{\bf X}_A\right),{\bf X}_A\right] \; ,\label{four}$$ where $|\partial X / \partial x|^{(3)}_{t = const}$ is the spatial Jacobian ${\rm det}(\partial X^a / \partial x^i)$ $(a,i =
1,2,3)$ computed when keeping fixed the value of $t$, and where $T_A(t,{\bf X}_A)$ denotes the solution of $ct = x^0(T_A,{\bf X}_A)$. From Eq. (\[three\]) (or Eqs. (A5) of Ref. [@dam2]), the latter quantity reads explicitly $$T_A(t,{\bf X}_A) = T_A^0(t) - {1 \over c^2} V_A^a X_A^a +
 {\cal O}(4)\; ,$$ so that by expanding $f_A[T_A(t,{\bf X}_A), {\bf X}_A]$ in powers of the small time shift $({\bf V}_A.{\bf X}_A)/c^2 =
{\cal O}(2)$ we can express it within a sufficient accuracy in terms of functions computed on a [*local-time*]{} simultaneity surface, namely $T_A = {\rm const.} = T_A^0(t)$: $$\begin{aligned}
 f_A\left[T_A\left(t,{\bf X}_A\right),{\bf X}_A\right] &=&
 f_A\left(T_A^0(t),{\bf X}_A\right) \nonumber \\
 & & - {1 \over c^2} V_A^a X_A^a
 \partial_T f_A\left(T_A^0(t),{\bf X}_A\right) \nonumber \\
 & & + {\cal O}(4) \; .  \label{five}\end{aligned}$$

As for the three-dimensional Jacobian entering Eq. (\[four\]) it is easy to see that it can be expressed as $$\begin{aligned}
 \left|{\partial X \over \partial x} \right|^{(3)} &= &
 \left({\partial t \over \partial T_A}\right)_{X^a = const}
 \left|{\partial X \over \partial x} \right|^{(4)} \nonumber \\
 &=& \left[e^0_{A 0}(T_A^0) + {1 \over c^2} 