by $\vec{N}$ and the inhomogeneity of the local time $d\tau = SdT$, are the main reason for introduction (3+1) ADT-representation of 4D-metric in Gravity Theory ([@MTW]). In addition, an adoption of this view leads to a very clear separation of the “physical” degrees of freedom in the canonical formalism and to the explicit hyperbolic formulations of Einstein Equations ([@ACY; @FM]).

Mass conservation law
=====================

The [**mass form**]{} $dM=\rho _{0}dV$ defined in P is introduced here, in addition to the volume form $dV_{G}$ of metric $G$. The reference mass density $\rho _{0}$, defined by this representation, satisfies the [**mass conservation law**]{} ([@MH],[@CQ]) $${\mathcal L}_{u_{\phi }}dM=d(i_{u_{\phi }}dM)=0.$$ Here ${\mathcal L}_{u_{\phi }}$ is the Lie (substantial) derivative of the exterior 4-form $dM$ in the direction of the vector field $u_{\phi}$. Recall that the Lie derivative of a differential form $\omega $ along a vector filed $u$ is defined as ${\mathcal L}_{u}\omega =\frac{d}{dt}\phi
^{*}_{t}\omega \vert _{t=0}, $ where $\phi ^{*}_{t}\omega $ is the pullback of the form $\omega $ by the flow $t \longrightarrow \phi
_{t}=exp(tu)$ of the vector field $u$ ([@MH]).

Equation (3.1) is equivalent to the condition $div_{G}(\rho
_{0}u_{\phi })=0$, where divergence is taken with respect to the volume form $dV$. In 