Derivation of Post-Newtonian (PN) expression for acceleration in Schwarzschild geometry

The expression for the acceleration of a near-earth satellite as presented in the IERS Technical note is given by $$\label{eq:problemeq} \tag{1} \frac{d^2\mathbf{r}}{dt^2} = \frac{GM_E}{c^2r^3} \left\{\left[2(\beta+\gamma)\frac{GM_E}{r} - \gamma \dot{\mathbf{r}} \cdot \dot{\mathbf{r}} \right] \mathbf{r} + 2(1+\gamma)(\mathbf{r}\cdot\dot{\mathbf{r}})\dot{\mathbf{r}} \right\}.$$

We are working in the Parametrised Post-Newtonian formalism, hence the dimensionless constants $\beta,\gamma$.

From what I can gather from here and here this expression can be derived from the "Schwarzschild isotropic one-body point mass metric".

Now, I know what the Schwarzschild metric looks like in isotropic coordinates but I can't see where Eq. (\ref{eq:problemeq}) comes from.

Another point to make is that in GR the dimensionless parameters $\beta,\gamma$ are equal to unity and when substituted above gives a well known formula for Schwarzschild precession which is given by $$\label{eq:problemeq2} \tag{1} \frac{d^2\mathbf{r}}{dt^2} = \frac{GM_E}{c^2r^3} \left\{\left[4\frac{GM_E}{r} - \dot{\mathbf{r}} \cdot \dot{\mathbf{r}} \right] \mathbf{r} + 4(\mathbf{r}\cdot\dot{\mathbf{r}})\dot{\mathbf{r}} \right\}.$$

Again, I don't recall this either.

Any suggestions?

• The derivation of that expression seems to be rather involved. I looked through the various references in IERS Technical Note No. 36 and AU Resolutions B1.3 and B1.4 (2000) and the best starting point I could find is [Klioner 2001; arxiv.org/abs/astro-ph/0107457] eq. (3) and the references for it [Will 1993] and [Klioner & Soffel 2000; arxiv.org/abs/gr-qc/9906123].
– N0va
Commented Feb 20, 2017 at 9:54
• It's definitely quite involved. I don't expect it to be done in one sitting :) On the hunt for some resources that can be understood. I find the old school PN guys quite difficult to follow! Commented Feb 20, 2017 at 11:41
• The eq of motion for your case is $\frac{d^2x^\sigma}{ds^2} = \Gamma^\sigma_{\mu\nu} \frac{dx^\mu}{ds}\frac{dx^\nu}{ds}$ . Just plug in the Schwarzschild metric tensor $g_{\mu\nu}$ into $\Gamma^{\sigma}_{\mu\nu} = 1/2 g^{\sigma\alpha}( \frac{\partial g_{\mu\alpha}}{\partial x_\nu} + \frac{\partial g_{\nu\alpha}}{\partial x_\mu} -\frac{\partial g_{\mu\nu}}{\partial x_\alpha})$ and do the math on the eq of motion. This is, i suppose, the simplest approach. Commented May 18, 2017 at 11:03
• @MihaiB. Unfortunately this wouldn't work. That what you would do in a completely relativistic framework. However, this is expressed in terms of the PPN formalism. Commented May 18, 2017 at 11:05
• That's very similar to the form used in Yeomans, D. K., et al. "Cometary orbit determination and nongravitational forces." Comets II (2004): 137-151, who reference Anderson J. D., et al. "Experimental test of general relativity using time-delay data from Mariner 6 and Mariner 7." Astrophys. J., 200, 221–233. The only difference is the use of the Gaussian gravitational constant $k$ instead of $GM_E$ Commented May 22, 2017 at 11:07