one mass.

Does anyone know the physical interpretation of the three extra terms? I would be happy if you tell me about it. I see there is one term that is basically "negative inverse cube" gravity pushing for instance planets away from the sun.

Now I found a paper, Third post-Newtonian dynamics of compact binaries:Equations of motion in the center-of-mass frame" by Blanchet and Iyer. The paper is outlining the post-Newtonian expansion to the third "3PN" order. The acceleration under Schwarzschild conditions is found in expression 3.9 and 3.10. Most of the terms become zero. I find the 3PN post-Newtonian accelerations under Schwarzshild conditions to be:

\begin{align} \frac{d\bar{v}}{dt} &=-\frac{GM}{r^2}\left(1-4\frac{GM}{rc^2} + 9\left(\frac{GM}{rc^2}\right)^2 - 16\left(\frac{GM}{rc^2}\right)^3\right)\hat{r} \\ &\qquad-\frac{GM}{r^2}\left(\frac{v^2}{c^2}-\frac{2GM}{rc^4}\left(\bar{v}\cdot\hat{r}\right)^2 +\frac{(GM)^2}{r^2c^6}(\bar{v}\cdot\hat{r})^2\right)\hat{r}\\ &\qquad-\frac{GM}{r^2}\left(-4\frac{(\bar{v}\cdot\hat{r})}{c^2}+2\frac{GM}{rc^4}(\bar{v}\cdot{\hat{r}})-4\frac{(GM)^2}{r^2c^6}(\bar{v}\cdot\hat{r})\right)\bar{v} \end{align}

Maybe I have made some mistake. The first four terms, the ones not depending on the velocity, looks like a converging series. Maybe also the rest of the terms converge to a known expression, do you know anything about that?

1. What is the physical interpretation of the various terms?


2. Does the accelerations of the post-Newtonian expansion, when applied to a case of only one spherically symmetric body, converge to an expression for the relativistic acceleration and if so, what is that expression?