the acceleration of the body $A$ frame. Here, $ G''_A = c^2 \ln(dT / d\tau_f)_A = v_A^2/2 -
c^2 \ln e^0_{A 0} + {\cal O}(2)$ measures the relative scaling, along the central worldline $X_A^a = 0$, between the local time $T_A$ and the global Minkowskian proper time $d\tau_f = \sqrt{-f_{\mu\nu}dz^\mu
dz^\nu}/c$ (see Sec. VI.E of [@dam1]).

Note that, in geometrical terms, the mathematical transformations we have just performed correspond to using a mapping between the $t={\rm const.}$ and $T_A = {\rm const.} = T_A^0(t)$ hypersurfaces by means of the congruence of worldlines ${\cal L}_{X_A^a}$ of constant spatial local coordinates (see Sec. III.D in [@dam1]).

Mass-energy integral
--------------------

The Blanchet-Damour post-Newtonian total mass-energy $m(t)$ of an isolated system can be written as [@bd] $$m(t) = \int d^3x\, \sigma(x) - {1 \over c^2} {d \over dt} \int d^3x
  \,(\sigma^i x^i) + {\cal O}(4) \; , \label{nine}$$ where $\sigma_\mu$ are densities of mass and mass currents \[$\sigma = (T^{00}+T^{ii})/c^2$ and $\sigma^i = T^{0i}/c$; $T^{\mu
\nu}$ denoting the components of the stress-energy tensor in the [*global*]{} coordinate system\]. The integration in (\[nine\]) is to be performed over a global-time $t={\rm const}.$ hypersurface spanning the whole $N$-body system. However, as $\sigma^\mu$ is nonzero only in 