Well-Posedness of the Einstein–Euler System in Asymptotically Flat Spacetimes

Well-Posedness of the Einstein–Euler System in Asymptotically Flat Spacetimes

Uwe Brauer and Lavi Karp
Abstract

We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein–Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for elliptic and hyperbolic equations in these spaces. The well posedness is obtained in these spaces. The results obtained are related to and generalize earlier works of Rendall [35] for the Euler-Einstein system under the restriction of time symmetry and of Gamblin [17] for the simpler Euler–Poisson system.

Departamento Matemática Aplicada Department of Mathematics
Universidad Complutense Madrid ORT Braude College
28040 Madrid, Spain P. O. Box 78, 21982 Karmiel, Israel
E-mail: oub@mat.ucm.es E-mail: karp@braude.ac.il

1 Introduction

This paper deals with the Cauchy problem for the Einstein-Euler system describing a relativistic self-gravitating perfect fluid, whose density either has compact support or falls off at infinity in an appropriate manner, that is, the density belongs to a certain weighted Sobolev space.

The evolution of the gravitational field is described by the Einstein equations

(1.1)

where is a semi Riemannian metric having a signature , is the Ricci curvature tensor, these are functions of and its first and second order partial derivatives and is the scalar curvature. The right hand side of (1.1) consists of the energy-momentum tensor of the matter, and in the case of a perfect fluid the latter takes the form

(1.2)

where is the energy density, is the pressure and is the four-velocity vector. The vector is a unit timelike vector, which means that it is required to satisfy the normalization condition

(1.3)

The Euler equations describing the evolution of the fluid take the form

(1.4)

where denotes the covariant derivative associated to the metric . Equations (1.1) and (1.4) are not sufficient to determinate the structure uniquely, a functional relation between the pressure and the energy density (equation of state) is also necessary. We choose an equation of state that has been used in astrophysical problems. It is the analogue of the well known polytropic equation of state in the non-relativistic theory, given by

(1.5)

The sound velocity is denoted by

(1.6)

The unknowns of these equations are the semi Riemannian metric , the velocity vector and the energy density . These are functions of and where are the Cartesian coordinates on . The alternative notation will also be used and Greek indices will take the values in the following.

The common method to solve the Cauchy problem for the Einstein equations consists usually of two steps. Unlike ordinary initial value problems, initial data must satisfy constraint equations intrinsic to the initial hypersurface. Therefore, the first step is to construct solutions of these constraints. The second step is to solve the evolution equations with these initial data, in the present case these are first order symmetric hyperbolic systems. As we describe later in detail, the complexity of our problems forces us to consider an additional third step, that is, after solving the constraint equations, we have to construct the initial data for the fluid equations.

The nature of this Einstein-Euler system (1.1), (1.4) and (1.5) forces us to treat both the constraint and the evolution equations in the same type of functional spaces. Under the above consideration, we have established the well posedness of this Einstein-Euler system in weighted Sobolev spaces of fractional order. Oliynyk has recently studied the Newtonian limit of this system in weighted Sobolev spaces of integer order [34]

We will briefly resume the situation in the mathematical theory of self gravitation perfect fluids describing compact bodies, such as stars: For the Euler-Poisson system Makino proved a local existence theorem in the case the density has compact support and it vanishes at the boundary, [29]. Since the Euler equations are singular when the density is zero, Makino had to regularize the system by introducing a new matter variable (). His solution however, has some disadvantages such as the fact they do not contain static solutions and moreover, the connection between the physical density and the new matter density remains obscure.

Rendall generalized Makino’s result to the relativistic case of the Einstein–Euler equations, [35]. His result however suffers from the same disadvantages as Makino’s result and moreover it has two essential restrictions: 1. Rendall assumed time symmetry, that means that the extrinsic curvature of the initial manifold is zero and therefore the Einstein’s constraint equations are reduced to a single scalar equation; 2. Both the data and solutions are functions. This regularity condition implies a severe restriction on the equation of state , namely .

Similarly to Makino and Rendall, we have also used the Makino variable

(1.7)

Our approach is motivated by the following observation. As it turns out, the system of evolution equations have the following form

(1.8)

where the unknown consists of the gravitational field the velocity of the fluid and the Makino variable , and the lower order term contains the energy density . Thus, we need to estimate by in the corresponding norm of the function spaces. Combining this estimation with the Makino variable (1.7), it results in an algebraic relation between the order of the functional space and the coefficient of the equation of state (1.5) of the form

(1.9)

This relation can be easily derived by considering , . Moreover, it can be interpreted either as a restriction on or on . Thus, unlike typical hyperbolic systems where often the regularity parameter is bounded from below, here we have both lower and upper bounds for differentiability conditions of the sort . A similar phenomenon for the Euler-Poisson equations was noted by Gamblin [17].

We want to interpret (1.9) as a restriction on rather than on . Therefore, instead of imposing conditions on the equation of state and in order to sharpen the regularity conditions for existence theorems, we are lead to the conclusion of considering function spaces of fractional order, and in addition, the Einstein equations consist of quasi linear hyperbolic and elliptic equations. The only function spaces which are known to be useful for existence theorems of the constraint equations in the asymptotically flat case, are the weighted Sobolev spaces , , , which were introduced by Nirenberg and Walker, [33] and Cantor [6], and they are the completion of under the norm

(1.10)

Hence we are forced to consider new function spaces , which generalize the spaces to fractional order. The well posedness of the Einstein-Euler system is obtained in these spaces. In order to achieve this, we have to solve both the constraint and the evolution equations in the spaces.

Another difficulty which arises from the non-linear equation of state (1.5) is the compatibility problem of the initial data for the fluid and the gravitational field. There are three types of initial data for the Einstein-Euler system:

  • The gravitational data is a triple , where is space-like manifold, is a proper Riemannian metric on and is a second fundamental form on (extrinsic curvature). The pair must satisfy the constrain equations

    (1.11)

    where is the scalar curvature with respect to the metric .

  • The matter variables, consisting of the energy density and the momentum density , appear in the right hand side of the constraints (1.11).

  • The initial data for Makino’s variable and the velocity vector of the perfect fluid.

The projection of the velocity vector , , on the tangent space of the initial manifold leads to the following relations

(1.12)

between the matters variable and . We cannot give , and by relations (1.5) and (1.12) solve for and , since this is incompatible with the conformal scaling (see Section 4.1). In order to overcome this obstacle, we let and , then (1.12) is equivalent to (4.10) and the last one is invertible.

The paper is organized as follows: In the next section we perform the reduction of the Einstein-Euler system into a first order symmetric hyperbolic system. Choquet-Bruhat showed that the choice of harmonic coordinates converts the field equations (1.1) into wave equations which then can be written as a first order symmetric hyperbolic system [10], [13], [19]. Reducing the Euler equations (1.4) to a first order symmetric hyperbolic system is not a trivial matter. We use a fluid decomposition and present a new reduction of the Euler equations. Beside having a very clear geometric interpretation, we give a complete description of the structure of the characteristics conformal cone of the system, namely, it is a union of a three-dimensional hyperplane tangent to the initial manifold and the sound cone.

In Section 3 we define the weighted Sobolev spaces of fractional order and present our main results. These include a solution of the compatibility problem, the construction of initial data and a solution to the evolution equations in the spaces. The announcement of the main results has been published in [5].

Section 4 deals with the constructions of the initial data. The common Lichnerowicz-York scaling method for solving the constraint equations cannot be applied here directly [13], [7], [42], since it violates the relations (1.12). We need to invert of (1.12) in order construct the initial data and there are two conditions which guarantee it: the dominate energy condition , this is invariant under scaling; and the causality condition, this is the speed of sound (1.6) less than speed of light. Unfortunately the last condition is not invariant under scaling. It is also necessary to restrict the matter variables to a certain region. We show the inversion of (1.12) exists provided that belong to a certain region. This fact enables us to construct initial data for the evolution equations.

The local existence for first order symmetric hyperbolic systems in is discussed in Section 5. The known existence results in the space [16], [24], [21], [38], [37], [28] cannot be applied to the spaces. The main difficulty here is the establishment of energy estimates for linear hyperbolic systems. In order to achieve it we have defined a specific inner-product in and in addition the Kato-Ponce commutator estimate [25], [38], [37] has an essential role in our approach. Once the energy estimates and other tools have been established in the space, we follow Majda’s [28] iteration procedure and show existence, uniqueness and continuity in that norm.

In Section 6 we study elliptic theory in which is essential for the solution of the constraint equations. We will extend earlier results in weighted Sobolev spaces of integer order which were obtained by Cantor [7], Choquet-Bruhat and Christodoulou [11], Choquet-Bruhat, Isenberg and York [12], and Christodoulou and O’Murchadha [14] to the fractional ordered spaces. The central tool is a priori estimate for elliptic systems in the spaces (6.21). Its proof requires first the establishment of analogous a priori estimate in Bessel potential spaces . Our approach is based on the techniques of pseudodifferential operators which have symbols with limited regularity and in order to achieve that we are adopting ideas being presented in Taylor’s books [38] and [39]. A different method was derived recently by Maxwell [30] who also showed existence of solutions to Einstein constraint equations in vacuum in with the best possible regularity condition, namely . The semi-linear elliptic equation is solved by following Cantor’s homotopy argument [7] and generalized to the spaces.

Finally, in the Appendix we deal with of the construction, properties and tools for PDEs in the weighted Sobolev spaces of fractional order . Triebel extended the spaces given by the norm (1.10) to a fractional order [40], [41]. We present three equivalent norms, one of which is a combination of the norm (1.10) and the norm of Lipschitz-Sobolevskij spaces [36]. This definition is essential for the understanding of the relations between the integer and the fractional order spaces (see (7.3)). However the double integral makes it almost impossible to establish any property needed for PDEs. Throughout the effort to solve this problem, we were looking for an equivalent definition of the norm: we let be a dyadic resolution of unity in and set

(1.13)

where . When is an integer, then the norms (1.10) and (1.13) are equivalent. Our guiding philosophy is to apply the known properties of the Bessel potential spaces term-wise to each of the norms in the infinite sum (1.13) and in that way to extend them to the spaces. Of course, this requires a careful treatment and a sound consideration of the additional parameter . Among the properties which we have extended to the spaces are the algebra, Moser type estimates, fractional power, the embedding to the continuous and an intermediate estimate.

2 The Initial Value Problem for the Euler-Einstein System

This section deals with the reduction of the coupled evolution equations (1.1), (1.4) and (1.5) into a first order symmetric hyperbolic system.

2.1 The Euler equations written as a symmetric hyperbolic system

It is not obvious that the Euler equations written in the conservative form are symmetric hyperbolic. In fact these equations have to be transformed in order to be expressed in a symmetric hyperbolic form. Rendall presented such a transformation of these equations in [35], however, its geometrical meaning is not entirely clear and it might be difficult to generalize it to the non time symmetric case. Hence we will present a different hyperbolic reduction of the Euler equations and discuss it in some details, for we have not seen it anywhere in the literature. The basic idea is to perform the standard fluid decomposition and then to modify the equation by adding, in an appropriate manner, the normalization condition (1.3) which will be considered as a constraint equation.

The fluid decomposition method consists of:

  1. The equation is once projected orthogonal onto which leads to

    (2.1)
  2. The equation is projected into the rest pace orthogonal to of a fluid particle gives us:

    (2.2)

Inserting this decomposition into (1.2) results in a system in the following form:

(2.3a)
(2.3b)

Note that we have beside the evolution equations (2.3a) and (2.3b) the following constraint equation: . We will show later, in subsection 2.1.1 that this constraint equation is conserved under the evolution equation, that is, if it holds initially at , then it will hold for . Note that in most textbooks, the equation (2.3b) is presented as , which is an equivalent form, since due to the normalization condition (1.3) we have .

In order to obtain a symmetric hyperbolic system we have to modify it in the following way. The normalization condition (1.3) gives that , so we add to equation (2.3a) and to (2.3b), which together with (1.6) results in,

(2.4a)
(2.4b)

where . As mentioned above we will introduce a new nonlinear matter variable which is given by (1.7). The idea which is behind this is the following: The system (2.4a) and (2.4b) is almost of symmetric hyperbolic form, it would be symmetric if we multiply the system by appropriate factors, for example, (2.4a) by and (2.4b) by . However, doing so we will be faced with a system in which the coefficients will either tend to zero or to infinity, as . Hence, it is impossible to represent this system in a non-degenerate form using these multiplications.

The central point is now to introduce a new variable which will regularize the equations even for . We do this by multiplying equation (2.4a) by . This results in the following system which we have written in matrix form:

(2.5)

In order to obtain symmetry we have to demand

(2.6)

where has been introduced in order to simplify the expression for . We choose so that

(2.7)

which gives the Makino variable (1.7). Taking into account the equation of state (1.5), we see that

(2.8)

Finally we have obtained the following system

(2.9)

which is both symmetric and non-degenerated. The covariant derivative takes in local coordinates the form which expresses the fact that the fluid is coupled to equations (1.1) for the gravitational field . In addition, from the definition of the Makino variable (1.7) we see that , so from the expression (1.6), is follows that and –which is given by (2.8)– are functions of . Thus the fractional power of the equation of state (1.5) does not appear in the coefficients of the system (2.9), and these coefficients are functions of the scalar , the four vector and the gravitational field .

Let us now recall a general definition of symmetric hyperbolic systems.

Definition (First order symmetric hyperbolic systems)

A quasilinear, symmetric hyperbolic system is a system of differential equations of the form

(2.10)

where the matrices are symmetric and for every arbitrary there exists a covector such that

(2.11)

is positive definite. The covectors for which (2.11) is positive definite, are spacelike with respect to the equation (2.10). Both matrices , satisfy certain regularity conditions, which are going to be formulated later.

Usually is chosen to be the vector which implies via the condition (2.11) that the matrix has to be is positive definite.

Now we want to show that of our system (2.9) is indeed positive definite. We do this in several steps.

  1. Explicit computation of the principle symbol (2.9);

  2. We show that is a space like covector with respect to the equations;

  3. Then we apply a deformation argument and show that the covector is a space like covector with respect to the equation.

For each the principle symbol is a linear map from to , where is a fiber in and is a fiber in the cotangent space . Since in local coordinates , the principle symbol of system (2.9) is

(2.12)

and the characteristics are the set of covectors for which is not an isomorphism. Hence the characteristics are the zeros of .

The geometric advantages of the fluid decomposition are the following. The operators in the blocks of the matrix (2.12) are , the projection on the rest hyperplane and , that is the reflection with respect to the same hyperplane. Therefore, the following relations hold:

which yields

(2.13)

It is now fairly easy to calculate the determinate of the right hand side of (2.13) and we have

Since is a projection,

(2.14)

and since is a reflection with respect to a hyperplane,

(2.15)

Consequently,

(2.16)

and therefore the characteristic covectors are given by two simple equations:

(2.17)
(2.18)
Remark (The structure of the characteristics conormal cone of )

The characteristics conormal cone is therefore a union of two hypersurfaces in . One of these hypersurfaces is given by the condition (2.17) and it is a three dimensional hyperplane with the normal . The other hypersurface is given by the condition (2.18) and forms a three dimensional cone the so called sound cone.

Remark

Equation (2.18) plays an essential role in determining whether the equations form a symmetric hyperbolic system.

Let us now consider the timelike vector and the linear combination , with from equation (2.9), we then obtain that

(2.19)

is positive definite. Indeed, is a reflection with respect to a hyperplane. The normal of this hyperplane is a timelike vector. Hence, is for the hydrodynamical equations a spacelike covector in the sense of partial differential equations. Herewith one has showed relatively elegant and elementary that the relativistic hydrodynamical equations are symmetric–hyperbolic.

Now we want however to show that the covector is spacelike with respect to the system (2.9). Since , the covector belongs to the sound cone

(2.20)

Inserting the right hand side of (2.20) yields

(2.21)

Since the sound velocity is always less than the light speed, that is , we conclude from (2.21) that also belongs to the sound cone (2.20). Hence, the vector can be continuously deformed to while condition (2.20) holds along the deformation path. Consequently, the determinant of (2.16) remains positive under this process and hence is also positive definite.

2.1.1 Conservation of the constraint equation

Now it will be shown that the condition , which acts as a constraint equation for the evolution equation, is conserved along stream lines . Because, if for the condition holds and if it is conserved a long stream lines, then holds also for . So let be a curve such that and set , then we need to establish

(2.22)

Multiplying the last four last rows of the Euler system (2.9) by and recalling that is the projection on the rest space orthogonal to , we have

2.2 The reduced Einstein field equations

In this paper we study the field equations (1.1) with the choice of the harmonic coordinate condition which takes the form

(2.23)

When (2.23) is imposed, then the Einstein equations (1.1) convert to

(2.24)

Hawking and Ellis proved the conservation of the harmonic coordinates for Einstein equations with matter including a perfect fluid [19]. Since (2.24) are quasi linear wave equations, the introducing auxiliary variables

(2.25)

reduce them into a first order symmetric hyperbolic system:

(2.26)

The object is a combination of Kronecker deltas with integer coefficients. We therefore conclude:

Conclusion (The evolution equations in a first order symmetric hyperbolic form)

The equations for Einstein gravitational field (1.1) coupled with the Euler equations (1.4) with the normalization conditions (1.3) and the equation of state (1.5), are equivalent to the system (2.26) and (2.9). The coupled systems (2.26) and (2.9) take the form of a first order symmetric hyperbolic system in accordance with Definition 2.1 and where is a positive definite matrix.

3 New Function Spaces and the Principle Results

Our principle results concern the solution of the compatibility of the initial data for the equations of the fluid and the gravitational field (1.12), solution to Einstein constraint equations (1.11) and solution to the coupled evolution equations (1.1) and (1.4).

The coupled evolution equations (1.1) and (1.4) are equivalent to the first order symmetric hyperbolic systems (2.9) and (2.26) as we have shown in section 2.1. The initial data for these coupled systems cannot be given freely, therefore they are constructed in the following way. Firstly the compatibility of the initial data for the fluid and the gravitational field (1.12) have to be solved and next the constraint equations (1.11), which lead to an elliptic system. The presence of an equation of state (1.5) compels us to treat both the elliptic and hyperbolic equations in the weighted Sobolev spaces of fractional order.

The Bessel potential spaces which are the natural choice for the hyperbolic systems are inappropriate for the solutions of the constraint equations in asymptotically flat manifolds. Roughly speaking, because the Laplacian is not invertible in these spaces.

As we explained in the introduction, the Nirenberg-Walker-Cantor weighted Sobolev spaces of integer order [6], [33] are suitable for the solutions of the constraints in asymptotically flat manifolds. Their norm is given by (1.10).

We first define the weighted fractional Sobolev spaces. We make a dyadic resolution of the unity in as follows. Let , and . Let be a sequence of such that on , , for and .

We denote by the Bessel potential spaces with the norm ()

where is the Fourier transform of . Also, for a function , .

Definition (Weighted fractional Sobolev spaces: infinite sum of semi norms)

For and ,

(3.1)

The space is the set of all temperate distributions with a finite norm given by (3.1).

3.1 The principle results

3.1.1 The compatibility of the initial data for the fluid and the gravitational field

The matter data (non-gravitational) which appear in the right hand side of (1.11) are coupled to the initial data of the perfect fluid (1.2) via the relations (1.12). Thus, an indispensable condition for obtaining a solution of the Einstein-Euler system is the inversion of (1.12). This system is not invertible for all , but the inverse does exist in a certain region.

Theorem (Reconstruction theorem for the initial data)

There is a real function such that if

(3.2)

then system (1.12) has a unique inverse. Moreover, the inverse mapping is continuous in norm.

Remark

The matter initial data for the Einstein-Euler system with the equation of state (1.5) cannot be given freely. They must satisfy condition (3.2). This condition includes the inequality

(3.3)

which is known as the dominate energy condition.

3.1.2 Solution to the constraint equations

The gravitational data is a triple , where is a space-like asymptotically flat manifold, is a proper Riemannian metric on , and is the second fundamental form on (extrinsic curvature). The metric and the extrinsic curvature must satisfy Einstein’s constraint equations (1.11). The free initial data is a set , where is a Riemannian metric, is divergence and trace free form, is a scalar function and is a vector.

Theorem (Solution of the constraint equations)

Given free data such that , , , and .

  • Then there exists two positive functions and such that , a vector field such that the gravitational data

    (3.4)

    satisfy the constraint equations (1.11) with and as the right hand side, here is the Killing vector field operator. In addition, the norms of depend continuously on the norms of .

  • Let , , and denote the inverse of relations (1.12). If satisfies (3.2), then the data for the four velocity vector and Makino variable are given by: , ,

    (3.5)

    and they satisfy the compatibility conditions (1.12). In addition, the norms of depend continuously on the norms of .

3.1.3 Solution to the evolution equations

The unknowns of the evolution equations are the gravitational field and its first order partial derivatives , the Makino variable and the velocity vector . We represent them by the vector , here denotes the Minkowski metric. The initial data for equation (2.26) are

(3.6)

where are given by (3.4), and (3.5) for equation (2.9). Theorem 3.1.2 guarantees that they satisfy the constraints (1.11) and the compatibility condition (1.12).

Theorem (Solutions of the evolution equations (2.26) and (2.9))

Let and . Given the solutions of the constraint equations as described in Theorem 3.1.2, then there exists a , a unique semi-Riemannian metric solution to (2.26) and a unique pair solution to (2.9)

(3.7)
(3.8)

4 The Initial Data

The Cauchy problem for Einstein fields equations (1.1) coupled with the Euler system (1.4) consists of solving coupled hyperbolic systems (2.26) and (2.9) with given initial data. There are two types of data for equations (1.1), gravitational and matter data.

The gravitational data is a triple , where is a space-like manifold, is a proper Riemannian metric on and is the second fundamental form on (extrinsic curvature).

Let be the unit normal to the hypersurface , be the projection on and define

(4.1)
(4.2)

The scalar is the energy density and the vector is the momentum density. These quantities are called matter variables and they appear as sources in the constraint equations (4.7) and (4.8) below.

In conjunction with these we must supply initial data for the velocity vector . So we apply the projection to and set . Then from the relation of the perfect fluid (1.2) , (4.1), and (4.2) we see that

(4.3)
(4.4)

The vectors and are tangent to the initial surface and so they can be identified with vectors and intrinsic to this surface. Recalling the normalization condition (1.3), we have . Thus the matter data can be identified with the initial data for the velocity vector as follow:

(4.5)
(4.6)

These two types of data cannot be given freely, because the hypersurface is a sub-manifold of , therefore the Gauss Codazzi equations lead to Einstein constraint equations

(4.7)
(4.8)

Here is the scalar curvature with respect to the metric .

We turn now to the conformal method which allows us to construct the solutions of the constraint equations (4.7) and (4.8). Before entering into details we have to discuss the relations between the initial data for the system of Einstein gravitational fields (1.1) and Euler equations (1.4) which are given by (4.5) and (4.6). As it turns out this relations is by no means trivial, and indeed they will force us to modify the conformal method.

4.1 The compatibility problem of the initial data for the fluid and the gravitational fields

On the one hand, the initial data for the Euler equations are and . On the other hand and , which are given by (4.5) and (4.6) respectively, appear as sources in the constraint equations (4.7) and (4.8). There we have the possibility of either to consider and as the fundamental quantities and construct then and or, vice verse, to consider and as the fundamental quantities and construct then and .

The first possibility does not work because the geometric quantities which occur on the left hand side of the constraint equations are supposed to scale with some power of a scalar function . So and , which are the sources in the constraint equations, must also scale with a definite power of . If is scaled with a certain power of , then would be scaled, according to the equation of state (1.5), to a different power. Hence, by (4.5) is a sum of different powers. Thus, the power which and are scaled would have to be zero and they would be left unchanged by the rescaling. Similarly it can be seen that would remain unchanged. So in fact would be unchanged and this is inconsistent with the scalding used in the conformal method.

Instead of constructing from it is more useful to introduce some auxiliary quantities. Beside the Makino variable , we set

(4.9)

Now we consider the following map