- metric), we have $u^{.}_{G}=[G_{00}]^{-\frac{1}{2}}\frac{\partial }{\partial T}.$

Let $u_{0}=\frac{\partial}{\partial T}$ be the flow vector associated with the metric $G_{0}$ and the corresponding 3D slicing $B_{0}$.

We require fulfillment of the following condition ensuring the irreversibility of the flow of time: $$<u_{\phi },u_{0}>_{G}\ >0.$$ Deformation history $\phi $ for which the condition (2.5) is satisfied is called [**admissible**]{}.

In coordinates $(X^{I})$ this condition reduces to the following simple inequality $$\phi ^{0}_{,0} >0$$ and, therefore is a restriction on the deformation history only.

For a synchronized deformation history $\phi $, this condition is trivially satisfied.

Time component $\phi ^{0}$ of the deformation history may be excluded from the list of dynamical variables by an appropriate “gauging”. Namely, we use the invariance of Lagrangian under the automorphisms of the bundle $(P,\pi ,B)$ to make the deformation history synchronized.

An automorphism $F: P \longmapsto P, X^{I}=F^{I}(Y^{A})$ of the bundle $(P,\pi ,B)$ determines the diffeomorphism of the base $B$ that can be considered as a change of variables $X^{I}=F^{I}(Y^{A})$.

In the new variables, the condition (2.6) takes the form $\frac{\partial \phi ^{0}}{\partial Y^{0}}=\phi
^{0}_{,I}F^{I}_{,0}=\phi ^{0}_{,0}F^{0}_{,0}>0$ (since $F^{I},\
I=1,2,3$ do not depend on $Y^{0}$). Thus, the class of admissible deformation histories $\phi $ is stable under 