## Abstract

We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.

## 1. Introduction

The initial value problem for the Einstein equations for perfect fluids, with suitable equations of state, is well understood in domains where the matter density is positive [1]. However, in physical models of isolated bodies in astrophysics one faces problems where the matter density has compact support and there are matter–vacuum interfaces. From the mathematical point of view, these physical situations can be treated as initial boundary-value problems for partial differential equations (PDEs). These cases arise frequently in studies of numerical relativity [2] and it is, therefore, important to have analytical results complementing the numerical frameworks.

Rendall [3] proved the existence of local (in time) solutions of an initial value problem for perfect fluid spacetimeswith vacuum interfaces. The fluids had polytropic equations of state and vanishing matter density at the interface. Initial boundary value problems (IBVPs), where the matter density vanishes at the interface, were also studied by Choquet-Bruhat & Friedrich for charged dust matter [4], where not only existence but also uniqueness of solutions have been proved. In both cases, the field equations were written in hyperbolic form in wave coordinates and no spacetime symmetries were required.

Problems where the matter density does not vanish at the interface bring a non-trivial discontinuity along the boundary. For the Einstein-fluid equations, Kind & Ehlers [5] use an equation of state for which the pressure vanishes for a positive value of the mass density. They proved that, for a given spherically symmetric perfect fluid distribution on a compact region of a space-like hypersurface, and for a given boundary pressure, there exists locally in time a unique spacetime that can be matched to a Schwarzschild exterior if and only if the boundary pressure vanishes. In turn, the only existing result along those lines without special symmetry assumptions is due to Andersson *et al.* [6] for elastic bodies also having a jump discontinuity in the matter across a vacuum boundary. Under those circumstances, they prove local existence and uniqueness of solutions of the IBVPs for the Einstein-elastic fluid system under some technical assumptions on the elasticity tensors. As in all cases above, in [6], the boundary is characterized by the vanishing of the normal components of the stress–energy tensor although, in that case, conditions on the continuity of the time derivatives of the metric are also imposed. These *compatibility conditions* arise naturally from the matching conditions across the matter–vacuum boundary and have to be imposed on the allowed initial data.

The present paper generalizes the Elhers–Kind approach to fluids with tangential pressure in spherical symmetry. This includes some cases of interest, such as particular cases of elastic matter. Unlike [5], we cannot ensure that, in general, the origin of the coordinates remains regular locally during the evolution. However, this can be ensured in some physically interesting cases.

The plan of the paper is as follows: in §2, we set up our IBVP specifying the initial and boundary data. In §3, we obtain a first-order symmetric hyperbolic (FOSH) system of PDEs and write our main result, which states existence and uniqueness of smooth solutions to the IBVP in a neighbourhood of the boundary. §4 contains an application of our results to elastic fluids with vanishing radial pressure and a regular centre.

We use units such that *c*=8*π*=1, Greek indices *α*,*β*,…=0,1,2,3 and Latin indices *a*,*b*,…=1,2,3.

## 2. The initial boundary value problem

Consider a spherically symmetric spacetime (*M*,*g*) with a boundary *S* and containing a fluid source. This gives rise to a fluid 4-velocity *u* and we define a time coordinate *T* such that *u* is normal to the surfaces of constant *T*. We also introduce a co-moving radial coordinate *R*.

The general metric for spherically symmetric spacetimes can be written, in co-moving spherical coordinates, as [7]
*T* and *R* coordinates which we fix by imposing
*R*_{0} will correspond to the boundary of the matter. For fluids with no heat flux, the components of the energy–momentum tensor *T*_{μν}, in the above coordinates, can be written as
*ρ* is the fluid energy density, *p*_{1} the *radial pressure* and *p*_{2} the *tangential pressure*.

### Assumption 2.1.

The equation of state for *p*_{1} and the energy conditions are such that

### Assumption 2.2.

The equation of state for *p*_{2} is such that

We note that although we do not assume that *p*_{1}=0, as in [5], the system of PDEs that we derive for the general case becomes singular for

### (a) Einstein and matter equations

The conservation of the energy–momentum tensor, ∇_{ν}*T*^{μν}=0, implies
*R* and *T*, respectively. The Einstein equations *G*_{μν}=*T*_{μν} lead to

(*μν*)=(*TT*):
*μν*)=(*RR*):
*μν*)=(*RT*):
*μν*)=(*θθ*)=(*ϕϕ*):

### Remark 2.3.

As an example, in the case of linear equations of state *p*_{1}=*γ*_{1}*ρ* and *p*_{2}=*γ*_{2}*ρ*, we simply get

Defining the radial velocity as
*R* as
*r*(*T*,0)=0, for regularity of the metric at the centre. Then, (2.10), (2.13) and (2.14), together with the evolution equations for *v* and *μ*, give
*r*,*v*,*ρ*,*Λ*,*μ* together with constraints (2.18) and (2.19). We note that although we could close the system without (2.18) and (2.24), those equations will be crucial to obtain a symmetric hyperbolic form. In §3, we shall apply suitable changes of variables in order to write our evolution system as FOSH system.

### (b) Initial data and boundary data

In spherical symmetry, the free initial data (at *T*=0 and *R*∈[0,*R*_{0}]) are expected to be^{1} *r*,*μ* and *Λ* can be obtained from (2.3), (2.18) and (2.19), respectively.

Note that the intrinsic metric and extrinsic curvature (i.e. the first and second fundamental forms) of the initial hypersurface

At the boundary of the fluid, we must specify the two smooth boundary functions

The fact that we need *r*) through the boundary. Therefore, *Φ*(*T*,*R*_{0})=0, this gives

In what follows, we will prove the existence and uniqueness of solutions to the evolution equations (2.20)–(2.24), on a neighbourhood of the boundary, for the variables *r*,*v*,*ρ*,*Λ*,*μ*, subject to the specified initial and boundary data. We will also show that, as the initial data obeys the constraints (2.18) and (2.19), the solutions will also satisfy the constraints. When the fluid boundary corresponds to characteristics, then the corner data will locally determine the boundary evolution and this will be the case when

## 3. Existence and uniqueness results on a neighbourhood of the boundary

We treat separately the cases

### (a) Case s 1 2 > 0

In this case, the boundary is non-characteristic and we will be able to use the theorem of Kind–Elhers [5] provided we write our evolution system as a FOSH system and give the appropriate data. We thus recall the theorem (whose proof uses results of Courant and Lax [8]):

### Theorem 3.1 (Kind–Elhers).

*Consider the system
**where F,G,H*_{j} *and A are C*^{k+1} *functions, for R>0, and A is always positive. Let C*^{k+1} *initial values* *on [R*_{1}*,R*_{0}*], R*_{1}*>0, and the C*^{k+1} *boundary value* *be given. Assume that* *equal the values of* *at (0,R*_{0}*), which are obtained from (3.1) and the initial data. Then, the system (3.1) has a unique C*^{k} *solution on a compact trapezoidal domain* *for small enough times.*

In order to apply this theorem, we will need to use new variables and write the evolution system (2.20)–(2.24) in quasi-linear symmetric hyperbolic form. We thus use the Kind–Elhers variables
*ρ* as a known function of *s*_{2} as known functions of *Φ*, it is not clear from (2.16) that it can be written as a smooth function of the new variables, so we need a further assumption.

### Assumption 3.2.

*Φ*, as obtained from (2.16), is a known smooth function of the variables {*X*,*Y*,*U*_{i},*R*}.

### Remark 3.3.

Fulfilling assumptions 2.1–3.2 depends on the type of matter and equations of state under consideration. For example, for the linear equations of state of remark 2.3 we get
*ρ*_{0}>0,*r*_{0}>0 and *γ*_{1}>0, satisfy the assumptions. As another example, there are cases where the coordinate system can be chosen to be synchronous and co-moving so that *Φ*≡0 and assumption 3.2 becomes trivial.

Then, taking into account the assumptions 2.1 and 2.2, and after a long calculation, our evolution system in terms of the new variables becomes
*p*_{1}≡*p*_{2}.

In our system, the equation (3.12) was obtained from (3.2) using (2.20), while (3.13) came from (3.3) together with (2.22). The evolution equation for *Λ* was derived from (2.23), and the evolution equation for *w* from (3.5) together with (2.21). Equation (3.16) came from (3.9). The evolution equations for *μ*,*X* and *Y* were obtained from (2.24), (3.6) and (3.7), respectively.

The constraints are given by the equations (3.4), (3.6), (3.7) and by the derivatives of (2.18) and (2.19) with respect to *R*. They can be expressed in the following way:
*C*^{1} solution {*X*,*Y*,*U*_{i}} of (3.12)–(3.17), the quantities *C*_{1},…,*C*_{5} satisfy a linear system of the form
*A*_{kl} are continuous functions of *X*,*Y*,*U*_{i}. We then conclude that the constraints *C*_{k}=0 are satisfied for all *T* if they are satisfied at *T*=0.

From the initial data *μ*,*Λ*,*ω*,*X*,*Y* are specified by (2.18), (2.19), (3.4), (3.6), (3.7), respectively. We impose that the quantities *μ*,*Λ*,*ω*,*X*,*Y* (i) satisfy the constraints initially, i.e. *C*_{k}(0,*R*)=0, and (ii) are smooth functions in a region [*R*_{1},*R*_{0}], for some *R*_{1}>0.

To apply theorem 3.1, we also need the necessary boundary function which we get from (3.13), considering *Φ*(*T*,*R*_{0})=0, as

We are now in the position of applying Kind–Elhers’ theorem to (3.12)–(3.19), given the above initial and boundary data, and this proves existence and uniqueness of solutions to the initial boundary value problem in a neighbourhood of the boundary. In detail, we get the following result.

### Theorem 3.4.

*Consider a fluid matter field satisfying assumptions 2.1, 2.2 and 3.2 with* *. Then, the system (3.12)–(3.19) is a FOSH system of the form (3.1) for the variables X,Y and* *. Suppose that the initial data for those variables are smooth on [R*_{1}*,R*_{0}*], for some R*_{1}*>0, and that the constraints C*_{1}*,…,C*_{5} *are satisfied at T=0. Suppose a spherically symmetric distribution of such matter is given together with the smooth boundary functions* *and* *(or* *). Suppose that the given initial and boundary data for Y and their time derivatives satisfy the corner conditions at (0,R*_{0}*). Then, there exists locally in time a unique smooth solution to the Einstein equations in*

A well-known case is that of a Kottler spacetime exterior (A 2) which can be attached to the fluid if and only if *m* is the Kottler mass.

### (b) Case p 1 = s 1 2 = 0

While the constraints equations *C*_{2},…,*C*_{5} remain the same in the case *s*_{2} does not appear in *H*_{j}. This means that we do not need to use assumptions 2.1 and 2.2 in order to close the system, as before. Instead, we make the following assumption.

### Assumption 3.5.

Both *ρ*>0 and *p*_{2} are smooth known functions of the variables

In this case, the characteristics of the system (3.29) are the vertical lines of constant *R* and the trapezoidal region

In our case, the integration along the characteristics gives, at each point (*T*,*R*), simply
*R*_{0}), one can integrate from that point to get uniquely the boundary functions

As mentioned in §2b, from the initial data *h*_{0} and *K*_{0} from the expressions (2.26). The corner conditions are the matching conditions evaluated at (0,*R*_{0}), and these give the components of *h*_{0} and *K*_{0} at *R*_{0}. In particular, this provides the corner data

### Theorem 3.6.

*Consider a matter field with* *and satisfying assumptions 3.2 and 3.5. Then, the system (3.12)–(3.19) reduces to the form (3.29) which corresponds to a semi-linear FOSH system for the variables* *. Consider the smooth initial data* *for the variables U*_{j} *on [R*_{1}*,R*_{0}*], for some R*_{1}*>0, and suppose that the matching conditions to an exterior spacetime are satisfied at (0,R*_{0}*). Suppose that the constraints C*_{2}*,…,C*_{5} *are satisfied at T=0. Then, there exists a unique smooth solution to the Einstein equations in the rectangle [R*_{1}*,R*_{0}*]×T*_{0} *for small enough T*_{0}*>0.*

## 4. An application to self-gravitating elastic bodies

In this section, we consider a simple application of the previous result to elastic matter. In this case, the matter does not necessarily satisfy (2.5) or (2.9). So, before proceeding, we recall some basic facts about elastic matter, adapting the presentation to the particular setting we shall consider in the context of spherical symmetry.

The material space *X* for elastic matter is a three-dimensional manifold with Riemannian metric *γ*. Points in *X* correspond to particles of the material and the material metric *γ* measures the distance between particles in the relaxed (or unstrained) state of the material. Coordinates in *X* are denoted by *y*^{A}, *A*=1,2,3. The spacetime configuration of the material is described by the map *ψ*:*M*→*X*, where *M* denotes the spacetime with metric *g*. The differential map *ψ*_{*}:*T*_{p}*M*→*T*_{ψ(p)}*X*, which is also called the relativistic deformation gradient, is represented by a rank 3 matrix with entries *x*^{μ} are coordinates in *M*. In turn, the matrix kernel is generated by the 4-velocity vector *u* satisfying

The push-forward of the contravariant spacetime metric from *M* to *X* is defined by
*X*. This metric contains information about the state of strain of the material, which can be described by comparing this metric with structures in *X*, e.g. the material metric. The material is said to be unstrained at an event *p*∈*M* if *G*_{AB}=*γ*_{AB} at *p*.

The dynamical equations for the material can be derived from the Lagrangian density *h*=*h*(*y*^{A},*G*^{AB}), which describes the dependence of the energy on the state of strain and specifies the material.

The stress–energy tensor for elastic matter can be written as [11]
*g* with line-element (2.1) gives
*η*=*e*^{−2Λ}, *β*=1/*r*^{2} and *Λ*=0 and *r*=*R*, so that *η*=1 and *β*=1/*R*^{2}.

The matter density of the material is, in our case, given by
*ϵ*_{0} is an arbitrary positive function. The energy–momentum tensor can then be expressed by
*ρ*=*ϵh*, *p*_{1}=2*ϵη*(∂*h*/∂*η*), *h*=*h*(*R*,*η*,*r*). Using (4.2), we get
*p*_{1}=*p*_{1}(*ρ*) as in assumption 2.1, so theorem 3.4 does not apply. We will now investigate a particular elastic fluid for which

### (a) Magli’s ansatz

Magli [11] found a class of non-static spherically symmetric solutions of the Einstein equations corresponding to anisotropic elastic spheres. These models have vanishing radial stresses and generalize the Lemaître–Tolman–Bondi dust models of gravitational collapse, by including tangential stresses.

Assuming that the equation of state for the elastic matter, prescribed by the function *h*, does not depend on the eigenvalue *η* in the material space, i.e. *h*=*h*(*R*,*r*), then it is possible to write the spherically symmetric metric as [12]
*c*(*T*) is an arbitrary function, reflecting the invariance with respect to time rescaling, which can be chosen such that *Φ*(*T*,*R*_{0})=0 for some *R*_{0}>0. One can then obtain a first integral to the Einstein equations as
*F*(*R*)>0 and *f*(*R*)>−1 are arbitrary functions which we assume to be smooth on [0,*R*_{0}]. With the assumption *h*=*h*(*R*,*r*), and using the first integral, we also get

A physically interesting example of a spacetime with *h*=*h*(*R*,*r*) is the non-static Einstein cluster in spherical symmetry, which describes a gravitational system of particles sustained only by tangential stresses [11,13,14].

To make contact with our formalism, in this case, we get a semi-linear system of the form (3.29), for the variables *R*_{1},*R*_{0}], *R*_{1}>0, as
*F* and *f* which come from the initial density *R*_{1},*R*_{0}], *R*_{1}>0, we are under the conditions of theorem 3.6 which ensures existence and uniqueness of solutions in a neighbourhood of the boundary.

In this case, we can also obtain similar results in a neighbourhood of the origin *R*=0, at least for some choices of initial data. This has to be done separately because our evolution system (3.29) is singular at the centre.

Consider an open region *r*(*T*,0)=0, smooth functions *f* and *F* such that [11]
*minimal stability requirement*, namely that *h* has a minimum at *r*=*R* [11]. Physically, this means that the centre is unstrained and has zero radial velocity as

In [11], Magli shows that there are open sets of *f* such that the regularity conditions at the centre are fulfilled. From the PDE point of view, this implies that for such data there exists a unique

### Proposition 4.1.

*Consider a spacetime with metric* (4.9) *and containing elastic matter satisfying* (4.10). *Consider smooth initial data* *on* [0,*R*_{0}] *satisfying the (corner) matching conditions to an exterior spacetime at* (0,*R*_{0}) *and the regularity conditions at the centre. Then, there exists a unique smooth solution to the Einstein equations on* [0,*R*_{0}]×*T*_{0} *for small enough* *T*_{0}.

A suitable exterior to such elastic spacetimes is given by the Schwarzschild solution, which can be matched initially at *R*_{0}. We omit the details of the matching conditions here as they were given in [11].

As a final remark, we note that the analysis of this particular IBVP can be taken much further if one has explicit solutions for *r*. These solutions can be obtained by assuming *h*=*h*(*r*)>0. In this case, equations (4.10) and (4.11) decouple and *Φ*′=−(1/*h*)(*dh*/*dr*)*r*′, yielding *Φ*(*T*,*R*_{0})=0. Consider a linear stress–strain relation *p*_{2}=*kρ*, where the adiabatic index *k* satisfies −1≤*k*≤1 in order to comply with the weak energy condition [15]. This choice of stress–strain is equivalent to set *h*(*r*)=*r*^{−2k}. In that case, one has *k*, such as *r*(0,*R*)=*R* and *r*(*T*,0)=0 and have a regular origin for certain choices of initial data [15]. This allows us, for example, to solve the matching conditions and obtain the boundary hypersurface [11], as well as to study of global properties of the matched spacetime such as the formation of horizons and spacetime singularities [12,15].

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

Both authors conceived the problem, derived the results and written the paper. They both gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This work was supported by CMAT, Univ. Minho, through FEDER funds COMPETE and FCT Projects Est-OE/MAT/UI0013/2014 and PTDC/MAT-ANA/1275/2014.

## Acknowledgements

We thank Robert Beig and Marc Mars for useful comments. We thank the referees for the comments and suggestions.

## Appendix A. Matching conditions in spherical symmetry

In appendix A, for completeness, we recall some basic facts about spacetime matching theory and write the matching conditions in spherical symmetry adapted to our context. These conditions are known but are of importance here because they provide compatibility conditions for our IBVP problem.

Let (*M*^{±},*g*^{±}) be spacetimes with non-null boundaries *S*^{±}. Matching them requires an identification of the boundaries, i.e. a pair of embeddings *Ω*^{±}: *S*→*M*^{±} with *Ω*^{±}(*S*)=*S*^{±}, where *S* is an abstract copy of any of the boundaries. Let *ξ*^{i} be a coordinate system on *S*. Tangent vectors to *S*^{±} are obtained by *M*^{+}, then *M*^{−} or vice versa. The first and second fundamental forms on *S*^{±} are simply
*M*^{±},*g*^{±}) across a non-null hypersurface *S*, are the equality of the first and second fundamental forms [16]:
*S*^{+} can be parametrized as *Ω*^{+}={*t*=*t*_{0}(*λ*),*ϱ*=*ϱ*_{0}(*λ*)}, where two dimensions were omitted, and we can assume

Considering our choice *Φ*(*T*,*R*_{0})=0 for the interior spacetime, we can take the following parametrization *Ω*^{−}={*T*=*λ*,*R*=*R*_{0}}, so that *S* gives
*S*, implies
*ϱ*_{0}(*λ*), *t*_{0}(*λ*) and imply the continuity of the mass

Generalizing this procedure, consider now the matching where the interior is still the anisotropic fluid (2.1) and the exterior is the spacetime given by a general spherically symmetric metric:
*S*^{±} are parametrized as before. Then, the equality of the first fundamental forms, for *Φ*(*T*,*R*_{0})=0, imply
*ρ*_{0}(*λ*), *t*_{0}(*λ*) and *S* [16]. With this boundary data, and knowing the exterior spacetime, one gets

## Footnotes

↵1 We use a ‘tilde’ for boundary data defined on (

*T*,*R*=*R*_{0}) and a ‘hat’ for initial data defined on (*T*= 0,*R*).

- Received February 17, 2017.
- Accepted July 10, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.