Brief introduction to the nonlinear stability of Kerr Sergiu Klainerman J er emie Szeftel October 31 2022

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Brief introduction to the nonlinear stability of Kerr

Sergiu Klainerman, J´er´emie Szeftel

October 31, 2022

Abstract

This a brief introduction to the sequence of works [63], [39], [61], [62] and [81]

which establish the nonlinear stability of Kerr black holes with small angular mo-

mentum. We are delighted to dedicate this article to Demetrios Christodoulou for

whom we both have great admiration. The ﬁrst author would also like to thank

Demetrios for the magic moments of friendship, discussions and collaboration he

enjoyed together with him.

1 Kerr stability conjecture

1.1 Kerr spacetime

Let (K(a, m),ga,m) denote the family of Kerr spacetimes depending on the parameters

m(mass) and a(with J=am angular momentum). In Boyer-Lindquist coordinates the

Kerr metric is given by

ga,m =−q2∆

Σ2(dt)2+Σ2(sin θ)2

|q|2dφ −2amr

Σ2dt2

+|q|2

∆(dr)2+|q|2(dθ)2,(1.1)

where (∆ = r2+a2−2mr, q =r+ia cos θ,

Σ2= (r2+a2)|q|2+ 2mra2(sin θ)2= (r2+a2)2−a2(sin θ)2∆.(1.2)

The asymptotically ﬂat1metrics ga,m verify the Einstein vacuum equations

Ric(g) = 0,(1.3)

1That is they approach the Minkowski metric for large r.

arXiv:2210.14400v2 [math.AP] 27 Oct 2022

are stationary and axially symmetric2, possess well-deﬁned event horizons r=r+(the

largest root of ∆(r) = 0), domain of outer communication r > r+and smooth future null

inﬁnity I+where r= +∞. The metric can be extended smoothly inside the black hole

region, see Figure 1. The boundary r=r−(the smallest root of ∆(r) = 0) inside the

black hole region is a Cauchy horizon across which predictability fails3.

Figure 1: Penrose diagram of Kerr for 0 <|a|< m. The surface r=r+, the larger root of ∆ = 0, is

the event horizon of the black hole, r > r+the domain of outer communication, I+is the future null

inﬁnity, corresponding to r= +∞.

Here are some of the most important properties of K(a, m):

•K(a, m) possesses a canonical family of null pairs, called principal null pairs, of the

form (λe4, λ−1e3), with λ > 0 an arbitrary scalar function, and

e4=r2+a2

|q|2∂t+∆

|q|2∂r+a

|q|2∂φ, e3=r2+a2

∆∂t−∂r+a

∆∂φ.(1.4)

•The horizontal structure, perpendicular to e3, e4, denoted H, is spanned by the

vectors

e1=1

|q|∂θ, e2=asin θ

|q|∂t+1

|q|sin θ∂φ.(1.5)

The distribution generated by His non-integrable for a6= 0.

2That is K(a, m) possess two Killing vectorﬁelds: the stationary vectorﬁeld T=∂t, which is time-like

in the asymptotic region, away from the horizon, and the axial symmetric Killing ﬁeld Z=∂φ.

3Inﬁnitely many smooth extensions are possible beyond the boundary.

•The horizontal structure (e3, e4,H) has the remarkable property that all components

of the Riemann curvature tensor R, decomposed relative to them, vanish with the

exception of those which can be deduced from R(ea, e3, eb, e4).

•K(a, m) possesses the Killing vectorﬁelds T,Zwhich, in BL coordinates, are given

by T=∂t,Z=∂φ.

•In addition to the symmetries generated by T,Z,K(a, m) possesses also a non-trivial

Killing tensor4, i.e. a symmetric 2-tensor Cαβ verifying the property D(γCαβ)= 0.

The tensor carries the name of its discoverer B. Carter, see [15], who made use of

it to show that the geodesic ﬂow in Kerr is integrable. Its presence, in addition to

Tand Z, as a higher order symmetry, is at the heart of what Chandrasekhar, see

[17], called the most striking feature of Kerr, “the separability of all the standard

equations of mathematical physics in Kerr geometry”.

•The Carter tensor can be used to deﬁne the Carter operator

C=Dα(CαβDβ),(1.6)

a second order operator which commutes with a,m. This property plays a crucial

role in the proof of our stability result, Theorem 1.1, more precisely in Part II of

[39].

1.2 Kerr stability conjecture

The discovery of black holes, ﬁrst as explicit solutions of EVE and later as possible

explanations of astrophysical phenomena5, has not only revolutionized our understanding

of the universe, it also gave mathematicians a monumental task: to test the physical

reality of these solutions. This may seem nonsensical since physics tests the reality of its

objects by experiments and observations and, as such, needs mathematics to formulate the

theory and make quantitative predictions, not to test it. The problem, in this case, is that

black holes are by deﬁnition non-observable and thus no direct experiments are possible.

4Given by the expression C=−a2cos2θg+O,O=|q|2e1⊗e1+e2⊗e2.

5According to Chandrasekhar “Black holes are macroscopic objects with masses varying from a few

solar masses to millions of solar masses. To the extent that they may be considered as stationary and

isolated, to that extent, they are all, every single one of them, described exactly by the Kerr solution.

This is the only instance we have of an exact description of a macroscopic object. Macroscopic objects, as

we see them around us, are governed by a variety of forces, derived from a variety of approximations to a

variety of physical theories. In contrast, the only elements in the construction of black holes are our basic

concepts of space and time. They are, thus, almost by deﬁnition, the most perfect macroscopic objects

there are in the universe. And since the general theory of relativity provides a single two parameter

family of solutions for their description, they are the simplest as well.”

Astrophysicists ascertain the presence of such objects through indirect observations6and

numerical experiments, but both are limited in scope to the range of possible observations

or the speciﬁc initial conditions in which numerical simulations are conducted. One can

rigorously check that the Kerr solutions have vanishing Ricci curvature, that is, their

mathematical reality is undeniable. But to be real in a physical sense, they have to

satisfy certain properties which, as it turns out, can be neatly formulated in unambiguous

mathematical language. Chief among them7is the problem of stability, that is, to show

that if the precise initial data corresponding to Kerr are perturbed a bit, the basic features

of the corresponding solutions do not change much8.

Conjecture (Stability of Kerr conjecture). Vacuum, asymptotically ﬂat, initial data sets,

suﬃciently close to Kerr(a, m),|a|/m < 1, initial data, have maximal developments with

complete future null inﬁnity and with domain of outer communication9which approaches

(globally) a nearby Kerr solution.

1.3 Resolution of the conjecture for slowly rotating black holes

1.3.1 Statement of the main result

The goal of this article is to give a short introduction to our recent result in which we

settle the conjecture in the case of slowly rotating Kerr black holes.

Theorem 1.1. The future globally hyperbolic development of a general, asymptotically

ﬂat, initial data set, suﬃciently close (in a suitable topology) to a Kerr(a0, m0)initial data

set, for suﬃciently small |a0|/m0, has a complete future null inﬁnity I+and converges in

its causal past J−1(I+)to another nearby Kerr spacetime Kerr(af, mf)with parameters

(af, mf)close to the initial ones (a0, m0).

6The physical reality of these objects was recently put to test by LIGO which is supposed to have

detected the gravitational waves generated in the ﬁnal stage of the in-spiraling of two black holes. Rainer

Weiss, Barry C. Barish and Kip S. Thorne received the 2017 Nobel prize for their “decisive contributions”

in this respect. The 2020 Nobel prize in Physics was awarded to R. Genzel and A. Ghez for providing

observational evidence for the presence of super massive black holes in the center of our galaxy, and to

R. Penrose for his theoretical foundational work: his concept of a trapped surface and the proof of his

famous singularity theorem.

7Other such properties concern the rigidity of the Kerr family, see [46] for a current survey, or the

dynamical formations of black holes from regular conﬁgurations, see the [23], [59] and the introduction

to [3] for an up to date account of more recent results.

8If the Kerr family would be unstable under perturbations, black holes would be nothing more than

mathematical artifacts.

9This presupposes the existence of an event horizon. Note that the existence of such an event horizon

can only be established a posteriori, upon the completion of the proof of the conjecture.

Figure 2: The Penrose diagram of the ﬁnal space-time in Theorem 1.1 with initial hypersurface Σ0,

future space-like boundary A, and I+the complete future null inﬁnity. The hypersurface H+is the

future event horizon of the ﬁnal Kerr.

The precise version of the result, all the main features of the architecture of its proof, as

well as detailed proofs for most of the main steps are to be found in [63]. The full proof

relies also on our joint work [39] with E. Giorgi, our papers [61], [62] on GCM spheres,

and the extension [81] to GCM hypersurfaces by D. Shen.

1.3.2 Brief comments on the proof

We will discuss the main ideas of the proof in more details in section 4. It pays however

to give already a graphic sense of the main building blocks of our approach, which we call

general covariant modulated(GCM), admissible spacetimes.

The main features of these spacetimes M=(ext)M ∪ (top)M ∪ (int)Mare as follows:

•The capstone of the entire construction is the sphere S∗, on the future boundary Σ∗

of (ext)M, which veriﬁes a set of speciﬁc extrinsic and intrinsic conditions denoted

by the acronym GCM.

•The spacelike hypersurface Σ∗, initialized at S∗, veriﬁes a set of additional GCM

conditions.

•Once Σ∗is speciﬁed the whole GCM admissible spacetime Mis determined by a

more conventional construction, based on geometric transport type equations10.

10More precisely (ext)Mcan be determined from Σ∗by a speciﬁed outgoing foliation terminating in

the timelike boundary T,(int)Mis determined from Tby a speciﬁed incoming one, and (top)Mis a

complement of (ext)M ∪ (int)Mwhich makes Ma causal domain.

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BriefintroductiontothenonlinearstabilityofKerrSergiuKlainerman,JeremieSzeftelOctober31,2022AbstractThisabriefintroductiontothesequenceofworks[63],[39],[61],[62]and[81]whichestablishthenonlinearstabilityofKerrblackholeswithsmallangularmo-mentum.WearedelightedtodedicatethisarticletoDemetriosChristod...

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Brief introduction to the nonlinear stability of Kerr Sergiu Klainerman J er emie Szeftel October 31 2022

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