Penrose diagram of Janus black holes

Question

Recently I'm interested in the Janus black holes, it's a solution of three dimensional Einstein-scalar action which can be embedded in ten dimensional type IIB supergravity with appropriate ansatz. And its metric is

$$ds^2=f(\mu){\rm cos}^2\mu ds_{BTZ}^2=f(\mu)(-d\tau^2+d\mu^2+r_0^2{\rm cos}^2\tau d\theta^2),$$

so it's time dependent, and function $f(\mu)$ is consisted of some Jacobi elliptic functions which have similar shape as $1/{\rm cos}^2\mu$ but with longer period than $\pi/2$, and its spatial asymptotic infinity is $\pm\mu_0>\pi/2$. So the penrose diagram is elongated horizontally:

But it confuses me that the coordinate transformation is unchanged as the old BTZ black hole ($t$,$r\rightarrow t,r^*\rightarrow U,V\rightarrow\mu,\tau$), so the original time $t$ and space $r$ can't reach $\pm\mu_0$. So what I want to ask is that how can we draw the penrose diagram above if there is no coordinate transformation to cover the whole region.

The original paper is https://arxiv.org/abs/hep-th/0701108, thank you for any help.

This post imported from StackExchange Physics at 2019-04-13 07:44 (UTC), posted by SE-user Jiahui Bao