# Penrose diagram of Janus black holes

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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
asked Feb 15, 2019

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