$$
d s^{2}=-(1+2 \Phi) d t^{2}+(1-2 \Phi)\left(d x^{2}+d y^{2}+d z^{2}\right)
$$
where $\Phi$ is the Newtonian potential $(-1 \ll \Phi<0)$. Consider a nearly Newtonian perfect fluid [stress-energy tensor
$$
T^{\alpha \beta}=(\rho+p) u^{\alpha} u^{\beta}+p g^{\alpha \beta}, \quad p \ll \rho ;
$$
$$
v^{j} \equiv d x^{j} / d t \ll 1
$$
Show that the equations $T_{; \nu}^{\mu \nu}=0$ for this system reduce to the familiar Newtonian law of mass conservation, and the Newtonian equation of motion for a fluid in a gravitational field:
$$
\frac{d \rho}{d t}=-\rho \frac{\partial v^{j}}{\partial x^{j}}, \quad \rho \frac{d v^{j}}{d t}=-\rho \frac{\partial \Phi}{\partial x^{j}}-\frac{\partial p}{\partial x^{j}},
$$
7CCMMS38 COURSE NOTES ：
Demand that the intervals $a \mathscr{B}$ and $\mathscr{P} \mathscr{Q}$ be sufficiently small compared to the scale of curvature of spacetime; or specifically,
$$
R^{(A B)}(\mathscr{G B})^{2} \ll 1 / N
$$
and
$$
R^{(P Q)}(\mathscr{P} \mathcal{Q})^{2} \ll 1 / N,
$$
where $R^{(A B)}$ and $R^{(P Q)}$ are the largest relevant components of the curvature tensor in the two regions in question.
Demand that the time scale, $\tau_{0}$, of the geodesic clocks employed be small compared to $\mathscr{B} B$ and $\mathscr{P} \mathcal{Q}$ individually; thus,
$$
\begin{aligned}
&\tau_{0} \ll \mathscr{Q} \mathscr{B} / N, \
&\tau_{0} \ll \mathscr{P} \mathscr{Q} / N
\end{aligned}
$$