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 $R-\pi <T <\pi -R$.

not(by the authors' own admission as one can see from the figure's caption) intended to capture the entire Kerr solution in the same way as the conformal diagrams of the R-N solution. What is being shown there, and in the corresponding figure in Reall's notes, is just a conformal diagram for the axis of symmetry of Kerr, which happens to be a trulytwo-dimensional, totally geodesic, submanifold! $\endgroup$8more comments