G^{-1}(C_{4}(\phi )-\gamma))\Pi =^{s}\frac{1}{2}\left(\begin{array}{cc} 0 & 0\\
g^{IK}\langle \phi ^{.}_{,K},\phi ^{.}_{,0}\rangle_{h}-N^{I}\
&g^{IK}C_{3}(\phi )_{KJ}-\delta ^{I}_{J}
\end{array}\right) ,$$ Here $\Pi $ is the projector on the slices $B_{\phi ,t}$ defined in (4.3). The last equality is valid in the synchronized case. Notice that the basic invariants $Tr(A^k )$ for the tensor (4.6) are the same as for the tensor $C_{3}(\phi )-g$.

For the simplicity we use the same symbol $E^{el}$ for the restriction of this tensor to the slices $B_{\phi ,t}$.

Tensor $E(\phi)^{el}$ is a measure of the deviation of the Cauchy metric $C_{3}(\phi )$ of the actual state from the “ground state” $G$. For a synchronized deformation $\phi $ and a material metric $G$ with the zero shift vector, $E^{el}=\frac{1}{2}(C(\phi )-g_{t})$ has the form of the conventional elastic strain tensor.

The deformation $\phi $ is [**essentially 3-dimensional**]{} in the sense that only the spatial Euclidean metric $g_{0}=h$ in $B$ is deformed. The 4D-tensor $C_{4}(\phi )=\phi ^{*}h$ defines the degenerate metric in the material space-time $P$. It is instructive to compare $C_{4}(\phi )$ with the (degenerate) tensor $\gamma
=G-u_{\phi }\otimes u_{\phi }$. The elastic strain tensor $E^{el}$ measures the deviation of $C_{4}(\phi )$ from $\gamma $ on the slices $B_{\phi ,t}$. Thus, the 