这是一份liverpool利物浦大学PHYS374的成功案例

Set
$$
f \equiv\left(e_{0}+e_{1}\right) R^{2-n}
$$
we have the equality:
$$
\frac{d}{d t} f=-n R^{-1} \partial_{t} R e_{0}+R^{2-n} \mathcal{R} .
$$
Therefore, since we have supposed $\partial_{t} R \geq 0$ :
$$
\frac{d}{d t} f \leq R^{2-n} \mathcal{R} \leq R^{2-n}|\mathcal{R}| .
$$
We now estimate $|\mathcal{R}|$.

PHYS374 COURSE NOTES ：

$$
\begin{aligned}
\theta^{0} &=d t \
\theta^{i} &=d x^{i}+\beta^{i} d t
\end{aligned}
$$
with $t \in \mathbb{R}$ and $x^{i}, i=1,2,3$ local coordinates on $M$. The Pfaff or convective derivatives $\partial_{\alpha}$ with respect to $\theta^{\alpha}$ are
$$
\begin{aligned}
&\partial_{0} \equiv \frac{\partial}{\partial t}-\beta^{i} \partial_{i} \
&\partial_{i} \equiv \frac{\partial}{\partial x^{i}}
\end{aligned}
$$
In this coframe, the metric $g$ reads
$$
d s^{2}=g_{\alpha \beta} \theta^{\alpha} \theta^{\beta} \equiv-N^{2}\left(\theta^{0}\right)^{2}+g_{i j} \theta^{i} \theta^{j}
$$