the time-dependent lapse function $N$ and the shift vector field $\vec{N}$ tangent to the slices $B_{\phi ,t}$ such that the metric is block-diagonalized in the moving coframe $(dT,dX^{A}+N^{A}dT)$: $$ds^{2}=G_{IJ}dX^{I}dX^{J}=g_{IJ}(dX^{I}+N^{I}dT)(dX^{J}+N^{J}dT)+S^{2}dT^{2}.$$

Matrix representation of the material metric tensor $G$ and inverse tensor $G^{-1}$ have in these notations, the forms $$\left(\begin{array}{cc} G_{00} & G_{0J}\\
G_{I0} & G_{IJ}\end{array}\right) =\left(\begin{array}{cc} N_{A}N^{A}+S^{2} & N_{J}\\
N_{I} & g_{IJ} \end{array}\right) ,\ \ \left(\begin{array}{cc} G^{00} & G^{0J}\\
G^{I0} & G^{IJ}\end{array}\right) =\left(\begin{array}{cc} \frac{1}{S^{2}} & -\frac{N^{J}}{S^{2}}\\
-\frac{N^{I}}{S^{2}} & g^{IJ}+\frac{N^{I}N^{J}}{S^{2}}
\end{array}\right) ,$$ where $g$ is the 3D-metric induced by $G$ on the slices $B_{\phi ,t}$ and $g^{-1}$ is the corresponding inverse tensor. In these notations $
\sqrt{\vert G\vert }=S\sqrt{\vert g\vert}.$

In what follows we assume that the 4D-deformation history $\phi $ is synchronized. Thus slices $B_{\phi ,t}$ has the form $T=t=const$.

In these notations, the flow vector $u_{G}$ and the corresponding 1-form have the form $$u_{G}  =  \frac{1}{S}\partial _{T}-\frac{N^{A}}{S}\partial
 _{X^{A}},\ \ u_{G}^{\flat} = SdT.$$

The last formula gives the “material time differential” $d\tau =SdT$ for the material metric $G$. In our context, the coordinate X dependence of lapse function S(T,X) accounts for heterogeneity of material aging in different points of the solid. On the Fig. 3 the local observer at different points 