time. The $\partial\over {\partial t}$ is a directional derivative along each of the classical spacetimes which can be viewed as classical ‘trajectories’ in the gravitational configuration space. In view of (11)-(13), we identify the variable $t$ as the classical time parameter. Using (31), we now write $${{\partial S} \over {\partial t}}=-{1\over {Ma}}{{\partial S_o}\over {\partial a}}
{{\partial S}\over {\partial a}}\,,$$ where $S$ is given in (9). Writing $S(a,\phi)=S_o(a)+S_1(a,\phi)$, we write (32) as $${{\partial S} \over {\partial t}}=-{1\over {Ma}}({{\partial S}\over {\partial a}})^2
+{1\over {Ma}}{{\partial S_1}\over {\partial a}}{{\partial S}\over {\partial a}}\,.$$ We now substitute $({{\partial S}\over {\partial a}})^2$ from (9) to get $${{\partial S} \over {\partial t}}=-{1\over {a^3}}({{\partial S}\over {\partial \phi}})^2
+Mka-2a^3V(\phi)
+{1\over {Ma}}{{\partial S_1}\over {\partial a}}{{\partial S}\over {\partial a}}\,.$$ In the region of interest i.e., when $V(\phi)\approx V_0$, $P_\phi^0=0$, we write $P_a\approx P_a^0$ and using (13) we find $$\frac {1}{Ma}(P_a^0)^2=2a[(-P_a^0)'+\frac {Mk}{2}-3a^2V_0].$$ Further neglecting $(\frac {\partial S_1}{\partial a})^2 $ term we get $$\begin{aligned}
\frac {1}{Ma}\frac {\partial S_1}{\partial a}\frac {\partial S}{\partial a}&\simeq&
\frac {1}{Ma}\frac {\partial S_1}{\partial a}\frac {\partial S_0}{\partial a}\nonumber\\&=&
\frac {1}{Ma}\frac {\partial S_0}{\partial a}\frac {\partial S}{\partial a} -
\frac {1}{Ma}(\frac {\partial S_0}{\partial a})^2 \nonumber \\&=&
\frac {1}{Ma}\frac {\partial S_0}{\partial a}\frac {\partial S}{\partial a}-
\frac {1}{Ma}(P_a^0)^2.\end{aligned}$$ Using (32) to replace the first term in 