Tài liệu Đề tài " The causal structure of microlocalized rough Einstein metrics " pptx

Annals of Mathematics The causal structure of microlocalized rough Einstein metrics By Sergiu Klainerman and Igor Rodnianski Annals of Mathematics, 161 (2005), 1195–1243 The causal structure of microlocalized rough Einstein metrics By Sergiu Klainerman and Igor Rodnianski Abstract This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. In this paper we develop the geometric analysis of the Eikonal equation for microlocalized rough Einstein metrics. This is a crucial step in the derivation of the decay estimates needed in the first paper. 1. Introduction This is the second in a series of three papers in which we initiate the study of very rough solutions of the Einstein vacuum equations. By very rough we mean solutions which cannot be dealt with by the classical techniques of energy estimates and Sobolev inequalities. In fact in this work we develop and take advantage of Strichartz-type estimates. The result, stated in our first paper [Kl-Ro1], is in fact optimal with respect to the full potential of such estimates. 1 We recall below our main result: Theorem 1.1 (Main Theorem). Let g be a classical solution 2 of the Einstein equations R αβ (g)=0(1) expressed 3 relative to wave coordinates x α ,  g x α = 1 |g| ∂ μ (g μν |g|∂ ν )x α =0.(2) 1 To go beyond our result will require the development of bilinear techniques for the Ein- stein equations; see the discussion in the introduction to [Kl-Ro1]. 2 We denote by R αβ the Ricci curvature of g. 3 In wave coordinates the Einstein equations take the reduced form g αβ ∂ α ∂ β g μν = N μν (g,∂g) with N quadratic in the first derivatives ∂g of the metric. 1196 SERGIU KLAINERMAN AND IGOR RODNIANSKI Assume that on the initial spacelike hyperplane Σ given by t = x 0 =0, ∇g αβ (0) ∈ H s−1 (Σ) ,∂ t g αβ (0) ∈ H s−1 (Σ) with ∇ denoting the gradient with respect to the space coordinates x i , i =1, 2, 3 and H s the standard Sobolev spaces. Also assume that g αβ (0) is a continuous Lorentz metric and sup |x|=r |g αβ (0) − m αβ |−→0 as r −→ ∞ , where |x| = (  3 i=1 |x i | 2 ) 1 2 and m αβ is the Minkowski metric. Then 4 the time T of existence depends in fact only on the size of the norm ∂g μν (0) H s−1 (Σ) = ∇g μν (0) H s−1 (Σ) + ∂ t g μν (0) H s−1 (Σ) , for any fixed s>2. In [Kl-Ro1] we have given a detailed proof of the theorem by relying heav- ily on a result, which we have called the Asymptotics Theorem, concerning the geometric properties of the causal structure of appropriately microlocal- ized rough Einstein metrics. This result, which is the focus of this paper, is of independent interest as it requires the development of new geometric and analytic methods to deal with characteristic surfaces of the Einstein metrics. More precisely we study the solutions, called optical functions, of the Eikonal equation H αβ (λ) ∂ α u∂ β u =0,(3) associated to the family of regularized Lorentz metrics H (λ) , λ ∈ 2 N , defined, starting with an H 2+ε Einstein metric g, by the formula H (λ) = P <λ g(λ −1 t, λ −1 x)(4) where 5 P <λ is an operator which cuts off all the frequencies above 6 λ. The importance of the eikonal equation (3) in the study of solutions to wave equations on a background Lorentz metric H is well known. It is mainly used, in the geometric optics approximation, to construct parametrices asso- ciated to the corresponding linear operator  H . In particular it has played a fundamental role in the recent works of Smith[Sm], Bahouri-Chemin [Ba-Ch1], [Ba-Ch2] and Tataru [Ta1], [Ta2] concerning rough solutions to linear and nonlinear wave equations. Their work relies indeed on parametrices defined with the help of specific families of optical functions corresponding to null 4 We assume however that T stays sufficiently small, e.g. T ≤ 1. This a purely technical assumption which one should be able to remove. 5 More precisely, for a given function of the spatial variables x = x 1 ,x 2 ,x 3 , the Littlewood Paley projection P <λ f =  μ< 1 2 λ P μ f, P μ f = F −1  χ(μ −1 ξ) ˆ f(ξ)  with χ supported in the unit dyadic region 1 2 ≤|ξ|≤2. 6 The definition of the projector P <λ in [Kl-Ro1] was slightly different from the one we are using in this paper. There P <λ removed all the frequencies above 2 −M 0 λ for some sufficiently large constant M 0 . It is clear that a simple rescaling can remedy this discrepancy. ROUGH EINSTEIN METRICS 1197 hyperplanes. In [Kl], [Kl-Ro], and also [Kl-Ro1] which do not rely on specific parametrices, a special optical function, corresponding to null cones with ver- tices on a timelike geodesic, was used to construct an almost conformal Killing vectorfield. The main message of our paper is that optical functions associated to Einstein metrics, or microlocalized versions of them, have better properties. This fact was already recognized in [Ch-Kl] where the construction of an opti- cal function normalized at infinity played a crucial role in the proof of the global nonlinear stability of the Minkowski space. A similar construction, based on two optical functions, can be found in [Kl-Ni]. Here, we take the use of the spe- cial structure of the Einstein equations one step further by deriving unexpected regularity properties of optical functions which are essential in the proof of the Main Theorem. It was well known (see [Ch-Kl], [Kl], [Kl-Ro]) that the use of Codazzi equations combined with the Raychaudhuri equation for the trχ, the trace of null second fundamental form χ, leads to the improved estimate for the first angular derivatives of the traceless part of χ. A similar observation holds for another null component of the Hessian of the optical function, η. The role of the Raychaudhuri equation is taken by the transport equation for the “mass aspect function” μ. In this paper we show, using the structure of the curvature terms in the main equations, how to derive improved regularity estimates for the undiffer- entiated quantities ˆχ and η. In particular, in the case of the estimates for η we are led to introduce a new nonlocal quantity μ/ tied to μ via a Hodge system. The properties of the optical function are given in detail in the statement of the Asymptotics Theorem. We shall give a precise statement of it in Section 2 after we introduce a few essential definitions. The paper is organized as follows: In Section 2 we construct an optical function u, constant on null cones with vertices on a fixed timelike geodesic, and describe our basic geometric entities associated to it. We define the surfaces S t,u , the canonical null pair L, L and the associated Ricci coefficients. This allows us to give a precise statement of our main result, the Asymptotic Theorem 2.5. In Section 3 we derive the structure equations for the Ricci coefficients. These equations are a coupled system of the transport and Codazzi equations and are fundamental for the proof of Theorem 2.5. In Section 4 we obtain some crucial properties of the components of the Riemann curvature tensor R αβγδ . The remaining sections are occupied with the proof of the Asymptotics Theorem. We give a detailed description of their content and the strategy of the proof in Section 5. The paper is essentially self-contained. From the first paper in this series [Kl-Ro1] we only need the result of Proposition 2.4 (Background Estimates) which in any case can be easily derived from the the metric hypothesis (5), the 1198 SERGIU KLAINERMAN AND IGOR RODNIANSKI Ricci condition (1), and the definition (4). We do however rely on the following results: • Isoperimetric and trace inequalities, see Proposition 6.16. • Calderon-Zygmund type estimates, see Proposition 6.20. • Theorem 8.1. The proof of the important Theorem 8.1 is delayed to our third paper in the series [Kl-Ro2]. The first two ingredients are standard modifications of the classical isoperimetric and Calderon-Zygmund estimates; see [Kl-Ro]. We recall our metric hypothesis (referred in [Kl-Ro1, §2] as the bootstrap hypothesis) on the components of g relative to our wave coordinates x α . Metric Hypothesis. ∂g L ∞ [0,T ] H 1+γ + ∂g L 2 [0,T ] L ∞ x ≤ B 0 ,(5) for some fixed γ>0. 2. Geometric preliminaries We start by recalling the basic geometric constructions associated with a Lorentz metric H = H (λ) . Recall, see [Kl-Ro1, §2], that the parameters of the Σ t foliation are given by n, v, the induced metric h and the second fundamental form k ij , according to the decomposition, H = −n 2 dt 2 + h ij (dx i + v i dt) ⊗ (dx j + v j dt),(6) with h ij the induced Riemannian metric on Σ t , n the lapse and v = v i ∂ i the shift of H. Denoting by T the unit, future oriented, normal to Σ t and k the second fundamental form k ij = −D i T,∂ j  we find, ∂ t = nT + v, ∂ t ,v =0,(7) k ij = − 1 2 L T H ij = − 1 2 n −1 (∂ t h ij −L v h ij ) with L X denoting the Lie derivative with respect to the vectorfield X. We also have the following; see [Kl-Ro1, §§2, 8]: c|ξ| 2 ≤ h ij ξ i ξ j ≤ c −1 |ξ| 2 ,c≤ n 2 −|v| 2 h (8) for some c>0. Also n, |v|  1. The time axis is defined as the integral curve of the forward unit normal T to the hypersurfaces Σ t . The point Γ t is the intersection between Γ and Σ t . ROUGH EINSTEIN METRICS 1199 Definition 2.1. The optical function u is an outgoing solution of the Eikonal equation H αβ ∂ α u∂ β u =0(9) with initial conditions u(Γ t )=t on the time axis. The level surfaces of u, denoted C u , are outgoing null cones with vertices on the time axis. Clearly, T (u)=|∇u| h (10) where h is the induced metric on Σ t , |∇u| 2 h =  3 i=1 |e i (u)| 2 relative to an orthonormal frame e i on Σ t . We denote by S t,u the surfaces of intersection between Σ t and C u . They play a fundamental role in our discussion. Definition 2.2 (Canonical null pair). L = bL  = T + N, L =2T − L = T − N.(11) Here L  = −H αβ ∂ β u∂ α is the geodesic null generator of C u , b is the lapse of the null foliation (or shortly null lapse) b −1 = −L  ,T = T(u),(12) and N is the exterior unit normal, along Σ t , to the surfaces S t,u . Definition 2.3. A null frame e 1 ,e 2 ,e 3 ,e 4 at a point p ∈ S t,u consists, in addition to the null pair e 3 = L,e 4 = L,ofarbitrary orthonormal vectors e 1 ,e 2 tangent to S t,u . All the estimates in this paper are in fact local and independent of the choice of a particular frame. We do not need to worry that these frames cannot be globally defined. Definition 2.4 (Ricci coefficients). Let e 1 ,e 2 ,e 3 ,e 4 be a null frame on S t,u as above. The following tensors on S t,u χ AB = D A e 4 ,e B ,χ AB = D A e 3 ,e B ,(13) η A = 1 2 D 3 e 4 ,e A ,η A = 1 2 D 4 e 3 ,e A , ξ A = 1 2 D 3 e 3 ,e A  are called the Ricci coefficients associated to our canonical null pair. We decompose χ and χ into their trace and traceless components. trχ = H AB χ AB , trχ = H AB χ AB ,(14) ˆχ AB = χ AB − 1 2 trχH AB , ˆχ AB = χ AB − 1 2 trχ H AB .(15) 1200 SERGIU KLAINERMAN AND IGOR RODNIANSKI We define s to be the affine parameter of L, i.e. L(s) = 1 and s =0on the time axis Γ t . In [Kl-Ro], where n = 1 we had s = t − u. Such a simple relation does not hold in this case; we have instead, along any fixed C u , dt ds = n −1 .(16) We shall also introduce the area A(t, u) of the 2-surface S(t, u) and the radius r(t, u) defined by A =4πr 2 .(17) Along a given C u we have 7 ∂A ∂t =  S ntrχ. Therefore, along C u , dr dt = r 2 ntrχ(18) where, given a function f, we denote by ¯ f(t, u) its average on S t,u .Thus ¯ f(t, u)= 1 4πr 2  S t,u f. The following Ricci equations can also be easily derived (see [Kl-Ro]). They express the covariant derivatives D of the null frame (e A ) A=1,2 ,e 3 ,e 4 relative to itself. D A e 4 = χ AB e B − k AN e 4 , D A e 3 = χ AB e B + k AN e 3 ,(19) D 4 e 4 = − ¯ k NN e 4 , D 4 e 3 =2η A e A + ¯ k NN e 3 , D 3 e 4 =2η A e A + ¯ k NN e 4 , D 3 e 3 =2ξ A e A − ¯ k NN e 3 , D 4 e A = D/ 4 e A + η A e 4 , D 3 e A = D/ 3 e A + η A e 3 + ξ A e 4 , D B e A = ∇/ B e A + 1 2 χ AB e 3 + 1 2 χ AB e 4 where, D/ 3 , D/ 4 denote the projection on S t,u of D 3 and D 4 , ∇/ denotes the induced covariant derivative on S t,u and, for every vector X tangent to Σ t , ¯ k NX = k NX − n −1 ∇ X n.(20) Thus ¯ k NN = k NN − n −1 N(n) and ¯ k AN = k AN − n −1 ∇ A n. Also, χ AB = −χ AB − 2k AB ,(21) η A v = − ¯ k AN , ξ A = k AN + n −1 ∇ A n − η A 7 This follows by writing the metric on S t,u in the form γ AB (s(t, θ),θ)dθ a dθ B , rela- tive to angular coordinates θ 1 ,θ 2 , and its area A(t, u)=  √ γdθ 1 ∧ dθ 2 . Thus, d dt A =  1 2 γ AB d dt γ AB √ γdθ 1 ∧ dθ 2 . On the other hand d ds γ AB =2χ AB and ds dt = n. ROUGH EINSTEIN METRICS 1201 and, η A = b −1 ∇/ A b + k AN .(22) The formulas (19), (21) and (22) can be checked in precisely the same manner as (2.45–2.53) in [Kl-Ro]. The only difference occurs because D T T no longer vanishes. We have in fact, relative to any orthonormal frame e i on Σ t , D T T = n −1 e i (n)e i .(23) To check (23) observe that we can introduce new local coordinates ¯x i =¯x i (t, x) on Σ t which preserve the lapse n while making the shift V to vanish identically. Thus ∂ t = nT and therefore, for an arbitrary vectorfield X tangent to Σ t , we easily calculate, D T T,X = n −2 X i D ∂ t ∂ t ,∂ i  = −n −2 X i ∂ t , D ∂ t ∂ i  = −n −2 X i ∂ t , D ∂ i ∂ t  = −n −2 X i 1 2 ∂ i ∂ t ,∂ t  = n −2 X i 1 2 ∂ i (n 2 )=n −1 X(n). Equations (21) indicate that the only independent geometric quantities, besides n, v and k are trχ, ˆχ, η. We now state the main result of our paper giving the precise description of the Ricci coefficients. Note that a subset of these estimates was stated in Theorem 4.5 of [Kl-Ro1]. Theorem 2.5. Let g be an Einstein metric obeying the Metric Hypothesis (5) and H = H (λ) be the family of the regularized Lorentz metrics defined according to (4). Fix a sufficiently large value of the dyadic parameter λ and consider, corresponding to H = H (λ) , the optical function u defined above. Let I + 0 be the future domain of the origin on Σ 0 . Then for any ε 0 > 0, such that 5ε 0 <γwith γ from (5), the optical function u can be extended throughout the region I + 0 ∩ ([0,λ 1−8ε 0 ] × R 3 ) and there the Ricci coefficients trχ,ˆχ, and η satisfy the following estimates:     trχ − 2 r     L 2 t L ∞ x + ˆχ L 2 t L ∞ x + η L 2 t L ∞ x  λ − 1 2 −3ε 0 ,(24)     trχ − 2 r  L q (S t,u ) +     ˆχ L q (S t,u ) + η L q (S t,u )  λ −3ε 0 ,(25) with 2 ≤ q ≤ 4. In the estimate (118) the function 2 r can be replaced with 2 n(t−u) . In addition, in the exterior region r ≥ t/2,     trχ − 2 s     L ∞ (S t,u )  t −1 λ −4ε 0 , ˆχ L ∞ (S t,u )  t −1 λ −ε 0 + ∂H(t) L ∞ x ,(26) η L ∞ (S t,u )  λ −1 + λ −ε 0 t −1 + λ ε ∂H(t) L ∞ x where the last estimate holds for an arbitrary positive ε, ε<ε 0 . Also, there exist the following estimates for the derivatives of trχ: 1202 SERGIU KLAINERMAN AND IGOR RODNIANSKI (27) sup r≥ t 2 L  trχ − 2 r   L 2 (S t,u )  L 1 t +  sup r≥ t 2 L  trχ − 2 n(t − u)   L 2 (S t,u )  L 1 t ≤ λ −3ε 0 ,  sup r≥ t 2 ∇/ trχ L 2 (S t,u )  L 1 t +  sup r≥ t 2 ∇/  trχ − 2 n(t − u)   L 2 (S t,u )  L 1 t ≤ λ −3ε 0 . (28) In addition, there are weak estimates of the form, sup u≤ t 2     (∇/,L )  trχ − 2 n(t − u)      L ∞ (S t,u )  λ C (29) for some large value of C. The inequalities  indicate that the bounds hold with some universal con- stants including the constant B 0 from (5). 3. Null structure equations In the proof of Theorem 2.5 we rely on the system of equations satisfied by the Ricci coefficients χ, η. Below we write down our main structural equa- tions. Their derivation proceeds in exactly the same way as in [Kl-Ro] (see Propositions 2.2 and 2.3) from the formulas (19) above. Proposition 3.1. The components trχ, ˆχ, η and the lapse b verify the following equations: 8 L(b)=−b ¯ k NN ,(30) L(trχ)+ 1 2 (trχ) 2 = −|ˆχ| 2 − ¯ k NN trχ − R 44 ,(31) D/ 4 ˆχ AB + 1 2 trχˆχ AB = − ¯ k NN ˆχ AB − ˆα AB ,(32) D/ 4 η A + 1 2 (trχ)η A = −(k BN + η B )ˆχ AB − 1 2 trχk AN − 1 2 β A .(33) Here ˆα AB = R 4A4B − 1 2 R 44 δ AB and β A = R 4A34 . Also, when μ = L (trχ) − 1 2 (trχ) 2 −  k NN + n −1 ∇ N n  trχ,(34) 8 which can be interpreted as transport equations along the null geodesics generated by L. Indeed observe that if an S tangent tensorfield Π satisfies the homogeneous equation D/ 4 Π = 0 then Π is parallel transported along null geodesics. ROUGH EINSTEIN METRICS 1203 there is the equality L(μ)+trχμ(35) =2(η A − η A )∇/ A (trχ) − 2ˆχ AB  2∇/ A η B +2η A η B + ¯ k NN ˆχ AB +trχ ˆχ AB +ˆχ AC ˆχ CB +2k AC χ CB + R B43A  −L (R 44 )+(2k NN − 4n −1 ∇ N n))  1 2 (trχ) 2 −|ˆχ| 2 − ¯ k NN trχ − R 44  +4 ¯ k 2 NN trχ + (trχ +4 ¯ k NN )(|ˆχ| 2 + R 44 ) −trχ  2(k AN − η A )n −1 ∇ A n − .2|n −1 N(n)| 2 + R 4343 +2k Nm k m N  . Remark 3.2. Equation (31) is known as the Raychaudhuri equation in the relativity literature; see e.g. [Ha-El]. Remark 3.3. Observe that our definition of μ differs from that in [Kl-Ro]. Indeed there we had, instead of μ, ˜μ = L (trχ) − 1 2 (trχ) 2 − 3 ¯ k NN trχ and the corresponding transport equation: L(˜μ)+trχ˜μ =2(η A − η A )∇/ A (trχ) − 2ˆχ AB  2∇/ A η B +2η A η B (36) + ¯ k NN ˆχ AB +trχ ˆχ AB +ˆχ AC ˆχ CB +2k AC χ CB + R B43A  −L (R 44 ) − L( ¯ k NN )trχ − 3L( ¯ k NN )trχ +4 ¯ k 2 NN trχ +(trχ +4 ¯ k NN )(|ˆχ| 2 + R 44 ). We obtain (35) from (36) as follows: The second fundamental form k verifies the equation (see formula (1.0.3a) in [Ch-Kl]), L nT k ij = −∇ i ∇ j n + n(R iT jT − k im k m j ). In particular, L nT k NN = −∇ 2 N n + n(R NTNT − k Nm k m N ). Exploiting the definition of the Lie derivative L nT , we obtain T (k NN )+2k(∇ N T,N)=−n −1 ∇ 2 N n +(R NTNT − k Nm k m N ). It then follows that 1 2 L (k NN )+ 1 2 L(k NN ) − 2(k NN ) 2 − 2(k AN ) 2 = −n −1 ∇ 2 N n +(R NTNT − k Nm k m N ). [...]... some large value of C 2 n(t − u) λC L∞ (St,u ) 1226 SERGIU KLAINERMAN AND IGOR RODNIANSKI Corollary 9.2 The estimates of Theorem 9.1 can be extended to the + + whole region I0 ∩ ([0, t∗ ] × R3 ), where I0 is the future domain of the origin on Σ0 Remark 9.3 The proof of Corollary 9.2 requires an extension argument The estimates of the Asymptotics Theorem, which are uniform with respect to the bootstrap... in Proposition 7.7 Using however the specific structure of the component R44 relative to the wave coordinates we can overcome this difficulty and prove the following: Theorem 8.1 On any null hypersurface Cu , t (110) ∇R44 (H) u L2 (Sτ,u ) dτ λ−1 Proof The proof of the theorem requires a rather long and tedious argument which we present in our paper [Kl-Ro2] 9 Asymptotics Theorem We start by recalling already... Section 8 Using the bootstrap assumptions and the results of Sections 6 and 7 we provide a detailed proof of the Asymptotics Theorem 1209 ROUGH EINSTEIN METRICS 6 Bootstrap assumptions and Basic Consequences Throughout this section we shall use only the following background property, see Proposition 2.4 in [Kl-Ro1], of the metric H in [0, t∗ ] × R3 : (48) ∂H λ− 2 −4ε0 1 L2 L∞ t x By the H¨lder inequality... the Asymptotics Theorem In this section we describe the main ideas in the proof of the Asymptotics Theorem (1) Section 6 We start by making some primitive assumptions, which we refer to as • Bootstrap assumptions They concern the geometric properties of the Cu and St,u foliations Based on these assumptions we derive further important properties, such as • Sharp comparisons between the functions u,... ≤ 2s, the equation L(n(t−u)−s) = −1 L(n)n(t − u) can be written in the form n d n(t − u) − s | s|∂H| ds Thus with the help of Lemma 6.9 we obtain | |n(t − u) − s| s2 M(∂H) The inequality (56) is an immediate consequence of (55) and Lemma 6.7 The estimate (57) follows from (56), (48), and the L2 estimate for the HardyLittlewood maximal function 1213 ROUGH EINSTEIN METRICS We shall now compare the values... for the second derivatives of the higher frequencies of G do in fact diverge badly 1225 ROUGH EINSTEIN METRICS where R∗ := L2 (Cu ) RABCD L2 (Cu ) + RABC4 L2 (Cu ) + RB43A L2 (Cu ) A,B,C,D Note that some of the above estimates hold only throughout the region Ω∗ Theorem 9.1 Throughout the region Ω∗ the quantities trχ − 2 , χ, and r ˆ η satisfy the following estimates: trχ − (118) trχ − (119) 2 r... 1224 SERGIU KLAINERMAN AND IGOR RODNIANSKI 8 A remarkable property of R44 While the spacetime metric g verifies the Einstein equations Rμν (g) = 0 this is certainly not true for the effective metric H = H(λ) This could create serious problems in the proof of the asymptotics theorem as the Ricci curvature appears as a source term in the null structure equations We have already established an improved estimate... ) Namely, the inequality trχ − λ−1−4ε0 + ∂H(t) + M4 (∂H)(t) 2 ∞ r Lx M3 trχ − 2 ∞ s Lx 1229 ROUGH EINSTEIN METRICS Now squaring and integrating (136) in time we infer from (111) and the just proved estimate (131) for trχ − 2 that r 1 2 λ− 2 −3ε0 , L2 L∞ + ∂H L2 L∞ t x t x r which is the estimate claimed in (118) of Theorem 9.1 On the other hand, application of the elliptic estimate (88) of Proposition... proposition concerning the estimates of Hodge systems on the surfaces St,u They are similar to the estimates of Lemma 5.5 in [Kl-Ro] We need however to make an important modification based on Corollary 4.4 L2 Proposition 6.19 Let ξ be an m+1 covariant, totally symmetric tensor, a solution of the Hodge system on the surface St,u ⊂ Ω∗ ; then div ξ = F, / curl ξ = G, / trξ = 0 Then ξ obeys the estimate |∇ξ|2... due to the nontriviality of the lapse function n 12 1211 ROUGH EINSTEIN METRICS Proof Consider U = n(t − u) − s and proceed as in the lemma above by noticing that du = 0 Therefore, ds d d U= n(t − u) − s = n−1 L(n)n(t − u) ds ds = n−1 L(n)s + n−1 L(n) n(t − u) − s Integrating from the axis Γt we find, (51) s n−1 L(n)ds + U (s) = γ U (s )n−1 L(n)ds γ where γ is the null geodesic starting on the axis . Annals of Mathematics The causal structure of microlocalized rough Einstein metrics By Sergiu Klainerman and Igor Rodnianski Annals of Mathematics,. occupied with the proof of the Asymptotics Theorem. We give a detailed description of their content and the strategy of the proof in Section 5. The paper is