An intrinsic hyperboloid approach for Einstein Klein-Gordon equations
In  Klainerman introduced the hyperboloidal method to prove the global existence results for nonlinear Klein-Gordon equations by using commuting vector fields. In this paper, we extend the hyperboloidal method from Minkowski space to Lorentzian spacetimes. This approach is developed in  for proving, under the maximal foliation gauge, the global nonlinear stability of Minkowski space for Einstein equations with massive scalar fields, which states that, the sufficiently small data in a compact domain, surrounded by a Schwarzschild metric, leads to a unique, globally hyperbolic, smooth and geodesically complete solution to the Einstein Klein-Gordon system.
In this paper, we set up the geometric framework of the intrinsic hyperboloid approach in the curved spacetime. By performing a thorough geometric comparison between the radial normal vector field induced by the intrinsic hyperboloids and the canonical , we manage to control the hyperboloids when they are close to their asymptote, which is a light cone in the Schwarzschild zone. By using such geometric information, we not only obtain the crucial boundary information for running the energy method in , but also prove that the intrinsic geometric quantities including the Hawking mass all converge to their Schwarzschild values when approaching the asymptote.
We introduce the intrinsic hyperboloid approach in the dynamic, Lorentzian spacetime. This approach is developed in  to prove, under the maximal foliation gauge, the global stability of Minkowski space for Einstein equations with massive scalar fields, which reads as
with the stress-energy tensor
where and R denote the Ricci curvature tensor and the scalar curvature of the Lorenzian metric respectively. Applying the conservation law , which is due to the Bianchi identity, gives the Einstein Klein-Gordon system
It is obvious that , with being Minkowski, trivially solves the system. To construct nontrivial global solutions of (1.1)-(1.2), it is natural to consider the Cauchy problems with the initial data set being small perturbations of the trivial one.
We first briefly review the framework for studying the Cauchy problem of the Einstein equations. Let be globally hyperbolic which means that there is a Cauchy hypersurface, which is a spacelike hypersurface with the property that any causal curve intersects it at precisely one point. This allows to be foliated by the level surfaces of a time function . Let T be the future directed unit normal to . Let be the second fundamental form of in defined by
where denotes the covariant differentiation of in .
Let be the induced metric of on . We decompose
where is the lapse function and is the shift vector field. Assuming , then the metric can be written as
and the Einstein equations are equivalent to the evolution equations
together with the constraint equations
where is the mean curvature of in , denotes the covariant differentiation of , and are the Ricci curvature and the scalar curvature of on .
The maximal foliation gauge imposes
This implies satisfies the elliptic equation
and the second fundamental form satisfies the Codazzi equation
where is the magnetic part of the Weyl curvature, defined in (4.8).
The first proof of the global stability of Minkowski spacetime for generic, asymptotically flat data is provided in the monumental work , where the Einstein vacuum Bianchi equation is thoroughly and systematically treated. Heuristically, we regard the nonlinear wave equation verifying the standard null condition as the vastly simplified model for the Einsteinian Bianchi equation. Then (1.1)-(1.2) is a coupled system between such nonlinear wave equations and the Klein-Gordon equation in the Einsteinian background. Due to the presence of the massive scalar field, the approach we introduce in this paper is to twist the hyperboloidal energy method devised in the flat spacetime in  for the Klein-Gordon equations to the Lorentzian spacetime, in the sense of incorporating it to the intrinsic energy scheme devised in . Such generalization triggers fundamental changes to the geometry of the intrinsic framework in  for the Einstein equations, which by itself is very challenging even merely for the vacuum case. Our approach is robust for treating both the scalar field and the Einstein part of the equation system. This will be fully confirmed in .
In what follows, we will use the linear Klein-Gordon equation to motivate the use of the intrinsic hyperboloids.
To begin with, let us recall some basics of the invariant vector fields for the free
This set of vector fields is named as commuting vector fields due to the fundamental property
with the second identity occurring only when .
In order to get the decay estimate by the energy approach, we rely on two ingredients: one is the boundedness of the energy or the generalized energy; the other is the Klainerman-Sobolev inequality.
The standard Klainerman-Sobolev inequality
relies on the full set of derivatives, where , and denotes the application of the differential operators in to up to times. For the free wave equation , by using as a multiplier, one can obtain the conserved energy
By using the canonical Morawetz vector field, as a multiplier, one can obtain the conserved generalized energy, which is uniformly comparable for to
which gives more information for than desired.
To see the difference in the treatment for the Klein-Gordon equation, we consider the linear Klein-Gordon equation
which stays conserved if is replaced by except . In contrast to the case of the free wave,
the boundedness of energy does not hold for the full set of the commuting vector fields in . The Klainerman-Sobolev
inequality (1.13) can not be used directly. To get the decay estimates for the Klein-Gordon equations,
in  the Klainerman-Sobolev inequality is applied on the canonical hyperboloids
which are the surfaces orthogonal to . The Klainerman-Sobolev
inequality on hyperboloids merely relies on the Lorentz boosts which are commuting
vector fields of (1.17) and tangent to the hyperboloids. By virtue of this tool, the standard sharp decay
can be derived from the boundedness of energies on hyperboloids. Thus, in order to get the sharp decay for the solutions of (1.17), the same order of commuting vector fields are applied and energies have to be controlled up to one order higher compared with the free wave case. This coincides with the case when we treat Klein-Gordon equation (1.2) coupled with the Einstein Bianchi equations, for which (1.17) and the free wave are the simplest toy models for each part.
We also observe that the two weighted multipliers, and , can not be used to obtain bounded generalized energy for (1.17). This fact together with the fact that the scaling is not a commuting vector field for (1.17), demonstrates that decomposing in terms of the null frame does not improve the decay along the good direction . This is another difference compared with the free wave. Contributed by the commuting vector fields , exhibits much stronger decay along the tangential directions of ; however, has the weakest decay along , the future directed unit normal of . The weakest decay is much weaker than that a free wave exhibits along its only bad direction .
Figure 1(a) depicts the method in , where the data with compact support in are given at . The energy argument is divided into two steps. The first step is the local energy propagation from to the initial hyperboloidal slice , with . The second step is to propagate energy on hyperboloids from to the last slice , in the region enclosed by a Minkowskian light cone as the boundary, along which the solution varnishes due to finite speed of propagation. This figure gives us the blue-print of treating the Einstein-Klein-Gordon system.
In order to set up the Cauchy problem for the Einstein-Klein-Gordon system (1.1)-(1.2) appropriately, to match with, in particular, the scenario that the data for Klein-Gordon equation have compact support, we consider the initial data set for (1.1)-(1.2), which verify the Einstein constraint equations (1.7) and is compactly supported within , the unit Euclidian ball. Outside of the co-centered Euclidean ball of radius , there glues a surrounding Schwarzschild metric specified at Theorem 4.6. See Figure 1(b). We will call the region with , exterior to the Schwarzschild outgoing light cone as the Schwarzschild zone , where initiates from the Euclidean sphere with the value of specified in Section 4.4. We still need to determine the foliation of hyperboloids in the curved spacetime.
There are two options at this point. One way is based entirely on the symmetry and geometry in Minkowski space. This method has been developed in  and  for the Einstein equations under the wave coordinates. The philosophy of the regime is to close the energy argument without aiming at achieving sharp decay for geometric quantities. This allows the stability result to be achieved within a much smaller framework compared with . However it is less precise on the asymptotic behavior of the solution (see [8, Page 47]).
In this paper and , we take the other option which constructs intrinsic hyperboloids adapted to the curved spacetimes. We not only prove the global nonlinear stability, but also give a comprehensive, analytic, global-in-time depiction of the solution. The goal of this paper is to introduce the geometric framework, which equips the nonlinear analysis with sets of tetrads, recovering the symmetry and playing the role of coordinates, all of which are adapted to the dynamical spacetime. The global existence of such tetrads will be justified simultaneously with the quantitative depiction of the spacetime.
When setting up the geometric framework, it is necessary to discriminate, among all the symmetry in the Minkowski space, the most crucial geometric information that needs precision from those allowing error to be controlled analytically. For this purpose, we run a simple energy argument for
by taking the approach as in , that is to consider
The error integral contributed by one term contained in the commutator on the right hand side of (1.20) takes the form
where the deformation tensor is defined by
with D denoting the connection induced by the metric . Here is the Lorentz boost in Minkowski space, since is not the Minkowski metric. With in , the derivative of contracted with this term is evaluated at the bad direction along which it does not decay strongly. Under local coordinates the expression of contains derivatives of the metric , paired with large weights. The best decay for expected by the approach from  is below the borderline for applying Gronwall inequality to control energies. To salvage the energy argument, we construct a set of approximate Lorentz boosts and the intrinsic hyperboloids , adapted to the Einstein background, so that , where denotes the unit normal of the constructed foliation of hyperboloids. These and can be viewed as the corresponding replacements of and in the curved spacetime. The construction of these and needs to preserve the following features:
and are tangent to .
exhausts the chronological future of an origin , with the origin to be appropriately chosen. All the hyperboloids are asymptotic to the outgoing light cone emanating from .
The origin is chosen at , which can be done due to the local extension of the solution. We may choose so that intersects outside of . The freedom of such choice is fixed in Section 2, which is crucial for the proof of the main results of this paper. We leave the details of the constructions to Section 2-3. See Proposition 2.3 for using the first feature to prove and more results on .
The geometric constructions equip us with the approximate Lorentz boost, scaling, and translation vector fields. With them we can run the commuting vector field approach to the Einstein Bianchi equations, which can be viewed as an extension of the approach in , where the regime is based on the construction of the rotation vector fields and the intrinsic null cones. The task, in our situation, is much more involved, due to the difficulties caused by the massive scalar field, the geometry of the hyperboloids, as well as the complexity of the analytic control on the Lorentz boosts. In what follows we focus on addressing the following two basic issues.
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For the Weyl part of curvature, we will run the regime of Bel-Robinson energies defined by the weighted multipliers and . For closing the top order energy, we encounter the issue of requiring higher order -energy for the massive scalar field. However, in order to close the energy estimates, we have to control the energies of the Weyl tensors and the massive scalar field, up to the same order in terms of the -derivatives.
The intrinsic hyperboloids, in principle, are defined from the Minkowskian counterparts by a global diffeomorphism, which needs to be justified simultaneously with the proof of the global existence of the solution. In Minkowski space, the density of the foliation of the canonical hyperboloids approaches infinity near the causal boundary. The control on the intrinsic foliation is considerably more delicate since, analytically, terms of type appear frequently, with when approaching the causal boundary .
To solve the first issue, it is crucial to use the Einstein Bianchi equations, see Lemma 4.1, which allows us to perform the integration by parts when treating the worst type of terms. We then take advantage of the null forms in the Einstein Bianchi equations, together with the expected strong decay from the scalar field. This enables the top order energies to be closed at a sharp level. We will sketch briefly the energy scheme in Section 4.
The second issue is connected to the set-up of the wave zone, the region where we run the energy estimates. We have to take account of the gravitational influence to the causal structure of the foliation of the intrinsic hyperboloids. In this paper, we focus on controlling the intrinsic geometry of the chronological future for and for all in the Schwarzschild zone. This geometric control is significant for dealing with the problem of leakage, for demonstrating that a constructed function is almost optical, and for justifying an excision procedure on the wave zone for the energy scheme. These aspects will be explained in the sequel.
In the Minkowskian set-up (Figure 1(a)), a light cone is used to cut the family of hyperboloids,
as the boundary of the wave zone. The cone needs to be uniformly away from the asymptote. The set-up of such
boundary in the curved spacetime is more subtle. First of all, this boundary should be chosen in the Schwarzschild
region, to guarantee the dynamical part of the solution is contained in its interior. It ought to be a canonical
Schwarzschild light cone
In Lorentzian spacetime the light cones are usually characterized by the level sets of an optical function (see ) which is defined as the solution of the eikonal equation
with prescribed boundary or initial conditions. Then the optical function naturally measures the distance to the causal boundary. To obtain the information of would require geometric controls on the foliations of light cones . However, since such light cones are not involved in our analysis, we do not use the actual optical function. In our framework, conceptually replaces the role usually played by . The geometric control on the hyperboloids lies at the core of our analysis. To achieve the desired analytic feature, we choose to be the proper time to , where verifies the eikonal equation
Throughout the chronological future , we define an alterative function, still denoted by , which does not verify (1.22), yet taking the role of measuring the distance to . In particular, we can show that this function , vanishing on , is sufficiently close to the canonical optical function near in . To show such property, we perform in Section 5 a full analytic comparison between the radial normal of the Schwarzschild frame and the normal vector field on , induced by the foliation of the intrinsic hyperboloid . The main estimates are established in Theorem 5.12 throughout , which is the major building block of this paper. These estimates and their higher order counterparts will be used in the main energy scheme in .
Next we address the issue of the leakage. Let be a point inside the wave zone, near the boundary . The distance maximizing timelike geodesic connecting and is not entirely contained in the wave zone. See Figure 1(b). This phenomenon can be easily seen in Minkowski space. In Minkowskian case the deformation tensors of the boosts vanish and the deformation tensor of the scaling vector field has a standard value. However, in the dynamical spacetime, deformation tensors and need to be analyzed, which is done by integrating along the aforementioned time-like geodesics with the help of the structure equations which contain both the curvature components under the hyperboloidal frame and the second fundamental forms; see Section 3.2. Whether the path of the integration is contained in the wave zone determines how to control the integrand. The geometric information outside of the wave zone can not be provided by the energy estimates. Such information is obtained simultaneously with the main estimates in Section 5 by geometric comparisons and bootstrap arguments.
Now we explain, as part of the energy scheme, the excision of a region which is related to the so-called last slice of hyperboloids, denoted by . As a standard method for proving global results of non-linear dynamical problem, one can suppose a set of bootstrap assumptions hold till certain maximal life-span. Due to various concerns, we set the maximal life-span in terms of the proper time, labeled by . Once the bootstrap assumptions can be improved for all , by the principle of continuation, the solution and the quantitative control can be extended beyond . The wave zone is a region which is enclosed by the initial slice , the last slice as well as the cone . Consider the energy estimates on , which are crucial for controlling . When where , we no longer expect a regular subset of within the wave zone to do the energy estimates (see Figure 2 in Section 4). The subset of wave zone with will be excised for obtaining the -energy. This may lead to the loss of control of in a region with large within the wave zone, which would fail the energy control on . Our strategy is to show that the region of excision is fully contained in , where and other geometric quantities can be controlled by the main estimates. This proof has to be done merely depending on local energy estimate, and the assumption that the foliation of exists up to , which is the case in this paper (see Section 6).
As the other application of the main estimates, we show that the Hawking mass is convergent to the ADM mass of the surrounding Schwarzschild metric along every hyperboloid.
Finally, we comment on the analysis of the intrinsic geometry in . This analysis is independent of the
long-time energy estimates in the wave zone. The idea is to use the transport equations to perform the long-time
geometric comparison. We define a set of quantities which encode the deviation between the intrinsic and the extrinsic
tetrads, and derive the transport equations for them along well-chosen paths. In order to prove the
function is almost optical, we uncover a series of cancelations, contributed by the Schwarzschild metric
and the structure equations of the hyerboloidal foliation. It necessitates
delicate bootstrap arguments and weighted estimates
The paper is organized as follows. In Sections 2-3 we carefully set up the analytic framework of the foliation of intrinsic hyperboloids, and provide the geometric construction of the intrinsic frame of the Lorentz boosts, since such set-up and construction have never appeared in the literature. In Section 4 we sketch the energy scheme in the proof of global stability of Minkowski space for (1.1)-(1.2). In Section 5, by assuming the foliation of the intrinsic hyperboloids and the maximal foliation exist till the last slice of hyperboloid, we provide a thorough depiction of the intrinsic geometry in the Schwarzschild zone, presented in Theorem 5.12, as the main estimates of this paper. The region considered there is the most sensitive region for having the geometric control on hyperboloids. The set of main estimates depends merely on local-in-time energy estimates and the smallness of the given data on the initial maximal slice. We then give applications of the main estimates. The one in Section 6 is to control the region of the excision. In Section 7, we give the asymptotic behavior of the Hawking mass along all hyperboloids.
2. Construction of the boost vector fields
By standard energy and iteration argument, we first solve the Cauchy problem of EKG back to the past to certain fixed . Let be the spacial origin of the given initial slice. We denote by the geodesic through with velocity , where T is the future-directed time-like unit normal of the initial slice . The geodesic is extended (back-in-time) within the radius of injectivity of , intersecting at . is chosen so that the given Cauchy data at the initial slice is fully contained in , where denotes the causal future of . Hence depends on the size of the support of Cauchy data, and is comparable to . To be more precise, is chosen such that intersects at outside of . Now by the shift of , as well as an abuse of notation, at and the initial data is prescribed at , according to the time coordinate after the shift.
We use to denote the chronological future of . Let be the time-like radius of injectivity of , which is defined to be the supremum over all the values for which the exponential map
is a global diffeomorphism from to its image in , where
is the canonical hyperboloid in and is the time-like geodesic with and . We use to denote the part of within the time-like radius of injectivity. In  we will prove that the time-like radius of injectivity is simultaneously when we prove the global well-posedness for EKG, provided the Cauchy data is sufficiently small. Thus we will have once this result is established.
For a point in , we use to denote its geodesic distance to . Then is a smooth function on satisfying with . We introduce the vector field
Then is geodesic, i.e. and . Using this we define the lapse function by
Clearly are the level sets of which give a foliation of in terms of hyperboloids. Moreover, by the Gauss lemma we can see that is the future directed normal to and
for any , where is the unique point in such that .
Using we may introduce the second fundament form of defined by
where are vector fields tangent to . Clearly is an tangent,
symmetric tensor. We will use and to denote the trace and traceless part of
According to the expression of , we can derive that the future directed unit normal T of takes the form
This together with and (2.3) implies that
For future reference, we set
According to the definition of , for any there corresponds a unique with such that
Then gives the geodesic normal coordinates for .
For any there hold
By using (2.3) we can consider the local expansion of at as follows
which, in view of (2.5), implies that
Therefore we can obtain which gives the first identity in (2.10) as . The last identity follows as a consequence of the first two. ∎
2.1. Construction of the boost vector fields
Recall that in Minskowski space, in terms of the geodesic coordinates introduced by (2.9), the boost vector fields are defined by
Note that and . It is straightforward to show that
By using the exponential map to lift vector fields, this leads to introduce boost vector fields
defined on .
The boost vector fields , are tangent to and
Let denote one of the boost vector fields , . Then
Now consider . For a symmetric tensor , suppose
we can obtain from the first equality in (2.14) that
3. Intrinsic hyperboloids
We will use to denote the induced metrics on and let be the covariant differentiation. It is known that
denote the tensor of projection to .
Let . Then for fixed , gives a foliation of . Let be the induced metric on and let be the associated covariant differentiation. Since is normal to , we have
where is the lapse function given by . By using and (2.3) we have
This shows that on and the lapse function is given by
Therefore , where
Let denote the outward unit normal of in . Then, according to (3.1) we have
Similarly let be the induced metric on and let be the corresponding covariant differentiation. Then
where denotes the tensor of projection to given by
Note that for fixed , gives the radial foliation of . Since T is normal to , we have
where for the second equality we used
The equation (3.4) follows from the fact
which also shows that on . Let denote the outward normal vector field of in . Then
3.1. Decomposition of frames
Using T and we define
It is easy to see that
Thus if is an orthonormal frame on , then form a null frame.
We define a pair of functions
which can be regarded as the counterparts for“” in the Minkowski spacetime.
Due to the construction, there hold the two fundamental facts:
in . if and only if , which holds only on , the causal boundary of .
Assuming for some fixed constant ,
9any is asymptotically approaching as . This can be seen by using (3.8)
and in .
This shows that and hence the component along is obtained. We therefore obtain the decomposition of in (3.9).