-\frac{\partial f^r}{\partial W}W'\theta^r\right]\right)\end{aligned}$$

Variation of the effective action with respect to $N$, $N^r$, $V$, $W$ yields the Einstein equations for the model with additional terms resulting from the gauge-fixing and ghost parts of the action: $$\label{g.Ein.eqs}
R_{\mu}^{\nu}-\frac12\delta_{\mu}^{\nu}R=
 \kappa\left(T_{\mu(mat)}^{\nu}+T_{\mu(obs)}^{\nu}+T_{\mu(ghost)}^{\nu}\right),$$ where $T_{\mu(mat)}^{\nu}$ stands for the energy-momentum tensor of matter fields which can be included into the model, $T_{\mu(obs)}^{\nu}$ and $T_{\mu(ghost)}^{\nu}$ are obtained by varying the gauge-fixing and ghost action, respectively, so that $T_{\mu(obs)}^{\nu}$ describes the observer in a reference frame. It is clear that $T_{\mu(obs)}^{\nu}$ and $T_{\mu(ghost)}^{\nu}$ are not true tensors since they depend on chosen gauge conditions. Equations (\[g.Ein.eqs\]) can be called [*the gauged Einstein equations*]{}. By adding ghost equations and gauge conditions (\[d.c.1\]), (\[d.c.2\]) to the gauged Einstein equations, we obtain [*the extended set of Lagrangian equations*]{} for our model which is presented in Appendix A. The explicit expressions of $T_{\mu(obs)}^{\nu}$, $T_{\mu(ghost)}^{\nu}$ are given in Appendix C. Below $\kappa=8\pi$ (in special units when $G=c=\hbar=1$).

Now we can find the momenta conjugate to all gravitational and ghost variables: $$\begin{aligned}
\label{PN}
P_N&=&\pi_N;\\
\label{PNr}
P_{N^r}&=&\pi_{N^r};\\
\label{PV}
P_V&=&\frac{W\dot W}N-\frac{W'W N^r}N-\pi_N\frac{\partial f}{\partial V}
   -\pi_{N^r}\frac{\partial f^r}{\partial V};\\
\label{PW}
P_W&=&\frac{W\dot V}N+\frac{V\dot W}N-\frac{W'V N^r}N-\frac{W V'N^r}N-\frac{V W(N^r)'}N
   -\pi_N\frac{\partial f}{\partial W}-\pi_{N^r}\frac{\partial f^r}{\partial W};\\
\label{Pbt0}
P_{\bar\theta_0}&=&N'\theta^r-\frac{\partial f}{\partial V}V'\theta^r
   -\frac{\partial f}{\partial W}W'\theta^r
   