The weak null condition, global existence and the asymptotic behavior of solutions to Einstein s equations

Transcription

1 The weak null condition, global existence and the asymptotic behavior of solutions to Einstein s equations

2 The Einstein vacuum equations determine a 4-d manifold M with a Lorentzian metric g, sign( 1, 1, 1, 1), with vanishing Ricci curvature: R µν = 0 The initial value problem: Given a 3-d manifold Σ, with Riemannian metric g 0, and a symmetric two-tensor k 0, we want to find a 4-d manifold (M, g) satisfying Einstein equations, and an imbedding Σ M such that g 0 is the restriction of g to Σ and k 0 is the second fundamental form of Σ. The initial data problem is over determined and data must satisfy the constraint equations: R 0 k i j 0j k 0 i + k 0 i i k 0 j j = 0, j k 0 ij j i k 0 j = 0. Here R 0 is the scalar curvature of g 0 and is covariant differentiation with respect to g 0. Einstein s equations are invariant under diffeomorphisms. Eliminate this freedom by fixing a gauge condition or system of coordinates.

3 Harmonic coordinates or wave coordinates are given as solutions of the wave equations g x µ = 0, where the geometric wave operator is g = α α = g αβ α β + g αβ Γαβ ν ν Here g αβ is the inverse of the metric, and Γ ν αβ are the Christoffel symbols for the metric. We can locally find wave coord. so x 0 =0 on Σ The metric in wave coordinates satisfy g αβ Γ ν αβ = gαβ β g αµ 1 2 gαβ µ g αβ = 0. (1) In wave coordinates, the vacuum Einstein equations are a system of nonlinear wave equations g αβ α β g µν = F µν (g)( g, g), (2) with F (u)(v, v) depending quadratically on v. ((1) is preserved under (2).) Initial data g t=0, t g t=0 (3) Local Existence (2)(3) Chouquet-Bruhat 1952 We use the summation convention over repeated indices. α = / x α and Greek indices α, β, µ, ν... = 0,..., 3

4 g αβ X α X β < 0, X-tangent vector Σ Causal curve x(s); g αβ ẋ α ẋ β 0, ẋ = dx/ds Future: ẋ 0 > 0, Past: ẋ 0 < 0. Globally hyperbolic space-times: Every inextendable causal curve intersects the initial surface Σ once and only once. (Any sol. constructed using an evolution eq.) Local solution: Globally hyperbolic, smooth Future casually geodesically complete Future geodesics x(s); g αβ ẋ α ẋ β = const 0, ẋ 0 > 0, can be extended forever 0 s <. Global Solution: Globally hyperbolic and future causally geodesically complete, smooth.

5 Global stability of Minkowski space Christodoulou-Klainerman (CK) Constructing a global solution from initial data which is close to and asymptotically approaching the Minkowski metric m = diag( 1, 1, 1, 1). Smallness assumption on data (Σ, g 0, k 0 ): Σ is diffeomorphic to R 3 and data are close to the data for Minkowski space (R 3, δ, 0). Initial data (R 3, g 0, k 0 ) are asymptotically flat: g 0 ij = (1+2M/r)δ ij+o(r 1 ε ), k 0 = o(r 2 ε ), ε > 0, when r = x. Here M > 0 by the positive mass theorem. CK assumed ε > 1/2. Other, restricted, global existence results: Friedrich, Klainerman-Nicolo All proofs avoid using wave coordinates; it was believed that these would be badly behaved in the large and possibly blow-up even if in a coordinate invariant formulation the curvature remained bounded. We will come back to this.

6 Global Existence in the wave coordinates L-Rodinanski (LR) Simpler shorter proof that works with matter. CK equation for curvature, no global coord. LR global equation for metric. CK no explicit null condition. LR weak null condition. (cancelation that makes it more likely it has global exist than generic eqns of the same form) CK constructs vector fields tangential to the curved light cones which a priori are unknown. LR use the vector fields of the Minkowski cones. (The vector fields are needed to get decay.) We will explain the role of the null condition and the vector fields below, which for Einstein s equations are both related to the geometry of the light cones.

7 Asymptotic behavior The metric approaches the Minkowski metric. CK-detailed asymptotic behavior for the different components of the curvature. In particular, CK show the existence of the Bondi mass, along light cones, and the Bondi mass law, that this decays to 0 in the interior. We can recover this but we need to use vector fields that are better adapted to the light cones, which diverge from the Minkowski ones The outgoing light cones approach those of the Schwarzwschild metric with the same ADM mass The Schwarzwschild cones are transformed the Minkowski ones with the Regge-Wheeler coord. We use the vector fields that commute with the flat wave operator in these coord. These only depend on the initial data through the ADM mass but do not depend on the solution.

8 Decay and the role of the Vector fields Light cones: union of null geodesics g αβ ẋ α ẋ β = 0 Geometry of light cones related to Behavior of solutions to PDE g αβ α β φ = F Singularities propagate along characteristics. Decay at infinity related to char. surfaces. φ = m αβ α β φ = 0 = φ C t, φ C t. (A wave starting from the origin will after time t be supported close to sphere S t of radius t with total energy φ 2 Area(S t ) φ 2 t 2 const.) t r t + r / φ=0 = φ C/t and /φ + ( t + r )φ C/t 2 Invariance of the wave operator under the Lorenz group. Generator Z = x a b x b a, [, Z] = 0, Zφ = 0 = Zφ C t, Z = r / = /φ C t 2. Generalize to a variable metric wave operator.

9 Einstein s equations in wave coordinates g αβ α β g µν = F µν (g)( g, g), (4) Stability around the Minkowski metric m = diag( 1, 1, 1, 1); h µν = g µν m µν is small. Generic systems of wave equations: φ I = c JK Iαβ φ J α β φ K + d JK Iαβ αφ J β φ K + cubic with small initial data.(here = m αβ α β.) Blow up for small data (John) e.g. φ = ( t φ) 2. Global existence if ciαβ JK = 0 and djk Iαβ satisfies the null condition (Christodoulou,Klainerman) e.g. φ = ( t φ) 2 x φ 2. Need εt 1 decay to handle general quadratic nonlinear terms. Problem:(4) does not satisfy the null condition (4) satisfy the weak null condition(lr). Essentially the system decouples in a null-frame φ 2 = ( t φ 1 ) 2, φ 1 = 0 φ 1 εt 1, φ 2 εt 1 ln t Global existence (L-radial case, Alinhac-general) φ = φ φ but solutions only decay like εt 1+cε. The weak null condition detect situations where the asymptotic behavior is not free.

10 The weak null condition for a generic system φ I = A JK I,αβ α φ J β φ K + cubic terms (5) is that the asymptotic system for Φ I =rφ I : ( t + r )( t r )Φ I r 1 A JK I,nm ( t r ) n Φ J ( t r ) m Φ K, (6) has global solutions for all small data. Here, A JK I,nm = 1 α =n, β =m ( 2) m+najk I,αβ ˆωαˆω β, ˆω =( 1, ω), ω S 2 The usual null condition is that AI,nm JK (ω) 0. The asymptotic system was introduced by Hörmander to find the time of blow-up. The asymptotic system (6) is obtained from the system (5) by neglecting derivatives tangential to the outgoing Minkowski light cones; t= x, and cubic terms, that are decaying faster φ=r 1 ( t + r )( t r )(rφ)+angular derivatives µ = 1 2ˆω µ( t r ) + tangential derivatives µ

11 t r t + r / φ=0 = φ C/t and /φ + ( t + r )φ C/t 2 A simple example of a system satisfying the weak null condition, violating the standard null condition and yet possessing global solutions is φ 1 = φ 3 2 φ 1 + ( φ 2 ) 2, φ 2 = 0, φ 3 = 0. Another example is provided by the equation (7) φ = φ φ. (8) The proof of small data global existence for this is very involved, [L-](radial case),[alinhac] The asymptotic system for Einstein s eq. can be modelled by that of (7). We will refer to φ 2 φ as the quasilinear terms and φ φ as the semilinear terms.

12 Einstein s eq. h = g m small g h µν = g αβ α β h µν = F µν (h)( h, h) = = P ( µ h, ν h)+q µν ( h, h)+g µν (h)( h, h), where Q µν are linear combinations of the standard null-forms and G µν (h)( h, h) is cubic, P (k, p) = 1 4 tr k tr p 1 2 tr tr(k, p), tr k = mαβ k αβ L A, B L tr tr(k, p) = m αβ m γδ k αγ p βδ Null-frame decomposition of Einstein s eq. We define a null-frame of vectors by L=( 1, ω), L=(1, ω) and A, B S 2 such that T ={L, A, B} span the tangent of the outgoing light cones and U ={A, B, L, L} span the full tangent space. In terms of the null-frame k UV = k αβ U α V β : P (l, k) = c ijkl l Ti U j k Tk U l 1 8 (l LLk LL + l LL k LL ) where the sum is over T i T and U i U. Parity cond. For each L comp. there is an L. m αβ k αβ = 1 2 (k LL + k LL ) + δ AB k AB Null cond: Q µν ( h, h) = m αβ Q µν ( α h, β h)

13 The asymptotic system for Einstein s eq. g h µν 1 4ˆω µˆω ν P ( h, h), = t r P ( h, h)= h T U h T U 1 4 h LL h LL Then T µˆω µ =0, for T T and L µˆω µ =2 so ( g h) L L P ( h, h) ( g h) T U 0, T T, U U Asymptotic form of wave coordinate cond. h LT 0, T T = {L, A, B}, (9) by expanding Γ µν µ = µ h µ ν+ 1 2 ν tr h + O(h h) in a null-frame: L h L ν + L h L ν + A h A ν ν tr h + O(h h) = 0 Hence as far as the semilinear terms it looks like the system φ 2 = ( φ 1 ) 2 where φ 1 = 0. The quasilinear part: Since g αβ =m αβ +h αβ it follows that the inverse g αβ =m αβ h αβ +O(h 2 ); g = g αβ α β 1 4 h LL 2, so ( h) LL 1 4 h LL 2 h LL. It appears as bad as φ=φ φ, but because of (9) h LL M/r.

14 For Einstein s eq. the whole energy tensor has to be estimated together but for the asymptotic system we can separate the components. What is used in the existence proof: Energy inequality φ(t, ) L 2 φ(0, ) L 2 + φ(s, ) 0 L 2 ds Since in our application φ = φ φ we see that if φ(t, x) Cε/t then the Energy E(t) = φ(t, ) L 2 satisfies E(t) E(0) + t t 0 Cε(1 + s) 1 E(s) ds which leads to E(t) (1 + t) Cε E(0). If instead φ(t, x) Cε ln t /t we would get E(t) (1 + t) Cε ln t E(0), which grows faster than any power of t. We really need ε/t decay for most components. Generalized energy inequality for curved space time with energies on light cones and weights

15 The Klainerman-Sobolev inequality (r = x ) φ(t, x) C I 2 ZI φ(τ, ) L 2 (1+t+r)(1+ t r ) 1/2, where Z I is a product of I vector fields of the form i, x i j x j i, t i + x i t that commute with and t t + x i i ; [, Z] = c Z. φ I =1 ZI φ /(1+t), φ Zφ /(1+ t r ) Sharp Decay estimate t (1+t) φ(t, ) L C (1+τ) g φ(τ, ) L dτ 0 t + C Z I φ(τ, ) dτ L 1+τ 0 I 2 This estimate is related to the asymptotic eq. It is obtained by regarding the angular derivatives as lower order and integrating the eq. 1 r ( t + r )( t r )(rφ) = φ + 1 r 2 ωφ L 1 L estimate (1+t) φ(t, ) L C t 0 I 2 g Z I φ(τ, ) L 1(1+τ) 1 dτ

16 Commute the equation with the vector fields to get similar equations and energy estimates for the vector fields applied to the solution. The commutator estimates If [Z, ]=0 then [Z, g ] = [Z, g αβ α β ]φ Expand in a nullframe Cε 1+t+r h αβ α β φ = h LL 2 L φ + h φ I 1 Z I φ, The decay, the energy and the commutator estimates for g hold under some weak decay obtained from the K-S ineq. and strong decay: h LT + Zh LL Cε(1 + t r )/(1 + t + r) obtained from the wave coordinate cond. C-K for Einstein s eq. and Alinhac for φ = φ φ had to modify the vector fields at infinity in order to get good commutators. For us the wave coordinate gauge makes the geometry of the characteristic surfaces be close that of the Minkowski, or rather the Schwarzschild, ones.

17 Structure of the proof: Energies E(t) = I N ZI φ(t, ) L 2. Assuming decay like φ Cε(1 + t) 1 some components the Gen. energy ineq. t E(t) E(0) + Cε(1 + 0 τ) 1 E(τ) dτ +... gives energy estimate E(t) (1 + t) Cε. for The energy est. and the K-S ineq. gives weak decay est. φ Cε(1+t) 1+Cε (1 + t r ) 1/2 Integrating the weak decay est. applied to Zφ gives full decay for derivatives tangential to the outgoing Minkowski light cones φ Cε(1+t) 2+Cε (1 + t r ) 1/2 Nonlinear Estimates using the equations and the weak decay estimates gives strong decay estimates φ Cε(1 + t) 1 for some comp. The strong decay estimates can then be feed back into the energy inequality.

18 Wave coordinate condition (h = g m) µ h µ ν = 1 2 ν tr h + h h Expand in a null-frame: L h L ν + L h L ν + A h A ν = 1 2 ν tr h + h h Hence L h LT h + h h so It follows that h LT C h + C h h h LT Cεt 2+Cε We further decompose the metric g = m + h 0 + h 1 where h 0 is an explicit function picking up the main decay at space-like infinity: h 0 αβ = χ( r 1+t ) M 1 + r δ αβ, where χ(s) = 1, s > 1/2 and χ(s) = 0, s < 1/4 and approximately satisfies the wave equations.

19 We get wave equations for h 1, but since h 1 decays more initially we can put exterior weights in the energy which give more exterior decay: h 1 (1 + t) 1+Cε (1 + (r t) + ) γ Since h 0 also approximately satisfies the wave coordinate condition so does h 1 and this gives better decay from the wave coord cond: h 1 LT Cεt 2+Cε (1 + (r t) + ) γ and integrating this gives h 1 LT Cεt 1 ((1 + (t r) + )/(1 + t)) γ We conclude that h LL = M ( ) r + O( ε 1 + (t r)+ γ) 1 + t 1 + t From this we conclude that g = +h αβ α β +h LL 2 L +M r 2 L S the wave operator in the Schwarzschild metric.

20 Expand in a nullframe h αβ α β φ = h LL 2 L φ + h φ In the Regge-Wheeler modified coordinates r = r + Mχ(r/t) ln r, x = r ω the Schwarzschild wave operator is close to the constant coefficient wave operator in the wave zone r > t/2: S and we get g = 2 t + x = 2 t + 2 r + 2 r r + 1 r 2 ω Use the vector fields Z that commutes with.

21 Bounds for the metric If T is tangential In general Z I h 1 where Z I h 1 T U ε (1 + t + r )(1 + q + )γ ε 2 S 0 (t, r ) (1 + t + r )(1 + q+ )1 Cε ε t + r (1 + q ) γ S 0 (t, r ) = t r ln ( 1 + (t + r ) (t r ) 2 ) + 1 For the component from the wave coordinates Z I h 1 LT ( ε 1 + q ) γ 1 + t + r 1 + t + r Here 0 < γ < 1 depending on decay of data.

22 Asymptotic for the characteristics surfaces and metric t = r + M ln r + O(r 1 ) = r + O(r 1 ) where the first term is the characteristic surfaces for the Schwarzschild metric. asymptotic for the metric along characteristics of the Schwarzschild metric.

23 The asymptotic system for Einstein s eq. h µν 1 4 L µl ν P ( h, h), = t r P ( h, h)= h T U h T U 1 4 h LL h LL ( h) T U 0, T T, U U If φ = 0 then φ(t, r ω) Φ(t r, ω)/r. H 1 T U (q, ω) = lim r r h 1 T U (r q, r ω) T tangential component. Moreover where P ( h, h) n(q, ω)/(r ) 2 n(q, ω) = P ( q H 1, q H 1 )(q, ω) = 1 2 δab δ A B q H1 AA (q, ω) q H1 BB (q, ω) 0. contains the space tangential components.

24 In conclusion h µν L µ L ν n(q, ω)/(r ) 2 Explicit formula Lµ (ω)l ν (ω)n(τ, ω) h µν r t t + τ x, ω The limit lim r exist, where N ( ) ds(ω) χ τ 2 t+r dτ. ) (r h 1 µν N q µν (τ, ω, r )dτ µν (q, ω, r ) = L µ (ω)l ν (ω)n(q, ω)s 0 (t, r ), where S 0 (t, r) = t r ln ( 1+(t+r) 2 1+(t r) 2 ) + 1,

25 The Trautman-Bondi mass Change the metric to the coordinates g µν. Einstein-vacuum equations R αβ (g) = 0 can be written in the form where λ αβµ = y ν g π αβ = λαβµ y µ, ( g (g αβ g µν g αµ g βν ) ) = λ αµβ and π αβ is the Landau-Lifshitz pseudo tensor... The tensor π αβ = π βα is symmetric and due to the anti-symmetry of λ αβγ satisfies the divergence free condition β ( g π αβ) = 0. We show that the limit exist (q = r t) m α T (q, ω) = lim r (r ) 2 ( λ αβγ ) L γ L β (r, q, ω)

26 and that the mass MT α (q ) = S 2 mα T (q, ω) ds(ω) satisfies the mass law M α T (q 2 ) = M a T (q 1 ) 1 4 where q 2 q 1 S 2 Lα q Ĥ 1 2 dω dq H 1 µν (q, ω) = lim r r h 1 µν (r q, r ω) and ĤAB 1 = H1 AB 1 2 δ ABδ CD HCD 1 s the traceless part of of the angular part: q Ĥ 1 2 = δ AB δ A B q H1 AA (q, ω) q H1 BB (q, ω)

27 Asymptotic for the curvature [CK]: R ALBL t 1, R LLBL t 2, R LLLL t, 3 and, if S, T, S, T are tangential: R ST UV t 2, R ST S T t 3. Some components decay even more t 4, t 5, but it requires additional decay of initial data. [LR]: Recover this up to t 3 using the vector fields of Schwarzschild metric (depending on M) R 2 g + g g t 1 (1 + t r ) 2 By the Bianchi identity 0 = D α R αβµν = L α D L R αβµν + L α D L R αβµν 2 A Ã α D A R αβµν, where we expressed the divergence in null frame. D L R Lβµν R t 2 (1 + t r ) 2 Integrate up R Lβµν t 2 (1 + t r ) 1 Similarly R LβLν t 3

28 Future work and the weak null condition in other contexts: More detailed asymptotics O(t 3 ) (L-Rodnianski) Einstein s eq. coupled to a charged scalar field. Asymtotics for Maxwell-Klein Gordon (L-Sterbenz) using generalized Morawetz space-time estimates. Einstein s equations coupled to Vlasov fluids. Global existence for general wave equations satisfying the weak null condition. In particular for Eisntein s equations without using the wave coordinate condition. I proved global existence for φ = c αβ φ α β φ without using modified vector fields. Weak null condition for other equations. Nonlinear Klein-Gordon, Schroedinger (L-Soffer)

29 Delort proved that for small initial data the nonlinear Klein-Gordon equations: v + v = βv 3, (t, x) R 1+1 have global solutions with asymptotics, as ϱ = (t 2 x 2 ) 1/2, with log-corrections ( v(t, x) ρ 1 2 e iφ(ϱ, x/ϱ) a(x/ϱ)+e iφ(ϱ, x/ϱ) a(x/ϱ) φ(ϱ, x/ϱ)=ϱ β a(x/ϱ ) 2 ln ϱ We [LSof] consider the inverse problem of scattering, we show that for any given smooth arbitrarily large asymptotic expansion of this form there is a solution agreeing with it at infinity. We also give a complete asymptotic expansion. For the proof we start by introducing the hyperbolic coordinates ϱ 2 = t 2 x 2, t = ϱ cosh y, x = ϱ sinh y, v(t, x) = ϱ 1/2 V (ϱ, y) and the equation become 2 ϱ V + ( 1 + βϱ 1 V 2 + ϱ 2 /4 ) V ϱ 2 2 y V = 0 By neglecting the last term we get and ODE and this gives the first term in the expansion. This idea is similar to the weak null condition. ),