I am trying to make sense of the Penrose diagram of a non extremal Reissner-Nordström spacetime, that is, the solution with two horizons. The coordinates are $$ v'=\text{exp}\left(\frac{r_+-r_-}{2r_+^2}v\right) \hspace{3cm} \tag 1$$
$$ w'=-\text{exp}\left(\frac{r_--r_+}{2r_+^2}w\right) \tag 2$$ where $r_+,r_-$ are the outermost and innermost horizons respectively, and $v$ and $w$ are defined as $$v\equiv \bar{t}+r \hspace{3cm} \tag 3$$ $$ w\equiv 2t-v \tag 4$$ and $\bar{t}$ is defined as $${\bar{t}=t+\frac{r_+^2}{r_+-r_-}\log{|r-r_+|}-\frac{r_-^2}{r_+-r_-}\log{|r-r_-|} }\tag 5$$ My source (Ray d'Inverno 's Introducing Einstein's Relativity) claims that $r$ is defined implicitly by the relation $$v'w'=-\text{exp}\left(\frac{r_+-r_-}{2r_+^2}r\right)(r-r_+)(r-r_-)^{-\frac{r_-^2}{r_+^2}}\tag 6$$ (although I wasn't able to verify this). My question is how to extract information from these coordinates? I get that the lines of constant $t$ and $r$ must be hyperbolas, but how do I see that the three regions ($r\in (0,r_-)$, $r\in (r_-,r_+)$, $r\in (r_+,\infty)$) repeat themselves infinitely in the diagram, and, further, how do i see that the horizons are represented by 45º lines? Using the relation above, the horizons are both characterized by $v'w'=0$. But neither $v',w'$ is anywhere zero, so I can't quite grasp what's going on.
