(paper) Stability properties of Kerr BH's

[malawi_glenn]

TL;DR

Stability properties of Kerr BH's

Have these articles been discussed here previously? I could not find it but my search skills suck.

Kerr stability for small angular momentum​

https://arxiv.org/abs/2104.11857 (just 800 pages)

This is our main paper in a series in which we prove the full, unconditional, nonlinear stability of the Kerr family Kerr(a,m) for small angular momentum, i.e. |a|/m≪1, in the context of asymptotically flat solutions of the Einstein vacuum equations (EVE). Three papers in the series, \cite{KS-GCM1} and \cite{KS-GCM2} and \cite{GKS1} have already been released. We expect that the remaining ones \cite{GKS2}, \cite{KS:Kerr-B} and \cite{Shen} will appear shortly. Our work extends the strategy developed in \cite{KS}, in which only axial polarized perturbations of Schwarzschild were treated, by developing new geometric and analytic ideas on how to deal with with general perturbations of Kerr. We note that the restriction to small angular momentum appears only in connection to Morawetz type estimates in \cite{GKS2} and \cite{KS:Kerr-B}

Recently (31th May this year)

Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes​

https://arxiv.org/abs/2205.14808 (just 900 pages)

This is the last part of our proof of the nonlinear stability of the Kerr family for small angular momentum, i.e |a|/m≪1, in which we deal with the nonlinear wave type estimates needed to complete the project. More precisely we provide complete proofs for Theorems M1 and M2 as well the curvature estimates of Theorem M8, which were stated without proof in sections 3.7.1 and 9.4.7 of \cite{KS:Kerr}. Our procedure is based on a new general interest formalism (detailed in Part I of this work), which extends the one used in the stability of Minkowski space. Together with \cite{KS:Kerr} and the GCM papers \cite{KS-GCM1}, \cite{KS-GCM2}, \cite{Shen}, this work completes proof of the Main Theorem stated in Section 3.4 of \cite{KS:Kerr}.

Worth reading?

 

[martinbn]

I don't think so. I don't think that the older, shorter, and easier work on the stability of Minkowski has been discussed either.

 

[malawi_glenn]

I don't think so

Referreing to "discussed earlier" or "worth reading" or both?

 

[martinbn]

Referreing to "discussed earlier" or "worth reading" or both?

discussed earlier

 

[PeterDonis]

Worth reading?

That would depend on what you were interested in. I don't think these results are surprising to anyone, since I don't think anyone expected that there would be an issue with treating Kerr black holes with very small angular momentum as perturbations of Schwarzschild black holes. But the details might be of interest if you're interested in those kinds of details.

 

[PAllen]

Do these papers say anything about stability of the interior? It is often argued that the CTC region inside a Kerr BH is an unstable artifact of perfect symmetry, but I have never actually seen a thorough argument for this position.

 

[PeterDonis]

Do these papers say anything about stability of the interior?

The "Kerr stability conjecture" stated in these papers, which is basically what they're attempting to prove as a theorem, refers to future null infinity and the "domain of outer communication", which indicates to me that the stability conjecture is only asserted regarding the exterior region (or possibly that plus some portion of the region between the outer and inner horizons).

I'm not sure how one would go about proving anything regarding stability of the Kerr deep interior, i.e., the region inside the inner horizon, since the inner horizon is a Cauchy horizon.

 

[martinbn]

That would depend on what you were interested in. I don't think these results are surprising to anyone, since I don't think anyone expected that there would be an issue with treating Kerr black holes with very small angular momentum as perturbations of Schwarzschild black holes. But the details might be of interest if you're interested in those kinds of details.

I don't understand. How does this make to conjecture (now a theorem) more believable?

Do these papers say anything about stability of the interior? It is often argued that the CTC region inside a Kerr BH is an unstable artifact of perfect symmetry, but I have never actually seen a thorough argument for this position.

I think it is expected that it is unstable in the sense that a small perturbation of the initial data would lead to a solution that is qualitatively different. The singularity should be more like a Schwarzschild's.

 

[PeterDonis]

The singularity

Is not even covered by this theorem, as far as I can tell, for the reason I gave in post #7.

 

[PeterDonis]

How does this make to conjecture (now a theorem) more believable?

If nobody expected anything different to begin with, why wouldn't the conjecture be believable?

If you are asking why nobody expected there to be an issue in the first place, remember that, as I said in post #7, we are (as far as I can tell) only talking about the exterior region (outside the horizon). I think most GR textbooks already discuss approximations in which the Kerr exterior with small angular momentum is treated as a perturbation of Schwarzschild. (I'm pretty sure I saw this first in MTW; I'll see if I can find it there.) This theorem basically just replaces "approximation" with "exact theorem".

 

[PeterDonis]

I think it is expected that it is unstable in the sense that a small perturbation of the initial data would lead to a solution that is qualitatively different. The singularity should be more like a Schwarzschild's.

I think this is correct in that most physicists in the field expect that the Kerr inner horizon is unstable because of the "infinite blueshift" effect there, and that a more realistic interior would have a spacelike singularity instead of the Kerr inner horizon. But, as I have said, I don't think this particular theorem touches on that, since I don't think it is meant to apply to the Kerr interior.

 

[malawi_glenn]

Popular description https://www.quantamagazine.org/black-holes-finally-proven-mathematically-stable-20220804/

 

[BWV]

Popular description https://www.quantamagazine.org/black-holes-finally-proven-mathematically-stable-20220804/

So were there some new mathematical techniques developed, if so can anyone give some general color?

In a 912-page paper posted online on May 30, Szeftel, Elena Giorgi of Columbia University and Sergiu Klainerman of Princeton University have proved that slowly rotating Kerr black holes are indeed stable. The work is the product of a multiyear effort. The entire proof — consisting of the new work, an 800-page paper by Klainerman and Szeftel from 2021, plus three background papers that established various mathematical tools — totals roughly 2,100 pages in all.