Nonlinear stability of Kerr for small
angular momentum

Jérémie Szeftel
CNRS & Laboratoire Jacques-Louis Lions,Sorbonne Université1

Einstein vacuum equations• (M, g) Lorentzian manifold• dim(M) = 4, and signature of g is (−, +, +, +)• Minkowski: M = R1+3 , g = −(dt)2 + (dx1 )2 + (dx2 )2 + (dx3 )2• Rα βγµ curvature tensor,

Ricβµ = Rγ βγµ Ricci tensor• Einstein vacuum equations:

Ricαβ = 0

• General covariance: Φ∗ g ∼ g for any diffeomorphism Φ2Evolution problem for Einstein vacuum equations
Initial data sets for EVE: (Σ(0) , g(0) , k(0) ) satisfying constraints
Evolution problem. For suitable (Σ(0) , g(0) , k(0) ), there exists a
unique maximal globally hyperbolic development satisfying EVE[Choquet-Bruhat 52’], [Choquet-Bruhat-Geroch 69’]

Explicit stationary solutions: Minkowski, Kerr black holes Ka,m3

The final state conjecture
What happens asymptotically to the maximal developments of[Choquet-Bruhat-Geroch 69’]?
Final state conjecture. Generic initial data sets lead to solutions
which behave, in the large, like a finite number of Kerr black holes
moving away from each other plus a radiative decaying term
In general, complicated transitory regime (back holes form, interact,merge,...) but striking asymptotic simplification of dynamic!

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The mathematical theory of black holes
Three problems intimately connected to the final state conjecture:
1. Collapse. Can black holes form from reasonable initial data
configurations? [Penrose 65’], [Christodoulou 09’], [Klainerman-Rodnianski12’], [Klainerman-Luk-Rodnianski 14’], [An-Luk 17’], [An 20’]...
2. Rigidity. Does the Kerr family of solutions exhaust all possible
stationary vacuum black holes? [Carter-Robinson 71’], [Hawking 73’],[Ionescu-Klainerman 09’], [Alexakis-Ionescu-Klainerman 10’],...
3. Stability. Is the Kerr family stable under arbitrary small
perturbations?
We will focus form now on on the third topic
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Kerr and the Schwarzschild solutions
Kerr metric given in Boyer-Lindquist (t, r, θ, ϕ) coordinates by2 ρ222(sinθ)2amrΣρ2 ∆222dϕ−(dr)+ρ(dθ)
dt+ga,m = − 2 (dt)2 +Σρ2Σ2∆∆ = r2 + a2 − 2mr, ρ2 = r2 + a2 (cos θ)2 , Σ2 = (r2 + a2 )2 − a2 (sin θ)2 ∆Kerr is stationary (∂t is Killing) and axisymmetric (∂ϕ is Killing)
The Schwarzschild metric is spherically symmetric and corresponds
to the particular case a = 0, m > 0
−1
2m2m
gm = − 1 −(dt)2 + 1 −(dr)2 +r2 (dθ)2 +(sin θ)2 (dϕ)2rr

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Kerr and Schwarzschild black holes
Schwarzschild spacetimes correspond to non rotating black holes
while for |a| ≤ m, Kerr spacetimes correspond to rotating black holes
In Schwarzschild, black hole in r < 2m and event horizon is r = 2m

r<2mExterior region

r>2m

r=2m

Black hole

Event Horizontr

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Stability conjecture for the Kerr family

In the context of asymptotically flat solutions to the Einstein vacuum
equation, we have the following conjecture:
Conjecture (Stability of the exterior region of Kerr). Small
perturbations of given initial conditions of an exterior Kerr ga,m with|a| < m have maximal future developments converging to anotherexterior Kerr metric gaf ,mf with |af | < mf

First stability result: Minkowski [Christodoulou-Klainerman 93’]8

The linearized gravity system (LGS)
To get insight into the problem, look at linearized Einstein equations
around a Kerr solution (which is the so called LGS), i.e.
δRic[ga,m ](h)αβ = 0,g = ga,m + h,and prove decay for its solutions
From 50’s to 90’s, a large physics literature devoted to the LGS[Regge-Wheeler 57’], [Teukolsky 73’], [Chandrasekhar 75’], [Whiting 89’]...
In the context of the stability of Minkowski, LGS reduces to theunderstanding of ψ = 0
In the case of Kerr, LGS is a large and entangled system, whoseanalysis is non trivial
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Teukolsky equations
To analyze LGS, identify a quantity which decouples from the rest ofthe solution and from which one can in principle reconstruct thewhole solution
For Kerr, Teukolsky (1973) exhibited such a quantity at the level ofcurvature satisfying a non conservative linear wave equation

ga,m + Aα± (r, θ)∂α + V± (r, θ) Ψ±=0

Goal: derive decay for the Teukolsky quantities Ψ± , and rely on it toderive decay for the rest of LGS by exhibiting a triangular structure10Decay of scalar waves and geometry of Kerr
Simplest model is scalar wave equation on Kerr: ga,m ψ = 0• Trapped null geodesics: a priori obstacle to decay of linear waves(but trapping is hyperbolic!)• Super-radiance: Killing vectorfield ∂t is timelike only for r large
enough (no global positive conserved energy for linear waves)• Redshift region: small neighborhood of the horizon experiences
good properties from point of view of decay of linear waves• Large r region: Kerr is asymptotically flat11

Decay for Teukolsky on Kerr
• Start with ga,m ψ = 0: [Blue-Soffer 03’], [Dafermos-Rodnianski 05’, 09’],[Tataru-Tohaneanu 08’], [Andersson-Blue 09’], [D-R-Shlapentokh Rothman 14’]
• Teukolsky has no exponentially growing modes [Whiting 89’], but
still far from quantitive decay rate for solutions• Chandrasekhar transform from Teukolsky to Regge-Wheeler
equation which is a conservative linear wave equation• a = 0: [Dafermos-Holzegel-Rodnianski 16’]
• |a|  m: [Ma 17’], [Dafermos-Holzegel-Rodnianski 17’]
• |a| < m: [Millet 23’], [Shlapentokh Rothman-Teixera Da Costa 20’-23’]

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Decay for LGS on KerrObstructions to decay for LGS: ∂m gm,a , ∂a gm,a and LX gm,a belong
to the kernel of LGS, for all vectorfields X⇒ At the nonlinear level, need to use modulation theory, i.e. need totrack change of (m, a) and of diffeomorphism ΦGeneral covariance requires to find strategy to implement modulation
in infinite dimensions

Decay estimates for LGS:
• a = 0: [Dafermos-Holzegel-Rodnianski 16’]
• |a|  m: [Andersson, Bäckdahl, Blue and Ma 19’], [Häfner, Hintz and Vasy 19’]13

Nonlinear stability of Kerr for |a|  m
Theorem: The stability conjecture holds true for |a|  m
Proof relies on:• Modulation procedure: [Klainerman-Sz 19’a], [Klainerman-Sz 19’b], and[Shen 22’]

• Statement of the result, complete strategy, gauge choices anddecay estimates (except for Ψ± ): [Klainerman-Sz 21’]
• Formalism for non integrable structures and wave estimates:[Giorgi-Klainerman-Sz 22’]

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The continuity argument and gauge fixing
In the context of a continuity argument, we initialize the outgoing"null cones" of the exterior region on a carefully chosen last slice

Last Slice

terio
Ex

r=r1

r=r0rRegion

Event Horizon

S*rio
te
Inieg
rRon

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Structure of the proof
1. Control of Teukolsky equations in nonlinear perturbations of Kerr
2. Control curvature, connection and metric components byexhibiting a triangular structure using:(a) Transport equations along the generator of the "null cones"(b) Elliptic systems of Hodge type on spheres
3. To deal with the presence of a kernel at the linearized level:(a) Construct spheres "at infinity"(b) Define suitable constants (m, a) converging to (mf , af )(c) Tracking of the "axis"
4. Top derivatives: energy-Morawetz estimates and Maxwell likecharacter of the Bianchi identities

16Modulation procedure
Modulation corresponds to tracking af , mf and Φf
Start with tracking of Φf : find analog of spheres S(t, r) of Kerr
Main building block: in [Klainerman-Sz 19’a] [Klainerman-Sz 19’b], we
design a procedure to construct a unique 2-parameter family ofpreferred codimension 2 surfaces in general perturbations of Kerr
General Covariant Modulated (GCM) spheres rely on fixing
well-chosen geometric quantities to agree with their Kerr value
Intrinsic definition of mass (given by Hawking mass) and of angularmomentum, and tracking of the "axis" for such GCM spheres using
an effective version of the uniformization theorem [Klainerman-Sz 19’b]17

Non integrability in the case a 6= 0Gauge choice specifies the connection coefficients associated to a nullpair (e3 , e4 ) and its associated horizontal bundle {e3 , e4 }⊥In Kerr, specific choice (e3 , e4 ) known as principal null directions
diagonalize the Riemann curvature tensor (and used to define Ψ± )The horizontal bundle {e3 , e4 }⊥ of the principal null frame of Kerr is
non integrable for a 6= 0 (in the sense of Frobenius)
All gauge choices used in the field so far were integrable
In perturbations of Kerr, requires: to make sense of non integrable
frames; to design geometric construction of such non integrable
gauges; and to derive estimates in this non integrable setting

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