# 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?
• Regarding (1), I doubt that there is a physical interpretation of the individual terms. However, I suggest checking either C. Will’s early papers on post-Newtonian gravity or his recent book “Theory and Experiment in Gravitational Physics”. – G. Smith Mar 28 at 17:33
• Why do you say “Maybe I have made some mistake”? The 3PN result looks consistent with the first 2PN one. – G. Smith Mar 28 at 17:39
• Regarding (2), these expansions are for $d^2\bar{r}/dt^2$ where $t$ is coordinate time. The relativistic acceleration would be $d^2\bar{r}/d\tau^2$ where $\tau$ is proper time. – G. Smith Mar 28 at 17:43
• Well you might get one kind of error when doing a "translation" to a euclidian system with coordinate time and another kind of error because you are truncating a series and only using the first few terms of the series. If you can find out if the series converges to some expression you could at least get rid of the error that is due to truncation. – Agerhell Mar 28 at 17:53
• This is probably the Einstein–Infeld–Hoffmann equations, extended to higher order: en.wikipedia.org/wiki/… . It is probably true that some of the lower-order terms have definite physical interpretations, such as effects of relativistic inertia or gravitomagnetic effects. But by the time you get to third order, I doubt that there is much you can do to find term-by-term physical interpretations. Note that this sort of thing is going to work only in specially chosen coordinate systems, so it doesn't have direct physical significance. – Ben Crowell Mar 29 at 2:02

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.

• From a list of series I find that $\sum_{k=1}^{\infty}k^2z^k=\frac{z(1+z)}{(1-z)^3}$. From this I get the non-velocity dependent term above to converge to $-\frac{GM}{r^2}(\frac{1-\frac{GM}{rc^2}}{(1+\frac{GM}{rc^2})^3})\hat{r}$, setting $z=-GM/(rc^2)$ and assuming that the series goes on as it started. this looks strange. Maybe you are supposed to get different expressions depending on the choice of metric and also that the expression for potential energy is different in the metric used which is described as "isotropic" but maybe differs someway from the most known isotropic coordinates. – Agerhell Apr 3 at 11:07

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).

• On page 2-9 in the above mentioned JPL documentation there is an expression numbered 2-16 for the metric that besides from a scaling term looks like: ${ds}^2=(1-\frac{2GM}{rc^2})c^2{dt}^2-(1+\frac{2GM}{rc^2})(dx^2+dy^2+dz^2)$ by which one is supposed to get to the first expression above. – Agerhell Apr 5 at 15:14