$v$ is the relative speed of the stars in units $c=1$, and is related through the virial theorem to the curvature generated by the whole binary system of total mass $M$.

We adopt the “mostly plus” convention $$\begin{gathered} {\eta }_{\mu \nu }\equiv \\ \text{diag}(-,+,+,+) \end{gathered}$$, and the Riemann and Ricci tensors are defined as $R_{\nu \rho \sigma }^{\mu }={\partial }_{\rho }{\mathrm{\Gamma }}_{\nu \sigma }^{\mu }+{\mathrm{\Gamma }}_{\alpha \rho }^{\mu }{\mathrm{\Gamma }}_{\nu \sigma }^{\alpha }-\rho \leftrightarrow \sigma$, $R_{\mu \nu }\equiv R_{\mu \alpha \nu }^{\alpha }$.

To be more precise, accelerations are traded for their equations of motion, which for the PM point of view means eliminating them as they will give rise to $\mathcal{O}(G^2)$ terms.

The 4PN order in the center-of-mass position is sufficient for the purposes of the present work; however, the derivation at 5PN in the PM limit is a very useful test of Lorentz invariance, and thus of the correctness of the calculation.

The use of either of the two equivalent operators for some $n$ brings us to different expressions for $V_{\mathrm{PM}}^{*}$.