non-linear stability of kerr de sitter black holesphintz/files/kdsstab-cambridge.pdf · 2020. 4....
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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
Data:
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
Initial value problem
Data:
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
Initial value problem
Data:
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
Initial value problem
Data:
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
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
Ω
Σ = t = 0
t = const.
H+ H+
r =r1
r =r2
i+
Kerr–de Sitter family
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
3
Valid for r ∈(r(H+, b0), r(H+, b0)
)⊂ [r1, r2].
Ω
r = r(H+, b0) r = r(H+, b0)
r =r1
r =r2
i+
Extension: t = t − F (r)
Kerr–de Sitter family
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
3
Valid for r ∈(r(H+, b0), r(H+, b0)
)⊂ [r1, r2].
Ω
r = r(H+, b0) r = r(H+, b0)
r =r1
r =r2
i+
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!
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!
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!
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!
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 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 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
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
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
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
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
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
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
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
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
Goal: Solve Ric(g) + Λg = 0.
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.
Reduced Einstein equation:
(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 Solve (*) (quasilinear wave equation) =⇒ L∂t W (g) = 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
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∗W (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∗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
Cσ
σ1
σ2
σN=σ = −α
(Wunsch–Zworski ’11, Vasy ’13, Dyatlov ’15, H. ’15)
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
Cσ
σ1
σ2
σN=σ = −α
(Wunsch–Zworski ’11, Vasy ’13, Dyatlov ’15, H. ’15)
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
Cσ
σ1
σ2
σN=σ = −α
(Wunsch–Zworski ’11, Vasy ’13, Dyatlov ’15, H. ’15)
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
Cσ
σ1
σ2
σN=σ = −α
(Wunsch–Zworski ’11, Vasy ’13, Dyatlov ’15, H. ’15)
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
Cσ
σ1
σ2
σN=σ = −α
(Wunsch–Zworski ’11, Vasy ’13, Dyatlov ’15, H. ’15)
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
Cσ
σ1
σ2
σN=σ = −α
(Wunsch–Zworski ’11, Vasy ’13, Dyatlov ’15, H. ’15)
Attempt #1
(Lr = 0, L = Dg (Ric + Λ)− δ∗gDgW .)
Hope: N = 1, σ1 = 0.
Cσσ1
=σ = −α
r =d
dsgb0+sb
∣∣s=0
+ r .
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
r =d
dsgb0+sb
∣∣s=0
+ r .
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
r =d
dsgb0+sb
∣∣s=0
+ r .
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
r =d
dsgb0+sb
∣∣s=0
+ r .
Then: Could solve
(Ric + Λ)(gb + g)− δ∗gW (gb + g) = 0
for g = O(e−αt), b ∈ R4. (H.–Vasy ’16)
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
)= 0.
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
)= 0.
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
)= 0.
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
)= 0.
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
)= 0.
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)︸ ︷︷ ︸
=:θ
Thus:Dg (Ric + Λ)(r)− δ∗g
(DgW (r) + r2θ
)= 0.
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)︸ ︷︷ ︸
=:θ
Thus:Dg (Ric + Λ)(r)− δ∗g
(DgW (r) + r2θ
)= 0.
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
)= 0.
Attempt #2, cont.
Then: Could solve
(Ric + Λ)(gb + g)− δ∗g(W (gb + g) + θ
)= 0
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
)= 0
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
)= 0
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
)= 0
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
)= 0
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!
Fix?
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).
Fix?
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).
Fix?
=⇒ δgGgδ∗(DgW (a2e−iσ2t)
)= 0.
Theorem (H.–Vasy ’16)
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
Fix?
=⇒ δgGgδ∗(DgW (a2e−iσ2t)
)= 0.
Theorem (H.–Vasy ’16)
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
Fix?
=⇒ δgGgδ∗(DgW (a2e−iσ2t)
)= 0.
Theorem (H.–Vasy ’16)
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
Fix?
=⇒ δgGgδ∗(DgW (a2e−iσ2t)
)= 0.
Theorem (H.–Vasy ’16)
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
Fix?
=⇒ δgGgδ∗(DgW (a2e−iσ2t)
)= 0.
Theorem (H.–Vasy ’16)
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
Fix?
=⇒ δgGgδ∗(DgW (a2e−iσ2t)
)= 0.
Theorem (H.–Vasy ’16)
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
Fix?
=⇒ δgGgδ∗(DgW (a2e−iσ2t)
)= 0.
Theorem (H.–Vasy ’16)
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
Attempt #3
Solve(Ric + Λ)(gb + g)− δ∗
(W (gb + g) + θ
)= 0
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
Solve(Ric + Λ)(gb + g)− δ∗
(W (gb + g) + θ
)= 0
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
Solve(Ric + Λ)(gb + g)− δ∗
(W (gb + g) + θ
)= 0
for g = O(e−αt) and (b, θ).
Almost: 1 : (Ω, gb)→ (Ω, gb0) is not a wave map: W (gb) 6= 0.
Final attempt
Solve
(Ric + Λ)(gb + g)− δ∗(W (gb + g)−χW (gb) + θ
)= 0.
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.
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.
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.
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.
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
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
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
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
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
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
Ω
ΣH+ H+
r =r1
r =r2
I +
CH+
i+
I analysis in the cosmological region (ongoing work by Schlue)
I Penrose’s Strong Cosmic Censorship Conjecture (work byDafermos–Luk for Kerr)
Outlook
Ω
ΣH+ H+
r =r1
r =r2
I +
CH+
i+
I analysis in the cosmological region (ongoing work by Schlue)
I Penrose’s Strong Cosmic Censorship Conjecture (work byDafermos–Luk for Kerr)
Outlook
Ω
ΣH+ H+
r =r1
r =r2
I +
CH+
i+
I analysis in the cosmological region (ongoing work by Schlue)
I Penrose’s Strong Cosmic Censorship Conjecture (work byDafermos–Luk for Kerr)
Thank you!