LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS
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1 LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS TSZ KIU AARON CHOW Abstract. In this paper, we treat the vacuum Einstein equations as a local Cauchy problem. We show that it reduces to a system of nonlinear wave equations in wave coordinates. We first discuss the local theory of nonlinear wave equations, and then we apply it to prove the local existence theorem of the vacuum Einstein equation which is originally due to Choquet-Bruhat[5]. Contents 1. Introduction 1 2. Initial Value Formulation Cauchy Problem and Constraint Equations Wave Gauge 3 3. Local Theory of Nonlinear Wave Equations Energy Estimates Local Existence Theorem 9 4. Local Existence of Einstein Equations 13 Acknowledgments 15 References Introduction In general relativity, a spacetime N 4, g is a Riemannian 4-manifold with a Lorentzian metric g. The evolution of spacetime is determined by the Einstein equations. In vacuum, these equations are equivalent to the Ricci flat condition 1.1 Ricg = Vacuum Einstein equations. This is the case when there are no matter fields, and everything is purely geometric and gravitational. In fact, this is a system of partial differential equations of the Lorentzian metric g, which can be reduced to wave equations in an appropriate setting. In this paper, we are going to formulate the vacuum Einstein equations as a Cauchy problem, and prove the existence of solutions. We will derive the initial value formulation in Section 2, which will be followed by a discussion of local theory of nonlinear wave equations in Section 3. We will prove the local existence theorem of vacuum Einstein equations in Section 4. More detailed discussions on the Einstein spacetime and the origins of Einstein equations can be found in [4]. 1
2 2 TSZ KIU AARON CHOW 2. Initial Value Formulation For the Cauchy problem of linear wave equations we are given the spacetime background and solutions evolve in the background in time from their initial values and time derivatives. However in General Relativity, we are solving for the spacetime itself. Thus, the Cauchy problem will be for solutions to a certain nonlinear wave equation Cauchy Problem and Constraint Equations. We consider a spacelike hypersurface M 3 N, g with a unit timelike vector field E defined normally on it.a vector v is timelike if gv, v <. Using this timelike vector field, then locally around M the spacetime N, g is topologically the same as the product manifold M ɛ.ɛ. Thus N, g is sliced by a family of diffeomorphic spacelike hypersurfaces M t = M t} locally around M. Now we identify time coordiates x = t with M x = E, and a coordinate system x 1, x 2, x 3 in M as the spatial coordinates. Thus by diffeomorphisms x, x 1, x 2, x 3 } forms a local coordinate system in M ɛ, ɛ. We adopt the convention that Greek letters α, β, µ, ν,.. run from 1 to 4 while letters i, j, k,.. denote indices from 1 to 3. By restricting the metric on M g t = g Mt, locally around M we can view the Einstein s equations as describing the time evolution of the metric g t. Now we set M as the initial hypersurface of evolution, and denote g := ĝ. Differentiating in t, for any X, Y T p M, denoting as the Levi-Civita connection of M, ĝ we get d dt t= gx, Y = ĝ E X, Y + ĝx, E Y = ĝ X E + [X, E ], Y + ĝ Y E + [Y, E ], X = ĝ X Y, E ĝ Y X, E = 2ĥX, Y, where we have used [X, E ] = [Y, E ] = by E T M, and ĥx, Y = ĝ X Y, E is the second fundamental form on M. Note that h is a symmetric, 2 tensor. The above calculations give us the local Cauchy problem for the Einstein equations: Ricg = on M ɛ, ɛ 2.1 g M = ĝ d t= g Mt = 2ĥ dt Since ĝ and ĥ are geometric quantities on the hypersurface M, they cannot be specified arbitrarily. They need to satisfy the compatibility conditions of a hypersurface, which are called the constraint equations. Recall the Gauss and Codazzi equations defined on an embedded hypersurface: ˆRijkl = R ijkl ĥikĥjl + ĥilĥjk Gauss R ijk = j ĥ ik k ĥ ij Codazzi
3 LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS 3 Summing over i, k and j, l respectively in the Gauss equation, we get Rĝ = ˆR ijij = R ijij ĥii 2 + ĥijĥij = R g + 2Ricg ĥii 2 + ĥijĥij = Rĝ + ĥii 2 ĥijĥij = R g + 2Ricg. Taking the trace of the Codazzi equation, we get Ricg j = R iji = j ĥ ii i ĥ ij. Thus the vacuum Einstein equations Ricg = imply Rĝ + ĥii ĥijĥij = constraint equations. j ĥ ii i ĥ ij = This motivates the following definition: Definition 2.3. M, ĝ, ĥ is an initial data set to 2.1 if i ĝ is a Riemannian metric on M and ĥ is a symmetric, 2 tensor on M, ii ĝ, ĥ satisfies the constraint equations 2.2. We remark that readers can refer to [2] for the existence of such initial data sets Wave Gauge. The Einstein equations are a system of second order partial differential equations. We are going to show in this subsection that if the wave coordinate condition is satisfied, then the Einstein equations reduce to a system of nonlinear wave equations which is treated in later sections. Firstly, we write Ricg in system of coordinates: Ricg µν = 1 2 gαβ 2 αβg µν 1 2 gαβ 2 µνg αβ gαβ 2 ανg βµ gαβ 2 βµg αν + F µν g, g, where F µν g, g involves terms only up to first order derivatives of g. In particular, if we can reduce the second, third and fourth terms to first order derivatives of g, then Ricg µν = resembles the form of nonlinear wave equations in 3.1. We now introduce wave coordinates. Applying the g-laplacian on the coordinate functions x µ, we get g x µ = 1 g g αβ xµ g x α x β = α g αµ gαµ g βρ α g βρ This calculation motivates us to introduce = g αβ g µρ α g βρ gαµ g βρ α g βρ. 2.4 H µ := g αβ g µρ α g βρ gαµ g βρ α g βρ.
4 4 TSZ KIU AARON CHOW We then compute that ν H α Thus, = ν g σβ g αρ σ g βρ g σβ g αρ σν g βρ νg σα g βρ σ g βρ gσα g βρ σν g βρ = g σβ g αρ σν g βρ gσα g βρ σν g βρ + lower order terms. g αµ ν H α + g αν µ H α = g αµ g σβ g αρ σν g βρ g αµg σα g βρ σν g βρ g αν g σβ g αρ σµ g βρ g ανg σα g βρ σµ g βρ + lower order terms = g σβ σν g βµ gβρ µν g βρ g σβ σµ g βν gβρ µν g βρ + lower order terms = g αβ 2 µνg αβ g αβ 2 ανg βµ g αβ 2 βµg αν + lower order terms. Therefore we have shown that Ricg µν = 1 2 gαβ 2 αβg µν 1 2 g αµ ν H α 1 2 g αν µ H α + F µν g, g for some function F µν g, g which contains terms only up to first order derivatives of g. Now we define the reduced Einstein tensor to be 2.5 Ric H g µν := Ricg µν g αµ ν H α g αν µ H α, whereas the reduced Einstein equations is 2.6 = Ric H g µν = 1 2 gαβ 2 αβg µν + F µν g, g. Note that the reduced Einstein equations is a system of nonlinear wave equations. Another important observation is that if H µ, then the reduced Einstein equations are equivalent to the Einstein equations. We will now develop the local theory of nonlinear wave equations that can be applied to give the local existence of solutions.
5 LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS 5 3. Local Theory of Nonlinear Wave Equations In this section, we consider quasilinear wave equations in R n+1 for φ : R + R n R of the form 3.1 a αβ φ αβ 2 φ = F φ, φ φ, t φ t= = φ, φ 1, where a αβ : R R and F : I R n R are smooth functions of their arguments. We require a αβ to be a n + 1 n + 1 symmetric matrix on I R n satisfying 3.2 a αβ m αβ < 1 1 α,β 1, α = β = for some constants m αβ = 1, α = β., otherwise By the smoothness of a, F there exists a constant C A,N such that 3.3 sup xa γ αβ x A,N and 3.4 α,β x A γ N sup x, p A γ N γ x,pf x, p A,N. We will follow [1] to prove the local existence theorem of quasilinear wave equations using energy estimates in this section. Moreover, we refer to chapter 5 in [1] for the proof of the existence and uniqueness of solutions to linear wave equations, and we will assume this result throughout the section. More detailed discussions can also be found in [3]. We denote n φ 2 := t φ 2 + xi φ 2. We now derive the important energy estimates which will be used in proving the main theorems Energy Estimates. Theorem 3.5. Let φ be a solution to 3.1. Then for some constant Cn >, we have the following energy estimate: sup t [,T ] φ L2 R n t φ, φ 1 H1 R n L 2 R n + Proof. We consider i=1 F L 2 R n tdt exp C a L R n tdt t φa αβ αβφ t t φ 2 g, α = β = = α t φ β φa αβ 1 2 t α φ β φa αβ t φ α a αβ β φ, α, β = 1, 2,..., n. 1 2 β t φ 2 a β 1 2 tφ 2 β a β α =, β = 1,..., n.
6 6 TSZ KIU AARON CHOW and so we have, for α = β = : = 1 2 t φa ttφdxdt 2 R n R n T } t φ 2 a dx 1 2 for α, β = 1, 2,..., n: for α =, β = 1, 2,..., n: t φa αβ αβφdxdt 2 R n R n } t φ 2 a dx 1 2 = 1 α φ β φa αβ dx R n T } 2 α a αβ t φ β φdxdt + R n Then we integrate the identity R n t φa β 2 βφdxdt = 1 2 R n } α φ β φa αβ dx R n t a αβ α φ β φdxdt; t φa αβ 2 αβφ F =, on R n [, T ] to get: t φ 2 a α φ β φg αβ dx R n T } 1 α,β n t φ 2 a α φ β φa αβ dx R n } + C 1 α,β n R n t φ 2 β a β dxdt. R n t φ 2 t a dxdt; a L R n t φ 2 L 2 R n t + φ L 2 R n t F L 2 R n t dt. By the assumption 3.2, the above implies that φ 2 L 2 R n T φ 2 L 2 R n + C a L R n t φ 2 L 2 R n t + φ L 2 R n t F L 2 R n t dt for some constant C >. Observe that since the above inequality holds for any T, we actually have sup φ 2 L 2 R n the right hand side. t [,T ]
7 LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS 7 We then use the inequality 2ab a 2 + b 2 to get φ L 2 R n t F L 2 R n tdt sup φ L 2 R n t [,T ] F L 2 R n tdt δ sup φ 2 L 2 R n + 1 T t [,T ] 2 δ 1 F L 2 R tdt n. By choosing δ to be small enough we get sup φ 2 L 2 R n t [,T ] φ 2 L 2 R n + C φ 2 L 2 R n + 2 F L2 R tdt n + C a L R n t φ 2 L 2 R n t dt. 2 F L 2 R tdt n + C The theorem follows from Gronwall s lemma. a L R n t sup φ 2 L 2 R n s [,t] Lemma 3.6. Gronwall s lemma Let f : R R be a positive continuous function and g : R R be a positive integrable function such that ft A + for some A for every t [, T ]. Then for every t [, T ]. t ft A exp fsgsds t gsds dt. The proof for this lemma can be found in a standard reference, such as chapter 4 in [1]. By bootstrapping Theorem 3.5, one can obtain Corollary 3.7. Let φ be a solution to 3.1 and k be a positive integer. Then there exists a positive constant Cn, k such that we have the following energy estimate: sup t [,T ] φ, t φ Hk R n H k 1 R n φ, φ 1 Hk R n H k 1 R n + + C exp C α + β k 1 α x a β x φ L 2 R n t + a L R n tdt. F H k 1 R n tdt α + β k 1 α x a β x φ L 2 R n t
8 8 TSZ KIU AARON CHOW Proof. We differentiate 3.1 by x to get x a αβ αβ φ + aαβ αβ 2 xφ = x F φ, φ φ, t φ t= = φ, φ 1 We now introduce a new function ψ := x φ, thus 3.8 becomes a αβ αβ ψ = F ψ, ψ ψ, t ψ, t= = x φ, x φ 1 for some F = x F ψ + p F ψ a αβ ψ. We then apply Theorem 3.5 to 3.9 to get sup t [,T ] ψ L 2 R n t x φ, x φ 1 H1 R n L 2 R n + We use Cauchy-Schwarz inequality to get F L2 R n tdt exp. C F L2 R n δ 1 F L 2 R n + Cδ ψ L 2 R n + a αβ ψ L 2 R n. We choose δ small to derive sup t [,T ] x φ, t x φ Ḣ1R n L 2 R n x φ, x φ 1 H1 R n L 2 R n + C F H1 R n t + a αβ T x φ L2 R n dt exp C a L R n tdt. The bounds for higher order terms sup xφ, γ t xφ Ḣ1R γ n L 2 R n, t [,T ] γ k a L R n tdt can be obtained similarly by differentiating 3.1 by γ x. Next, it remains to control sup φ L 2 R n. t [,T ] Note that the Fundamental Theorem of Calculus gives φt = φ + t φtdt. Thus, by triangle inequality and Minkowski integral inequality, φt L 2 R n φ L 2 R n + In fact the arbitrariness of T further implies that sup φ L2 R n φ L 2 R n + t [,T ] Therefore the conclusion of the Corollary follows. t φ L 2 R n dt. t φ L 2 R n dt..
9 LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS Local Existence Theorem. The local existence theorem for quasilinear wave equations builds upon the case for linear wave equations. We will state without proof the existence and uniqueness of solutions to the linear wave equations. Theorem 3.1. Consider the system of linear wave equations a αβ 2 αβ φ = F φ, t φ t= = φ, φ 1 with a αβ satisfying 3.2. If φ, φ 1 H k R n H k 1 R n and F L 1 [, T ]; H k 1 R n, then there is a unique solution solving the system φ, t φ L [, T ]; H k R n L [, T ]; H k 1 R n Before we proceed to the main theorem, let s recall the Sobolev embedding theorem. Theorem Given s > n 2, there exists Cn, s > such that φ H s R n. φ L R n φ H s R n Proof. The H s norm is given by 1 φ H s R n = 1 + y 2 s ˆφy 2 2 dy, R n where ˆφ is the Fourier transform of φ. We then use the Holder s inequality to get φx = ˆφye i2πxy dy R n ˆφy dy R n y 2 s ˆφy 2 2 dy 1 + y 2 s 2 dy. R n Rn When s > n 2, the integral R n 1 + y 2 s dy is finite, and so the conclusion follows. We now state the main theorem of this section. Theorem Suppose that with k n + 2, then there exists φ, φ 1 H k R n H k 1 R n T = T φ Hk R n, φ 1 H k 1 R n > such that there is a unique solution φ solving 3.1 with φ, t φ L [, T ]; H k R n L [, T ]; H k 1 R n.
10 1 TSZ KIU AARON CHOW To prove this theorem, we define iteratively a sequence of functions φ i for i 1 with φ 1 = such that for i 2, φ i is the unique solution to a αβ φ i 1 αβ φi = F φ i 1, φ i 1 φ i, t φ i t= = φ, φ 1 Note that by linearity this sequence is well defined by Theorem 3.1. We want to prove Theorem 3.12 by showing that φ i converges to a solution of 3.1. We will show two properties of φ i : 1 It is uniformly bounded; 2 It is a Cauchy sequence. Lemma Under the assumptions of Theorem 3.12, for T > sufficiently small, there is C > independent of i such that the sequence defined by 3.12 is uniformly bounded φ i, t φ i L [,T ];H k R n L [,T ];H k 1 R n. Proof. We will prove the lemma by induction, and the idea is to use the energy estimates. First assume that there is A > to be chosen such that 3.15 φ i 1, t φ i 1 L [,T ];H k R n L [,T ];H k 1 R n A. Our goal is to show that we can choose A appropriately to be a uniform bound of the sequence. For simplicity, we denote ft := φ i, t φ i H k R n H k 1 R n t We apply the energy estimates Corollary 3.7 to 3.13 to get 3.16 sup ft φ, φ 1 H k R n H k 1 R n + t [,T ] + C α + β k 1 α + β k 1 α x aφ i 1 β x φ i L 2 R n t+ F φ i 1, φ i 1 H k 1 R n tdt x α aφ i 1 x β φ i L 2 R n tdt exp C For instance, let s consider the second term on the right hand side. Using chain rule and Leibnitz rule, we have F φ i 1, φ i 1 H k 1 R n F γ α1 φ i 1 αr φ i 1 β1 x φ i 1 βs φ i 1 L2 R n αl + β l k 1 αl + β l k 1 α1 φ i 1 αr φ i 1 β1 φ i 1 βs φ i 1 L2 R n, where we have used 3.4 and absorbed the constants into C in the last inequality. For each term above, at most one of the α l and β l can have order larger than k 1 2. On the other hand, we have k 1 2 n+1 2. Hence for those α l, β l k 1 2, we apply the Sobolev embedding Theorem 3.11 with s = k 1 2 to get aφ i 1 L R n tdt. α l φ i 1 L R n α l φ i 1 H k 1 2 R n φi 1, t φ i 1 H k R n H k 1 R n,
11 LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS 11 and β l φ i 1 L R n β l φ i 1 H k 1 2 R n φi 1, t φ i 1 H k R n H k 1 R n. Therefore, F φ i 1, φ i 1 Hk R n H k 1 R n φ i 1, t φ i 1 αl + β l k 1 αl + β l H k R n H k 1 R n k φ i 1, t φ i 1 Hk R n H k 1 R n B, where we have used the induction hypothesis in the last inequality. For the third term in 3.16, we can bound x α aφ i 1 by a similar argument with 3.3. Moreover, either α or β can be larger than k 1 2. Thus we can again apply the Sobolev embedding. If α k 1 2, then α x aφ i 1 β x φ i L2 R n β x φ i L R n α x aφ i 1 L2 R n If β k 1 2, then φ i Hk R n α x aφ i 1 L 2 R n Bft. α x aφ i 1 β x φ i L 2 R n α x aφ i 1 L R n β x φ i L 2 R n aφ i 1 H k R n ft Bft. By a similar argument on the fourth term, we can now show that x α aφ i 1 x β φ i L2 R n t + x α aφ i 1 x β φ i L2 R n t Bft. α + β k 1 Then 3.16 implies that ft φ, φ 1 Hk R n H k 1 R n + BT + C Bftdt expc BT. We choose T sufficiently small such that BT φ, φ 1 Hk R n H k 1 R n and expc BT 2. Thus, ft 2C 2 φ, φ 1 H k R n H k 1 R n + By Gronwall s lemma we then have C Bftdt. ft 4C φ, φ 1 H k R n H k 1 R n exp2c 2 BT. We can choose T to be sufficiently small that exp2c 2 BT 2. With T so chosen as above, we thus have ft 8C φ, φ 1 Hk R n H k 1 R n.
12 12 TSZ KIU AARON CHOW Since this inequality holds for any t T, we actually have a stronger result φ i, t φ i L [,T ];H k R n L [,T ];H k 1 R n = sup ft t [,T ] 8C φ, φ 1 Hk R n H k 1 R n. Now we can choose A = 8C φ, φ 1 Hk R n H k 1 R, and the induction is completed for this A provided that T > is sufficiently small depending on A. n Lemma Under the assumptions of Theorem 3.12, for T > chosen to be smaller if necessary, φ i, t φ i is a Cauchy sequence in L [, T ]; H 1 R n L [, T ]; L 2 R n Proof. For i 3, we consider the equation for φ i φ i 1 : 3.18 a αβ φ i 1 2 αβφ i φ i 1 = a αβ φ i 1 a αβ φ i 2 2 αβφ i 1 + F φ i 1 F φ i 2. We observe that by 3.4, F φ i 1, φ i 1 F φ i 2, φ i 2 L2 R n 1 d = dt F tφi tφ i 2, t φ i t φ i 2 dt 1 = F x φ i 1 φ i 2 + F p φ i 1 φ i 2 dt 1 sup x <2A L 2 R n L 2 R n F x φ i 1 φ i 2 L 2 R n + sup F p φ i 1 φ i 2 L 2 R n dt x <2A φ i 1 φ i 2, φ i 1 φ i 2 H1 R n L 2 R n. Similarly, we use Lemma 3.14, 3.3 and the Sobolev embedding to get a αβ φ i 1 a αβ φ i 2 2 αβφ i 1 L 2 R n 2 αβφ i 1 L R n a αβ φ i 1 a αβ φ i 2 L2 R n φ i 1 Hk R n a αβ φ i 1 a αβ φ i 2 L 2 R n φ i 1 φ i 2, φ i 1 φ i 2 H 1 R n L 2 R n. Thus applying Theorem 3.5 to 3.18 yields φ i φ i 1, φ i φ i 1 L [,T ];H 1 R n L [,T ];L 2 R n T φ i 1 φ i 2, φ i 1 φ i 2 L [,T ];H 1 R n L [,T ];L 2 R n, for some positive constant C = C φ, φ 1 H 2 R n H 1 R n >. Therefore if we choose T sufficiently small, then the Cauchy property follows straightforwardly. Proof of Theorem By Lemma 3.17, since φ i, t φ i is a Cauchy sequence in L [, T ]; H 1 R n L [, T ]; L 2 R n, it converges to a limit φ, t φ L [, T ]; H 1 R n L [, T ]; L 2 R n. Moreover, by the uniform bound in L [, T ]; H k R n L [, T ]; H k 1 R n obtained in Lemma 3.14, and by Banach-Alaoglu there is a subsequence which has a weak limit in L [, T ]; H k R n L [, T ]; H k 1 R n. Therefore by the uniqueness of limit, the two limits agree. This completes the proof of Theorem 3.12.
13 LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS Local Existence of Einstein Equations We will follow [1] and [2] to establish the local existence theorem of vacuum Einstein equation due to Choquet-Bruhat in this section. Theorem 4.1. Let M 3, ĝ, ĥ be an initial data set which satisfies the constraint equations 2.2 with i,j ĝ ij δ ij < 1 2. Then there exists a metric g on M [, ɛ] that solves the vacuum Einstein equation such that the induced metric and induced second fundamental form on M are ĝ ij and ĥij respectively. Note that the reduced Einstein equation Ric H g = can be solved by Theorem So it suffices to show that H µ. We will prove it in three steps: a We can choose H µ = at t = with ĝ, ĥ being the first and second fundamental forms on M M [, ɛ]. b We then use the constraint equations 2.2 to show that t H µ = at t =. c We finally use the second Bianchi identity to show that H µ in M [, ɛ]. Lemma 4.2. Given M, ĝ, ĥ as in Theorem 4.1, we can choose Hµ = at t =. Proof. The Lorentz metric of M [, ɛ] is 1 O g µν =. O ĝ ij So that at t = the metric g µν and its spatial derivatives i g µν are determined. The second fundamental form also determines the spatial parts of t g µν at t = t g t g i t g µν t= =. t g i 2ĥij Then from 2.4 we have at t = : H µ = g αβ g µρ α g βρ gαµ g βρ α g βρ = g β g µρ t g βρ gµ g βρ t g βρ + known terms = g µρ t g ρ gµ g βρ t g βρ + known terms. The spatial components i = 1, 2, 3 give H i = g ij t g j + known terms, thus we can choose t g j such that H i = at t =. For the time component µ =, we have H = 1 2 tg g ij t g ij + known terms = 1 2 tg + known terms, we then fix t g to make H =. Lemma 4.3. If g solves Ric H g = with initial data set M, ĝ, ĥ, then th µ = at t =.
14 14 TSZ KIU AARON CHOW Proof. At t =, by 2.5 the vanishing of Ricg H gives = Ricg i + 1 2ĝα i H α + 1 2ĝαi H α, then the constraint equations Ricg i = and the vanishing of spatial derivatives of H µ gives = t H α. Lemma 4.4. If g solves Ric H g = with initial data set M, ĝ, ĥ, then Hµ satisfies the wave equation for a smooth function G. g H µ = GH µ, H µ Proof. By contracting the second Bianchi identity, we have the following identity 4.5 µ Ricg µν 1 2 R gg µν =. Then by 2.5, if Ric H g = holds we have and Therefore by 4.5 we have Ricg µν = 1 2 g αµ ν H α 1 2 g αν µ H α R = µ H µ. = g βµ µ g µν α H α g αµ ν H α g αν µ H α + lower order terms = g βµ g µρ g µν αρ H α g µρ g αµ νρ H α g µρ g αν µρ H α + lower order terms = g βµ αν H α g βµ αν H α g βµ g αν g µρ µρ H α + lower order terms = g H β + lower order terms Proof of Theorem 4.1. Firstly we solve the nonlinear wave equation Ric H g = with initial data set M, ĝ, ĥ. Then for this metric g, by Lemma 4.2 to Lemma 4.4 we have g H µ = GH µ, H µ H µ, t H µ. t= =, Hence H µ by the uniqueness of the solution to quasilinear wave equations. This shows that for this metric, we have = Ric H g = Ricg.
15 LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS 15 Acknowledgments. It is a pleasure to thank my mentor, Casey Rodriguez, for his excellent guidance and prompt reply to me questions throughout the summer. I would like to thank Peter May for hosting the REU program and his enlightening teaching. I would also like to thank Min Yan, Frederick Fong and Tianling Jin in HKUST for introducing to me the REU program. References [1] Jonathan Luk. Introduction to Nonlinear Wave Equations. jl845/nwnotes.pdf [2] Richard Schoen. Topics in Differential Geometry. schoen/math286/gr lectures.pdf [3] C. D. Sogge. Lectures on Nonlinear Wave Equations. International Press, [4] Robert M. Wald. General Relativity. The University of Chicago Press, [5] Yvonne Choquet-Bruhat. General Relativity and the Einstein Equations. Oxford University Press, 29.
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