3PN and higher order post-Newtonian Schwarzschild approximation

Question

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?

Answer 1

Although I previously commented that I doubt that there is a physical interpretation of the individual terms, I realized that there is a hand-wavy interpretation of the terms that don’t involve the velocity, beginning with the “negative inverse cube” term that is repulsive.

Your expansion is for the acceleration of a test mass $m$, but there is an equivalent expansion of the potential energy,

$$U=-\frac{GMm}{r}\left(1-2\frac{GM}{rc^2}+\dots\right).$$

This can be interpreted by thinking about how gravitational potential energy gravitates. The Newtonian PE,

$$U_0=-\frac{GMm}{r}$$

can be considered to “live” in the Newtonian gravitational field. (This can be actually be made precise for Newtonian gravity.) It is spatially distributed but is primarily in the region between $M$ and $m$.

Since we’re considering relativistic corrections to Newtonian gravity, it makes sense to consider the effective negative mass of this negative field energy,

$$m_{U_0}=\frac{U_0}{c^2}=-\frac{GMm}{rc^2},$$

and then consider the gravitational potential energy between this mass and $M$, assuming that they are separated by roughly $r$:

$$U_1=-\frac{GMm_{U_0}}{r}=\frac{G^2M^2m}{r^2c^2}$$

This is, up to a multiplicative constant of order 1 reflecting the non-localization of the field energy, the second term in the PE expansion.

It is repulsive because gravitational potential energy is negative.

You can continue to play the same game, thinking about the third term in the expansion as an attractive correction due to how the energy correction we just considered gravitates.

This interpretation is not to be taken too seriously. It is more just for intuition. However, the “gravity of gravitational energy” is a real thing in the post-Newtonian approach to GR. For example, if you read here about the $\beta_2$ parameter in Will’s original PPN formalism, it parameterizes “how much gravity is produced by unit gravitational potential energy”, and is nonzero in GR.

Another example of the gravity of gravitational potential energy is the Nordtvedt effect.

I don’t have any similar interpretation of the velocity-dependent terms, because there is no velocity dependence in Newtonian gravity.

I doubt that the series converges to any known function, because if it did, physicists would use it rather than the expansion.

Answer 2

If you set the secondary mass to zero (or more specifically the mass-ratio), what you are left with is a PN expansion of the geodesic equation in Schwarzschild space-time (in some particular coordinates (harmonic I think).