# Kerr solution for finite collapse time

The Kerr black hole solutions gives an analytic continuation that is asymptotically flat. Some people have argued that this is another universe, but others state that the analytic continuation represent the same space-time region at different times.

The Kerr solution is an exact, analytic solution for an eternal black hole where formation is infinitely in the past.

Is the Kerr solution amenable to perturbative analysis to include the collapse itself in the solution for finite times? How does the analytic continuation looks when considering a non-eternal solution? If there exists one and it is understood well enough, How does it map to the external space-time?

The maximal analytic continuation of the Kerr metric extends the $r$ coordinate from the $r=0$ "ellipsoid" to negative values of $r$ which involves literally making another copy of your coordinates and switching from one to another when you pass through the disk $r=0.$

But there is another sense where there is another universe for the Kerr solution. As long as there isn't too much angular momentum per unit mass there are two horizons at $r_+$ and $r_-.$ If you start out bigger than $r_+$ but go across the surface $r=r_+$ then it is an event horizon in the sense that there is then no way to avoid reaching the surface $r=r_-$ however once you reach that surface you have a choice you can head back towards larger $r$ or keep decreasing your $r$. If you increase your $r$ at this point you can even increase it to be larger than $r=r_+.$ But if you do so you end up in a different external universe.

Lets call the region of spacetime where we originally were when you were first in the region of $r>r_+$ the name Kansas (Hawking and Ellis called it region I/1). The most persuasive reason that you aren't in Kansas again is that first you left Kansas and then before you reached the surface $r=r_-$ absolutely everything in Kansas had its chance to affect you no matter how late in time it happened in Kansas. So if you returned to Kansas it would be after things happening arbitrarily late already affected you.

But when you reach the surface $r=r_-$ you also see a singularity and if you actually cross $r=r_-$ then you are in a region were Hawking and Ellis say that every point has a closed timelike curve (CTC) and maybe even touching the $r=r_-$ surface allows CTCs. So maybe the argument that they are different universes isn't very convincing. Now everyone always cites Hawking and Ellis as a place to get details about these closed timelike curves and while I see them mention the CTCs on page 164 I don't see any details and in particular I don't know how to read CTCs from a conformal diagram and I don't know if those CTCs require going through the $r=0$ disk, but it looks possible to go through the disk and back out again.

It also looks like you can go from Kansas ($r>r_+$) to the region with $r_-<r<r_+$ then at least touch the surface $r=r_-$ then travel through a region that is directly affected by two singularities until you finally emerge into a region that is directly affected by only one singularity, another Kansas.

Now if you look at the conformal diagram in Hawking and Ellis the entire negative $r$ section is simple with no horizons of any kind at $r=-r_-$ or $r=-r_+$ so maybe I'm reading it badly. In the picture link I give below it doesn't really show anything interesting beyond the disk $r=0$ but even in Hawking and Ellis the picture just looks like the conformal diagram of Minkowski space.

All that was to bring these two different types of alternate universes into play. After you touch the surface $r=r_-$ then at least one singularity is visible and also regions of negative $r$ beyond the disk that have the singularity on the edge of the disk. And if you climb back to regions of $r_-<r<r_+$ (a region where you are now directly affected by two separate ring singularities and two totally different regions of negative $r$) then eventually you end up back in a region of $r>r_+$ a region that most say is not your original Kansas. But I want to point out that there were already two regions of $r_-<r<r_+$ one where you have to have $r$ decrease (the first one you went through where there were two different ring singularities you have to avoid hitting each connected to different regions of $r<0$ but by avoiding both you can also avoid going into regions if negative $r$ as long as you fully go down to $r=r_-$). Then after you touch or pass through $r=r_-$ you can increase you $r$ and thus get back to $r>r_-$ but now when you enter this new region of $r_-<r<r_+$ and this time your $r$ can only increase until it reaches $r=r_+$. So when you enter one of the two regions with $r_-<r<r_+$ whether $r$ can increase or decrease depends on whether you were increasing or decreasing when you entered it.

OK so there are a lot of different regions. Multiple Kansases for each time you dunked yourself through the outer horizon. Two different regions (neither of which can affect each other) of negative $r$ for each time you pulled yourself outwards (bigger $r$) from a region of $r_-<r<r_+$ and each of them affecting you during that journey. And actually when you finally cross out of $r_-<r<r_+$ with increasing $r$ then you have two different (both new) Kansases to choose from each one only directly influenced by one of the singularities. See the rightwards image at http://en.m.wikipedia.org/wiki/Penrose_diagram#/media/File:PENROSE2.PNG

OK, now about collapse. I don't think we understand collapse for rotating bodies well. For instance in Kerr Spacetime http://arxiv.org/abs/0706.0622 Visser writes

note that although the causal pathologies [closed timelike curves] in the Kerr spacetime have their genesis in the maximally extended $r < 0$ region, the effects of these causal pathologies can reach out into part of the $r > 0$ region, in fact out to the inner horizon at $r = r_−$ [31] — so the inner horizon is also a chronology horizon for the maximally extended Kerr spacetime. Just what does go on deep inside a classical or semiclassical black hole formed in real astrophysical collapse is still being debated — see for instance the literature regarding “mass inflation” for some ideas [42]. For astrophysical purposes it is certainly safe to discard the $r < 0$ region, and almost all relativists would agree that it is safe to discard the entire region inside the inner horizon $r < r_-$

And it isn't just a popularity thing. Since there is an outer event horizon the things inside it don't have to affect anything. You can start with some slice of "now" that goes up to the event horizon and never have to worry about anything that crosses it. And if it hasn't collapsed to form that event horizon then those parts that get closer to it are time dilated so it is like the red hole where you never see it collapse.