Problem

Solve equation (2.16) and obtain (2.22). If $E>0$, the expansion law is given by

$$t=\left(\frac{R^2}{2E}\right)^{1/2}\left(\frac{\dot{R}^2R}{2GM_r(R)}\right)^{-1} \left(\frac{\sinh 2\eta}{2}-\eta\right)$$ (2.23)

and

$$x=\left(\frac{E}{GM_r(R)}\right)r=\sinh^2 \eta,$$ (2.24)

where $E$ represents the total energy

$$E= \frac{\dot{R}^2}{2}-\frac{GM_r(R)}{R}>0.$$ (2.25)