that it can be written as an integral on a local-time $T_A=T_A^0(t)={\rm const}.$ simultaneity surface, say $${\cal I}_A(T_A^0(t)) = \int_A d^3X_A\, {\tilde F}
 (T_A^0(t), {\bf X}_A)\; .\label{two}$$ Here, $T_A^0(t)$ denotes the value of the local time (in the reference system attached to body $A$) corresponding to the event on the [*central*]{} worldline of the body-$A$ reference system (${\bf X}_A
= 0$, also quoted as ${\cal L}_A$), whose global time coordinate is equal to $t$. In equations, if we write the spacetime coordinate transformation between the global coordinate system $x^\mu=(ct,x^i)$ and the body-$A$ local one $X_A^\alpha = (cT_A,X_A^a)$ as $$\begin{aligned}
 x^\mu = x^\mu(X_A^\alpha)& =& z_A^\mu(T_A) + e^\mu_{A a}(T_A)
  \bigl[X^a_A \nonumber \\
 & & \;\; + {1 \over 2c^2} A^a_A {\bf X}_A^2 - {1 \over c^2}
 ({\bf A}_A. {\bf X}_A) X_A^a \bigr] \; ,\label{three}\end{aligned}$$ $T_A^0(t)$ is defined as the unique solution of $ct =
x^0(T_A,{\bf 0})$, i.e. $ct \equiv z_A^0(T_A^0(t))$.

First, let us denote by $f_A(T_A, {\bf X}_A)$ the original function $f(t,{\bf x})$ reexpressed in terms of the local spacetime variables $X_A^\alpha$: $f_A(X_A^\alpha) \equiv f(x^\mu(X_A^\alpha))$. By mathematically transforming the variables of integration in Eq. (\[one\]) we get $${\cal I}_A(t) = \int_A d^3X_A \left( \left|{\partial X \over
 \partial x}\right|^{(3)}_{t = const}\right)^{-1}\!\!\! f_A\left[
 T_A\left(t, 