processes (in comparison to sound wave velocity).

Using the deformation $\phi $, we introduce the slicing of the material space-time $P$ by the level surfaces of the zeroth component of $\phi $ $$B_{\phi }=\{B_{\phi ,t}=\phi ^{0\ -1}(t)\} =\{(T,X)\in P\vert \phi
^{0}(T,X)=t\}.$$

For a synchronized deformation $B_{\phi ,t}=\{(T,X)\in P\vert T=t\}.$

There is a [**time flow vector field**]{} $u_{\phi }$ in $P$, associated with the slicing $B_{\phi ,t}$ of the space-time $P$ ([@MTW], [@CQ]). This (future directed) vector field represent the flow of “intrinsic” (proper in Relativity Theory) time in the material. Lifting the index in the 1-form $d\phi ^{0}$ with the help of the metric $G$ and normalizing obtained vector field, we define the time flow vector field as $$u^{.}_{\phi }=\frac{(d\phi ^{0})^{\#}}{\Vert d\phi ^{0}\Vert
}_{G}.$$

The norm of the 1-form $d\phi ^{0}$ is defined as $\Vert d\phi
^{0}\Vert ^2=(G^{AB}\phi ^{0}_{,A}\phi ^{0}_{,B})^{1/2}$ (summation agreement by repeating indices is used here). Thus, $u_{\phi }$ is the unit vector $G$-orthogonal to the slices $B_{\phi ,t}$. In the local coordinates $X^I$, $$u^{.}_{\phi }=\frac{G^{AB}\phi ^{0}_{,B}}{\sqrt{G^{AB}\phi
^{0}_{,A}\phi ^{0}_{,B}}}\frac{\partial}{\partial X^{A}}.$$

For the synchronized deformations does not depend on $\phi $: $$u^{.}_{\phi
}=u^{.}_{G}=\frac{G^{I0}}{\sqrt{G^{00}}}\frac{\partial}{\partial
X^{I}}.$$ Additionally, if the metric $G$ has the block-diagonal form in the coordinates $(X^{0}=T,X^I)$ (shortly, BD 