Maximal analytic extension of Robinson-Bertotti geometry
I want to find a maximal analytic extension of the Robinson-Bertotti metric given by $$ds^2 = - \lambda^2 dt^2 +Q^2 \frac{d\lambda^2}{\lambda^2}+Q^2d\Omega_{S^2}^2.$$ Where $Q$ is a constant charge and $d\Omega_{S^2}^2$ is the line element of the $2$-sphere. Here is what I did so far. I state the coordinate transformations and try to keep track of the ranges of the coordinates.
1. 394
Coordinate transformation $$u=t+\frac{Q}{\lambda} ~~~\Rightarrow u \in (-\infty,\infty]$$ $$v=t-\frac{Q}{\lambda} ~~~\Rightarrow v \in [-\infty,\infty)$$ This transforms the line element to $$ds^2=-\Big(\frac{2Q}{u-v}\Big)^2 du dv+Q^2 d\Omega_{S^2}^2.$$ 2. Coordinate transformation $$u=\tan\Big(\frac{U}{2}\Big)~~~\Rightarrow U \in (-\pi,\pi]$$ $$v=-\cot\Big(\frac{V}{2}\Big)~~~\Rightarrow V \in (-\pi,\pi)$$ This transforms the line element to $$ds^2=-\frac{Q^2}{\cos\big(\frac{U-V}{2}\big)}dUdV + Q^2 d\Omega_{S^2}^2.$$ 3. Coordinate transformation $$T=\frac{1}{2}\Big(U+V\Big) ~~~\Rightarrow u \in (-\pi,\pi)$$ $$R=\frac{1}{2}\Big(U-V\Big) ~~~\Rightarrow u \in (-\pi,\pi)$$ This transforms the line element to $$ds^2=\Phi^2 \Big[-dT^2+dR^2+\cos^2(R) d\Omega_{S^2}^2\Big].$$ With the overall conformal factor $\Phi^2 = \Big(\frac{Q}{\cos(R)}\Big)^2.$
Now to my questions:
* Regarding the first transformation: Is it correct to include $\infty$ and $-\infty$ in the ranges of $u$ and $v$ respectively? I think so because of the singularity at $\lambda=0$ but I can't justify it mathematically.
* Regarding the second transformation: Do I have to take something into account because of the discontinuity of $\arctan(-\frac{1}{v})$ at $v=0$. Also I think I should be done here regarding finding the maximal analytic extension but I want to deduce the Carter Penrose diagram, that's why I did another transformation.
* Is it correct that the Carter-Penrose diagram of this metric looks exactly the same as that from the <PERSON> spacetime? If not why? It seems strange to me because this spacetime is the near horizon limit of the extremal Reissner-Nordström black hole, where I wouldn't expect the same causal structure as in flat space. 818