Penrose conformal diagram of Morris-Thorne wormhole Consider the classical Morris-Thorne wormhole solution:
$$\tag{1}
ds^2 = dt^2 - dr^2 - (r^2 + a^2) \,d\Omega^2,
$$
where $a$ is a positive constant, $r > 0$ for one asymptoticaly flat spacetime, and $r < 0$ for a second one. This metric is an exact solution to Einstein’s equation with an exotic (negative) energy density and describes a simple static wormhole (the parameter $a$ defines the throat size). I would like to draw its conformal Penrose diagram, but my early attemps in finding proper coordinates failed. So the question is simple:
How can we build the Penrose diagram of the spacetime described by the metric (1)?  What are the coordinates transformations that brings this metric in a proper conformal form, so we could draw the Penrose diagram?
 A: Making a Penrose conformal diagram of a Morris-Thorne wormhole is challenging since the fundamental properties of such a spacetime can't really be represented in two dimensions (there isn't any two dimensional static wormhole due to topological issues), but Visser does offer an example of what it would look like :

This is pretty much just two Minkowski space diagrams, with the vertical dotted line representing the wormhole throat. The process for this is, following the standard construction of a Penrose diagram, to consider the spacetime in null radial coordinates :
\begin{eqnarray}
u, v &=& \frac{1}{\sqrt{2}} (t \pm r)\\
t, r &=& \frac{1}{\sqrt{2}} (u \pm v)
\end{eqnarray}
so that our metric is
\begin{equation}
ds^2 = -2 dudv - r^2(u, v) \,d\Omega^2,
\end{equation}
with $r(u, v) = \sqrt{(\frac{1}{2} (u - v)^2 + a^2)}$. From the spherical symmetry of the spacetime, we can simply drop the entire angular part of the metric to obtain the projective Penrose metric :
\begin{equation}
ds^2 = -2 dudv
\end{equation}
This is identical to the Minkowski case, except for the fact that Minkowski space had $r \geq 0$, so that $u - v \geq 0$, while here we have $r$ spanning $\mathbb{R}$, hence the doubling. We can then apply the same process as Minkowski space to obtain this diagram, although in the process we are losing the angular informations that tell us of the local properties of the wormhole, but we still have the global informations : on each side of the throat, there is an independent universe with its own conformal infinities.
