the neighbourhood of the bodies, we can directly use the results of the previous sub-section to decompose $m(t)$ as a sum of $N$ terms integrated over local simultaneity surfaces $T_A^0(t) = {\rm const.}$.

Let us recall the transformation law between the global and local coordinate systems of the mass densities pertaining to a given point of body A $$\begin{aligned}
 \sigma(x) & =& \left|{\partial X \over \partial x} \right|^{(4)}
  \Bigl[\left(1+2{v_A^2 \over c^2}\right)\Sigma(X) \nonumber \\
 & & \qquad\quad\quad + {4 \over c^2}
  V_A^a \Sigma^a_A(X)\Bigr] + {\cal O}(4) \; , \label{ten}\end{aligned}$$ as given in [@dam1].

Putting together the definition (\[nine\]) and the method explained in the previous sub-section II.A \[where we note that the four dimensional Jacobian cancels between Eqs. (\[seven\]) and (\[ten\])\] we arrive after some algebra to $$\begin{aligned}
  m(t) &=& \sum_A \biggl[ M^A\left(1+{1 \over 2c^2}
   v_A^2\right) + {1 \over c^2} {d \over dT_A}\left(M^A_a V_A^a\right)
  \nonumber \\
  & & \qquad + {1 \over 2c^2} \sum_l {2l+1 \over l!} M^A_L
   \left(G_L^A + G_L^{A\prime\prime} \right)\biggr] + {\cal O}(4) \; ,
 \nonumber \\ \label{eleven}\end{aligned}$$

where all quantities on the right-hand side must be evaluated at the intersection of the $t={\rm const.}$ hypersurface with the central worldline of the corresponding body (i.e. for $T_A = T_A^0(t)$). 