Initial boundary-value problem for the spherically symmetric Einstein equations with fluids with tangential pressure
IInitial boundary-value problem for the spherically symmetric Einsteinequations with fluids with tangential pressure
Irene Brito ∗ , Filipe C. Mena † , Centro de Matem´atica, Universidade do Minho, 4710-057 Braga, Portugal.
November 10, 2018
Abstract
We prove that, for a given spherically symmetric fluid distribution with tangential pressure onan initial spacelike hypersurface with a timelike boundary, there exists a unique, local in time solu-tion to the Einstein equations in a neighbourhood of the boundary. As an application, we considera particular elastic fluid interior matched to a vacuum exterior.Keywords: Einstein equations; General Relativity; Initial boundary value problems; Self-gravitatingsystems; Spacetime matching, Elasticity
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 [4]. However, in physical models ofisolated bodies in astrophysics one faces problems where the matter density has compact support andthere are matter-vacuum interfaces. From the mathematical point of view, these physical situationscan be treated as initial boundary value problems for partial differential equations (PDEs). Thesecases arise frequently in studies of numerical relativity (see e.g. [13]) and it is, therefore, importantto have analytical results complementing the numerical frameworks.Rendall [12] proved the existence of local (in time) solutions of an initial value problem for perfectfluid spacetimes with vacuum interfaces. The fluids had polytropic equations of state and vanishingmatter density at the interface. Initial boundary value problems (IBVP), where the matter densityvanishes at the interface, were also studied by Choquet-Bruhat and Friedrich for charged dust matterin [5], where not only existence but also uniqueness of solutions have been proved. In both cases, thefield equations were written in hyperbolic form in wave coordinates and no spacetime symmetries wererequired.Problems where the matter density does not vanish at the interface bring a non-trivial discontinuityalong the boundary. For the Einstein-fluid equations, Kind and Ehlers [8] use an equation of state forwhich the pressure vanishes for a positive value of the mass density. They proved that, for a givenspherically symmetric perfect fluid distribution on a compact region of a spacelike hypersurface, and ∗ e-mail: [email protected] † e-mail: [email protected] a r X i v : . [ g r- q c ] J u l or a given boundary pressure, there exists locally in time a unique spacetime that can be matched toa Schwarzschild exterior if and only if the boundary pressure vanishes. In turn, the only existing resultalong those lines without special symmetry assumptions is due to Andersson, Oliynyk and Schmidt[1] for elastic bodies also having a jump discontinuity in the matter across a vacuum boundary.Under those circunstances, they prove local existence and uniqueness of solutions of the IBVP forthe Einstein-elastic fluid system under some technical assumptions on the elasticity tensors. As in allcases above, in [1], the boundary is characterised by the vanishing of the normal components of thestress-energy tensor although, in that case, conditions on the continuity of the time derivatives of themetric are also imposed. These compatibility conditions arise naturally from the matching conditionsacross the matter-vacuum boundary and have to be imposed on the allowed initial data.The present paper generalises the Elhers-Kind approach to fluids with tangential pressure in spher-ical symmetry. This includes some cases of interest, such as particular cases of elastic matter. Unlike[8], we cannot ensure that, in general, the origin of the coordinates remains regular locally during theevolution. However, this can be ensured in some physically interesting cases.The plan of the paper is as follows: In Section 2, we setup our IBVP specifying the initial andboundary data. In Section 3, we obtain a first order symmetric hyperbolic (FOSH) system of PDEsand write our main result, which states existence and uniqueness of smooth solutions to the IBVPin a neighbourhood of the boundary. Section 4 contains an application of our results to elastic fluidswith vanishing radial pressure and a regular centre.We use units such that c = 8 π = 1, greek indices α, β, .. = 0 , , , a, b, .. = 1 , , 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 thesurfaces of constant T . We also introduce a comoving radial coordinate R .The general metric for spherically symmetric spacetimes can be written, in comoving sphericalcoordinates, as [16] g = − e T,R ) dT + e T,R ) dR + r ( T, R ) d Ω , (1)where d Ω = dθ + sin θdφ , and the components of the 4-velocity are written as u µ = ( e − Φ , , , . (2)There is freedom in scaling the T and R coordinates which we fix by imposingΦ( T, R ) = 0 , r (0 , R ) = R, (3)where R will correspond to the boundary of the matter. For fluids with no heat flux, the componentsof the energy-momentum tensor T µν , in the above coordinates, can be written as T T T = ρe , T RR = p e , T θθ = p r , T φφ = p r sin θ, (4)where ρ is the fluid energy density, p the radial pressure and p the tangential pressure .2 ssumption 1 The equation of state for p and the energy conditions are such that p = p ( ρ ) ∈ C ∞ (5) ρ > ρ + p > s ( ρ ) := d p ( ρ ) d ρ ≥ . (8) Assumption 2
The equation of state for p is such that p = p ( ρ ) ∈ C ∞ (9) and we use the notation s ( ρ ) := d p ( ρ ) d ρ . We note that although we do not assume that s necessarily remains positive when p = 0, as in [8],the system of PDEs that we derive for the general case becomes singular for s = 0, so we will haveto treat this case separately. The conservation of the energy-momentum tensor, ∇ ν T µν = 0, implies r ˙ ρ + r ˙Λ( ρ + p ) + 2 ˙ r ( ρ + p ) = 0 , for µ = T, (10) r Φ (cid:48) ( ρ + p ) + rp (cid:48) + 2 r (cid:48) ( p − p ) = 0 , for µ = R, (11)where the prime and dot indicate derivatives with respect to R and T , respectively. The Einsteinequations G µν = T µν lead to( µν ) = ( T T ): ρ = 1 r (cid:104) − r (cid:48) e − + ˙ r e − + 2 r ˙ r ˙Λ e − − r ( r (cid:48)(cid:48) − r (cid:48) Λ (cid:48) ) e − (cid:105) (12)( µν ) = ( RR ): p = − r (cid:104) − r (cid:48) e − + ˙ r e − − rr (cid:48) Φ (cid:48) e − + 2 r (¨ r − ˙ r ˙Φ) e − (cid:105) (13)( µν ) = ( RT ): ˙ r Φ (cid:48) + r (cid:48) ˙Λ − ˙ r (cid:48) = 0 (14)( µν ) = ( θθ ) = ( φφ ): p = e − r (cid:104) ˙ r ( ˙Φ − ˙Λ) − ¨ r (cid:105) + e − (cid:104) ˙Λ( ˙Φ − ˙Λ) − ¨Λ (cid:105) + e − r (cid:2) r (cid:48) (Φ (cid:48) − Λ (cid:48) ) + r (cid:48)(cid:48) (cid:3) + e − (cid:2) Φ (cid:48) (Φ (cid:48) − Λ (cid:48) ) + Φ (cid:48)(cid:48) (cid:3) , (15)3hile the contracted Bianchi identities are identically satisfied. We note that (15) can be obtainedfrom (10)-(14). Integrating (11), with (3), one getsΦ( T, R ) = − (cid:90) ρρ s ( ¯ ρ )¯ ρ + p ( ¯ ρ ) d ¯ ρ − (cid:90) RR ( p − p ) ρ + p r (cid:48) r dR. (16) Remark 1
As an example, in the case of linear equations of state p = γ ρ and p = γ ρ , we simplyget Φ( T, R ) = − γ γ ln (cid:18) ρρ (cid:19) − γ − γ γ ln (cid:18) rr (cid:19) , w ith r = r ( T, R ) , which will happen for a particular case of elastic matter that we will consider in Section 4. Defining the radial velocity as v := e − Φ ˙ r (17)and the mean density of the matter within a ball of coordinate radius R as µ := 3 r (cid:90) R ρr r (cid:48) d ¯ R, (18)one obtains from (12) r (cid:48) e − = 1 + v − µr , (19)where we also used the condition r ( T,
0) = 0, for regularity of the metric at the center. Then, (10),(13) and (14), together with the evolution equations for v and µ , give˙ r = ve Φ , (20)˙ v = (cid:20) − r (cid:16) µ p (cid:17) − r (cid:48) p (cid:48) ρ + p e − − p − p ρ + p r (cid:48) r e − (cid:21) e Φ , (21)˙ ρ = (cid:20) − ρ (cid:18) v (cid:48) r (cid:48) + 2 vr (cid:19) − (cid:18) p v (cid:48) r (cid:48) + 2 p vr (cid:19)(cid:21) e Φ , (22)˙Λ = v (cid:48) r (cid:48) e Φ , (23)˙ µ = − vr ( µ + p ) e Φ . (24)To summarize, the Einstein equations resulted in the system of evolution equations (20)-(24) for thefive variables r, v, ρ, Λ , µ together with constraints (18) and (19). We note that although we could closethe system without (18) and (24), those equations will be crucial to obtain a symmetric hyperbolicform. In Section 3, we shall apply suitable changes of variables in order to write our evolution systemas FOSH system. 4 .2 Initial data and boundary data In spherical symmetry, the free initial data (at T = 0 and R ∈ [0 , R ]) is expected to be ˆ ρ ( R ) := ρ (0 , R ) and ˆ v ( R ) := v (0 , R ) , satisfying ˆ v (0) = 0, and constrained by1 R (cid:90) R ˆ ρ ( ¯ R ) ¯ R d ¯ R ≤ v ( R ) , (25)as a consequence of (3), (18) and (19). The initial data for the remaining variables r, µ and Λ can beobtained from (3), (18) and (19), respectively.Note that the intrinsic metric and extrinsic curvature (i.e. the first and second fundamental forms)of the initial hypersurface h = e dR + R d Ω , K = ˆ v (cid:48) e dR + R ˆ vd Ω (26)are fully known once ˆ ρ ( R ) and ˆ v ( R ) are known.At the boundary of the fluid, we must specify the two smooth boundary functions ˜ p ( T ) := p ( T, R ) and ˜ w ( T ) := v ( T, R ) /r ( T, R ) (or ˜ µ ( T ) := µ ( T, R ), via (24)) which should satisfy thecorner conditions ˜ p (0) = ˆ p ( R ) and ˜ w (0) = ˆ w ( R ) (or ˜ µ (0) = ˆ µ ( R )). In terms of the initial dataset { ˆ ρ ( R ) , ˆ v ( R ) } , we note that from (24) we get ( ˙ µ/ ( µ + p )) (0 , R ) = − v ( R ) /R , at the corner.The fact that we need ˜ w ( T ) at the boundary is reminiscent of the compatibility conditions of[1] arising from the matching conditions, since ˜ w ( T ) is related to the time derivative of the metricat the boundary. In fact, in spherical symmetry, the matching conditions (see the appendix) implythe continuity of the areal radius (here r ) through the boundary. Therefore, ˙ r ( T, R ) also has to becontinuous and, for Φ( T, R ) = 0, this gives ˜ w ( T ), for the interior spacetime from the data of theexterior.In what follows, we will prove existence and uniqueness of solutions to the evolution equations(20)-(24), on a neighbourhood of the boundary, for the variables r, v, ρ, Λ , µ, subject to the specifiedinitial and boundary data. We will also show that, since the initial data obeys the constraints (18)and (19), the solutions will also satisfy the constraints. When the fluid boundary corresponds tocharacteristics, then the corner data will locally determine the boundary evolution and this will bethe case when s = p = 0, as we will show in Section 3.2. We treat separately the cases s > s = 0. s > In this case, the boundary is non-characteristic and we will be able to use the theorem of Kind-Elhers[8] provided we write our evolution system as a FOSH system and give the appropriate data. We thusrecall the theorem (whose proof uses results of Courant and Lax [6]): We use a ”tilde” for boundary data defined on (
T, R = R ) and a ”hat” for initial data defined on ( T = 0 , R ). heorem 1 [Kind-Elhers] Consider the system ˙ X + A ( U i ) Y (cid:48) = F ( X, Y, U i , R )˙ Y + A ( U i ) X (cid:48) = G ( X, Y, U i , R ) (27)˙ U j = H j ( X, Y, U i , R ) , i, j = 1 , ..., p, where F, G, H j and A are C k +1 functions, for R > , and A is always positive. Let C k +1 initialvalues ˆ X, ˆ Y , ˆ U i on [ R , R ] , R > , and the C k +1 boundary value ˜ Y ( T ) be given. Assume that ˜ Y (0) , ˙˜ Y (0) , ..., ˜ Y ( k +1) (0) equal the values of Y, ˙ Y , ..., Y ( k +1) at (0 , R ) , which are obtained from (27) and the initial data. Then, the system (27) has a unique C k solution on a compact trapezoidal domain T , for small enough times. Figure 1: Compact trapezoidal domain T with small T .In order to apply this theorem we will need to use new variables and write the evolution system(20)-(24) in quasi-linear symmetric hyperbolic form. We thus use the Kind-Elhers variables Q = ln (cid:18) Rr (cid:19) , (28) L = (cid:90) ρρ s ( ρ ) ρ + p ( ρ ) d ρ, (29) ω = r (cid:48) , (30) w = vr , (31) X = e − Λ L (cid:48) , (32) Y = v (cid:48) ω + 2 vr . (33)Our system will then have the 8 variables X, Y and U i = {Q , L , ω, w, µ, Λ } . (34)6efore proceeding, note that, from (20), one obtains˙ ω = ( v (cid:48) + v Φ (cid:48) ) e Φ . (35)Now, since (29) is invertible, we can consider ρ as a known function of L . Moreover, for given equationsof state, we can also consider p , p , s and s as known functions of L . Regarding Φ, it is not clearfrom (16) that it can be written as a smooth function of the new variables, so we need a furtherassumption: Assumption 3 Φ , as obtained from (16) , is a known smooth function of the variables { X, Y, U i , R } . Remark 2
Fulfilling Assumptions 1, 2 and 3 depends on the type of matter and equations of stateunder consideration. For example, for the linear equations of state of Remark 1 we get: ρ ( L ) = ρ e (1+ γ ) L / √ γ (36)Φ( L , Q , R ) = −√ γ L − γ − γ )1 + γ (cid:20) ln Rr + Q (cid:21) , (37) which for ρ > , r > and γ > satisfy the assumptions. As another example, there are cases wherethe coordinate system can be chosen to be synchronous and comoving so that Φ ≡ and Assumption3 becomes trivial. Then, taking into account the Assumptions 1 and 2, and after a long calculation, our evolution systemin terms of the new variables becomes:˙ Q = − we Φ , (38)˙ L = − s Y e Φ − s we Φ p − p ρ + p , (39)˙Λ = ( Y − w ) e Φ , (40)˙ w = − e Φ (cid:20) s ωe Q− Λ XR + w + 12 (cid:16) µ p (cid:17) + 2 ω R e Q− Λ) p − p ρ + p (cid:21) , (41)˙ ω = e Φ (cid:20) ω ( Y − w ) − s wXe Λ − Q R − ωR e Q p − p ρ + p (cid:21) , (42)˙ µ = − we Φ ( µ + p ) , (43)˙ X + s e Φ − Λ Y (cid:48) = e Φ (cid:20) XY (cid:18) s − d s d L − (cid:19) + 2 wX (cid:21) − wXe Φ p − p ρ + p + e Φ − Λ p − p ρ + p (cid:20) wXe Λ d s d L + 2 s ωR e Q (2 Y − w ) − s wXe Λ − s ωR we Q p − p ρ + p (cid:21) + 2 s wXe Φ (cid:18) s − s s (cid:19) , (44)˙ Y + s e Φ − Λ X (cid:48) = e Φ (cid:20) X (cid:18) s − d s d L (cid:19) − s ωe Q − Λ XR − ( Y − w ) − w − ρ + 3 p (cid:21) + 2 XR e Φ − Λ+ Q (cid:18) s s − s (cid:19) + e Φ p − p ρ + p (cid:20)(cid:18) s + 2 s (cid:19) ω XR e Q− Λ − µ + p +2 ρ − ω R e Q− Λ) (cid:18) − p − p ρ + p (cid:19) + 2 Y w (2 w − (cid:21) , (45)7hich has the symmetric hyperbolic form (27), under Assumption 3. Note that the system reduces tothe one of [8] for p ≡ p .In our system, the equation (38) was obtained from (28) using (20), while (39) came from (29)together with (22). The evolution equation for Λ was derived from (23), and the evolution equationfor w from (31) together with (21). Equation (42) came from (35). The evolution equations for µ, X and Y were obtained from (24), (32) and (33), respectively.The constraints are given by the equations (30), (32), (33) and by the derivatives of (18) and (19)with respect to R . They can be expressed in the following way C := L (cid:48) − e Λ X = 0 , (46) C := R Q (cid:48) + ωe Q − , (47) C := Re − Q w (cid:48) − ω ( Y − w ) = 0 , (48) C := Re − Q µ (cid:48) + 3 ω ( µ − ρ ) = 0 , (49) C := e − ( ω (cid:48) − Λ ω ) + Re − Q (cid:20)
16 (3 ρ − µ ) − w ( Y − w ) (cid:21) = 0 , (50)and, as in [8], it can be shown that for a given C solution { X, Y, U i } of (38)-(43), the quantities C , .., C satisfy a linear system of the form˙ C k = (cid:88) l =1 A kl C l , (51)where A kl are continuous functions of X, Y, U i . We then conclude that the constraints C k = 0 aresatisfied for all T if they are satisfied at T = 0 . From the initial data ˆ ρ ( R ) and ˆ v ( R ) , using (3), one obtains L (0 , R ) = L ( ˆ ρ ( R )) , w (0 , R ) = ˆ v ( R ) R , Q (0 , R ) = 0 , (52)and the initial data for µ, Λ , ω, X, Y are specified by (18), (19), (30), (32), (33), respectively. Weimpose that the quantities µ, Λ , ω, X, Y : (i) satisfy the contraints initially, i.e. C k (0 , R ) = 0, and (ii)are smooth functions in a region [ R , R ], for some R > T, R ) = 0, as:˜ Y ( T ) = Y ( T, R ) = − (cid:34) ˙ L s ( L ) (cid:35) ( T, R ) − w ( T, R ) (cid:20) p ( L ) − p ( L ) ρ ( L ) − p ( L ) (cid:21) ( T, R ) , (53)which, using (43), can be rewritten as˜ Y ( T ) = − (cid:34) ˙ L s ( L ) + 23 (cid:18) ˙ µµ + p ( L ) (cid:19) p ( L ) − p ( L ) ρ ( L ) − p ( L ) (cid:35) ( T, R ) . (54)This function will be known given the two smooth boundary functions ˜ p ( T ) and ˜ w ( T ) (or ˜ µ ( T ))which should satisfy the corner conditions, as described in Section 2.2.8e are now in the position of applying Kind-Elhers’ theorem to (38)-(45), given the above initialand boundary data, and this proves existence and uniqueness of solutions to the initial boundary valueproblem in a neighbourhood of the boundary. In detail, we get the following result: Theorem 2
Consider a fluid matter field satisfying Assumptions 1, 2 and 3 with s > . Then, thesystem (38) - (45) is a FOSH system of the form (27) for the variables X, Y and U j = { µ, Q, ω, w, Λ , L} .Suppose that the initial data for those variables is smooth on [ R , R ] , for some R > , and that theconstraints C , .., C are satisfied at T = 0 . Suppose a spherically symmetric distribution of suchmatter is given together with the smooth boundary functions ˜ p ( T ) and ˜ w ( T ) (or ˜ µ ( T ) ). Suppose thatthe given initial and boundary data for Y and their time derivatives satisfy the corner conditions at (0 , R ) . Then, there exists locally in time a unique smooth solution to the Einstein equations in T . A well known case is that of a Kottler spacetime exterior (74) which can be attached to the fluid if andonly if ˜ p ( T ) = − Λ, where Λ ∈ R . A way to see this using our framework is as follows: The conditionsof the continuity of the first and second fundamental forms across the boundary (see the appendix)imply the continuity of the normal component of the pressure ˜ p ( T ) = − Λ (this is also a well knownconsequence of the so-called Israel conditions). Due to the Einstein equations and the continuity ofthe areal radius, this condition turns out to be equivalent to the continuity of the mass through theboundary ˜ µ ( T )˜ r ( T ) = 6 m, where m is the Kottler mass. p = s = 0 While the constraints equations C , .., C remain the same in the case p = s = 0, the quasi-linearsystem (38)-(43) is modified since X ≡ L ≡
0. In that case, we get a semi-linear symmetric hyperbolicPDE system of the form ˙ U j = H j ( U i , R ) , i, j = 1 , .., , (55)for the variables U j = { Y, ω, w, Q , µ, Λ } . An important aspect of this system, compared to the previousone, is that the quantity s does not appear in H j . This means that we do not need to use Assumptions1 and 2 in order to close the system, as before. Instead, we make the following assumption: Assumption 4
Both ρ > and p are smooth known functions of the variables { Y, ω, w, Q , µ, Λ , R } . In this case, the characteristics of the system (55) are the vertical lines of constant R and the trape-zoidal region T of Figure 1 is now a rectangle. In particular, the boundary is now characteristic. IBVPwith characteristic boundaries were investigated, in more generality, by Chen [3] and Secchi [14] forquasi-linear systems.In our case, the integration along the characteristics gives, at each point ( T, R ), simply U j ( T, R ) = U j (0 , R ) + (cid:90) T H j ( U i , R ) d ¯ T . (56)which, using the methods of Courant and Lax [6, 8], gives a smooth local (in time) solution to thesystem (55), once smooth initial data ˆ U j ( R ) is prescribed. In our case, provided suitable data froman exterior spacetime at the corner (0 , R ), one can integrate from that point to get uniquely theboundary functions ˜ U j ( T ) := U j ( T, R ). In this sense, the spacetime boundary is completely fixed bythe initial corner conditions. 9s mentioned in Section 2.2, from the initial data { ˆ ρ, ˆ v } we can get h and K from the expressions(26). The corner conditions are the matching conditions evaluated at (0 , R ), and these give thecomponents of h and K at R . In particular, this provides the corner data ˆΛ := ˆΛ( R ) , ˆ v := ˆ v ( R )and ˆ v (cid:48) := ˆ v (cid:48) ( R ). In turn, this gives the remaining corner data for the systemˆ ω ( R ) = ˆ v R , ˆ Y ( R ) = ˆ v (cid:48) + 2 ˆ v R , ˆ Q ( R ) = 0 , ˆ w ( R ) = 1 , ˆ µ ( R ) = − R ( e − − − ˆ v ) . (57)We summarize this discussion as: Theorem 3
Consider a matter field with p = s = 0 and satisfying Assumptions 3 and 4. Then,the system (38) - (45) reduces to the form (55) which corresponds to a semi-linear FOSH system forthe variables U j = { Y, ω, w, Q , µ, Λ } . Consider the smooth initial data { ˆ ρ, ˆ v } for the variables U j on [ R , R ] , for some R > , and suppose that the matching conditions to an exterior spacetime aresatisfied at (0 , R ) . Suppose that the constraints C , .., C are satisfied at T = 0 . Then, there existsa unique smooth solution to the Einstein equations in the rectangle [ R , R ] × T for small enough T > . In this section, we consider a simple application of the previous result to elastic matter. In this case,the matter does not necessarily satisfy (5) or (9). So, before proceeding, we recall some basic factsabout elastic matter adapting the presentation to the particular setting we shall consider in the contextof 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 distancebetween particles in the relaxed (or unstrained) state of the material. Coordinates in X are denotedby y A , A = 1 , , . 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 isalso called relativistic deformation gradient, is represented by a rank 3 matrix with entries (cid:0) y Aµ (cid:1) = ∂y A ∂x µ ,where x µ are coordinates in M . In turn, the matrix kernel is generated by the 4-velocity vector u satisfying y Aµ u µ = 0 . The push-forward of the contravariant spacetime metric from M to X is defined by G AB = ψ ∗ g µν = g µν y Aµ y Bν , (58)which is symmetric positive definite and, therefore, a Riemannian metric on X . This metric containsinformation about the state of strain of the material, which can be described by comparing this metricwith 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 L = −√− det gρ, where ρ = (cid:15)h (59)is the rest frame energy per unit volume. The density of the matter, measured in the material restframe, is given by (cid:15) = ˜ (cid:15) ( y a ) √ det G AB , (60)10here ˜ (cid:15) ( y a ) is an arbitrary positive function. The equation of state is defined by the function h = h ( y A , G AB ) , which describes the dependence of the energy on the state of strain and specifies thematerial.The stress-energy tensor for elastic matter can be written as [9] T µν = (cid:15) (cid:18) − hg δ µ g ν + 2 ∂h∂G AB δ Aν g µB (cid:19) . (61)In spherical symmetry, the push-forward of the metric g with line-element (1) gives G AB = ηδ A δ B + βδ A δ B + ˜ βδ A δ B , (62)where η = e − , β = 1 /r and ˜ β = β/ sin θ . In the unstrained state, one has Λ = 0 and r = R, sothat η = 1 and β = 1 /R . The matter density of the material is, in our case, given by (cid:15) = (cid:15) ( R ) β √ η, (63)assuming that ˜ (cid:15) ( y A ) = (cid:15) ( R ) sin θ, where (cid:15) is an arbitrary positive function. The energy-momentumtensor can then be expressed by T µν = diag( − ρ, p , p , p ) , (64)where ρ = (cid:15)h, p = 2 (cid:15)η ∂h∂η , p = − (cid:15)r ∂h∂r and h = h ( R, η, r ). Using (59), we get p = 2 ηh ∂h∂η ρ, p = − rh ∂h∂r ρ. (65)This T µν for elastic matter falls in the class given by (4) although, in general, we will not have p = p ( ρ )as in Assumption 1, so Theorem 2 does not apply. We will now investigate a particular elastic fluidfor which p = s = 0 and, in that case, we may use Theorem 3. Magli [9] found a class of non-static spherically symmetric solutions of the Einstein equations corre-sponding to anisotropic elastic spheres. These models have vanishing radial stresses and generalizethe Lemaˆıtre-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 notdepend on the eigenvalue η in the material space, i.e. h = h ( R, r ) , then it is possible to write thespherically symmetric metric as [10] ds = − e dt + r (cid:48) h f dR + r d Ω , (66)where Φ( T, R ) = − (cid:90) h ∂h∂r r (cid:48) d R + c ( T ) , (67)11nd c ( T ) is an arbitrary function, reflecting the invariance with respect to time rescaling, which can bechosen such that Φ( T, R ) = 0, for some R >
0. One can then obtain a first integral to the Einsteinequations as e − ˙ r = − Fr + 1 + fh , (68)where F ( R ) > f ( R ) > − , R ].With the assumption h = h ( R, r ), and using the first integral, we also get ρ ( T, R ) = F (cid:48) πr r (cid:48) ,p ( T, R ) = 0 , (69) p ( T, R ) = − r h ∂h∂r ρ := H ρ, where H = − r h ∂h∂r is called adiabatic index (see [10], cf. (65)).A physically interesting example of a spacetime with h = h ( R, r ) is the non-static Einstein clusterin spherical symmetry, which describes a gravitational system of particles sustained only by tangentialstresses [7, 2, 9].To make contact with our formalism, in this case, we get a semi-linear system of the form (55),for the variables U j = { Y, Q , Λ , w, ω, µ } with smooth functions ρ ( Q , ω, R ) = F (cid:48) ( R ) e Q πωR (70) p ( Q , ω, R ) = H ( Q , R ) ρ ( Q , ω, R ) , (71)which clearly satisfy Assumption 4, and smooth initial data on [ R , R ], R >
0, asˆ Y = ˆΩ (cid:48) + 2 ˆΩ R , where ˆΩ = (cid:115) f ˆ h − (cid:18) − FR (cid:19) , ˆ Q = 0 , ˆΛ = 12 ln (cid:32) ˆ h f (cid:33) , ˆ w = 1 R (cid:115) f ˆ h − (cid:18) − FR (cid:19) , ˆ ω = 1 , ˆ µ = 3( F ( R ) − F (0))4 πR . The free initial data here is given by a smooth function ˆ h >
0, i.e. the initial equation of state,and smooth functions F and f which come from the initial density ˆ ρ and radial velocity ˆ v profiles,respectively, as in Section 2.2. For smooth ˆ h, F, f on some interval [ R , R ], R >
0, we are under theconditions of Theorem 3 which ensures existence and uniqueness of solutions in a neighbourhood ofthe boundary. 12n this case, we can also obtain similar results in a neighbourhood of the origin R = 0, at least forsome choices of initial data. This has to be done separately since our evolution system (55) is singularat the center.Figure 2: Domains D and T , where in D ∩ T the solutions agree due to uniqueness.Consider an open region D including the center, as in Figure 2, by considering the original variablesof Magli. In that case, the center is regular for r ( T,
0) = 0, smooth functions f and F such that [9] f (0) = h (0 , − F ( R ) = R ϕ ( R ) , with ϕ (0) is finite , (72)and equations of state satisfying the minimal stability requirement , namely that h has a minimumat r = R [9]. Physically, this means that the centre is unstrained and has zero radial velocity aslim R → ˙ r = 0.In [9], Magli shows that there are open sets of C ∞ data ˆ h, F and f such that the regularityconditions at the centre are fulfilled. From the PDE point of view, this implies that for such datathere exists a unique C ∞ solution to (68), for small enough time in a neighbourhood D of the centre.Uniqueness of the spherically symmetric initial data gives the uniqueness of solutions in the region D ∩ T . We have then proved:
Proposition 1
Consider a spacetime with metric (66) and containing elastic matter satisfying (67) .Consider smooth initial data
F, f, ˆ h on [0 , R ] satisfying the (corner) matching conditions to an exteriorspacetime at (0 , R ) and the regularity conditions at the center. Then, there exists a unique smoothsolution to the Einstein equations on [0 , R ] × T , for small enough T . A suitable exterior to such elastic spacetimes is given by the Schwarzschild solution, which can bematched initially at R . We omit the details of the matching conditions here since they were given in[9]. As a final remark, we note that the analysis of this particular IBVP can be taken much further ifone has explicit solutions for r . These solutions can be obtained by assuming h = h ( r ) >
0. In thiscase, equations (67) and (68) decouple and Φ (cid:48) = − h dhdr r (cid:48) , yielding Φ( T, R ) = − ln h + ln h ( r ( T, R )),which satisfies Φ( T, R ) = 0 . Consider a linear stress-strain relation p = kρ, where the adiabatic13ndex k satisfies − ≤ k ≤ h ( r ) = r − k . In that case, one has ˙ r = 2 F r k − + (1 + f ) r k − r k which is now integrable. Remarkably, for some values of k , such as k = − / k = 1 /
4, onecan obtain explicit solutions which satisfy r (0 , R ) = R and r ( T,
0) = 0 and have a regular origin forcertain choices of initial data [15]. This allows e.g. to solve the matching conditions and obtain theboundary hypersurface [9], as well as to study of global properties of the matched spacetime such asthe formation of horizons and spacetime singularities [15, 10].
Funding:
This work was supported by CMAT, Univ. Minho, through FEDER Funds COMPETEand FCT Projects Est-OE/MAT/UI0013/2014 and PTDC/MAT-ANA/1275/2014.
Acknowledgments:
We thank Robert Beig and Marc Mars for useful comments. We thank thereferees for the comments and suggestions.
Appendix: Matching conditions in spherical symmetry
In this appendix, for completeness, we recall some basic facts about spacetime matching theory andwrite the matching conditions in spherical symmetry adapted to our context. These conditions areknown but are of importance here since they provide compatibility conditions for our IBVP problem.Let ( M ± , g ± ) be spacetimes with non-null boundaries S ± . Matching them requires an identificationof the boundaries, i.e. a pair of embeddings Ω ± : S −→ M ± with Ω ± ( S ) = S ± , where S is an abstractcopy of any of the boundaries. Let ξ i be a coordinate system on S . Tangent vectors to S ± are obtainedby f ± αi = ∂ Ω α ± ∂ξ i though we shall work with orthonormal combinations e ± αi of the f ± αi . There are alsounique (up to orientation) unit normal vectors n α ± to the boundaries. We choose them so that if n α + points into M + then n α − points out of M − or viceversa. The first and second fundamental forms on S ± are simply q ± ij = e ± αi e ± βj g αβ | S ± , K ± ij = − n ± α e ± βi ∇ ± β e ± αj . The matching conditions (in the absence of shells), between two spacetimes ( M ± , g ± ) across a non-nullhypersurface S , are the equality of the first and second fundamental forms (see [11]): q + ij = q − ij , K + ij = K − ij . (73)We shall now specify the matching conditions for the case where the interior has the anisotropic fluidmetric (1) and the exterior is the Kottler spacetime given by metric: g + = − (cid:18) − m(cid:37) − Λ3 (cid:37) (cid:19) dt + dρ − m(cid:37) − Λ3 (cid:37) + (cid:37) d Ω . (74)The boundary S + can be parametrized as Ω + = { t = t ( λ ) , (cid:37) = (cid:37) ( λ ) } , where two dimensions wereomitted, and we can assume ˙ t > T, R ) = 0 for the interior spacetime, we can take the following parametriza-tion Ω − = { T = λ, R = R } , so that ˙ T = 1 and ˙ R = 0 . Then, the equality of the first fundamental14orms on S gives 1 = (cid:18) − m(cid:37) − Λ3 (cid:37) (cid:19) ˙ t − ˙ (cid:37) − m(cid:37) − Λ3 (cid:37) , (75) r ( λ, R ) = (cid:37) ( λ ) (76)and the equality of the second fundamental forms, on S , impliesΦ (cid:48) e − Λ = (cid:32) − ˙ t ¨ (cid:37) + ¨ t ˙ (cid:37) + 3 m ˙ (cid:37) ˙ t (cid:37) (cid:0) (cid:37) − m − Λ3 (cid:37) (cid:1) − m(cid:37) (cid:18) − m(cid:37) − Λ3 (cid:37) (cid:19) ˙ t (cid:33) , (77) rr (cid:48) e − Λ = − ˙ t (cid:18) (cid:37) − m − Λ3 (cid:37) (cid:19) . (78)Then, the matching conditions give expressions for (cid:37) ( λ ), t ( λ ) and imply the continuity of the mass˜ µ ˜ r = 6 m through the boundary, as is well known. By further substituting the Einstein equations,those conditions also imply that ˜ p = − Λ. In fact, from ˜ p = − Λ we also recover ˜ µ ˜ r = 6 m asmentioned at the end of Section 3.1.Generalising this procedure, consider now the matching where the interior is still the anisotropicfluid (1) and the exterior is the spacetime given by a general spherically symmetric metric: g + = − e µ ( t,(cid:37) ) dt + e ν ( t,(cid:37) ) d(cid:37) + ¯ r ( t, (cid:37) ) d Ω . (79)The boundaries S ± are parametrized as before. Then, the equality of the first fundamental forms, forΦ( T, R ) = 0, imply − − ˙ t e µ + ˙ (cid:37) e ν , (80) r ( λ, R ) = ¯ r ( t ( λ ) , (cid:37) ( λ )) (81)and the equality of the second fundamental forms gives − Φ (cid:48) e − Λ = e µ + ν (cid:0) ˙ ρ ˙ t ˙ µ + 2 ˙ ρ ˙ t µ (cid:48) + ˙ ρ ˙ νe ν − µ − ˙ t µ (cid:48) e µ − ν − t ˙ ρ ˙ ν − ˙ ρ ˙ t ν (cid:48) (cid:1) , (82) rr (cid:48) e − Λ = e µ + ν ¯ r (cid:0) ˙ t e − ν ¯ r (cid:48) + ˙ ρ e − µ ˙¯ r (cid:1) . (83)The above equations provide ρ ( λ ), t ( λ ) and p − ( λ ) = p +1 ( λ ), at the boundary. Another way to getthis last relation is to use the Israel conditions n µ − T − µν = n µ + T + µν at S (see e.g. [11]). With this boundarydata, and knowing the exterior spacetime, one gets ˜ r ( λ ) := r ( λ, R ) from (81) and, therefore, ˜ w ( λ ) asmentioned in Section 3.1. References [1] Andersson, L., Oliynyk, T. A. and Schmidt, B. G., Dynamical compact elastic bodies in generalrelativity,
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