Global stability problems in General Relativity

Transcription

1 Global stability problems in General Relativity Peter Hintz with András Vasy Murramarang March 21, 2018

2 Einstein vacuum equations Ric(g) + Λg = 0. g: Lorentzian metric (+ ) on 4-manifold M Λ R: cosmological constant Examples. Minkowski (Λ = 0): M = R 4 = R t R 3 x, g = dt 2 dx 2. Schwarzschild (Λ = 0) and Schwarzschild de Sitter (Λ > 0): M = R t I r S 2, g = f (r)dt 2 1 f (r) dr 2 r 2 g S 2, where f (r) = 1 2M r Λr 2 3, M > 0 black hole mass.

3 Initial value problem for the Einstein vacuum equation Data: Σ: 3-manifold h: Riemannian metric on Σ k: symmetric 2-tensor on Σ Find spacetime (M, g), Σ M, 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

4 Kerr de Sitter family: Ric(g) + Λg = 0, Λ > 0 manifold: M = [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 r = r1 Ω r = r2 Σ r = r r = r + Special case. b 0 = (M, 0): Schwarzschild de Sitter

5 Black hole stability (Λ > 0) 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 M 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!

6 Setup for Minkowski stability: Ric(g) = 0 (Λ = 0) Minkowski spacetime. M = R 4 = R t R 3 x, g = g Mink = dt 2 dx 2. Polar coordinates in R 3 x: g Mink = dt 2 dr 2 r 2 g S 2 Outgoing null direction: t + r Light cones: s 0 := t r = const t s 0 = 1 s 0 = 0 x

7 Schwarzschild spacetime: far field of system with mass M > 0. M = {(t, x): r = x R}, R 1; g = g M S = (1 2M r )dt 2 (1 2M r ) 1 dr 2 r 2 g S 2. Perturbation of Minkowski: g Mink g M S = O(r 1 ). Outgoing null direction: t + (1 2M r ) r Light cones: s := t r = const, r r 2M log r t s = 1 s = 0 r

8 Stability of Minkowski space (Λ = 0) Theorem (H. Vasy 17; Christodoulou Klainerman 93, Klainerman Nicolò 03, Lindblad Rodnianski 05, 10, Bieri 09, Lindblad 17,... ) Given: M R and smooth data (γ, k) on Σ = R 3 such that: γ γ M S δ(1 + r) 1 b, r 1, k k M S δ(1 + r) 2 b, M δ, and γ γ Mink δ, r 1, where δ > 0 small, b > 0 fixed. Then: geodesically complete solution g of the IVP for Ric(g) = 0 on M = R 4, g g Mink δ(1 + t + r) 1+ɛ ɛ > 0, with a precise asymptotic description on a compactification of R 4.

9 Related work Non-linear stability: de Sitter: Friedrich ( 80s), Ringström ( 08),... Hyperbolic space: Graham Lee ( 91),... Schwarzschild+symmetry: Klainerman Szeftel (in progress) Linear (mode) stability of black holes: Λ > 0: Kodama Ishibashi ( 04) Λ = 0: Regge Wheeler ( 57), Whiting ( 89), Shlapentokh- Rothman ( 14), Dafermos Holzegel Rodnianski ( 16) Compactification and precise asymptotics on Minkowski: Wang 14 Baskin Vasy Wunsch 15, 16

10 Gauge fixing: generalities Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (M, g) (M, g b0 ) be a wave map W (g) = Reduced Einstein equation: ( ) 1-form only = 0. involving g, g (Ric + Λ)(g) δ g 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.

11 Gauge fixing: generalities Goal: Solve Ric(g) + Λg = 0. Friedrich: Demand that W (g) = Reduced Einstein equation: ( ) 1-form only = θ. involving g, g (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.

12 Gauge fixing: generalities Goal: Solve Ric(g) + Λg = 0. DeTurck: Demand that 1: (M, g) (M, g b0 ) be a wave map W (g) = Reduced Einstein equation: ( ) 1-form only = 0. involving g, g (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.

13 Constraint damping Why choose δ := δg + l.o.t. carefully in P(g) := (Ric + Λ)(g) δ W (g) = 0? Numerical relativity. want: solution W of wave-type equation δ g G g δ W = 0 decays when W is initially small. (Gundlach et al 05)

14 Constraint damping, P(g) = (Ric + Λ)(g) δ W (g) Show that one can choose δ so that: Solutions W of δ g G g δ W = 0 decay (fast). Analysis. Use Newton iteration, D g P(h) = P(g), g g + h. Example 1 (Minkowski). If (Ric + Λ)(g) 0, then δ g G g δ (D g W (h)) 0 parts of h have better decay. Example 2 (black holes). Control of mode solutions: (Ric + Λ)(g) = 0 and D g P(e ct h(x)) = 0, Re c 0 D g W (e ct h(x)) 0 D g (Ric + Λ)(e ct h(x)) 0, so e ct h(x) satisfies a geometric equation!

15 Black hole stability (Λ > 0) Theorem Given C initial data (h, k) on Σ close to the data of the Schwarzschild de Sitter metric g b0, there exist a C metric g on M 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. Use Newton iteration for IVP for P(g) = (Ric + Λ)(g) δ W (G) = 0. Solve linear equation globally at each step!

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

17 In an ideal world... (Lh = 0, L = D g (Ric + Λ) δ W.) Hope: N = 1, σ 1 = 0. σ 1 C σ h = d ds g b 0 +sb s=0 + h. Iσ = α

18 In an ideal world... (Lh = 0, L = D g (Ric + Λ) δ W.) Hope: N = 1, σ 1 = 0. σ 1 C σ Iσ = α Then: Could solve h = d ds g b 0 +sb s=0 + h. (Ric + Λ)(g b + g) δ W (g b + g) = 0 for g = O(e αt ), b R 4. (H. Vasy 16)

19 In an ideal world... (Lh = 0, L = D g (Ric + Λ) δ W.) Hope: N = 1, σ 1 = 0. σ 1 C σ Iσ = α Then: Could solve h = d ds g b 0 +sb s=0 + h. (Ric + Λ)(g b + g) δ W (g b + g) = 0 for g = O(e αt ), b R 4. (H. Vasy 16) Newton iteration; solve linearized equation globally at each step. Automatically find final black hole parameters b and tail g.

20 In an ideal world... (Lh = 0, L = D g (Ric + Λ) δ W.) Hope: N = 1, σ 1 = 0. σ 1 C σ Iσ = α Then: Could solve h = d ds g b 0 +sb s=0 + h. (Ric + Λ)(g b + g) δ W (g b + g) = 0 for g = O(e αt ), b R 4. (H. Vasy 16) Newton iteration; solve linearized equation globally at each step. Automatically find final black hole parameters b and tail g. Difficult to verify!

21 Dealing with reality... (Lh = 0, L = D g (Ric + Λ) δ W.) Say N = 2, Im σ 2 > 0, and (ignoring linearized KdS family) h = a 2 (x)e iσ2t + h(t, x). Recall: a 2 (x)e iσ2t ker D g (Ric + Λ) (constraint damping!) Kodama Ishibashi 04 a 2 (x)e iσ2t = L V g is pure gauge

22 Dealing with reality... (Lh = 0, L = D g (Ric + Λ) δ W.) Say N = 2, Im σ 2 > 0, and (ignoring linearized KdS family) h = a 2 (x)e iσ 2t + h(t, x). Recall: a 2 (x)e iσ 2t ker D g (Ric + Λ) (constraint damping!) Kodama Ishibashi 04 a 2 (x)e iσ 2t = L V g is pure gauge, so h = L χv g + h. Find equation for interesting bit h: Thus: L h = L(L χv g) = δ W (L χv g) }{{} θ D g (Ric + Λ)( h) δ ( W ( h) + θ ) = 0.

23 Improved setup Then: Can solve (Ric + Λ)(g b + g) δ ( W (g b + g) + θ ) = 0 for g and (b, θ) (finite-dimensional parameters). 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.

24 How to arrange constraint damping? Theorem (H. Vasy 16) 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. Take δ w = δ g w + 1 dt s w w( t)g, 0 < 1. Idea of proof. 2 2 δ g G g δ 2 g il, L i grad t. High frequencies: propagation along null-geodesic flow Low frequencies: L i 2 g, transport along grad t; exponential decay due to sign of subprincipal symbol of L

25 Further applications of the global point of view Non-linear stability of: slowly rotating Kerr Newman de Sitter black holes (H., 16) Minkowski space (H. Vasy, 17)

26 Thank you!