Global regularity for Einstein-Klein-Gordon system with U(1)×R isometry group, I
aa r X i v : . [ m a t h . A P ] M a y GLOBAL REGULARITY FOR EINSTEIN-KLEIN-GORDON SYSTEMWITH U p q ˆ R ISOMETRY GROUP, I
Haoyang Chen and Yi Zhou Abstract.
This is the first of the two papers devoted to the study of global regularityof the 3+1 dimensional Einstein-Klein-Gordon system with a U p qˆ R isometry group.In this first part, we reduce the Cauchy problem of the Einstein-Klein-Gordon systemto a 2+1 dimensional system. Then, we will give energy estimates and construct thenull coordinate system, under which we finally show that the first possible singularitycan only occur at the axis. Keywords:
Einstein-Klein Gordon system, Cauchy problem, energy estimates, nullcoordinate, first singularity.
Introduction
Introduction and previous results.
Let p p q M, p q g q be a 3+1 dimensional glob-ally hyperbolic Lorentzian manifold which satisfies the following Einstein-scalar fieldequations:(1.1) p q G µν “ p q T µν “ B µ φ B ν φ ´ p q g µν B λ φ B λ φ ´ p q g µν V p φ q l p q g φ “ V p φ q where p q g is the Lorentzian metric, p q G µν is the Einstein tensor of p q g , p q T µν is thestress-energy tensor given as above, φ is the scalar field, and we take the potential V p φ q “ m φ .Then, the equation that the scalar field satisfies is a Klein-Gordon equation, whichmakes the system an Einstein-Klein-Gordon system. The Einstein-Klein-Gordon system(1.1) is equivalent to the following equations,(1.2) p q R µν “ p q ρ µν fi B µ φ B ν φ ` p q g µν m φ l p q g φ “ m φ where p q R µν is the Ricci curvature tensor. Date : May 23, 2019. School of Mathematical Sciences, Fudan University, Shanghai, China.
Email:[email protected] *Corresponding Author: School of Mathematical Sciences, Fudan University, Shanghai, China. Email: [email protected]
One fundamental open problem in the field of general relativity is the cosmic censorshipconjectures by Penrose. Roughly speaking, the weak cosmic censorship may be formu-lated as follows: For generic asymptotically flat Cauchy data, of the vacuum equationsor suitable Einstein-matter systems, the maximal development possesses a complete fu-ture null infinity. While the strong cosmic censorship states that the maximal Cauchydevelopment is inextendible for generic initial data. This question is partially related tothe study of the formation of event horizons. Although still open in general, there is aseries of results in spherical symmetric case by Christodoulou for the Einstein equationscoupled with a massless scalar field. On the other hand, of our interest, it is related tothe global well-posedness of the Cauchy problem in large.We aim to show the global regularity of the above 3+1 dimensional Einstein-Klein-Gordon systems with a U p q ˆ R isometry group, while in this paper we will firstly giveseveral results in preparation for the subsequent study on the global well-posedness.Our study is motivated by research on the vacuum Einstein equations with spacelikeKilling vector fields which is related to the study of wave map systems. We first reviewsome related results on the vacuum case and the wave map systems.For the 3+1 dimensional vacuum Einstein equations with one spacelike Killing field,as we can see in [6] and [20], the Einstein equations can be reduced to a 2+1 dimen-sional Einstein-wave map system on a 2+1 dimensional Lorentzian manifold where thetarget manifold is the hyperbolic space H . Yet the global existence problem of theEinstein-wave map system is still open. As a first step towards this global existence con-jecture, Andersson, Gudapati and Szeftel proved that the global regularity holds for theequivariant case in [2], by reference to some pioneering work on equivariant wave maps.Shatah and Tahvildar-Zadeh have proved the global regularity for 2+1 dimensionalequivariant wave maps with the target geodesically convex in [24]. This condition waslater relaxed by Grillakis to include a certain class of nonconvex targets, see [11]. Theirproof of regularity was also simplified later by Shatah and Struwe in [23]. They gaveseveral more results on equivariant wave maps in the areas of existence and uniqueness,regularity, asymptotic behavior, development of singularities, and weak solutions, see[25]. Further, as an improvement of these above results, for target manifolds that do notadmit nonconstant harmonic spheres, global existence of smooth solutions to the Cauchyproblem for corotational wave maps with smooth equivariant data was shown by Struwein [26].Then, for the 3+1 dimensional vacuum Einstein equations with G symmetry, it wasshown in [3] that the system reduce to a spherically symmetric wave map u : R ` Ñ H , where R ` is the 2+1 dimensional Minkowski spacetimes and the target H is thehyperbolic space. Thus the global regularity can be proved by the work of Christodoulouand Tahvildar-Zadeh [8] on 2+1 dimensional spherically symmetric wave maps. In [8],the range of the wave map u should be contained in a convex part of the target N . Thisrestriction was later shown unnecessary by Struwe in [28] as the target is the standardsphere. Further, Struwe give a more general result in [27], where the target is any smooth,compact Riemannian manifold without boundary. We refer to [10] for more results andreferences of wave maps. HE EINSTEIN-KLEIN-GORDON SYSTEM 3
The 3+1 dimensional spacetime with U p q ˆ R isometry group. In thispaper, we work on the Lorentzian manifold p q M “ R ˆ R ˆ R with a Lorentzian metric p q g on it, and we consider the polarized case where the Killing fields of p p q M, p q g q arehypersurface orthogonal. Then, with the existence of the translational Killing vector, themetric can be written in the following form, p q g “ e ´ γ p q g ` e γ p dx q where B x is the translational spacelike Killing vector field.As we mentioned before in [6] and [20], the 3+1 dimensional vacuum Einstein equationswith spacelike Killing field reduce to a 2+1 dimensional Einstein-wave map system withthe target manifold H . We will give a similar reduction to a 2+1 dimensional Einstein-wave-Klein-Gordon system in section 2, where γ in p q g satisfies a wave equation and thescalar field φ satisfies a Klein-Gordon equation.In the vacuum case with G symmetry, the wave maps equations reduced from theEinstein equations is a semilinear wave equations, for instance, special solutions of thiscase are the Einstein-Rosen waves, see [15] and references therein. While the majordifficulty in our problem is that the wave equations are coupled with Einstein equations,which make the system quasilinear. However, we develop a new way to solve this problemin 2+1 dimension with symmetry.Particularly, if the scalar field is massless, we can remove the condition that the Killingvector field B x is hypersurface orthogonal, and the metric will take the general form p q g “ e ´ γ p q g ` e γ p dx ` A α dx α q . We will mention in section 2 that the system reduce to a wave map equations coupledwith a linear wave equation on the Minkowski spacetimes, of which the problem left isto study the wave map system, same as in the vacuum case.Now we assume that the reduced spacetime p p q M, p q g q is a globally hyperbolic 2+1dimensional spacetime with Cauchy surface diffeomorphic to R , on which the reducedEinstein-wave-Klein-Gordon system is radially symmetric. And we assume that the U p q action on M is generated by a hypersurface orthogonal Killing field B θ . In particular, wewrite the metric p q g in the following form in this paper p q g “ ´ e α p t,r q dt ` e β p t,r q dr ` r dθ (1.3) fi ˇ g ` r dθ where ˇ g is a metric on the orbit space Q “ M { S and r is the radius function, definedsuch that 2 πr p p q is the length of the S orbit through p .1.3. The Cauchy data.
As we mentioned before, to study the Cauchy problem of the3+1 dimensional Einstein-Klein-Gordon system, we can equivalently consider the Cauchyproblem of the reduced 2+1 dimensional Einstein-wave-Klein-Gordon system.Now we introduce the definition of the Cauchy data set for the 2+1 dimensionalEinstein-wave-Klein-Gordon systems with U p q isometry group as follows, Definition 1.1 (Cauchy data set for the 2+1 dimensional Einstein-wave-Klein-Gordonsystem with U p q isometry group) . A Cauchy data set for the 2+1 dimensional Einstein-wave-Klein-Gordon system with a U p q isometry group is a 7-tuple p M , g , K, γ , γ , φ , φ q THE EINSTEIN-KLEIN-GORDON SYSTEM consisting of a Remannian 2-manifold p M , g q with a spacelike rotational Killing vectorfield and a 2-tensor K which is the second fundamental form and symmetric under thesame action, γ , γ are initial data for the wave equation that γ satisfies, φ , φ are initialdata for the Klein-Gordon equation that the scalar field satisfies. g , K are functions of r only and the following constraints equations hold: (1.4) R ´ K αβ K αβ ` p trK q “ p q T αβ n α n β (1.5) D β K αβ ´ D α K ββ “ p q T αµ n µ where n α p t, r q is the future directed unit normal, R is the scalar curvature on M , D α isthe intrinsic covariant derivative on M , and p q T αβ is the stress-energy tensor for thereduced system. For a smooth solution of the 2+1 dimensional Einstein-wave-Klein-Gordon system with U p q symmetry, it must hold that α p t, r q , β p t, r q are even functions of r . And we givesome normalisation of the metric functions α p t, r q and β p t, r q on the axis. It must holdthat β p t, q “
0, in order to avoid a conical singularity at the axis Γ, which means thatthe perimeter of a circle of radius r grows like 2 πcr at the axis, instead of 2 πr in theEuclidean metric. This condition can be realized by appropriately choosing the Cauchydata such that β p , q “
0, see section 2. Further, α p t, q is determined only up to achoice of time parametrization. We shall choose a time coordinate such that α p t, q “ . Finding solutions to the constraint equations is a research area in itself. Note thatCecile has proved the existence of such constraint equations in vacuum with translationalKilling vector field in [13][14], which is used in [15] to prove stability in exponential timeof the Minkowski spacetime in this setting. In our case, we will just briefly show thatsuch data exists, without going further into the study of the constraints.We will construct an asymptotically flat Cauchy data set which satisfies the followinginitial conditions(1.6) γ t p , r q “ γ p r q ě ,γ p , r q “ γ p r q ě , γ r p , r q “ γ p r q ą ´ r ´ . Remark 1.2.
The following initial condition in (1.6) γ r p , r q “ γ p r q ą ´ r ´ can be replaced by γ r p , r q “ γ p r q ą ´ Cr ´ with C an arbitrary positive constant. Here, we take C “ just for simplicity. For metrics of the form (1.3), it can be calculated that the second fundamental form K of a Cauchy t -level takes the following form K “ K rr dr , where K rr “ e ´ α ` β β t . We say the Cauchy data is ’asymptotically flat’ in the sense of Andersson[2] here and hereafter.
HE EINSTEIN-KLEIN-GORDON SYSTEM 5
For the reduced 2+1 dimensional Einstein-wave-Klein-Gordon system(see section 2),the constraint equations (1.4)(1.5) take the following form in local coordinates, e ´ β r ´ β r “ p e ´ α γ t ` e ´ β γ r ` e ´ α φ t ` e ´ β φ r ` m e ´ γ φ q ,r ´ e ´ α β t “ e ´ α γ t γ r ` e ´ α φ t φ r . If we take β t “ γ t “
0, the constraint equations become e ´ β r ´ β r “ p e ´ β γ r ` e ´ α φ t ` e ´ β φ r ` m e ´ γ φ q ,e ´ α φ t φ r “ . Remark 1.3.
Noting that β t “ means that M is a totally geodesic submanifold. More-over, what is more interesting, a direct calculation shows that the additional condition γ t “ makes the corresponding Cauchy hypersurface of the 3+1 Einstein-Klein-Gordonsystem a totally geodesic submanifold. Noting that all quantities above are functions of r only. Thus, we can properly choosethe initial data sets for γ and φ which is compatible with (1.6) such that the secondequation holds. Here, we set φ t “
0. What left to be done is solving the first equation,which is just an ordinary differential equation. Noting that it is equivalent to the followingequation ´B r p e ´ β q “ r p e ´ β γ r ` e ´ β φ r ` m e ´ γ φ q which is a first order linear differential equation of e ´ β , and with the condition β p , q “
0, the solution takes the form β p r q “ e ş r p ξγ r ` ξφ r q dξ ´
12 ln p ´ ż r ξm e ´ γ φ e ş ξ p ηγ r ` ηφ r q dη dξ q . Thus, properly choose the initial data sets for γ and φ such that(1.7) ż r ξm e ´ γ φ e ş ξ p ηγ r ` ηφ r q dη dξ ă , and(1.8) φ “ O p r ´ q , γ “ O p r ´ q as r Ñ 8 . The first condition (1.7) guarantees that the initial energy is finite. Thesecond condition (1.8) on asymptotic behavior guarantees the asymptotic flatness, whichis to say that β “ β ` ˜ β. Here, β is a constant and p ˜ β, K q P H s ` δ ˆ H sδ ` with H s ` δ and H sδ ` the weightedSobolev space(see [2][13]) where we take δ “ ´ and s ą
1. Thus, we finally give anon-trivial solution of the constraint equations.
Remark 1.4.
If we eliminate the condition γ t “ , it is also easy to give a constructionin a similar way by properly choosing γ t and φ t which have the same asymptotic behavioras above. THE EINSTEIN-KLEIN-GORDON SYSTEM
The problem of global well-posedness.
The proof by Choquet-Bruhat and Ge-roch(see [4][5]) of existence and uniqueness of maximal solutions to the Cauchy problemfor the vacuum Einstein equations, together with the equivalence of the Cauchy data,can be generalized to our case as follows, which guaranteed the local well-posedness.
Theorem 1.5.
Let p M , g , k, γ , γ , φ , φ q be the Cauchy data set for the 2+1 dimen-sional Einstein-wave-Klein-Gordon system with U p q isometry group. Then there is aunique, maximal Cauchy development p p q M, p q g, γ, φ q satisfying the the 2+1 dimensionalEinstein-wave-Klein-Gordon system. Our goal is to prove the following theorem about the global regularity,
Theorem 1.6.
Let p p q M, p q g q be the maximal Cauchy development of a regular Cauchydata set aforementioned in section 1.3. Then there is a global in time smooth solutionfor the Cauchy problem of the equations. As preparation for the above theorem, in this paper we give several results that willbe useful in the proof of the global regularity. The paper is organized as follows. InSection 2, we reduce the 3+1 dimensional Einstein-Klein-Gordon system with U p q ˆ R isometry group to a 2+1 dimensional Einstein-wave-Klein-Gordon system on p p q M, p q g q with U p q symmetry. In Section 3, using vector field methods, we give energy estimatesof the systems. In Section 4, we give a null coordinate system. In Section 5, we will showthat the first possible singularity must occur at the axis.2. Reduction to the 2+1 dimensional Einstein-wave-Klein-Gordon system
Consider the Einstein equations (1.2), where the spacetime p p q M, p q g q admits a space-like translational Killing vector field. The Einstein-Klein-Gordon system can be reducedto a 2+1 dimensional Einstein-wave-Klein-Gordon system in a similar way as we knowin [12]. For demonstration purposes, let us go ahead and perform the reduction. Givena spacelike hypersurface orthogonal Killing vector field, the metric can be written in thefollowing form: p q g “ e ´ γ p q g ` e γ p dx q where B x is the translational spacelike Killing vector field. In a similar way as in [4], the3+1 dimensional Einstein-scalar field system (1.2) on p p q M, p q g q can be rewritten as(2.1) p q R αβ “ ˜ R αβ ´ ˜ ∇ α B β γ ´ B α γ B β γ “ B α φ B β φ ` ˜ g αβ m φ (2.2) p q R α “ p q R “ ´ e γ ´ ˜ g αβ B α γ B β γ ` ˜ g αβ ˜ ∇ α B β γ ¯ “ e γ m φ . where ˜ g “ e ´ γ p q g and (2.2) is trivial in polarized case.Then, we give the following formulas about the conformal transformations of Riccitensors and the wave operator,(2.4) b | det p q g | “ e γ a | det ˜ g | HE EINSTEIN-KLEIN-GORDON SYSTEM 7 (2.5) l p q g u “ a | det p q g | B β ˆb | det p q g | p q g αβ B α u ˙ “ e ´ γ ` l ˜ g u ` ˜ g αβ B β γ B α u ˘ (2.6) p q R αβ “ ˜ R αβ ´ ˜ g αβ ˜ ∇ σ ˜ ∇ σ γ ´ ˜ ∇ α ˜ ∇ β γ ` ˜ ∇ α γ ˜ ∇ β γ ´ ˜ g αβ ˜ ∇ σ γ ˜ ∇ σ γ where α, β, σ “ , , p q ∇ σ B σ γ “ ´ e ´ γ m φ which is a wave equation.Equations (2.1) rewritten in a same way results in a 2+1 dimensional Einstein equations p q R αβ “ B α γ ¨ B β γ ` B α φ B β φ ` e ´ γ p q g αβ V p φ q fi p q ρ αβ This is equivalent to(2.8) p q G αβ “ p q T αβ fi p q ρ αβ ´ tr p p q ρ q p q g αβ . where p q T αβ is the stress-energy tensor. And the wave equation that the scalar fieldsatisfies can be rewritten as(2.9) l p q g φ “ e ´ γ m φ When the metric takes the form (1.3), we give the computation of the Einstein tensor p q G αβ , p q G “ r e α ´ β β r , p q G “ r β t , p q G “ r α r , p q G “ r ` e ´ β α rr ´ e ´ α β tt ` e ´ β α r p α r ´ β r q ` e ´ α β t p α t ´ β t q ˘ , p q G “ , p q G “ . and the stress-energy tensor p q T αβ , p q T “ γ t ` e α ´ β γ r ` φ t ` e α ´ β φ r ` m e α ´ γ φ “ e α e , p q T “ γ t γ r ` φ t φ r “ e p α ` β q m , p q T “ e β ´ α γ t ` γ r ` e β ´ α φ t ` φ r ´ m e β ´ γ φ , p q T “ r e ´ α γ t ´ r e ´ β γ r ` r e ´ α φ t ´ r e ´ β φ r ´ m r e ´ γ φ , p q T “ , p q T “ . THE EINSTEIN-KLEIN-GORDON SYSTEM
Thus, using the translational Killing vector field, we have reduced the 3+1 dimensionalEinstein-Klein-Gordon system (1.1) to a 2+1 dimensional Einstein-wave-Klein-Gordonsystem (2.7)(2.8)(2.9).Now we write the 2+1 dimensional radially symmetric Einstein-wave-Klein-Gordonsystem in local coordinates,(2.10) β r “ re β ´ α γ t ` rγ r ` r e β ´ α φ t ` r φ r ` m re β ´ γ φ (2.11) β t “ rγ t γ r ` rφ t φ r (2.12) α r “ re β ´ α γ t ` rγ r ` r e β ´ α φ t ` r φ r ´ m re β ´ γ φ e ´ β α rr ´ e ´ α β tt ` e ´ β α r p α r ´ β r q ` e ´ α β t p α t ´ β t q (2.13) “ ´ m e ´ γ φ ` e ´ α p γ t ` φ t q ´ e ´ β p γ r ` φ r q l p q g γ “ ´ e ´ α p γ tt ` p β t ´ α t q γ t q ` e ´ β p γ rr ` γ r r ` p α r ´ β r q γ r q (2.14) “ ´ m e ´ γ φ l p q g φ “ ´ e ´ α p φ tt ` p β t ´ α t q φ t q ` e ´ β p φ rr ` φ r r ` p α r ´ β r q φ r q (2.15) “ m e ´ γ φ. Therefore, as we mentioned in section 1, we can set β p , q “
0. Then, by (2.11), we have β p t, q “ V “
0, then we can remove the condition that the Killing vector field B x is hypersurface orthogonal, and the metric will take the general form p q g “ e ´ γ p q g ` e γ p dx ` A α dx α q . Thus, the reduction in [12] yields a wave map equations coupled with a wave equation.Noting that if V “
0, then α r “ β r by the 2+1 dimensional Einstein equations, then e α ´ β is independent of r . Let T p t q “ ż t e α ´ β p τ q dτ. Therefore, we get a coordinate p T, r q in which p Q , ˇ g q is conformally flat. For simplicity,we still denote T by t . Thus, rewritting the Einstein-wave map-scalar field system in thenew coordinates, we can obtain a spherically symmetric wave map to the hyperbolic space p H , h q coupled with a linear wave equation on the Minkowski spacetimes p R ` , m q .Noting that regularity for the linear wave equation l m φ “ p R ` , m q , which can be proved applying [8] in the system. HE EINSTEIN-KLEIN-GORDON SYSTEM 9 Energy estimates
As a preliminary part of later work, we will give the energy estimates of the Einstein-wave-Klein-Gordon system in this section. The energy estimates are performed in awell-known way as in [1]. The notations here will follow what were given in [2]. Forsimplicity, we denote p p q M, p q g q by p M, g q from now on.Let us define the energy on the Cauchy surface Σ t , E p t q : “ ż Σ t e ¯ µ q “ π ż e p t, r q re β p t,r q dr, the energy in a coordinate ball B r , E p t, r q : “ ż B r e ¯ µ q “ π ż r e p t, r q r e β p t,r q dr , the energy inside the causal past J ´ p O q of O , E O p t q : “ ż Σ t X J ´ p O q e ¯ µ q with O appears at the axis.3.1. Energy conservation.
We start by proving the energy is conserved.
Proposition 3.1.
The energy E p t q is conserved, ddt E p t q “ . Proof.
Consider two Cauchy surfaces Σ s and Σ τ at t “ s and t “ τ respectively. First, letus construct a divergence free vector field P T . Consider the Einstein’s equations (2.10)and (2.11). They can be rewritten as follows(3.1) ´ B r p e ´ β q “ re β e (3.2) ´ B t p e ´ β q “ re α m From the smoothness of β , we have ´B rt p e ´ β q “ ´B tr p e ´ β q . Together with (3.1)(3.2), we infer(3.3) ´ B t p re β e q ` B r p re α m q “ . Now we define a vector field P T : “ ´ e ´ α e B t ` e ´ β m B r . The divergence of P T is given by ∇ ν P νT “ a | g | B ν ´a | g | P νT ¯ (3.4) “ re β ` α ` ´B t p re β e q ` B r p re α m q ˘ . By (3.3), we know that P T is divergence free. Next, let us apply Stokes’ theorem inthe space-time region whose boundary is Σ s Y Σ τ . Due to the asymptotic flatness, theboundary terms at r “ 8 do not contribute. Thus, we have(3.5) ż Σ s e α P tT ¯ µ q ´ ż Σ τ e α P tT ¯ µ q “ . Therefore, it follows that(3.6) E p τ q “ E p s q which proves that the energy is conserved. (cid:3) With the conservation of the energy, we can prove that the metric functions β p t, r q and α p t, r q are uniformly bounded during the evolution of the Einstein-wave-Klein-Gordonsystem. Proposition 3.2.
There exists constants c ´ β , c ` β , c ´ α , c ` α depending only on the initialdata and the universal constants such that the following uniform bounds on the metricfunctions β p t, r q and α p t, r q hold c ´ β ď β p t, r q ď c ` β ,c ´ α ď α p t, r q ď c ` α . Proof.
For simplicity of notation, we use a generic constant c for the estimates on β p t, r q and α p t, r q . Integrating (3.1) with respect to r and noting the normalisation that β | Γ “ ´ e ´ β “ ż r e r e β dr “ π E p t, r q , which implies e β “ ˆ ´ π E p t, r q ˙ ´ . Now we define β p t q “ lim r Ñ8 β p r, t q . Then we have e β p t q “ ˆ ´ π E p t q ˙ ´ . Since E p t, r q is a nondecreasing function of r , then so is β p t, r q , therefore,1 “ e β p t, q ď e β p t,r q ď e β p t q . Moreover, since the energy is conserved E p t q “ E p q , β p t q is also conserved during theevolution of the Einstein-wave-Klein-Gordon system, β p t q “ β p q . HE EINSTEIN-KLEIN-GORDON SYSTEM 11
Thus, 0 ď β p t, r q ď β p q . Now we introduce f : “ m e ´ γ φ Thus the Einstein’s equation (2.12) for α r takes the form α r “ re β p e ´ f q . Similarly, we integrate the above equation with respect to r and noting the normalisationthat α | Γ “
0, we obtain α p t, r q ď c ż r p e ´ f q r e β dr ď c ż r e r e β dr ď c and α p t, r q ě ´ c ż r f r e β dr ě ´ c ż r e r e β dr ě ´ c. This finishes the proof of the proposition. (cid:3)
The Vector field Method.
Let X be a vector field on M . Set the correspondingmomentum P X as follows(3.7) P µX “ T µν X ν , then, we have(3.8) ∇ ν P νX “ X µ ∇ ν T νµ ` T νµ ∇ ν X µ . Since the stress-energy tensor T µν satisfies ∇ µ T µν “ , the first term in the right hand side vanishes, hence ∇ ν P νX “ T µν ∇ µ X ν “ p X q π µν T µν , where the deformation tensor p X q π µν is defined by p X q π µν : “ ∇ µ X ν ` ∇ ν X µ “ g σν B µ X σ ` g σµ B ν X σ ` X σ B σ g µν . For instance, consider T “ e ´ α B t , the corresponding momentum P T is(3.9) P T “ ´ e ´ α e B t ` e ´ β m B r . Then, we have the non-zero components of the deformation tensor are p T q π “ e α α r “ p T q π , p T q π “ e β ´ α β t . Thus, using the Einstein equations (2.11)(2.12), we have that the divergence of P T is, ∇ ν P νT “ e ´ α β t p e ´ f q ´ e ´ β α r m “ . This is compatible with (3.3)(3.4).Now let J ´ p O q be the causal past of the point O on the axis and I ´ p O q the chronologicalpast of O . Compared to the flat case, we give the following definitionsΣ Ot : “ Σ t X J ´ p O q ,K p t q : “ Y ď t ď t ă t O Σ t X J ´ p O q ,C p t q : “ Y ď t ď t ă t O Σ t X ` J ´ p O qz I ´ p O q ˘ ,K p t, s q : “ Y ď t ď t ă s Σ t X J ´ p O q ,C p t, s q : “ Y ď t ď t ă s Σ t X ` J ´ p O qz I ´ p O q ˘ for 0 ď t ă s ă t O .Then, the volume 3-form of p M, g q is given by¯ µ g “ re β ` α dt ^ dr ^ dθ and the area 2-form of p Σ , q q is given by¯ µ q “ re β dr ^ dθ. Let us define 1-forms ˜ l, ˜ n and ˜ m as follows˜ l : “ ´ e α dt ` e β dr, ˜ n : “ ´ e α dt ´ e β dr, ˜ m : “ rdθ, therefore, ¯ µ g “ ´ ˜ l ^ ˜ n ^ ˜ m ¯ . Then we introduce the 2-forms ¯ µ ˜ l and ¯ µ ˜ n such that¯ µ ˜ l : “ ´
12 ˜ n ^ ˜ m, ¯ µ ˜ n : “
12 ˜ l ^ ˜ m, so we have ¯ µ g “ ´ ˜ l ^ ¯ µ ˜ l , ¯ µ g “ ´ ˜ n ^ ¯ µ ˜ n . Now, let us apply the Stokes’ theorem for the ¯ µ g -divergence of P X in the region K p τ, s q .We have(3.10) ż K p τ,s q ∇ ν P νX ¯ µ g “ ż Σ Os e α P tX ¯ µ q ´ ż Σ Oτ e α P tX ¯ µ q ` F lux p P X qp τ, s q HE EINSTEIN-KLEIN-GORDON SYSTEM 13 where
F lux p P X qp τ, s q “ ´ ż C p τ,s q ˜ n p P X q ¯ µ ˜ n . Monotonicity of Energy.Proposition 3.3.
There holds E O p τ q ě E O p s q for ď τ ă s ă t O .Proof. Applying Stokes’ theorem (3.10) to the vector field P T , we have(3.11) 0 “ ´ ż Σ Os e ¯ µ q ` ż Σ Oτ e ¯ µ q ` F lux p P T qp τ, s q where F lux p P T qp τ, s q “ ´ ż C p τ,s q ˜ n p P T q ¯ µ ˜ n “ ´ ż C p τ,s q p e ´ m q ¯ µ ˜ n . Noting that we have e ě | m | , we obtain F lux p P T qp τ, s q ď E O p τ q ´ E O p s q ě , @ ď τ ď s ă t O . This concludes the proof of the proposition. (cid:3) Null coordinates
In this section, we introduce a null coordinate system, in which the wave equationsmay be written in a classical form in the flat case. This coordinate system will be ofgreat importance in our work on global well-posedness. In the following part, we assumethat all objects are smooth, unless otherwise stated.4.1.
Existence of null coordinates.
Let p Q , ˇ g q be the orbit space, where Q “ M { S and ˇ g “ ´ e α dt ` e β dr . As discussed in Section 1, the orbit space p Q , ˇ g q is a 2-dimensional globally hyperbolicLorentzian space and thus in particular locally conformally flat. Hence, as noted in [22],we may introduce a null coordinate system with respect to which ˇ g takes the formˇ g “ ´ e λ p u, v q dudv which means that the 3-dimensional manifold p M, g q admits a coordinate system p u, v, θ q such that g takes the form g “ ´ e λ p u, v q dudv ` r p u, v q dθ where now dθ is the line element on the S symmetry orbit. Then we can define T “ u ` v , R “ v ´ u . A direct calculation gives the following equations that the null coordinates satisfy,(4.1) u t ` e α ´ β u r “ ,v t ´ e α ´ β v r “ . Obviously, such coordinate system exists locally. Now we impose the following initialboundary conditions,(4.2) t “ u “ ´ r, v “ rr “ u “ v We will show that the solution of the above initial-boundary problem (4.1)(4.2) give aglobal null coordinate system in the next part, where a prior estimate for the Jacobianis given.4.2. L Estimate for the Jacobian.
Based on the energy estimates in section 3,we aim to derive uniform bounds for the Jacobian transformation between p t, r, θ q and p u, v, θ q coordinates in this section, which can lead to a global null coordinate system.In view of the definitions of the 1-forms ˜ l and ˜ n in section 3, their corresponding vectorfields are null, given by ˜ l “ e ´ α B t ` e ´ β B r , ˜ n “ e ´ α B t ´ e ´ β B r . Lemma 4.1.
There exists two scalar functions F and G such that (4.3) B v “ e F ˜ l, B u “ e G ˜ n. Moreover, F and G satisfy the following equations (4.4) 2 B v p G q “ e F re β p e ` m ´ f q , (4.5) 2 B u p F q “ ´ e G re β p e ´ m ´ f q . Proof.
In view of (4.1) and (4.2), on the initial surface, there holds ˆ B t u B r u B t v B r v ˙ “ ˆ e α ´ β ´ e α ´ β ˙ which implies ˆ B u t B v t B u r B v r ˙ “ ˆ e β ´ α e β ´ α ´ ˙ on t “ B v r “ , B u r “ ´ , ˜ l p r q “ e ´ β , ˜ n p r q “ ´ e ´ β on t “ B u and B v are null vectors, and B u , B v , ˜ l and ˜ n are all future directed.Thus, we infer that there exists two scalar functions F and G such that B v “ e F ˜ l, B u “ e G ˜ n, HE EINSTEIN-KLEIN-GORDON SYSTEM 15 with the normalization on the initial Cauchy surface F “ G “ β. Now we derive the equations that F and G satisfy. We have r ˜ l, ˜ n s “ e ´p β ` α q p´ α r B t ` β t B r q . Then, rB v , B u s “ e p F ` G q ´ r ˜ l, ˜ n s ` ˜ l p G q ˜ n ´ ˜ n p F q ˜ l ¯ “ e p F ` G q e ´p β ` α q p´ α r B t ` β t B r q ` e G B v p G qp e ´ α B t ´ e ´ β B r q´ e F B u p F qp e ´ α B t ` e ´ β B r q . Since rB v , B u s “ F and G are such that e ´ F B v p G q ´ e ´ G B u p F q “ re β p e ´ f q ,e ´ F B v p G q ` e ´ G B u p F q “ re β m , and hence 2 B v p G q “ e F re β p e ` m ´ f q , B u p F q “ ´ e G re β p e ´ m ´ f q . This concludes the proof of the lemma. (cid:3)
Now, with respect to the null coordinate system, let us revisit Stokes’ theorem for¯ µ g -divergence for P X in K p τ, s q . We have dv “ ´ e ´ F ˜ n, du “ ´ e ´ G ˜ l. And the volume 3-form of p M, g q takes the form¯ µ g “ re λ du ^ dv ^ dθ. Next, we introduce the 2-forms ¯ µ v and ¯ µ u as follows¯ µ g “ dv ^ ¯ µ v , ¯ µ g “ du ^ ¯ µ u . From the above two formulas, we infer¯ µ v “ ´ re λ p du ^ dθ q , ¯ µ u “ re λ p dv ^ dθ q . Therefore,
F lux p P X qp τ, s q “ ż C p τ,s q dv p P X q ¯ µ v , for instance, F lux p P T qp τ, s q “ ż C p τ,s q dv p P T q ¯ µ v , “ ´ ż C p τ,s q e ´ F p e ´ m q ¯ µ v . Lemma 4.2.
There exists constants c ´ G , c ` G , c ´ F and c ` F depending only on the initial dataand the universal constants such that the following uniform bounds hold c ´ G ď G ď c ` G c ´ F ď F ď c ` F . Proof.
Integrating (4.5) with the fact that F “ G “ β on the initial Cauchy surface, weobtain 2 F p u, v q ´ β p´ v, v q “ ż u ´ v e G re β p e ´ m ´ f qp u , v q du . Next, note that ´ e λ “ g pB u , B v q “ e F ` G g p ˜ n, ˜ l q “ ´ e F ` G λ “ F ` G . Thus, we have 2 F p u, v q ´ β p´ v, v q “ ż u ´ v e ´ F re β p e ´ m ´ f q e λ du . Noting the fact that | e ˘ m ´ f | ď e ˘ m and ´ v ď u , using Proposition 3.2, we obtain | F p u, v q| À c ` ż u ´ v e ´ F p e ´ m q re λ du . Since du “ ´ e ´ G ˜ l , we obtain | F p u, v q| À c ` ż u ´ v dv p P T q re λ du . After integration in θ , the right-hand sides are bounded by fluxes which in turn arebounded by the energy, and hence | F | ď c. For G , if u ď
0, we can integrate (4.4) from the initial surface and get the uniformbound in a similar way as we did for F .If u ą
0, we need some normalisation on Γ to estimate G . Noting that we have u “ v ,which infers R “ r “ u “ v “ T . In view of (4.1) and (4.2), on Γ, there holds ˆ B t u B r u B t v B r v ˙ “ ˆ T t T t T t ´ T t ˙ which implies ˆ B u t B v t B u r B v r ˙ “ ˆ T t ´ T t ´ T t ´ ´ T t ´ ˙ on Γ.Thus, we have B v r “ ´B u r, ˜ l p r q “ , ˜ n p r q “ ´ B u r “ ´ e G ´ β , B v r “ e F ´ β . HE EINSTEIN-KLEIN-GORDON SYSTEM 17
Therefore, we obtain that on Γ there holds G “ F . Now we can integrate (4.4) from the axis,2 G p u, v q ´ G p u, u q “ G p u, v q ´ F p u, u q“ ż vu e F re β p e ` m ´ f qp u, v q dv “ ż vu e ´ G re β p e ` m ´ f q e λ dv Noting the fact that | e ˘ m ´ f | ď e ˘ m and u ď v , using Proposition 3.2, we deduce | G p u, v q| À c ` ż vu e ´ G p e ` m q re λ dv , Since dv “ ´ e ´ F ˜ n , we obtain | G p u, v q| À c ` ż vu du p P T q re λ dv ď c. This finishes the proof of the lemma. (cid:3)
Corollary 4.3.
There exist constants c ´ λ and c ` λ depending only on the initial energyand the universal constants such that the following uniform bounds hold on the metricfunction λ in null coordinates (4.7) c ´ λ ď λ ď c ` λ . Proof.
This follows immediately from (4.6) and Lemma 4.2. (cid:3)
Now, by (4.3), we can write the Jacobian J of the transition functions between p t, r, θ q and p v, u, θ q , J : “ ¨˝ B v t B u t B θ t B v r B u r B θ r B v θ B u θ B θ θ ˛‚ “ ¨˝ e F ´ α e G ´ α e F ´ β ´ e G ´ β
00 0 2 ˛‚ then the inverse Jacobian J ´ is given by J ´ “ ¨˝ B t v B r v B θ v B t u B r u B θ u B t θ B r θ B θ θ ˛‚ “ ¨˝ e ´ F ` α e ´ F ` β e ´ G ` α ´ e ´ G ` β
00 0 1 ˛‚ Therefore, dv “ e p´ F ` α q dt ` e p´ F ` β q dr, du “ e p´ G ` α q dt ´ e p´ G ` β q dr. Corollary 4.4.
There exist constants c ´ µν , c ` µν and C ´ µν , C ` µν depending only on the initialdata and the universal constants such that all the entries of the Jacobian J and its inverse J ´ are uniformly bounded c ´ µν ď J µν ď c ` µν C ´ µν ď J ´ µν ď C ` µν for µ, ν “ , , .Proof. The proof follows from Proposition 3.2 and Lemma 4.2. (cid:3)
Corollary 4.5.
For the scalar functions r, R , there exist constants c , c such that thefollowing pointwise estimates hold r ě c R, r ď c R. Proof.
By the L estimate for the Jacobian and its inverse we have done in Corollary4.4, we can obviously get the following estimates,(4.8) |B R r | “ |B v r ´ B u r | “ | e F ´ β ` e G ´ β | ď c , (4.9) |B r R | “ |B r v ´ B r u | “ | e ´ G ` β ` e ´ F ` β | ď c , The proof then follows by applying the fundamental theorem of calculus. (cid:3)
With the use of Corollary 4.4, we can now construct a global null coordinate system,and we will consider our problem under this coordinate.4.3.
The Einstein equations in null coordinates.
In null coordinates, the compo-nents of the Einstein tensor take the following form G “ ´ e λ r ´ B u p e ´ λ B u r q ,G “ r ´ B u B v r,G “ ´ e λ r ´ B v p e ´ λ B v r q ,G “ ´ r e ´ λ B u B v λ. Other components are zero.Thus, rewritting the Einstein-wave-Klein-Gordon system (2.10)-(2.15) in null coordi-nates, we can get(4.10) B u p e ´ λ B u r q “ ´ e ´ λ r p γ u ` φ u q (4.11) r uv “ m re λ ´ γ φ (4.12) B v p e ´ λ B v r q “ ´ e ´ λ r p γ v ` φ v q (4.13) λ uv “ ´ γ u γ v ´ φ u φ v ` m e λ ´ γ φ (4.14) B u p r B v γ q ` B v p r B u γ q “ m re λ ´ γ φ (4.15) B u p r B v φ q ` B v p r B u φ q “ ´ m re λ ´ γ φ with stress-energy tensor, T “ γ u ` φ u ,T “ m e λ ´ γ φ ,T “ γ v ` φ v ,T “ r e ´ λ γ u γ v ` r e ´ λ φ u φ v ´ r e ´ γ m φ . We can also rewrite the system in p T, R q coordinates which reads(4.16) r T r λ T ` r R r λ R ´ r RR r “ γ T ` γ R ` φ T ` φ R ` e λ ´ γ m φ (4.17) ´ r T R r ` r T r λ R ` r R r λ T “ γ T γ R ` φ T φ R (4.18) r T r λ T ` r R r λ R ´ r T T r “ γ T ` γ R ` φ T ` φ R ´ e λ ´ γ m φ (4.19) e ´ λ λ RR ´ e ´ λ λ T T “ ´ e ´ γ m φ ` e ´ λ γ T ´ e ´ λ γ R ` e ´ λ φ T ´ e ´ λ φ R l g γ “ ´ e ´ λ γ T T ` e ´ λ γ RR ´ e ´ λ r T r γ T ` e ´ λ r R r γ R (4.20) “ ´ e ´ γ m φ l g φ “ ´ e ´ λ φ T T ` e ´ λ φ RR ´ e ´ λ r T r φ T ` e ´ λ r R r φ R (4.21) “ e ´ γ m φ with the stress-energy tensor T µν , T “ γ T ` γ R ` φ T ` φ R ` e λ ´ γ m φ “ e λ ˜ e ,T “ γ T γ R ` φ T φ R ,T “ γ T ` γ R ` φ T ` φ R ´ e λ ´ γ m φ ,T “ r e ´ λ γ T ´ r e ´ λ γ R ` r e ´ λ φ T ´ r e ´ λ φ R ´ r e ´ γ m φ . Let U “ p γ, φ q , rewrite equations p . qp . q in the following form on the Minkowskispacetimes p R ` , m q , l m U “ ´ U T T ` U RR ` R U R (4.22) “ e λ F p U q φ ` e λ G p U q φ ` r T r U T ´ ˆ r R r ´ R ˙ U R fi h. Estimates on Energy Flux.
In view of the L estimates of the Jacobian and themonotonicity of the energy , we give the following estimates on energy fluxes, Proposition 4.6.
For ď T ď T , consider the following space-time region t T ď T ď T u Ă Q , where Q denotes the maximal development. There holds the following estimates ż min p v, T ´ v q T ´ v ´ pB u γ q ` pB u φ q ` m e ´ γ φ ¯ p u , v q rdu À E p q , ż T max p u, T ´ u q ´ pB v γ q ` pB v φ q ` m e ´ γ φ ¯ p u, v q rdv À E p q . Proof.
Noting that the vector field P T is divergence free, using Stokes’ theorem, we canget ˇˇˇˇˇż min p v, T ´ v q T ´ v ż πθ “ e ´ F p e ´ m q ¯ µ v ˇˇˇˇˇ ď E p q and ˇˇˇˇˇż T max p u, T ´ u q ż πθ “ e ´ G p e ` m q ¯ µ u ˇˇˇˇˇ ď E p q . where we use the notations and computations given in Section 4.2. In view of the defini-tion of ¯ µ v and ¯ µ u , and the rotation invariance, we deduce ż min p v, T ´ v q T ´ v e ´ F p e ´ m q re λ du À E p q and ż T max p u, T ´ u q e ´ G p e ` m q re λ dv À E p q . Then, by the definition of e and m and (4.3) (4.6), we obtain ż min p v, T ´ v q T ´ v e G ´ e ´ G pB u γ q ` e ´ G pB u φ q ` m e ´ γ φ ¯ rdu À E p q and ż T max p u, T ´ u q e F ´ e ´ F pB v γ q ` e ´ F pB v φ q ` m e ´ γ φ ¯ rdv À E p q . Therefore, using Lemma 4.2, we finally obtain the estimates on fluxes, ż min p v, T ´ v q T ´ v ´ pB u γ q ` pB v φ q ` m e ´ γ φ ¯ rdu À E p q and ż T max p u, T ´ u q ´ pB v γ q ` pB v φ q ` m e ´ γ φ ¯ rdv À E p q . (cid:3) HE EINSTEIN-KLEIN-GORDON SYSTEM 21
Lower bound of γ . In this part, we aim to show that γ has a lower bound under thegiven initial conditions. We may achieve this by some translation. An easy observationtells that the 4-tuple p λ ` , r, γ ` , φ q also satisfy (4.16)-(4.21). Then, we define˜ λ “ λ ` , ˜ γ “ γ ` . Now we will prove that ˜ γ is nonnegative by a bootstrap argument. Firstly, we make abootstrap assumption(4.23) r ˜ γ ě . The goal will be to improve this bootstrap assumption.
Proposition 4.7.
Under the given initial conditions (1.6) and the bootstrap assumption (4.23) , for any T ą , R ą , there holds r ˜ γ ą . Proof.
Multiplying (4.20) by r , we can verify that p r ˜ γ q T T ´ p r ˜ γ q RR “ e λ ´ γ m r φ ´ r ´ p r u ¨ r v q ˜ γ In view of the normalisation on the initial surface in 4.2, we see that r ˜ γ is the solutionof the above 1-dimensional wave equation with initial-boundary conditions T “ r ˜ γ “ R p γ ` q , p r ˜ γ q T “ R e β ´ α γ ,R “ r ˜ γ “ . where γ , γ are the initial data of γ and γ t .Then, when R ě T , we have p r ˜ γ qp T, R q “ p R ` T q p γ p R ` T q ` q ` p R ´ T q p γ p R ´ T q ` q` ż R ` TR ´ T ξ e β ´ α p , ξ q γ p ξ q dξ ` ż T ż R `p T ´ τ q R ´p T ´ τ q e λ ´ γ m r φ dξdτ ´ ż T ż R `p T ´ τ q R ´p T ´ τ q r ´ p r u ¨ r v q ˜ γdξdτ, and when R ă T , we have p r ˜ γ qp T, R q “ p R ` T q p γ p R ` T q ` q ´ p T ´ R q p γ p T ´ R q ` q` ż T ` RT ´ R ξ e β ´ α p , ξ q γ p ξ q dξ ` ż T ż R `p T ´ τ q| R ´p T ´ τ q| e λ ´ γ m r φ dξdτ ´ ż T ż R `p T ´ τ q| R ´p T ´ τ q| r ´ p r u ¨ r v q ˜ γdξdτ. Firstly, we consider the lower bound of the case R ě T . By Proposition 3.2, the initialconditions (1.6), the bootstrap assumption (4.23), the calculation of the Jacobian, and the equalities above, we have r ˜ γ ě p R ` T q p γ p R ` T q ` q ` p R ´ T q p γ p R ´ T q ` q` ż R ` TR ´ T ξ e β ´ α p , ξ q γ p ξ q dξ ` ż T ż R `p T ´ τ q R ´p T ´ τ q e λ ´ γ m r φ dξdτ ě p R ` T q p γ p R ` T q ` q ` p R ´ T q p γ p R ´ T q ` qą R ă T , we can estimate similarly, r ˜ γ ě p R ` T q p γ p R ` T q ` q ´ p T ´ R q p γ p T ´ R q ` q Noting that by (1.6), on the initial surface, we have B r p r p γ ` qq “ r γ r ` r ´ p γ ` qě r γ r ` r ´ ą , if r ą
0. Therefore, r ˜ γ ą . when R ă T .Thus, we have shown that r ˜ γ ą . This finishes the proof of the proposition. (cid:3)
The above proposition close the bootstrap argument and yields the nonnegativity of˜ γ , which shows that γ has a lower bound γ ě ´ Existence of null coordinates in a light cone.
Now we consider a cone J ´ p O q with O a point on the axis. As is done before, we can similarly construct a null coordinatesystem in J ´ p O q with different initial boundary conditions,(4.24) t “ u “ ´ ˜ vr “ u “ t, ˜ v “ t The process of constructing such a coordinate system is just similar to what we havedone in 4.1 and 4.2, and obviously the equations take the same form in this coordinateas in 4.3. The estimates we did before in section 4 still hold in the new null coordinatesystem.This coordinate will be used only in the second part of our work of proving the globalexistence of the Einstein-wave-Klein-Gordon system. For simplicity, we still denote p ˜ u, ˜ v q by p u, v q . Here, we give two more properties. HE EINSTEIN-KLEIN-GORDON SYSTEM 23
Corollary 4.8.
There exist constants c , c , c , c such that r ě c R, t ě c T,r ď c R, t ď c T hold in J ´ p O q .Proof. Similar to Corollary 4.5, using (4.8)(4.9), we can obtain c R ď r ď c R. Moreover, by the L Estimate for the Jacobian, we have(4.25) |B T t | “ |B v t ` B u t | “ | e F ´ α ` e G ´ α | ď c , (4.26) |B R t | “ |B v t ´ B u t | “ | e F ´ α ´ e G ´ α | ď c , (4.27) |B t T | “ |B t v ` B t u | “ | e ´ F ` α ` e ´ G ` α | ď c , (4.28) |B r T | “ |B r v ` B r u | “ | e ´ F ` β ´ e ´ G ` β | ď c . Noting that in J ´ p O q there holds R ď T , and r À t owing to the fact that α and β are bounded, then, as in Corollary 4.5, the proof follows by applying the fundamentaltheorem of calculus in the region J ´ p O q . (cid:3) To work in the new coordinate system later, we need an energy estimate in p T, R q coordinates. We can similarly define energy in p T, R q coordinates,˜ E O p T q : “ ż ˜Σ T X J ´ p O q ˜ e ¯ µ ˜ q where p ˜Σ T , ˜ q q denotes the Cauchy surface. Then, we have Proposition 4.9.
For ď T ď T O , we have that in J ´ p O q , the following estimateshold ˜ E O p T q À E p q . Proof.
Firstly, the proposition holds at the initial time. Then, for T ą
0, we will useStokes’ theorem for P T on the region˜ K p T , s q : “ t T ď T, t ď s u X J ´ p O q . Using Stokes’ theorem and the estimates on energy flux, we can get(4.29) ´ ż ˜Σ T X J ´ p O q x P T , n y ¯ µ ˜ q À ε with n the unit normal vector, n “ ´ e ´ λ B T “ ´ e ´ λ pB u ` B v q Then, by Lemma 4.1, we have n “ ´ p e ´ λ ` F ´ α ` e ´ λ ` G ´ α q B t ´ p e ´ λ ` F ´ β ´ e ´ λ ` G ´ β q B r . Therefore, ´x P T , n y “ e ´ λ ` F p e ` m q ` e ´ λ ` G p e ´ m q . In view of the definition of e and m and (4.3) (4.6), the L Estimate for the Jacobian,Corollary 4.8, we imply ´x P T , n y “ e ´ λ ` F p e ´ F pB v γ q ` e ´ F pB v φ q ` m e ´ γ φ q` e ´ λ ` G p e ´ G pB u γ q ` e ´ G pB u φ q ` m e ´ γ φ qě C ppB v γ q ` pB v φ q ` pB u γ q ` pB u φ q ` m e ´ γ φ qě C ˜ e . Thus, by (4.29), we have ˜ E O p T q À ε. which concludes the proof of the proposition. (cid:3) The first singularity occurs on the axis
In this section, we prove that the first possible singularity must occur on the axis. Thisis equivalent to show that the Cauchy problem admits a regular solution away from theaxis.
Proposition 5.1.
For any p ¯ u, ¯ v q away from the axis, the solution of the system (4.16) - (4.21) is regular at p ¯ u, ¯ v q .Proof. Away from the axis, the equations become a 1 ` Q , without loss of generality, we consider p ¯ u, ¯ v q P ¯ Q z Q . For p ¯ u, ¯ v q P ¯ Q z Q , there exists a small light cone C p ¯ u, ¯ v q away from the axis withvertex at p ¯ u, ¯ v q satisfying T ď T ď T , ă R ď R ď R , @p T, R q P C p ¯ u, ¯ v q , and T ´ T ăă , R ´ R ăă . and p C p ¯ u, ¯ v q ztp ¯ u, ¯ v quq Ă Q . We denote the small scale of the light cone by e .Now, by estimates on energy flux, we have | U p ¯ u, ¯ v q| ď | U p´ ¯ v, ¯ v q| ` ż ¯ u ´ ¯ v | U u | du À C ` p ż ¯ u ´ ¯ v | U u | du q À C ` p ż ¯ u ´ ¯ v | U u | Rdu q ď C p ¯ u, ¯ v q . HE EINSTEIN-KLEIN-GORDON SYSTEM 25
Then,we rewrite the Einstein equations (4.14) and (4.15),(5.1) γ uv “ ´ r u r γ v ´ r v r γ u ` m e λ ´ γ φ , (5.2) φ uv “ ´ r u r φ v ´ r v r φ u ´ m e λ ´ γ φ. Let X “ sup C p ¯ u, ¯ v q | U v | . Integrating (5.1) from u to u , where p u , v q on the initial hypersurface of C p ¯ u, ¯ v q and p u, v q P C p ¯ u, ¯ v q , we obtain | γ v | ď M ` X ż uu | r u r | du ` ż uu | r v r γ u | du ` ż uu m e λ ´ γ φ du À M ` X ż uu du ` ż uu | γ u | du ` ε À M ` eX ` ˆż uu | γ u | rdu ˙ À M ` eX. Then we integrate (5.2) from u to u , noting that γ has a lower bound, we obtain | φ v | ď M ` X ż uu | r u r | du ` ż uu | r v r φ u | du ` ż uu m e λ ´ γ φdu À M ` X ż uu du ` ż uu | φ u | ` | e ´ γ φ | du À M ` eX ` ˆż uu p| φ u | ` e ´ γ φ q rdu ˙ À M ` eX ` E p qÀ M ` eX. Thus, X À M ` eX which implies X À M. We can similarly get upper bounds for B u U . Therefore, |B U | À M. Now we integrate (4.13) along the ingoing light ray, we get | λ v | À M ` ż uu | γ u || γ v | du ` ż uu | φ u || φ v | du ` ż uu m e λ ´ γ φ du À M ` M ˆż uu p| γ u | ` | φ u | q rdu ˙ ` E p qÀ M. The same estimate holds for B u λ . Then, |B λ | À M. Thus, it must hold that p ¯ u, ¯ v q P Q . This finished the proof of Proposition 5.1. (cid:3) Acknowledgement
Both authors are grateful to Prof. Naqing Xie for fruitful discussions. The first authorespecially thanks him for his kind guidance.Y. Zhou was supported by Key Laboratory of Mathematics for Nonlinear Sciences(Fudan University), Ministry of Education of China, P.R.China. Shanghai Key Labora-tory for Contemporary Applied Mathematics, School of Mathematical Sciences, FudanUniversity, P.R. China, NSFC (grants No. 11421061, grants No.11726611, grants No.11726612), 973 program (grant No. 2013CB834100) and 111 project.
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