A^a_A X^a\right]
 \left|{\partial X \over \partial x} \right|^{(4)} + {\cal O}(4)
 \; .\label{six}\end{aligned}$$ where $|\partial X / \partial x|^{(4)} = {\rm det}(\partial X^\alpha
/ \partial x^\mu)$ is the full four-dimensional Jacobian associated with the coordinate transformation (\[three\]). The time derivative $(\partial t/\partial T_A)$ in Eq. (\[six\]) is obtained from Eq. (\[three\]) or from putting $V_S = 0$ in expression (A6) in Appendix A of [@dam4].

Finally, we get $$\begin{aligned}
 {\cal I}_A(t) &=& \int_A d^3X_A\, \left[ \left|{\partial X
 \over \partial x}\right|^{(4)}\right]^{-1}\! \left[e^0_{A 0}\left(
 T_A^0 \right) + {1 \over c^2} A_A^a X_A^a\right]^{-1} \nonumber \\
 & &\quad \times \left[f_A\left(T_A^0,{\bf X}_A\right) - {1 \over
 c^2} V_A^a X_A^a \partial_T f_A\left(T_A^0,{\bf X}_A\right)\right]
 \; , \nonumber \\ & & \label{seven}\end{aligned}$$ where it is convenient to leave unexplicated the four-dimensional Jacobian because it will be directly cancelled when using the transformation laws of the mass and mass current densities $\sigma^\mu$ entering $f_A(t,{\bf x})$. For completeness, let us however mention its value $$\left| {\partial X \over \partial x} \right|^{(4)} = 1 -
 {2 \over c^2} W''(T,X) + {\cal O}(4) \; , \label{eight}$$ where $W''(T,X) = G''_A - A_A^a X^a + {\cal O}(2)$ is the inertial contribution to the local potential due to the change of the time scale and 