Non-linear stability of Kerr de Sitter black holesphintz/files/kdsstab-cambridge.pdf · 2020. 4....

Non-linear stability of Kerr–de Sitter black holes

Peter Hintz 1

(joint with Andras Vasy 2)

1Miller Institute, University of California, Berkeley

2Stanford University

Geometric Analysis and PDE Seminar

University of Cambridge

October 14, 2016

Einstein vacuum equations

Ein(g)− Λg = 0

⇐⇒ Ric(g) + Λg = 0

I g : Lorentzian metric (+−−−) on 4-manifold Ω

I Λ > 0: cosmological constant

I Ein(g) = GgRic(g), Gg r = r − 12 (trg r)g (trace-reversal)

Einstein vacuum equations

Ein(g)− Λg = 0 ⇐⇒ Ric(g) + Λg = 0

I g : Lorentzian metric (+−−−) on 4-manifold Ω

I Λ > 0: cosmological constant

I Ein(g) = GgRic(g), Gg r = r − 12 (trg r)g (trace-reversal)

Initial value problem

I Σ: 3-manifold

I h: Riemannian metric on Σ

I k: symmetric 2-tensor on Σ

Find spacetime (Ω, g), Σ → Ω, solving Ric(g) + Λg = 0, with

h = −g |Σ, k = IIΣ.

Theorem (Choquet-Bruhat ’53)

Necessary and sufficient for local well-posedness: constraintequations for (h, k).

Key difficulty: diffeomorphism invariance ⇒ need for gauge fixing

I Σ: 3-manifold

h = −g |Σ, k = IIΣ.

I Σ: 3-manifold

h = −g |Σ, k = IIΣ.

I Σ: 3-manifold

h = −g |Σ, k = IIΣ.

Kerr–de Sitter family

I manifold: Ω = [0,∞)t × [r1, r2]r × S2

I Cauchy surface: Σ = t = 0I C∞ family of stationary metrics gb, b = (M,~a) ∈ R× R3

I M: mass of the black hole

I ~a: angular momentum

Σ = t = 0

t = const.

Example. b0 = (M0,~0), Schwarzschild–de Sitter space

Metric:

gb0 = f d t2 − f −1 dr 2 − r 2 dσ2, f (r) = 1− 2M0

r− Λr 2

Valid for r ∈(r(H+, b0), r(H+, b0)

)⊂ [r1, r2].

r = r(H+, b0) r = r(H+, b0)

Extension: t = t − F (r)

Example. b0 = (M0,~0), Schwarzschild–de Sitter space

Metric:

gb0 = f d t2 − f −1 dr 2 − r 2 dσ2, f (r) = 1− 2M0

r− Λr 2

Valid for r ∈(r(H+, b0), r(H+, b0)

)⊂ [r1, r2].

r = r(H+, b0) r = r(H+, b0)

Extension: t = t − F (r)

Black hole stability

Theorem (H.–Vasy ’16)

Given C∞ initial data (h, k) on Σ

I satisfying the constraint equations,

I close (in H21) to the initial data induced by gb0 ,

there exist

I a C∞ metric g on Ω solving Ric(g) + Λg = 0 with initial data(h, k) at Σ,

I parameters b ∈ R4 close to b0 such that

g = gb + g , |g | = O(e−αt), α > 0.

Exponential decay towards a Kerr–de Sitter solution!

there exist

g = gb + g , |g | = O(e−αt), α > 0.

there exist

g = gb + g , |g | = O(e−αt), α > 0.

there exist

g = gb + g , |g | = O(e−αt), α > 0.

Related work

Non-linear stability:

I de Sitter: Friedrich (’80s), Anderson (’05), Ringstrom (’08),Rodnianski–Speck (’09)

I Minkowski: Christodoulou–Klainerman (’93), Lindblad–Rodnianski (’00s), Bieri–Zipser (’09), Speck (’14), Taylor(’15), Huneau (’15), LeFloch–Ma (’15)

I Hyperbolic space: Graham–Lee (’91)

Linear (mode) stability of black holes:

I Λ > 0: Kodama–Ishibashi (’04)

I Λ = 0: Regge–Wheeler (’57), Chandrasekhar (’83), Whiting(’89), with Andersson–Ma–Paganini (’16), Shlapentokh-Rothman (’14), Dafermos–Holzegel–Rodnianski (’16)

Related work

(Non-)linear fields on black hole spacetimes:

I Λ > 0: Sa Barreto–Zworski (’97), Bony–Hafner (’08), Vasy(’13), Melrose–Sa Barreto–Vasy (’14), Dyatlov (’10s), Schlue(’15), H.–Vasy (’10s), . . .

I Λ = 0: Wald (’79), Kay–Wald (’87), Andersson–Blue (’10s),Tataru, with Marzuola, Metcalfe, Sterbenz, and Tohaneanu(’10s), Luk (’10s), Dafermos–Rodnianski–Shlapentokh-Rothman (’14), Lindblad–Tohaneanu (’16), . . .

Gauge fixing

Goal: Solve Ric(g) + Λg = 0.

DeTurck: Demand that 1 : (Ω, g)→ (Ω, gb0) be a wave map

⇐⇒W (g) = gklgij(Γ(g)kij − Γ(gb0)kij)dx l = 0.

Reduced Einstein equation:

(Ric + Λ)(g)− δ∗gW (g) = 0. (*)

I (h, k) 7→ Cauchy data for g in (*) with W (g) = 0 at Σ.

I Solve (*) (quasilinear wave equation) =⇒ L∂t W (g) = 0 at Σ.

I Second Bianchi: δgGgδ∗gW (g) = 0. Wave equation!

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

(Ric + Λ)(g)− δ∗gW (g) = 0. (*)

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

(Ric + Λ)(g)− δ∗gW (g) = 0. (*)

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

(Ric + Λ)(g)− δ∗gW (g) = 0. (*)

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

(Ric + Λ)(g)− δ∗gW (g) = 0. (*)

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

(Ric + Λ)(g)− δ∗gW (g) = 0. (*)

I Solve (*) (quasilinear wave equation)

=⇒ L∂t W (g) = 0 at Σ.

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

(Ric + Λ)(g)− δ∗gW (g) = 0. (*)

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

(Ric + Λ)(g)− δ∗gW (g) = 0. (*)

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

(Ric + Λ)(g)− δ∗gW (g) = 0. (*)

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

DeTurck/Friedrich: Demand that 1 : (Ω, g)→ (Ω, gb0) be a forcedwave map

⇐⇒W (g) = gklgij(Γ(g)kij − Γ(gb0)kij)dx l = −θ.

(Ric + Λ)(g)− δ∗g (W (g) + θ) = 0. (*)

I (h, k) 7→ Cauchy data for g in (*) with W (g) + θ = 0 at Σ.

I Second Bianchi: δgGgδ∗g (W (g) + θ) = 0. Wave equation!

=⇒W (g) + θ ≡ 0, and (Ric + Λ)(g) = 0.

Gauge fixing

(Ric + Λ)(g)− δ∗W (g) = 0. (*)

I Second Bianchi: δgGgδ∗W (g) = 0. Wave equation!

=⇒W (g) ≡ 0, and (Ric + Λ)(g) = 0.

Linearization around g = gb0

Asymptotics for equation Lr = Dg (Ric + Λ)(r)− δ∗gDgW (r) = 0:

r =N∑j=1

rjaj(x)e−iσj t + r(t, x), t 0,

I σj ∈ C resonances (quasinormal modes)

I aj(x)e−iσt resonant states, L(aj(x)e−iσj t) = 0

I rj ∈ C, r = O(e−αt), α > 0 fixed, small

σN=σ = −α

(Wunsch–Zworski ’11, Vasy ’13, Dyatlov ’15, H. ’15)

r =N∑j=1

σN=σ = −α

r =N∑j=1

σN=σ = −α

r =N∑j=1

σN=σ = −α

r =N∑j=1

σN=σ = −α

r =N∑j=1

σN=σ = −α

Attempt #1

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

Hope: N = 1, σ1 = 0.

Cσσ1

=σ = −α

dsgb0+sb

∣∣s=0

Then: Could solve

(Ric + Λ)(gb + g)− δ∗gW (gb + g) = 0

for g = O(e−αt), b ∈ R4. (H.–Vasy ’16)

False!

Attempt #1

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

Hope: N = 1, σ1 = 0.

Cσσ1

=σ = −α

dsgb0+sb

∣∣s=0

Then: Could solve

(Ric + Λ)(gb + g)− δ∗gW (gb + g) = 0

False!

Attempt #1

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

Hope: N = 1, σ1 = 0.

Cσσ1

=σ = −α

dsgb0+sb

∣∣s=0

Then: Could solve

(Ric + Λ)(gb + g)− δ∗gW (gb + g) = 0

False!

Attempt #1

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

Hope: N = 1, σ1 = 0.

Cσσ1

=σ = −α

dsgb0+sb

∣∣s=0

Then: Could solve

(Ric + Λ)(gb + g)− δ∗gW (gb + g) = 0

False!

Attempt #2

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

Say N = 2, Imσ2 > 0

, and (ignoring linearized KdS family)

r = r2a2(x)e−iσ2t + r(t, x).

Hope: a2(x)e−iσ2t = LV g is pure gauge, so r = r2LχV g + r .

Lr = −L(r2LχV g) = r2δ∗g DgW (LχV g)︸︷︷︸

=:θThus:

Dg (Ric + Λ)(r)− δ∗g(DgW (r) + r2θ

Attempt #2

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

Say N = 2, Imσ2 > 0, and (ignoring linearized KdS family)

r = r2a2(x)e−iσ2t + r(t, x).

=:θThus:

Attempt #2

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

r = r2a2(x)e−iσ2t + r(t, x).

Hope: a2(x)e−iσ2t = LV g is pure gauge

, so r = r2LχV g + r .

=:θThus:

Attempt #2

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

r = r2a2(x)e−iσ2t + r(t, x).

=:θThus:

Attempt #2

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

r = r2a2(x)e−iσ2t + r(t, x).

=:θThus:

Attempt #2

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

r = r2a2(x)e−iσ2t + r(t, x).

Thus:Dg (Ric + Λ)(r)− δ∗g

(DgW (r) + r2θ

Attempt #2

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

r = r2a2(x)e−iσ2t + r(t, x).

Thus:Dg (Ric + Λ)(r)− δ∗g

(DgW (r) + r2θ

Attempt #2

(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)

r = r2a2(x)e−iσ2t + r(t, x).

=:θThus:

Attempt #2, cont.

Then: Could solve

(Ric + Λ)(gb + g)− δ∗g(W (gb + g) + θ

for g and (b, θ) (finite-dimensional parameters).

Hope is false!

Not to worry: Have not yet used the constraint equations.

L = Dg (Ric + Λ)− δ∗gDgW

Delicate!

Attempt #2, cont.

Then: Could solve

(Ric + Λ)(gb + g)− δ∗g(W (gb + g) + θ

Hope is false!

Delicate!

Attempt #2, cont.

Then: Could solve

(Ric + Λ)(gb + g)− δ∗g(W (gb + g) + θ

Hope is false!

Delicate!

Attempt #2, cont.

Then: Could solve

(Ric + Λ)(gb + g)− δ∗g(W (gb + g) + θ

Hope is false!

Delicate!

Attempt #2, cont.

Then: Could solve

(Ric + Λ)(gb + g)− δ∗g(W (gb + g) + θ

Hope is false!

Delicate!

L = Dg (Ric + Λ)− δ∗DgW , where δ∗ = δ∗g + 0th order.

Apply linearized second Bianchi identity to

0 = L(a2(x)e−iσ2t)

= Dg (Ric + Λ)(a2e−iσ2t)− δ∗DgW (a2e−iσ2t).

L = Dg (Ric + Λ)− δ∗DgW , where δ∗ = δ∗g + 0th order.

Apply linearized second Bianchi identity to

0 = L(a2(x)e−iσ2t)

= Dg (Ric + Λ)(a2e−iσ2t)− δ∗DgW (a2e−iσ2t).

=⇒ δgGgδ∗(DgW (a2e−iσ2t)

For g the Schwarzschild–de Sitter metric, one can choose δ∗ suchthat δgGgδ

∗ has no resonances with Imσ ≥ 0. (‘Constraintdamping,’ Gundlach et al (’05).) Take

δ∗w = δ∗gw + γ1 dt ⊗s w − γ2w(∇t)g , γ1, γ2 0.

=⇒ DgW (a2e−iσ2t) = 0

=⇒ Dg (Ric + Λ)(a2e−iσ2t) = 0

Mode stability for linearized Einstein equation (Kodama–Ishibashi’04) =⇒ a2e−iσ2t = LV g

∗ has no resonances with Imσ ≥ 0. (‘Constraintdamping,’ Gundlach et al (’05).)

=⇒ DgW (a2e−iσ2t) = 0

=⇒ Dg (Ric + Λ)(a2e−iσ2t) = 0

=⇒ DgW (a2e−iσ2t) = 0

=⇒ Dg (Ric + Λ)(a2e−iσ2t) = 0

=⇒ DgW (a2e−iσ2t) = 0

=⇒ Dg (Ric + Λ)(a2e−iσ2t) = 0

=⇒ DgW (a2e−iσ2t) = 0

=⇒ Dg (Ric + Λ)(a2e−iσ2t) = 0

=⇒ DgW (a2e−iσ2t) = 0

=⇒ Dg (Ric + Λ)(a2e−iσ2t) = 0

Mode stability for linearized Einstein equation (Kodama–Ishibashi’04)

=⇒ a2e−iσ2t = LV g

=⇒ DgW (a2e−iσ2t) = 0

=⇒ Dg (Ric + Λ)(a2e−iσ2t) = 0

Attempt #3

Solve(Ric + Λ)(gb + g)− δ∗

(W (gb + g) + θ

for g = O(e−αt) and (b, θ).

Almost: 1 : (Ω, gb)→ (Ω, gb0) is not a wave map: W (gb) 6= 0.

Attempt #3

(W (gb + g) + θ

Almost: 1 : (Ω, gb)→ (Ω, gb0) is not a wave map

: W (gb) 6= 0.

Attempt #3

(W (gb + g) + θ

Almost: 1 : (Ω, gb)→ (Ω, gb0) is not a wave map: W (gb) 6= 0.

Final attempt

(Ric + Λ)(gb + g)− δ∗(W (gb + g)−χW (gb) + θ

Solution of the non-linear equation

Nash–Moser iteration scheme:

I Solve linearized equation globally at each step.

I Use linear solutions to update g , b and θ.

Automatically find

I final black hole parameters b,

I finite-dimensional modification θ of the gauge.

Automatically find

Outlook

I more precise asymptotic expansion, ringdown

I stability for large a (mode stability?)

I couple Einstein equation with matter

I Non-linear stability of the Kerr–Newman–de Sitter family (forsmall a) for the Einstein–Maxwell system: work in progress

I Einstein–(Maxwell–)scalar field, Einstein–Vlasov, . . .

I higher-dimensional black holes (Myers–Perry, Reall, . . . )

Outlook

ΣH+ H+

I analysis in the cosmological region (ongoing work by Schlue)

I Penrose’s Strong Cosmic Censorship Conjecture (work byDafermos–Luk for Kerr)

Outlook

ΣH+ H+

Outlook

ΣH+ H+

Thank you!

Non-linear stability of Kerr de Sitter black holesphintz/files/kdsstab-cambridge.pdf · 2020. 4....

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