Rosenau solutions or Rosenau Metrics in Ricci flow

Question

In the book “Hamilton Ricci flow" written by Bennet Chow, Peng Lu and Lei Ni. The Rosenau solution is $$g = u \cdot h = \frac{\sinh(-t)}{cosh(x) + cosh(t)} (dx^2 + d \theta^2), x \in \mathbb{R}, \theta \in \mathbb{S^{1}(2)}.$$ But in another book “The Ricci flow in Riemannian Geometry” written by Hopper and Andrews, the metric is defined by $$g_{\alpha} = \frac{1}{1 - \alpha^2 x^2} \bar{g},$$where $\bar{g}$ is the standard metric on $\mathbb{S^{2}}$ and $\alpha = \alpha(k)$. and there is a note:"Up to a rescaling by a factor depending on $k$, these are exactly the metrics arising in an important explicit solution of the Ricci flow on $\mathbb{S^{2}}$" known as the Rosenau solution. But I don't understand how I can get a Rosenau solution from $g_{\alpha}$? Is there any references or some points I ignored?