Non-linear stability of Kerr de Sitter black holes

Transcription

1 Non-linear stability of Kerr de Sitter black holes Peter Hintz 1 (joint with András Vasy 2 ) 1 Miller Institute, University of California, Berkeley 2 Stanford University Geometric Analysis and PDE Seminar University of Cambridge October 14, 2016

2 Einstein vacuum equations Ein(g) Λg = 0 g: Lorentzian metric (+ ) on 4-manifold Ω Λ > 0: cosmological constant Ein(g) = G g Ric(g), G g r = r 1 2 (tr g r)g (trace-reversal)

3 Einstein vacuum equations Ein(g) Λg = 0 Ric(g) + Λg = 0 g: Lorentzian metric (+ ) on 4-manifold Ω Λ > 0: cosmological constant Ein(g) = G g Ric(g), G g r = r 1 2 (tr g r)g (trace-reversal)

4 Initial value problem Data: Σ: 3-manifold h: Riemannian metric on Σ k: symmetric 2-tensor on Σ

5 Initial value problem Data: Σ: 3-manifold h: Riemannian metric on Σ k: symmetric 2-tensor on Σ Find spacetime (Ω, g), Σ Ω, solving Ric(g) + Λg = 0, with h = g Σ, k = II Σ.

6 Initial value problem Data: Σ: 3-manifold h: Riemannian metric on Σ k: symmetric 2-tensor on Σ Find spacetime (Ω, g), Σ Ω, solving Ric(g) + Λg = 0, with h = g Σ, k = II Σ. Theorem (Choquet-Bruhat 53) Necessary and sufficient for local well-posedness: constraint equations for (h, k).

7 Initial value problem Data: Σ: 3-manifold h: Riemannian metric on Σ k: symmetric 2-tensor on Σ Find spacetime (Ω, g), Σ Ω, solving Ric(g) + Λg = 0, with h = g Σ, k = II Σ. Theorem (Choquet-Bruhat 53) Necessary and sufficient for local well-posedness: constraint equations for (h, k). Key difficulty: diffeomorphism invariance need for gauge fixing

8 Kerr de Sitter family manifold: Ω = [0, ) t [r 1, r 2 ] r S 2 Cauchy surface: Σ = {t = 0} C family of stationary metrics g b, b = (M, a) R R 3 M: mass of the black hole a: angular momentum i + r = r 1 Ω r = r 2 t = const. H + Σ = {t = 0} H +

9 Kerr de Sitter family Example. b 0 = (M 0, 0), Schwarzschild de Sitter space Metric: g b0 = f d t 2 f 1 dr 2 r 2 dσ 2, f (r) = 1 2M 0 r Λr 2 3 Valid for r ( r(h +, b 0 ), r(h +, b 0 ) ) [r 1, r 2 ]. i + r = r 1 Ω r = r 2 r = r(h +, b 0 ) r = r(h +, b 0 )

10 Kerr de Sitter family Example. b 0 = (M 0, 0), Schwarzschild de Sitter space Metric: g b0 = f d t 2 f 1 dr 2 r 2 dσ 2, f (r) = 1 2M 0 r Λr 2 3 Valid for r ( r(h +, b 0 ), r(h +, b 0 ) ) [r 1, r 2 ]. i + r = r 1 Ω r = r 2 Extension: t = t F (r) r = r(h +, b 0 ) r = r(h +, b 0 )

11 Black hole stability Theorem (H. Vasy 16) Given C initial data (h, k) on Σ satisfying the constraint equations, close (in H 21 ) to the initial data induced by g b0,

12 Black hole stability Theorem (H. Vasy 16) Given C initial data (h, k) on Σ satisfying the constraint equations, close (in H 21 ) to the initial data induced by g b0, there exist a C metric g on Ω solving Ric(g) + Λg = 0 with initial data (h, k) at Σ,

13 Black hole stability Theorem (H. Vasy 16) Given C initial data (h, k) on Σ satisfying the constraint equations, close (in H 21 ) to the initial data induced by g b0, there exist a C metric g on Ω solving Ric(g) + Λg = 0 with initial data (h, k) at Σ, parameters b R 4 close to b 0 such that g = g b + g, g = O(e αt ), α > 0.

14 Black hole stability Theorem (H. Vasy 16) Given C initial data (h, k) on Σ satisfying the constraint equations, close (in H 21 ) to the initial data induced by g b0, there exist a C metric g on Ω solving Ric(g) + Λg = 0 with initial data (h, k) at Σ, parameters b R 4 close to b 0 such that g = g b + g, g = O(e αt ), α > 0. Exponential decay towards a Kerr de Sitter solution!

15 Related work Non-linear stability: de Sitter: Friedrich ( 80s), Anderson ( 05), Ringström ( 08), Rodnianski Speck ( 09) Minkowski: Christodoulou Klainerman ( 93), Lindblad Rodnianski ( 00s), Bieri Zipser ( 09), Speck ( 14), Taylor ( 15), Huneau ( 15), LeFloch Ma ( 15)

16 Related work Non-linear stability: de Sitter: Friedrich ( 80s), Anderson ( 05), Ringström ( 08), Rodnianski Speck ( 09) Minkowski: Christodoulou Klainerman ( 93), Lindblad Rodnianski ( 00s), Bieri Zipser ( 09), Speck ( 14), Taylor ( 15), Huneau ( 15), LeFloch Ma ( 15) Hyperbolic space: Graham Lee ( 91)

17 Related work Non-linear stability: de Sitter: Friedrich ( 80s), Anderson ( 05), Ringström ( 08), Rodnianski Speck ( 09) Minkowski: Christodoulou Klainerman ( 93), Lindblad Rodnianski ( 00s), Bieri Zipser ( 09), Speck ( 14), Taylor ( 15), Huneau ( 15), LeFloch Ma ( 15) Hyperbolic space: Graham Lee ( 91) Linear (mode) stability of black holes: Λ > 0: Kodama Ishibashi ( 04) Λ = 0: Regge Wheeler ( 57), Chandrasekhar ( 83), Whiting ( 89), with Andersson Ma Paganini ( 16), Shlapentokh- Rothman ( 14), Dafermos Holzegel Rodnianski ( 16)

18 Related work (Non-)linear fields on black hole spacetimes: Λ > 0: Sá Barreto Zworski ( 97), Bony Häfner ( 08), Vasy ( 13), Melrose Sá Barreto Vasy ( 14), Dyatlov ( 10s), Schlue ( 15), H. Vasy ( 10s),... 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),...

19 Gauge fixing Goal: Solve Ric(g) + Λg = 0.

20 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (Ω, g) (Ω, g b0 ) be a wave map

21 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (Ω, g) (Ω, g b0 ) be a wave map W (g) = g kl g ij (Γ(g) k ij Γ(g b0 ) k ij)dx l = 0.

22 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (Ω, g) (Ω, g b0 ) be a wave map W (g) = g kl g ij (Γ(g) k ij Γ(g b0 ) k ij)dx l = 0. Reduced Einstein equation: (Ric + Λ)(g) δg W (g) = 0. (*)

23 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (Ω, g) (Ω, g b0 ) be a wave map W (g) = g kl g ij (Γ(g) k ij Γ(g b0 ) k ij)dx l = 0. Reduced Einstein equation: (Ric + Λ)(g) δg W (g) = 0. (*) (h, k) Cauchy data for g in (*) with W (g) = 0 at Σ.

24 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (Ω, g) (Ω, g b0 ) be a wave map W (g) = g kl g ij (Γ(g) k ij Γ(g b0 ) k ij)dx l = 0. Reduced Einstein equation: (Ric + Λ)(g) δg W (g) = 0. (*) (h, k) Cauchy data for g in (*) with W (g) = 0 at Σ. Solve (*) (quasilinear wave equation)

25 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (Ω, g) (Ω, g b0 ) be a wave map W (g) = g kl g ij (Γ(g) k ij Γ(g b0 ) k ij)dx l = 0. Reduced Einstein equation: (Ric + Λ)(g) δg W (g) = 0. (*) (h, k) Cauchy data for g in (*) with W (g) = 0 at Σ. Solve (*) (quasilinear wave equation) = L t W (g) = 0 at Σ.

26 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (Ω, g) (Ω, g b0 ) be a wave map W (g) = g kl g ij (Γ(g) k ij Γ(g b0 ) k ij)dx l = 0. Reduced Einstein equation: (Ric + Λ)(g) δg W (g) = 0. (*) (h, k) Cauchy data for g in (*) with W (g) = 0 at Σ. Solve (*) (quasilinear wave equation) = L t W (g) = 0 at Σ. Second Bianchi: δ g G g δg W (g) = 0. Wave equation! = W (g) 0, and (Ric + Λ)(g) = 0.

27 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (Ω, g) (Ω, g b0 ) be a wave map W (g) = g kl g ij (Γ(g) k ij Γ(g b0 ) k ij)dx l = 0. Reduced Einstein equation: (Ric + Λ)(g) δg W (g) = 0. (*) (h, k) Cauchy data for g in (*) with W (g) = 0 at Σ. Solve (*) (quasilinear wave equation) = L t W (g) = 0 at Σ. Second Bianchi: δ g G g δg W (g) = 0. Wave equation! = W (g) 0, and (Ric + Λ)(g) = 0.

28 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck/Friedrich: Demand that 1: (Ω, g) (Ω, g b0 ) be a forced wave map W (g) = g kl g ij (Γ(g) k ij Γ(g b0 ) k ij)dx l = θ. Reduced Einstein equation: (Ric + Λ)(g) δ g (W (g) + θ) = 0. (*) (h, k) Cauchy data for g in (*) with W (g) + θ = 0 at Σ. Solve (*) (quasilinear wave equation) = L t W (g) = 0 at Σ. Second Bianchi: δ g G g δ g (W (g) + θ) = 0. Wave equation! = W (g) + θ 0, and (Ric + Λ)(g) = 0.

29 Gauge fixing Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (Ω, g) (Ω, g b0 ) be a wave map W (g) = g kl g ij (Γ(g) k ij Γ(g b0 ) k ij)dx l = 0. Reduced Einstein equation: (Ric + Λ)(g) δ W (g) = 0. (*) (h, k) Cauchy data for g in (*) with W (g) = 0 at Σ. Solve (*) (quasilinear wave equation) = L t W (g) = 0 at Σ. Second Bianchi: δ g G g δ W (g) = 0. Wave equation! = W (g) 0, and (Ric + Λ)(g) = 0.

30 Linearization around g = g b0 Asymptotics for equation Lr = D g (Ric + Λ)(r) δ g D g W (r) = 0:

31 Linearization around g = g b0 Asymptotics for equation Lr = D g (Ric + Λ)(r) δg D g W (r) = 0: N r = r j a j (x)e iσ j t + r(t, x), t 0, j=1

32 Linearization around g = g b0 Asymptotics for equation Lr = D g (Ric + Λ)(r) δ g D g W (r) = 0: r = N r j a j (x)e iσ j t + r(t, x), t 0, j=1 σ j C resonances (quasinormal modes) a j (x)e iσt resonant states, L(a j (x)e iσ j t ) = 0

33 Linearization around g = g b0 Asymptotics for equation Lr = D g (Ric + Λ)(r) δ g D g W (r) = 0: r = N r j a j (x)e iσ j t + r(t, x), t 0, j=1 σ j C resonances (quasinormal modes) a j (x)e iσt resonant states, L(a j (x)e iσ j t ) = 0 r j C, r = O(e αt ), α > 0 fixed, small

34 Linearization around g = g b0 Asymptotics for equation Lr = D g (Ric + Λ)(r) δ g D g W (r) = 0: r = N r j a j (x)e iσ j t + r(t, x), t 0, j=1 σ j C resonances (quasinormal modes) a j (x)e iσt resonant states, L(a j (x)e iσ j t ) = 0 r j C, r = O(e αt ), α > 0 fixed, small σ 2 C σ σ 1 σ N Iσ = α

35 Linearization around g = g b0 Asymptotics for equation Lr = D g (Ric + Λ)(r) δ g D g W (r) = 0: r = N r j a j (x)e iσ j t + r(t, x), t 0, j=1 σ j C resonances (quasinormal modes) a j (x)e iσt resonant states, L(a j (x)e iσ j t ) = 0 r j C, r = O(e αt ), α > 0 fixed, small σ 2 C σ σ 1 σ N Iσ = α (Wunsch Zworski 11, Vasy 13, Dyatlov 15, H. 15)

36 Attempt #1 (Lr = 0, L = D g (Ric + Λ) δ g D g W.) Hope: N = 1, σ 1 = 0. σ 1 C σ Iσ = α

37 Attempt #1 (Lr = 0, L = D g (Ric + Λ) δ g D g W.) Hope: N = 1, σ 1 = 0. σ 1 C σ Iσ = α r = d ds g b 0 +sb s=0 + r.

38 Attempt #1 (Lr = 0, L = D g (Ric + Λ) δ g D g W.) Hope: N = 1, σ 1 = 0. σ 1 C σ Iσ = α Then: Could solve r = d ds g b 0 +sb s=0 + r. (Ric + Λ)(g b + g) δ g W (g b + g) = 0 for g = O(e αt ), b R 4. (H. Vasy 16)

39 Attempt #1 (Lr = 0, L = D g (Ric + Λ) δ g D g W.) Hope: N = 1, σ 1 = 0. σ 1 C σ Iσ = α Then: Could solve r = d ds g b 0 +sb s=0 + r. (Ric + Λ)(g b + g) δ g W (g b + g) = 0 for g = O(e αt ), b R 4. (H. Vasy 16) False!

40 Attempt #2 (Lr = 0, L = D g (Ric + Λ) δ g D g W.) Say N = 2, Im σ 2 > 0

41 Attempt #2 (Lr = 0, L = D g (Ric + Λ) δ g D g W.) Say N = 2, Im σ 2 > 0, and (ignoring linearized KdS family) r = r 2 a 2 (x)e iσ 2t + r(t, x).

42 Attempt #2 (Lr = 0, L = D g (Ric + Λ) δg D g W.) Say N = 2, Im σ 2 > 0, and (ignoring linearized KdS family) r = r 2 a 2 (x)e iσ2t + r(t, x). Hope: a 2 (x)e iσ2t = L V g is pure gauge

43 Attempt #2 (Lr = 0, L = D g (Ric + Λ) δg D g W.) Say N = 2, Im σ 2 > 0, and (ignoring linearized KdS family) r = r 2 a 2 (x)e iσ2t + r(t, x). Hope: a 2 (x)e iσ2t = L V g is pure gauge, so r = r 2 L χv g + r.

44 Attempt #2 (Lr = 0, L = D g (Ric + Λ) δg D g W.) Say N = 2, Im σ 2 > 0, and (ignoring linearized KdS family) r = r 2 a 2 (x)e iσ2t + r(t, x). Hope: a 2 (x)e iσ2t = L V g is pure gauge, so r = r 2 L χv g + r. L r = L(r 2 L χv g) = r 2 δ g D g W (L χv g) }{{}

45 Attempt #2 (Lr = 0, L = D g (Ric + Λ) δg D g W.) Say N = 2, Im σ 2 > 0, and (ignoring linearized KdS family) r = r 2 a 2 (x)e iσ2t + r(t, x). Hope: a 2 (x)e iσ2t = L V g is pure gauge, so r = r 2 L χv g + r. L r = L(r 2 L χv g) = r 2 δ g D g W (L χv g) }{{} =:θ

46 Attempt #2 (Lr = 0, L = D g (Ric + Λ) δg D g W.) Say N = 2, Im σ 2 > 0, and (ignoring linearized KdS family) r = r 2 a 2 (x)e iσ2t + r(t, x). Hope: a 2 (x)e iσ2t = L V g is pure gauge, so r = r 2 L χv g + r. L r = L(r 2 L χv g) = r 2 δ g D g W (L χv g) }{{} =:θ

47 Attempt #2 (Lr = 0, L = D g (Ric + Λ) δg D g W.) Say N = 2, Im σ 2 > 0, and (ignoring linearized KdS family) r = r 2 a 2 (x)e iσ2t + r(t, x). Hope: a 2 (x)e iσ2t = L V g is pure gauge, so r = r 2 L χv g + r. Thus: L r = L(r 2 L χv g) = r 2 δ g D g W (L χv g) }{{} =:θ D g (Ric + Λ)( r) δ g ( Dg W ( r) + r 2 θ ) = 0.

48 Attempt #2, cont. Then: Could solve (Ric + Λ)(g b + g) δ g ( W (gb + g) + θ ) = 0 for g and (b, θ) (finite-dimensional parameters).

49 Attempt #2, cont. Then: Could solve (Ric + Λ)(g b + g) δg ( W (gb + g) + θ ) = 0 for g and (b, θ) (finite-dimensional parameters). Hope is false!

50 Attempt #2, cont. Then: Could solve (Ric + Λ)(g b + g) δg ( 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.

51 Attempt #2, cont. Then: Could solve (Ric + Λ)(g b + g) δg ( 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 = D g (Ric + Λ) δg D g W

52 Attempt #2, cont. Then: Could solve (Ric + Λ)(g b + g) δg ( 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 = D g (Ric + Λ) δg D g W Delicate!

53 Fix? L = D g (Ric + Λ) δ D g W, where δ = δ g + 0th order.

54 Fix? L = D g (Ric + Λ) δ D g W, where δ = δg + 0th order. Apply linearized second Bianchi identity to 0 = L(a 2 (x)e iσ2t ) = D g (Ric + Λ)(a 2 e iσ2t ) δ D g W (a 2 e iσ2t ).

55 Fix? = δ g G g δ ( D g W (a 2 e iσ 2t ) ) = 0.

56 Fix? Theorem (H. Vasy 16) = δ g G g δ ( D g W (a 2 e iσ 2t ) ) = 0. For g the Schwarzschild de Sitter metric, one can choose δ such that δ g G g δ has no resonances with Im σ 0. ( Constraint damping, Gundlach et al ( 05).)

57 Fix? Theorem (H. Vasy 16) = δ g G g δ ( D g W (a 2 e iσ 2t ) ) = 0. For g the Schwarzschild de Sitter metric, one can choose δ such that δ g G g δ has no resonances with Im σ 0. ( Constraint damping, Gundlach et al ( 05).) Take δ w = δ g w + γ 1 dt s w γ 2 w( t)g, γ 1, γ 2 0.

58 Fix? Theorem (H. Vasy 16) = δ g G g δ ( D g W (a 2 e iσ 2t ) ) = 0. For g the Schwarzschild de Sitter metric, one can choose δ such that δ g G g δ has no resonances with Im σ 0. ( Constraint damping, Gundlach et al ( 05).) Take δ w = δ g w + γ 1 dt s w γ 2 w( t)g, γ 1, γ 2 0. = D g W (a 2 e iσ 2t ) = 0

59 Fix? Theorem (H. Vasy 16) = δ g G g δ ( D g W (a 2 e iσ 2t ) ) = 0. For g the Schwarzschild de Sitter metric, one can choose δ such that δ g G g δ has no resonances with Im σ 0. ( Constraint damping, Gundlach et al ( 05).) Take δ w = δ g w + γ 1 dt s w γ 2 w( t)g, γ 1, γ 2 0. = D g W (a 2 e iσ 2t ) = 0 = D g (Ric + Λ)(a 2 e iσ 2t ) = 0

60 Fix? Theorem (H. Vasy 16) = δ g G g δ ( D g W (a 2 e iσ 2t ) ) = 0. For g the Schwarzschild de Sitter metric, one can choose δ such that δ g G g δ has no resonances with Im σ 0. ( Constraint damping, Gundlach et al ( 05).) Take δ w = δ g w + γ 1 dt s w γ 2 w( t)g, γ 1, γ 2 0. = D g W (a 2 e iσ 2t ) = 0 = D g (Ric + Λ)(a 2 e iσ 2t ) = 0 Mode stability for linearized Einstein equation (Kodama Ishibashi 04)

61 Fix? Theorem (H. Vasy 16) = δ g G g δ ( D g W (a 2 e iσ 2t ) ) = 0. For g the Schwarzschild de Sitter metric, one can choose δ such that δ g G g δ has no resonances with Im σ 0. ( Constraint damping, Gundlach et al ( 05).) Take δ w = δ g w + γ 1 dt s w γ 2 w( t)g, γ 1, γ 2 0. = D g W (a 2 e iσ 2t ) = 0 = D g (Ric + Λ)(a 2 e iσ 2t ) = 0 Mode stability for linearized Einstein equation (Kodama Ishibashi 04) = a 2 e iσ 2t = L V g

62 Attempt #3 Solve (Ric + Λ)(g b + g) δ ( W (g b + g) + θ ) = 0 for g = O(e αt ) and (b, θ).

63 Attempt #3 Solve (Ric + Λ)(g b + g) δ ( W (g b + g) + θ ) = 0 for g = O(e αt ) and (b, θ). Almost: 1: (Ω, g b ) (Ω, g b0 ) is not a wave map

64 Attempt #3 Solve (Ric + Λ)(g b + g) δ ( W (g b + g) + θ ) = 0 for g = O(e αt ) and (b, θ). Almost: 1: (Ω, g b ) (Ω, g b0 ) is not a wave map: W (g b ) 0.

65 Final attempt Solve (Ric + Λ)(g b + g) δ ( W (g b + g) χw (g b ) + θ ) = 0.

66 Solution of the non-linear equation Nash Moser iteration scheme:

67 Solution of the non-linear equation Nash Moser iteration scheme: Solve linearized equation globally at each step.

68 Solution of the non-linear equation Nash Moser iteration scheme: Solve linearized equation globally at each step. Use linear solutions to update g, b and θ.

69 Solution of the non-linear equation Nash Moser iteration scheme: Solve linearized equation globally at each step. Use linear solutions to update g, b and θ. Automatically find final black hole parameters b, finite-dimensional modification θ of the gauge.

70 Outlook more precise asymptotic expansion, ringdown

71 Outlook more precise asymptotic expansion, ringdown stability for large a (mode stability?)

72 Outlook more precise asymptotic expansion, ringdown stability for large a (mode stability?) couple Einstein equation with matter

73 Outlook more precise asymptotic expansion, ringdown stability for large a (mode stability?) couple Einstein equation with matter Non-linear stability of the Kerr Newman de Sitter family (for small a) for the Einstein Maxwell system: work in progress

74 Outlook more precise asymptotic expansion, ringdown stability for large a (mode stability?) couple Einstein equation with matter Non-linear stability of the Kerr Newman de Sitter family (for small a) for the Einstein Maxwell system: work in progress Einstein (Maxwell )scalar field, Einstein Vlasov,...

75 Outlook more precise asymptotic expansion, ringdown stability for large a (mode stability?) couple Einstein equation with matter Non-linear stability of the Kerr Newman de Sitter family (for small a) for the Einstein Maxwell system: work in progress Einstein (Maxwell )scalar field, Einstein Vlasov,... higher-dimensional black holes (Myers Perry, Reall,... )

76 Outlook CH + i + I + r = r 1 Ω r = r 2 H + Σ H +

77 Outlook CH + i + I + r = r 1 Ω r = r 2 H + Σ H + analysis in the cosmological region (ongoing work by Schlue)

78 Outlook CH + i + I + r = r 1 Ω r = r 2 H + Σ H + analysis in the cosmological region (ongoing work by Schlue) Penrose s Strong Cosmic Censorship Conjecture (work by Dafermos Luk for Kerr)

79 Thank you!