local coordinates the Mass Conservation Law has the form $$\left(\frac{G^{IB}\phi
^{0}_{,B}}{\Vert d\phi ^{0}\Vert }\rho _{0}\sqrt{\vert G\vert
}\right) _{,I}=0.$$

Due to the properties of the metric $G$ and the deformation $\phi $, the material space-time manifold $P$ is foliated by the phase curves of the flow vector field $u_{\phi }$ and thus the value of the reference mass density $\rho _{0}(0,X)$ at $T=0$ uniquely defines its values for all $T>0$.

In the synchronized case $\phi ^{0}=T, \Vert d\phi ^{0}\Vert
=\sqrt{G^{00}}=S^{-1},\ G^{M0}=-\frac{N^{M}}{S^{2}},\
G^{00}=S^{-2}$ and (3.2) take the form of the following balance law $$(\rho _{0}\sqrt{\vert g\vert })_{,0}=\sum
_{A=1}^{3}\left(N^{A}\rho _{0}\sqrt{\vert g\vert }\right) _{,A}.$$ From (3.3) we note that the shift vector field $\vec{N}$ can describe the [**matter (density) flow due to the some internal processes**]{} such as the phase or chemical transformations.

If, in addition to being synchronized, the material metric $G$ is also in the BD-form ($\vec{N}=0$), the flow term in (3.3) disappears and the mass conservation law is equivalent to the following representation of the reference mass density in terms of its initial value $\rho _{0}(0,X)=\rho _{0}$: $$\rho _{0}(T,X^{I})=\rho _{0}(0,X)\sqrt{\frac{G_{00}}{\vert
G\vert}}=\frac{\rho _{0}(0,X)}{\sqrt{\vert g(T,X)\vert }},$$ where $G(0,X)$ is assumed to be Euclidean metric. If metric $G$ does not changes with time $T$, 