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

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  • $\begingroup$ What do you mean by "calculate the conformal compactification"? Let $(M,g)$ be any asymptotically flat space time and let $(M,\Omega^2 g)$ be a conformal compactification, then for any function $f:M\to \mathbb{R}$ that is bounded both above and below, $(M,f\Omega^2 g)$ is another conformal compactification. Can you edit to tell us what you think is the answer for Minkowski, and what sort of information you want to get from such a computation? $\endgroup$
    – Willie Wong
    Commented Oct 13, 2015 at 5:44
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    $\begingroup$ Hmm, I guess the literature is really encouraging you to go through the exercise yourself. :-) Still doesn't have the explicit answer, but these notes on Black Holes by Harvey Reall contain some more detail about how the Reissner-Nordstrom case works. $\endgroup$
    – Igor Khavkine
    Commented Oct 13, 2015 at 13:41
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    $\begingroup$ Just checking in here to point out a small fact which might shed some light on the matter. The diagrams in Figure 28 in Sec 5.6 of Hawking and Ellis' book mentioned by @IgorKhavkine are 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 truly two-dimensional, totally geodesic, submanifold! $\endgroup$
    – Umberto Lupo
    Commented Oct 15, 2015 at 18:46
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    $\begingroup$ This is quite different from the situation in Reissner-Nordström where, thanks to the spherical symmetry of the solution, one can rather easily capture the entire (four-dimensional) spacetime by means of a Penrose diagram in which every point represents a two-sphere. So one might wonder how much Figure 28 in Hawking and Ellis' book actually tells us about the global conformal structure of the Kerr spacetime. If you are somewhat flexible about what you'd like your (conformal) diagram of Kerr to do then consider consulting Section 3.7 in Piotr Chruściel's notes... $\endgroup$
    – Umberto Lupo
    Commented Oct 15, 2015 at 19:04
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    $\begingroup$ ... where "projection diagrams" are defined and discussed (see also this paper). The remarks towards the end of page 146 there are particularly relevant $\endgroup$
    – Umberto Lupo
    Commented Oct 15, 2015 at 19:08

2 Answers 2

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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.

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Although the focus of the original question was on conformal compactification, a necessary step along the way is an introduction of double-null coordinates that are regular on the horizons and bifurcation spheres of Kerr (where the standard Boyer-Lindquist coordinates become singular). An alternative set of such coordinates, which are obtained in a briefer and more elementary way than the classic Pretorius-Israel references cited in Willie's answer can be found in

Hayward, Sean A., Kerr black holes in horizon-generating form, Phys. Rev. Lett. 92, No. 19, Article ID191101, 4 p. (2004). [arXiv:gr-qc/0401111] ZBL1267.83059.

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