I'm looking for a book/paper where the conformal compactification of Kerr spacetime is calculated. I've seen plenty of reference for the Minkowski, but none (explicitly calculated) for Kerr.


Thank you.


PS: prior to writing this post, I was already referred to "Large scale structure of spacetime" by Hawking and Ellis, but there they only discuss the maximal extension of Kerr (and other solutions), not conformal compactification calculation.


EDIT on what I meant by "compactification for Minkowski": The Minkowski metric is


$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\Omega^{2}$


where $-\infty<t<\infty$ and $0\leq r<\infty$. (We now want to get the coordinates with finite ranges)


First go to the double-null coordinates $u=t-r$ and $v=t+r$, and then change to $T=\arctan u+\arctan v$, $R=\arctan v-\arctan u$. We then obtain


$ds^{2}=\omega^{-2}(T,R)(-dT^{2}+dR^{2}+\sin^{2}Rd\Omega^{2})$


with $\omega(T,R)=2\cos U \cos V=\cos T+\cos R$.


Therefore, the original metric is related to the new one by (explicitly given) conformal transf. $\omega^{2}$ as


$d\tilde{s}^{2}=\omega^{2}(T,R)ds^{2}=-dT^{2}+dR^{2}+\sin^{2}R d\Omega^{2}$


where $R,T$ have the *finite* ranges $0\leq R<\pi$ and $R-\pi <T <\pi -R$.



Conformal compactification of Kerr spacetime
See Section 8 of Pretorius and Israel, "Quasi-spherical light cones of the Kerr geometry". <http://arxiv.org/abs/gr-qc/9803080> The paper contains quite a bit more of course. But I want to point out that in the expression you find there, $R^2$ is fairly reasonable as a quantity in the usual (say, Boyer-Lindquist) coordinates. But the function $r\_\*$ is quite difficult to get a handle on. 


