Disembodied boundary data for Einstein's equations
aa r X i v : . [ g r- q c ] N ov Disembodied boundary data for Einstein’s equations
Jeffrey Winicour , Department of Physics and AstronomyUniversity of Pittsburgh, Pittsburgh, PA 15260, USA Max-Planck-Institut f¨ur Gravitationsphysik,Albert-Einstein-Institut,14476 Golm, Germany
A strongly well-posed initial boundary value problem based upon constraint-preserving bound-ary conditions of the Sommerfeld type has been established for the harmonic formulation of thevacuum Einstein’s equations. These Sommerfeld conditions have been previously presented in a 4-dimensional geometric form. Here we recast the associated boundary data as 3-dimensional tensorfields intrinsic to the boundary. This provides a geometric presentation of the boundary data analo-gous to the 3-dimensional presentation of Cauchy data in terms of 3-metric and extrinsic curvature.In particular, diffeomorphisms of the boundary data lead to vacuum spacetimes with isometric ge-ometries. The proof of well-posedness is valid for the harmonic formulation and its generalizations.The Sommerfeld conditions can be directly applied to existing harmonic codes which have been usedin simulating binary black holes, thus ensuring boundary stability of the underlying analytic system.The geometric form of the boundary conditions also allows them to be formally applied to any metricformulation of Einstein’s equations, although well-posedness of the boundary problem is no longerensured. We discuss to what extent such a formal application might be implemented in a constraintpreserving manner to 3+1 formulations, such as the Baumgarte-Shapiro-Shibata-Nakamura systemwhich has been highly successful in binary black hole simulation.
PACS numbers: PACS number(s): 04.20.-q, 04.20.Cv, 04.20.Ex, 04.25.D-
I. INTRODUCTION
Previous work has established the strong well-posedness of the initial-boundary value problem (IBVP) for Einstein’sequations expressed as a hyperbolic system in harmonic coordinates. The result was first obtained using pseudo-differential theory, i.e. using a Fourier-Laplace expansion, which established well-posedness in the generalized sense [1].The result was subsequently obtained using standard energy estimates [2, 3]. This places the IBVP on the sameanalytic footing as the Cauchy problem, whose well-posedness was also established using harmonic coordinates inthe classic work of Choquet-Bruhat [4]. The geometric formulation of the boundary conditions and boundary datafor the IBVP is more complicated than for the Cauchy problem. Recently, this boundary data was presented in a4-dimensional geometric form [5]. Here we recast the boundary data as 3-dimensional tensor fields intrinsic to theboundary, analogous to the presentation of Cauchy data in terms of the 3-metric and extrinsic curvature of the initialCauchy hypersurface. The spacetime metric which solves the harmonic IBVP with this data is uniquely determinedup to a diffeomorphism.In the Cauchy problem, initial data on a spacelike hypersurface S determine a solution in the future domain ofdependence D + ( S ) (which consists of those points whose past directed characteristics all intersect S ). In the IBVP,data on a timelike boundary T which meets S in a surface B are used to further extend the solution to the domain ofdependence D + ( S ∪T ). In practical applications B is topologically a sphere surrounding some system of interest buthere we only consider the local problem in a neighborhood of a point of intersection between the Cauchy hypersurfaceand the boundary. For hyperbolic systems, the global solution in the spacetime manifold M can be obtained bypatching together local solutions. The setting for this local problem is depicted in Fig. I.There are no natural boundaries for the gravitational field analogous to the conducting boundaries that play a majorrole in electromagnetism. Consequently, the IBVP for Einstein’s equations only received widespread attention afterits importance to the artificial outer boundaries used in numerical relativity was pointed out [6]. The first well-posedIBVP was achieved for a tetrad formulation of Einstein’s theory in terms of a first differential order system whichincluded the tetrad, the connection and the curvature tensor as evolution fields [7]. A strongly well-posed IBVP waslater established for the harmonic formulation of Einstein’s equations as a system of second order quasilinear waveequations for the metric [1, 2]. Strong well-posedness guarantees the existence of a unique solution which dependscontinuously on both the Cauchy data and the boundary data. The results were further generalized in [3] to apply togeneral quasilinear symmetric hyperbolic systems whose boundary conditions have a certain hierarchical form.The initial data for the Cauchy problem can be formulated in terms of two symmetric tensor fields ˜ h ab (a Euclidean FIG. 1: Data on the 3-manifolds S and T , which intersect in the 2-surface B , locally determine a solution in the spacetimemanifold M . k ab on a 3-manifold ˜ S subject to the (Hamiltonian and momentum) constraints0 = (3) ˜ R + (˜ k aa ) − ˜ k ab ˜ k ab (1.1)and 0 = ˜ ∇ a (˜ k ab − δ ab ˜ k cc ) , (1.2)where ˜ ∇ a is the covariant derivative and (3) ˜ R is the curvature scalar associated with ˜ h ab . As characterized in [9], viaan embedding in a 4-dimensional manifold M , the data ˜ h ab and ˜ k ab on the disembodied S determine aLorentzian metric satisfying Einstein’s equations which is unique up to diffeomorphism and whose restriction to theembedding S of ˜ S gives rise to its intrinsic 3-metric h ab and extrinsic curvature k ab .Here we present an analogous result for the IBVP. The difficulties underlying the IBVP, which have most recentlybeen discussed in [5] and [8], make the formulation of disembodied boundary data more difficult than for the Cauchyproblem. There are three main complications. • The first complication stems from a well-known property of the IBVP for the flat-space scalar wave equation( ∂ t − ∇ )Φ = 0in the region x ≤
0. Although the initial Cauchy data consist of Φ | t =0 and ∂ t Φ | t =0 , only half as much boundarydata can be freely prescribed at x = 0, e.g the Dirichlet data ∂ t Φ | x =0 , or the Neumann data ∂ x Φ | x =0 or theSommerfeld data ( ∂ t + ∂ x )Φ | x =0 (based upon the derivative in the outgoing characteristic direction). For a givenphysical problem, this implies that the boundary data cannot be prescribed before the boundary condition isspecified, i.e. the correct boundary data depend upon the boundary condition, unlike the situation for theCauchy problem. The analogue in the gravitational case is the inability to prescribe both the metric and itsnormal derivative on a timelike boundary, which implies that you cannot freely prescribe both the intrinsicmetric of the boundary and its extrinsic curvature. This leads to a further complication regarding constraintenforcement on the boundary, i.e. the Hamiltonian and momentum constraints (1.1) and (1.2) cannot be enforceddirectly because they couple the metric and its normal derivative. Here we restrict our attention to Sommerfeldboundary data which we prescribe in a constraint free manner. • A Sommerfeld boundary condition for a metric component supplies the value of the derivative K c ∂ c g ab in anoutgoing null direction K c . Such boundary conditions are very beneficial for numerical work since they allowdiscretization error to propagate across the boundary (whereas Dirichlet and Neumann boundary conditionsreflect the error and trap it in the numerical grid). However, the boundary does not pick out a unique outgoingnull direction at a given point (but, instead, essentially a half null cone). This complicates the geometric for-mulation of a Sommerfeld boundary condition. In addition, constraint preservation does not allow specificationof Sommerfeld data for all components of the metric, as will be seen in formulating the Sommerfeld conditions(3.21) - (3.23). • The third major complication arises from gauge freedom. In the evolution of the Cauchy data it is necessary tointroduce a foliation of the spacetime by Cauchy hypersurfaces S t , with unit timelike normal n a . The evolutionof the spacetime metric g ab = − n a n b + h ab (1.3)is carried out along the flow of an evolution vector field t a related to the normal by the lapse α and shift β a according to t a = αn a + β a , β a n a = 0 . (1.4)The choice of foliation is part of the gauge freedom in the resulting solution but does not enter into thespecification of the initial data. In the IBVP, the foliation is unavoidably coupled with the formulation of theboundary condition. Thus some gauge information must be incorporated in the formulation of the boundarycondition and boundary data.In order to resolve these complications, we include the specification of a foliation B t of the boundary T as partof the boundary data. This supplies the gauge information which determines a unique outgoing null direction fora Sommerfeld condition. In Section II, we formulate the 3-dimensional prescription of the boundary data and thestatement of our main result, a Theorem establishing the existence of a solution satisfying a version of the geometricuniqueness property proposed in [8]. The 3-dimensional description of the data in terms of scalar, vector and tensorfields intrinsic to the boundary is quite abstract at this stage. Their physical significance only becomes clear inSection III, where the proof of the Theorem is given. The proof is based upon the well-posedness of the 4-dimensionalgeometrical formulation of the harmonic IBVP given in [5]. Constraint preservation is established by incorporatingthe harmonic conditions into the boundary conditions.The motivation for this work stems from the need for an improved understanding and implementation of boundaryconditions in the computational codes being used to simulate binary black holes. We discuss the applicability of ourresults to numerical relativity in Sec. IV.Much of the presentation in the paper is coordinate independent and we use Latin letters ( a, b, c, ... ) as abstractindices [10] to denote the types of vector and tensor fields and to indicate their manipulations. This notation servesto describe either 4-dimensional tensor fields on the spacetime M or 3-dimensional tensor fields intrinsic to S or T .When spacetime coordinates x µ = ( t, x i ) are introduced, we use Greek letters ( µ, ν, ρ, ... ) to describe the corresponding4-dimensional tensor components and Latin letters ( i, j, k, ... ) to denote the spatial components. II. DISEMBODIED DATA FOR THE IBVP
We state our main result concerning Sommerfeld boundary data for Einstein’s equations.
Geometric Uniqueness Theorem:
Consider the 3-manifolds ˜ T and ˜ S meeting in an edge ˜ B . On ˜ S prescribethe smooth, symmetric tensor fields ˜ h ab and ˜ k ab , subject to the Hamiltonian and momentum constraints and thecondition that ˜ h ab be a Riemannian metric . On ˜ B prescribe the smooth scalar field ˜Θ. On ˜ T prescribe a smoothfoliation ˜ B t parametrized by a scalar function ˜ t , where (for convenience) ˜ t = 0 on ˜ B . In addition, on ˜ T , prescribethe scalar field ˜ q , the vector field ˜ q a and the rank-2 symmetric tensor field ˜ σ ab , which are all smooth and vanish on˜ B . Here the rank-2 property of ˜ σ ab is defined with respect to the foliation ˜ B t by the requirement˜ σ ab ∂ a ˜ t = 0 . (2.1)Then, after embedding ˜ S ∪ ˜ T as the boundary S ∪ T of a 4-manifold M , as depicted in Fig. I, this data providesSommerfeld boundary data for a vacuum spacetime, in a region including a neighborhood of the embedded edge B ,which is unique up to diffeomorphism.The geometrical interpretation of the data involves the metric of the embedded spacetime, whose existence is thecontent of the Theorem. Before proceeding to the proof in the next section, it is useful to supply some intuitivemeaning to the data. As in the Cauchy problem, ˜ h ab and ˜ k ab are identified with the 3-metric and the extrinsiccurvature of the embedding S of ˜ S . The scalar field ˜Θ determines the hyperbolic angle describing the initial velocityof the embedded boundary T relative to the inertial frame picked out by S . The fields ˜ q and ˜ q a supply informationconcerning the subsequent dynamics of the boundary and its foliation. Together ˜ q and ˜ q a determine the componentsof a 4-vector q a describing the curvature of the outgoing null geodesics normal in M to the embedding B t of ˜ B t . Thefield ˜ σ ab determines the optical shear σ of the outgoing null hypersurface through B t .Note that the data contains no metric information about the embedded boundary T , not even that it is a timelike3-manifold. Such structure only emerges from the construction of a solution for the spacetime metric. Thus fieldsderived algebraically from the metric, such as the unit normal to the boundary, are to be considered as subsidiaryunknowns.The requirement that the boundary data vanish at ˜ B stems from its interpretation relative to the initial Cauchydata. This requirement ensures the continuity of the resulting spacetime metric and its first derivatives. The fullcompatibility conditions necessary for a C ∞ spacetime metric involve enforcing the Einstein equations and theirderivatives on B . It is not clear how to implement C ∞ compatibility conditions in terms of disembodied data. III. THE EXISTENCE OF A GEOMETRICALLY UNIQUE SOLUTION
Our task is to show that harmonic evolution of the data prescribed in the Geometric Uniqueness Theorem of Sec. IIlocally determine a vacuum spacetime which is unique up to diffeomorphism. We embed ˜ S , ˜ T and ˜ B in a 4-manifold M with boundary consisting of the corresponding pieces S , T and B . The unknown is a spacetime metric g ab on M which satisfies Einstein’s equations and is determined up to a diffeomorphism by the data specified in the Theorem.We follow the construction given in [5].The first step is to give initial data for g ab . Let h ab and k ab be the fields induced by ˜ h ab and ˜ k ab on S ; Θ be thefield induced by ˜Θ on B ; and ˆ q , ˆ q a and ˆ σ ab be the fields induced by ˜ q , ˜ q a and ˜ σ ab on T . Let B t be the foliation of T corresponding to the embedding of ˜ B t , where t is the parametrization induced by ˜ t . On S prescribe a transversefield n a and construct the Lorentzian metric g ab = − n a n b + h ab , so that n a is the future directed unit normal to S .Require that n a satisfy n a N a | B = sinh Θ, where N a is the outward unit normal to T .We introduce the boundary decomposition of the metric g ab = N a N b − T a T b + Q ab , (3.1)where T a is the future directed unit normal in T to B t . (Here the boundary fields N a , T a and Q ab are unknownssubsidiary to g ab .) This leads to an orthonormal tetrad ( T a , N a , Q a , ¯ Q a ) on T , where Q a is a complex null vectortangent to B t with normalization Q ab = Q ( a ¯ Q b ) , Q a ¯ Q a = 2 , Q a Q a = 0 . (3.2)(The tetrad is unique up to the spin freedom Q a → e iθ Q a which does not enter our construction in any essential way.)Uniquely associated with this tetrad (independent of the choice of Q a ) are the outgoing and ingoing null vector fields K a = T a + N a and L a = T a − N a , respectively, which lie in the null directions normal to B t . They form a null tetrad( K a , L a , Q a , ¯ Q a ) with metric decomposition g ab = − K ( a L b ) + Q ( a ¯ Q b ) . (3.3)Next we tie down the gauge freedom by introducing an evolution field t a on M . We require that t a be tangent to T and that it generate the foliation B t according to L t t = 1 , (3.4)where L t is the Lie derivative with respect to t a . We extend t a to M such that it generates a foliation S t , withparametrization satisfying (3.4) and unit normal n a . The gauge freedom is then pinned down by introducing spatialcoordinates x i on S and extending them to M according to L t x i = 0 . (3.5)The scalars x µ = ( t, x i ) serve as coordinates for M which are adapted to the evolution. Note that t a and the adaptedcoordinates x µ are explicitly constructed fields on M with no metric properties. In the relationship (1.4) between t a and the unit normal n a to S t , it is n a which contains metric information and is a subsidiary unknown.As part of the initial data, we prescribe L t n a on S . Along with the initial choice of n a , h ab and k ab , this determinesthe initial data g ab | t =0 and L t g ab | t =0 . We use the evolution field t a to provide a background metric ˚g ab on M whichis uniquely and geometrically determined by the Lie transport of the initial data according to ˚g ab | t =0 = g ab | t =0 , L t ˚g ab | t =0 = L t g ab | t =0 , L t L t ˚g ab = 0 . (3.6)In the coordinates x µ = ( t, x i ) adapted to the evolution, this reduces to ˚g µν = g µν | t =0 + t ( ∂ t g µν | t =0 ) . (3.7)The connection ∇ ˚ a and curvature tensor ˚R dcab associated with the background ˚g ab have the same transformationproperties as the corresponding quantities ∇ a and R dcab associated with g ab . In particular, the difference ∇ a − ∇ ˚ a defines a tensor field C dab according to ( ∇ a − ∇ ˚ a ) v d = C dab v b , (3.8)for any vector field v b . In terms of the (nonlinear) perturbation f ab = g ab − ˚g ab (3.9)of the metric from the background, we have C dab = 12 g dc (cid:16) ∇ ˚ a f bc + ∇ ˚ a f bc − ∇ ˚ c f ab (cid:17) . (3.10)We take f ab to be the evolution variable for solving Einstein’s equations. Since ˚g ab is explicitly known, a solutionfor f ab is equivalent to a solution for g ab . By construction, the initial data for f ab is homogeneous, i.e. f ab | t =0 = L t f ab | t =0 = 0 . (3.11)The boundary data on T consist of the vector field q a = ˆ qN a + ˆ q a (3.12)and the tensor field σ ab = ˆ σ ab − Q ab Q cd ˆ σ cd . (3.13)Here, and elsewhere, the physical metric g ab is used to raise and lower indices and to normalize the tetrad vectors.Now, just as the Cauchy data h ab and k ab must be identified as the intrinsic metric and extrinsic curvature of S in order to construct a solution of the initial value problem, we give a geometric identification of the boundary data.We identify this data as the geodesic curvature and shear, relative to their background metric values, of the outgoingnull vector K a on T according to the formulae q a = K b ( ∇ b − ∇ ˚ b ) K a , (3.14) σ ab = 12 ( Q ac Q bd − Q ab Q cd )( ∇ c − ∇ ˚ c ) K d . (3.15)The last equation can be re-expressed in the spin-weight-2 form σ := Q a Q b σ ab = 12 Q a Q b ( ∇ a − ∇ ˚ a ) K b . (3.16)The use of the shear in posing geometrical boundary conditions for the harmonic formulation was suggested earlierin [11].Using (3.8) and (3.10), we recast (3.14) and (3.16) as Sommerfeld boundary conditions which determine the com-ponents of the outgoing null derivatives K a ∇ ˚ a f bc according to12 K b K c K a ∇ ˚ a f bc = q a K a , (3.17)( Q b K c K a − K b K c Q a ) ∇ ˚ a f bc = q a Q a , (3.18)( L b K c K a − K b K c L a ) ∇ ˚ a f bc = q a L a , (3.19)( 12 Q b Q c K a − Q b K c Q a ) ∇ ˚ a f bc = 2 σ. (3.20)In addition to these six Sommerfeld conditions, we impose the four additional boundary conditions that C d := g ab C dab = 0 on T , i.e. the harmonic constraints. In terms of the null tetrad decomposition, they take the Sommerfeldform − C a K a = (cid:0) Q b ¯ Q c K a + K b K c L a − K b ¯ Q c Q a − K b Q c ¯ Q a (cid:1) ∇ ˚ a f bc = 0 , (3.21) − C a Q a = (cid:0) L b Q c K a + K b Q c L a − K b L c Q a + Q b Q c ¯ Q a (cid:1) ∇ ˚ a f bc = 0 , (3.22) − C a L a = (cid:0) L b L c K a + Q b ¯ Q c L a − ¯ Q b L c Q a − Q b L c ¯ Q a (cid:1) ∇ ˚ a f bc = 0 . (3.23)Together, (3.17) - (3.23) provide Sommerfeld boundary conditions for the components of K a ∇ ˚ a f bc in the sequentialorder ( KK ) , ( QK ) , ( LK ) , ( QQ ) , ( Q ¯ Q ) , ( LQ ) , ( LL ) in terms of the boundary data and the derivatives of precedingcomponents in the sequence. Such a hierarchy of Sommerfeld boundary conditions satisfy the requirements of Theorem1 of [3] which establishes a strongly well-posed IBVP for a quasilinear hyperbolic system.In order to apply this theorem, we reduce Einstein’s equations to a quasilinear wave system by modifying theEinstein tensor by the harmonic constraints C d = g ab C dab = 12 g ab g dc (cid:16) ∇ ˚ a f bc + ∇ ˚ a f bc − ∇ ˚ c f ab (cid:17) (3.24)according to E ab := G ab − ∇ ( a C b ) + 12 g ab ∇ d C d . (3.25)We eliminate the coordinate freedom, up to our choice of evolution field t a on M and coordinates x i on S , by workingin the coordinates x µ adapted to the evolution. The coordinate components of (3.24) then take the form C ρ = 12 g µν g ρσ (cid:16) ∇ ˚ µ f νσ + ∇ ˚ µ f νσ − ∇ ˚ σ f µν (cid:17) = g µν (Γ ρµν − Γ˚ ρµν ) . (3.26)The requirement that Γ ρ := g µν Γ ρµν = 0 is the standard harmonic coordinate condition that (cid:3) g x µ = 0. In the presentcase, setting C ρ = 0 implies Γ ρ = g µν Γ˚ ρµν , which is an example of harmonic coordinates with a forcing term, asdiscussed in [12]. (In numerical relativity, these have been called generalized harmonic coordinates [13].) For moregeneral forcing terms the harmonic constraint takes the formΓ ρ = H ρ ( x, g ) , (3.27)where restriction of the forcing term H ρ to depend only upon x µ and g µν ensures that the system remains well-posed.In the adapted coordinates, the reduced Einstein equations E µν = 0 form the desired quasilinear wave system for f µν , g ρσ ∇ ˚ ρ ∇ ˚ σ f µν = 2 g λτ g ρσ C λµρ C τ νσ + 4 C ρσ ( µ g ν ) λ C λρτ g στ − g ρσ ˚R λρσ ( µ g ν ) λ . (3.28)With the hierarchy of Sommerfeld boundary conditions (3.17) - (3.23), Theorem 1 of [3] now applies and ensures that(3.28) has a well-posed IBVP and, in particular, determines a unique solution f µν . In addition, the resulting metric g µν must satisfy the harmonic constraints C ρ = 0 because they are built into the initial data and boundary conditions.(See Sec. IV for details concerning constraint preservation.) Therefore g µν solves Einstein’s equations.The solution has been obtained in coordinates which are harmonic with respect to the background ˚g µν , i.e. g µν (Γ ρµν − Γ˚ ρµν ) = 0. The resulting spacetime metric g ab and background metric ˚g ab determined by the data are in a gauge whichdepends upon the choice of evolution field t a . Under a diffeomorphism Ψ of M which reduces to the identity mapon S and T , we have ( t a , g ab , ˚g ab ) → (Ψ ∗ t a , Ψ ∗ g ab , Ψ ∗ ˚g ab ). Thus Ψ ∗ g ab is a diffeomorphic solution of Einstein’sequations which is harmonic with respect to the background Ψ ∗ ˚g ab .A more important question, which concerns the issue of geometric uniqueness raised in [8], is the behavior of thespacetime under a diffeomorphism ˜Ψ of the disembodied boundary ˜ T . It is known for the Cauchy problem thatdata ( ˜Ψ ∗ ˜ h ab , ˜Ψ ∗ ˜ k ab ) on ˜ S lead to a spacetime which is isometric to the spacetime with Cauchy data (˜ h ab , ˜ k ab ). Theanalogous result holds for the boundary data. Under a diffeomorphism ˜Ψ of ˜ T the boundary data maps accordingto (˜ t, ˜ q, ˜ q ab , ˜ σ ab ) → ( ˜Ψ ∗ ˜ t, ˜Ψ ∗ ˜ q, ˜Ψ ∗ ˜ q ab , ˜Ψ ∗ ˜ σ ab ). After the embedding in M , this induces data ( ˜Ψ ∗ t, ˜Ψ ∗ ˆ q, ˜Ψ ∗ ˆ q ab , ˜Ψ ∗ ˆ σ ab )on T . Now consider any smooth extension of ˜Ψ to a diffeomorphism of M . Suppose the original data leads bythe above construction to the spacetime metric g ab with the evolution field t a , background ˚g ab and relative geodesiccurvature and relative shear ( q a , σ ab ) of the outgoing null vector K a normal to the t -foliation of T . Then, by the sameconstruction, the mapped data lead to the metric ˜Ψ ∗ g ab with the evolution field ˜Ψ ∗ t a , background ˜Ψ ∗ ˚g ab and relativegeodesic curvature and relative shear ( ˜Ψ ∗ q a , ˜Ψ ∗ σ ab ) of the outgoing null vector ˜Ψ ∗ K a normal to the ˜Ψ ∗ t -foliation of T . In this way, diffeomorphisms of ˜ S and ˜ T , which map their intersection ˜ B into itself, generate an isometry classof vacuum spacetimes.We have thus shown that the resulting vacuum spacetimes are diffeomorphic if the disembodied boundary data andinitial Cauchy data are diffeomorphic. This comprises a version of geometric uniqueness, which has been pointed outas a missing ingredient in prior formulations of the IBVP [8]. However, the result is not as strong as for the pureCauchy problem for which the converse is also true: the resulting vacuum spacetimes are diffeomorphic only if theinitial data are diffeomorphic. The converse is more complicated for the IBVP and does not hold in the disembodiedsetting because the data ˜ σ ab are superfluous. It is only the trace-free part of the embedded data ˆ σ ab which enters theshear σ ab . But there is no way to pick out the trace free part without knowledge of the metric, which is an unknownat the stage of specifying data. (For linear perturbations, a background metric could be used but this constructiondoes not extend to the nonlinear case.) This is perhaps an unavoidable feature of disembodied Sommerfeld data forthe IBVP.As a result, disembodied boundary data which are not related by a diffeomorphism can lead to isometric spacetimes.However, it follows by direct geometrical construction that an isometry class of spacetimes does uniquely determinethe embedded boundary data q a and σ ab up to a diffeomorphism. It is only upon the transition from σ ab to ˆ σ ab that the ambiguity of adding a trace enters. In this sense, a stronger version of geometric uniqueness applies to theembedded data.The Sommerfeld boundary conditions were based upon the background metric obtained by the Lie transport (3.6)of the Cauchy data along the streamlines of t a . Modification of this transport law would lead to a different backgroundand the spacetime generated by the same disembodied data would in general not be diffeomorphic. This is similarto the scalar wave problem, for which the same boundary data would lead to different solutions if, say, a Dirichletboundary condition were used instead of a Sommerfeld condition. In the present case, a new background metricimplies a different form of the Sommerfeld condition and, for fixed boundary data, the resulting spacetime will (ingeneral) not be isometric to the original spacetime. The effect of the boundary data is changed by the differencebetween the two background connections. IV. NUMERICAL APPLICATION
The motivation for this work has been the treatment of the outer boundary in the numerical simulation of theinspiral and merger of binary black holes. The boundary conditions (3.17) - (3.23), which lead to a strongly well-posed IBVP, can be applied directly to any of the harmonic evolution codes which have been used to simulate thisbinary problem [14, 15, 16, 17, 18]. At present, none of these harmonic codes incorporate boundary conditions thatensure strong well-posedness. The closest example is the pseudo-spectral harmonic code described in [16, 17] whichincorporates a second differential order boundary condition which freezes the Ψ Weyl component and was shown tobe well-posed in the generalized sense in the high frequency limit [11].An important attribute of strong well-posedness is the estimate of the boundary values of the solution and itsderivatives which are provided by an energy conservation law obeyed by the principle part of the equations. Thisboundary stability extends to the semi-discrete system of ordinary differential equations in time which are obtained byreplacing spatial derivatives by finite differences obeying summation by parts (the discrete counterpart of integrationby parts), so that energy conservation caries over to the semi-discrete problem. This stability then extends to thenumerical evolution algorithm obtained by applying an appropriate time integrator, such as Runge-Kutta.The advantage of boundary stability in numerical applications is that the smoothness of the solution is unaffectedby reflections off the boundary. This avoids the long timescale instabilities which might otherwise arise from multiplereflections. For the harmonic Einstein problem, each component of the metric obeys a quasilinear wave equation sothat it is straightforward to develop a summation by parts algorithm based upon the standard energy expression fora scalar wave in a curved spacetime [19]. A code incorporating such an algorithm was applied, using a version ofthe well-posed Sommerfeld boundary conditions presented here, to the test problem of a highly nonlinear gauge wavepropagating inside a cubic boundary [20]. Although the proof of strong well-posedness given in [2] was based upon ascalar wave energy differing by a small boost from the standard energy expression, the successful results for this testproblem confirm the robustness of the underlying approach.Strong well-posedness extends to more general quasi-harmonic formulations for which the reduced Einstein equationshave the form E µν := G µν − ∇ ( µ C ν ) + 12 g µν ∇ ρ C ρ + A µνσ C σ = 0 , (4.1)where C ρ = g µν Γ ρµν − H ρ ( x, g ) are the generalized harmonic constraints (3.27) and the coefficients A µνσ have the de-pendence A µνσ ( x, g, ∂g ). (This includes constraint modified versions of the harmonic system.) Constraint preservationfollows from the Bianchi identity ∇ µ G µν = 0 which implies a homogeneous wave equation for C µ , ∇ ρ ∇ ρ C µ + R µρ C ρ − ∇ ρ ( A µρσ C σ ) = 0 . (4.2)If the boundary conditions enforce C ρ | T = 0 and the initial data enforces C ρ | S = ∂ t C ρ | S = 0 then the unique solutionof (4.2) is C ρ = 0. As a result, the Sommerfeld boundary conditions in the geometrical form (3.17) - (3.20), alongwith (3.21) - (3.23) which enforce C ρ | T = 0, lead to a well-posed harmonic IBVP in which the harmonic constraints C ρ = 0 are satisfied everywhere. In turn, (4.1) implies that the Hamiltonian and momentum constraints G µν n ν = 0are also satisfied.Beyond the geometrical aspects of the harmonic IBVP, there are practical concerns that arise in astrophysicalapplications. The linear time dependence of the background metric (3.7) could lead to deleterious long time scaleeffects in numerical simulations. For that reason, it is preferable to modify the prescription (3.6) for the backgroundmetric so that (3.7) changes to ˚g µν = g µν | t =0 + te − λt ( ∂ t g µν | t =0 ) (4.3)and the time dependence damps on a time scale determined by λ . Alternatively, the purpose of tying the backgroundmetric to the initial Cauchy data was to make clear that it did not affect the geometric uniqueness of the solution.Otherwise, the Minkowski metric in the coordinates adapted to the evolution could have been chosen as the backgroundmetric. This would leave intact the well-posedness of the IBVP and might be the most expedient approach in somenumerical applications, although (4.3) has the advantage of suppressing nonlinear effects in the early stage of anevolution.Another practical concern is that the proper boundary data q a and σ are not normally known in an astrophysicalapplication. The practice in simulating an isolated system is to assume homogeneous data on the artificial outerboundary, i.e. q a = σ = 0. Such homogeneous data in general leads to some spurious back reflection from theboundary. However, as discussed in [3] for the harmonic IBVP, similar boundary conditions on a round sphericalouter boundary of large surface area radius R lead to reflected waves whose amplitude falls off asymptotically as 1 /R for both quadrupole gauge waves and quadrupole gravitational waves. (The calculation assumes that the linearizedapproximation is applicable. Modifications of the boundary conditions by lower differential order terms involvingfactors of R lead to a faster 1 /R falloff for the reflected quadrupole gravitational waves. [3])The boundary conditions (3.17) - (3.23) are slightly different than those considered in [3] and, in particular, thehomogeneous boundary condition σ = 0 leads to reflected waves with only a 1 /R falloff, as previously noted in [11].However, improved performance can be obtained by taking advantage of the sensitivity of the boundary conditionsto the location of the indices. If (3.16) and (3.20) are replaced by2 σ = 12 Q a Q b ( ∇ a − ∇ ˚ a ) K b + 12 Q a Q b ( ∇ a − ∇ ˚ a ) K b = 12 ( Q b Q c K a − Q b K c Q a ) ∇ ˚ a f bc , (4.4)then agreement with the corresponding boundary condition (94) of [3] is obtained. In that case, for the boundaryconditions (3.17) - (3.19) and (4.4), along with the harmonic constraints (3.21) - (3.23), the amplitudes of the reflectedquadrupole gauge waves and quadrupole gravitational waves again fall off asymptotically as 1 /R . Thus the applicationof these Sommerfeld boundary conditions with homogeneous data results in small back reflection from the boundaryof an isolated system.While the Sommerfeld boundary conditions considered here were developed for the harmonic IBVP, their geometricnature allows them to be formally applied to any metric version of the reduced Einstein equations, in particularthe alternative “3 + 1” formulations upon which much numerical work has been based. However, in that case, well-posedness and constraint preservation do not necessarily follow. One approach to dealing with these issues would beto re-express the 3 + 1 formulations in terms of the covariant 4-dimensional Z C µ replaced by a more general vector field Z µ , which can be used to change the principle part of the resulting evolution system. The Z Z x µ = ( t, x i ) = ( t, x, y.z ), in which the metric takes the form (1.3) where, in terms of the lapse and shift (1.4), g tt = − α + h ij β i β j , g ti = h ij β j , g ij = h ij . (4.5)Although there is a different decomposition (3.1) intrinsic to the boundary, the outgoing null direction used informulating a Sommerfeld condition is intrinsic to both decompositions. Let ˆ N µ be the unit normal to the boundaryfoliation B t , which lies in the t = const Cauchy hypersurface, so that the metric (1.3) has the further decomposition g µν = − n µ n ν + ˆ N µ ˆ N ν + Q µν , (4.6)where again Q µν = Q ( µ ¯ Q ν ) is the 2-metric intrinsic to the foliation B t of the boundary. Then the outgoing null vector K µ normal to the foliation has components K µ = T µ + N µ = e − Θ ( n µ + ˆ N µ ) where Θ is the hyperbolic angle arisingfrom the velocity of the boundary relative to the Cauchy hypersurfaces. In terms of the lapse and normal componentof the shift, tanh Θ = − α β i ˆ N i , (4.7)where β i ˆ N i < β i ˆ N i determines whether the advective derivative n µ ∂ µ is outward ( β i ˆ N i <
0) or inward ( β i ˆ N i >
0) at the boundary and,as a result, can affect the number of required boundary conditions.The Sommerfeld derivative takes the simple 3+1 form K µ ∂ µ = e − Θ ( n µ ∂ µ + ˆ N µ ∂ µ ) , ˆ N µ ∂ µ = ˆ N i ∂ i , (4.8)where n µ ∂ µ is the part containing the time derivative ∂ t . Thus the Sommerfeld boundary conditions (3.17) - (3.20)can be used to supply boundary values for the time derivatives of 6 metric components, or equivalently to supplyboundary values for 5 components of the extrinsic curvature k µν of the Cauchy foliation and for the time derivativeof one metric component. In particular, (3.17) - (3.18) and (3.20) supply boundary values for ˆ N µ ˆ N ν k µν , Q µ ˆ N ν k µν and Q µ Q ν k µν and (3.19) supplies the boundary value for the time derivative of the normal component of the shiftˆ N i ∂ t β i , i.e. ( h zz ) − / ∂ t β z for a boundary aligned with the z -coordinate.The remaining Sommerfeld boundary conditions (3.21) - (3.23), which enforce the harmonic constraints, requiremodification depending upon the particular 3 + 1 formulation and gauge conditions. Only 6 components of Einstein’sequations are used in a 3 + 1 evolution, with the gauge conditions determining the evolution of the lapse and shift.As a first example, consider the ADM formulation in which the 6 Einstein equations h ρµ h σν R ρσ = 0 (4.9)are evolved. The evolution of the Hamiltonian and momentum constraints H := G µν n µ n ν and P µ := h µν n γ G νγ is governed by the contracted Bianchi identity ∇ ν G νµ = 0, which gives rise to the symmetric hyperbolic constraintpropagation system n γ ∂ γ H − ∂ j P j = B γ G νγ n ν (4.10) n γ ∂ γ P i − h ij ∂ j H = B iγ G νγ n ν , (4.11)where the coefficients B γ and B µγ arise from Christoffel symbols and do not enter the principle part. An analysisof this system shows that only one boundary condition is allowed provided β i ˆ N i ≤
0, i.e provided the boundary ismoving inward relative to the Cauchy hypersurfaces. The theory of symmetric hyperbolic systems then guaranteesthat all the constraints will be preserved if H + P i N i = G µν n µ K ν = 0 (4.12)is satisfied at the boundary. (Additional boundary conditions are necessary for constraint preservation if β i ˆ N i > G µν K µ K ν = 0 , (4.13)i.e. the Raychaudhuri equation (cf. [10]) K µ ∂ µ θ + 12 θ + σ ¯ σ = 0 , (4.14)where θ = Q µν ∇ µ K ν is the expansion of the outgoing null rays tangent to K µ . Thus, for the ADM system, constraintpreservation can be enforced by the Sommerfeld boundary condition (4.14) for θ , which supplies the boundary valuesfor the remaining component Q µν k µν of the extrinsic curvature. Unfortunately, although the subsidiary constraintsystem is symmetric hyperbolic, the ADM evolution system is only weakly hyperbolic and consequently leads tounstable evolution.0In terms of astrophysical applications, the most important 3 + 1 formulation is the BSSN system, which has beenused by the majority of groups [23, 24, 25, 26, 27] carrying out binary black hole and neutron star simulations. Thedevelopment of the BSSN formulation has proceeded through an interplay between educated guesses and feedbackfrom code performance. Only in hindsight has its success spurred mathematical analysis, which has shown thatcertain versions are strongly hyperbolic and thus have a well-posed Cauchy problem [32, 33, 34]. Although significantprogress has been made in establishing some of the necessary conditions for well-posedness and constraint preservationof the IBVP [35, 36, 37, 38], there is still no satisfactory mathematical theory on which to base numerical work. Incurrent numerical practice, the boundary conditions for BSSN evolution systems are applied in a naive, homogeneousSommerfeld form to each evolution variable (cf. [27]).The geometric nature of the Sommerfeld boundary conditions (3.17) - (3.20) and their role in a well-posed harmonicIBVP suggest that they might lead to improved performance over the present boundary treatment of the BSSN system.However, there are two complications. The first involves the sign of the normal component of the shift β i ˆ N i at theboundary, which also entered the above discussion of the ADM constraint system. A recent analysis [38] of the BSSNevolution system shows that the number of incoming fields at the boundary, and therefore the number of requiredboundary conditions, depends upon the sign of β i ˆ N i . A practical scheme for dealing with this would require theDirichlet boundary condition β i ˆ N i = 0 (or some similar Dirichlet condition to control the sign) rather than theSommerfeld condition (3.19) for the normal component of the shift.The other complication for the BSSN system involves constraint preservation. The BSSN evolution system enforcesthe 6 Einstein equations h ρµ h σν R ρσ − h µν H = 0 , (4.15)for which the constraint system implied by the Bianchi identity takes the form n γ ∂ γ H − ∂ j P j = B γ G νγ n ν n γ ∂ γ P i + 13 h ij ∂ j H = B iγ G νγ n ν . (4.16)This is no longer symmetric hyperbolic and would not lead to stable constraint preservation even for the Cauchyproblem. In order to remedy this, the BSSN evolution system modifies (4.15) by mixing in a set of auxiliary constraints,which combine with the constraint system (4.16) to form a larger symmetric hyperbolic constraint system. The freedomin the constraint-mixing parameters and gauge conditions complicates a general treatment. Here the discussion willbe limited to a particular choice [38] for which the linearization off Minkowski space leads to a symmetric hyperbolicsystem with a well-posed IBVP for the case of a Dirichlet boundary condition β i ˆ N i = 0 on the normal componentof the shift. Although the nonlinear evolution system is no longer symmetric hyperbolic, the boundary conditionsfor the linearized theory can be formally applied and lead to a symmetric hyperbolic constraint system. Constraintpreservation then follows for the parameter range ( b ≤ , b ≤
1) in the boundary conditions given in equation (97)of [38]. The particular choice b = 0, leads to the boundary condition [39] H − P i N i = G µν n µ ( n ν − N ν ) = Z , (4.17)where Z represents contributions from the auxiliary constraints, or, by using the evolution system (4.15), G µν L µ L ν = Z . (4.18)It is a bizarre feature of the 3+1 problem that the constraint preserving boundary conditions switch from the outgoingRaychaudhuri form (4.13) to the ingoing Raychaudhuri form (4.18) in going from the ADM to the BSSN system. TheRaychaudhuri equation for the outgoing null direction cannot be imposed in the allowed range of ( b , b ). Nevertheless,(4.18) can still be used to supply boundary values for the remaining Q µν k µν component of extrinsic curvature.It appears from the above discussion that the formal application of the Sommerfeld boundary conditions to theBSSN system must be restricted to (3.17), (3.18) and (3.20) which supply boundary values for 5 components of theextrinsic curvature. Boundary values for the remaining Q µν k µν component must be obtained in accord with constraintpreservation, e.g. from (4.18 ) or some variant depending upon the particular formulation. The normal componentof the shift requires a Dirichlet boundary condition that determines its sign, e.g. β i N i = 0. Appropriate boundaryconditions for the lapse and tangential components of the shift depend upon the specific gauge conditions (see [38]for an example). In the spirit of the BSSN formalism, computational experiments would be necessary to determinewhether (3.17), (3.18) and (3.20) lead to improved performance.1 V. SUMMARY
We have shown that a geometrically unique spacetime can be locally constructed from Sommerfeld data determinedby the fields (˜ t, ˜ q, ˜ q a , ˜ σ ab ) on a disembodied boundary ˜ T , along with the initial data prescribed on ˜ S . The boundarydata specify a foliation ˜ B t of ˜ T but involve no metric or other geometric properties of the boundary. After theembedding of ˜ T ∪ ˜ S as the boundary T ∪ S of a 4-manifold M , the induced fields ( t, ˆ q, ˆ q a , ˆ σ ab ) supply the necessaryboundary data for an isometry class of spacetime metrics which satisfy Einstein’s equations. Under diffeomorphismsof the disembodied boundary ˜ T and Cauchy hypersurface ˜ S , the mapped data determine diffeomorphic vacuumspacetimes.The gauge in which a particular metric g ab is constructed via a solution of the harmonic IBVP depends upon thechoice of evolution field t a , which also determines an associated background metric ˚g ab by Lie transport of the initialdata. Together g ab and ˚g ab supply the geometric interpretation of the boundary data in terms of the outgoing nullvector K a normal to the foliation B t of the boundary. The field q a , constructed from ˆ q and ˆ q a via (3.12), is thegeodesic curvature of K a , relative to its geodesic curvature computed with the background metric. The field σ ab (orequivalently σ ) computed from ˆ σ ab via (3.15) (or (3.16)) is the shear of K a , relative to its background value. Theresulting metric is harmonic with respect to the background metric, according to C ρ = 0 (see (3.26)). All possiblechoices of t a are related by diffeomorphism, so that all possible gauges are included.The Sommerfeld boundary conditions have direct application to harmonic evolution codes used in simulating binaryblack holes, where they would provide a numerical algorithm based upon a strongly well-posed IBVP. Furthermore,for harmonic evolution, homogeneous Sommerfeld data gives rise to asymptotically small back reflection of quadrupolewaves from an asymptotically large spherical outer boundary of an isolated system.The geometric nature of the 6 boundary conditions (3.17) - (3.20) suggests that they might also be applicableto codes based upon a 3 + 1 formulation, e.g. the BSSN system, as a way to reduce spurious boundary effectsgenerated by the naive Sommerfeld conditions now in practice. There are two caveats. First, strong well-posednessof the IBVP does not directly apply to any present 3 + 1 system. Second, the additional Sommerfeld boundaryconditions (3.21) - (3.23) only guarantee preservation of the Hamiltonian and momentum constraints for harmonic(or quasi-harmonic) formulations and would have to be replaced in accord with constraint preservation. In the caseof the particular BSSN formulation shown in [38] to posses a well-posed IBVP in the linearized approximation, thereare further complications. Instead of the Sommerfeld condition (3.19), a Dirichlet boundary condition is required tocontrol the sign of the normal component of the shift. The 5 other Sommerfeld conditions (3.17), (3.18) and (3.20)can be used to supply boundary values for 5 components of the extrinsic curvature. In addition, the boundary valuesof the remaining component of extrinsic curvature can be supplied by a constraint preserving boundary condition,but there is no apparent way to do this in a Sommerfeld form. These complications would, at the least, lead to morespurious reflection from the boundary than for a harmonic code. The results of this paper can perhaps guide furtherexperimentation and investigation towards a modification of the BSSN system that adds to its successful features. Acknowledgments
This research was supported by NSF grants PHY-0553597 and PHY-0854623 to the University of Pittsburgh. Muchof this work is based upon previous collaborations with H-O. Kreiss, O. Reula and O. Sarbach and I have benefitedfrom their continued input. The motivation and guidance for the new ideas developed here have come from discussionswith H. Friedrich. I am particular grateful to B. Schmidt for reading the manuscript and suggesting improvements. [1] H.O. Kreiss and J. Winicour, “Problems which are well-posed in a generalized sense with applications to the Einsteinequations”,
Class. Quantum Grav. , S405–S420 (2006).[2] “Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates”, H.-O. Kreiss, O.Reula, O. Sarbach, J. Winicour, Class. Quantum Grav. , 5973 (2007).[3] H-O. Kreiss, O. Reula, O. Sarbach and J. Winicour “Boundary conditions for coupled quasilinear wave equations withapplication to isolated systems”, Commun. Math. Phys. , 1099 (2009).[4] Y. Foures-Bruhat, “Theoreme d’existence pour certain systemes d’equations aux derive´es partielles nonlinaires”,
ActaMath. , 141 (1952).[5] J. Winicour, “Geometrization of metric boundary data for Einstein’s equations”, Gen. Rel. Grav. , 1909 (2009).[6] J. M. Stewart, “The Cauchy problem and the initial boundary value problem in numerical relativity”, Class. QuantumGrav. , 2865 (1998). [7] H. Friedrich and G. Nagy, “The initial boundary value problem for Einstein’s vacuum field equation”, Commun. Math.Phys. , 619 (1999).[8] H. Friedrich, “Initial boundary value problems for Einstein’s field equations and geometric uniqueness”,
Gen. Rel. Grav. , 1947 (2009).[9] S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space Time , (Cambridge University Press, 1973).[10] R. M. Wald,
General Relativity , p. 23 (University of Chicago Press, 1984).[11] M. Ruiz, O. Rinne and O. Sarbach, “Outer boundary conditions for Einstein’s field equations in harmonic coordinates”,
Class. Quantum Grav , , 6349 (2007).[12] H. Friedrich, “Hyperbolic reductions for Einstein’s equations”, Class. Quant. Grav. , 1451 (1996).[13] F. Pretorius, “Numerical relativity using a generalized harmonic decomposition”, Class.Quant.Grav. , 425 (2005).[14] F. Pretorius,“Evolution of Binary Black-Hole Spacetimes”, Phys. Rev. Lett. , , 121101(2005).[15] F. Pretorius, “Simulation of binary black hole spacetimes with a harmonic evolution scheme”, Class. Quantum Grav. , ,S529 (2006).[16] L. Lindblom, M.A. Scheel, L.E. Kidder, R. Owen and O. Rinne, “A new generalized harmonic evolution system”, Class.Quantum Grav. , , S447 (2006).[17] O. Rinne, L. Lindblom and M.A. Scheel, “Testing outer boundary treatments for the Einstein equations”, Class. QuantumGrav. , , 4053 (2007).[18] B. Szil´agyi, D. Pollney, L. Rezzolla, J. Thornburg and J. Winicour, “An explicit harmonic code for black-hole evolutionusing excision”, Class. Quantum Grav. , , S275 2007.[19] M. C. Babiuc, B. Szil´agyi and J. Winicour, “Harmonic initial-boundary evolution in general relativity”, Phys. Rev. D , ,064017 (2006).[20] M. C. Babiuc, H-O. Kreiss and J. Winicour, “Constraint-preserving Sommerfeld conditions for the harmonic Einsteinequations”, Phys. Rev. D , , 044002 (2007).[21] M. Shibata and T. Nakamura, “Evolution of three-dimensional gravitational waves: Harmonic slicing case”, Phys. Rev. D , 5428 (1995).[22] T. Baumgarte and S. L. Shapiro, “On the numerical integration of Einstein’s field equations”, Phys. Rev. D , 024007(1999).[23] “M. Campanelli, C. O. Lousto, P. Marronetti and Y. Zlochower”, ”Accurate evolutions of orbiting black-hole binarieswithout excision”, Phys. Rev. Lett. , , 111101 (2006).[24] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz and J. van Meter, “Gravitational-wave extraction from an inspiralingconfiguration of merging black holes”, Phys. Rev. Lett. , , 111102 (2006).[25] P. Diener, F. Herrmann, D. Pollney, E. Schnetter, E. Seidel, R. Takahashi, J. Thornburg and J. Ventrella, “Accurateevolution of orbiting binary black holes”, Phys. Rev. Lett. , , 121101 (2006).[26] J. A. Gonzalez, U. Sperhake, B. Bruegmann, M. Hannam and S. Husa, “Total recoil: the maximum kick from nonspinningblack-hole binary inspiral”, Phys. Rev. Lett. , , 091101 (2007).[27] Z. B. Etienne, J. A. Faber, Y. T. Liu, S. L. Shapiro, K. Taniguchi, T. W. Baumgarte, “Fully general relativistic simulationsof black hole-neutron star mergers”, Phys. Rev. D , 084002 (2008).[28] C. Bona, T. Ledvinka, C. Palenzuela and M. ˘Z´a˘cek, “General-covariant evolution formalism for numerical relativity”, Phys.Rev. D , 104005 (2003).[29] C. Bona, T. Ledvinka, C. Palenzuela and M. ˘Z´a˘cek, “A symmetry-breaking mechanism for the Z4 general-covariantevolution system”, Phys.Rev. D , 064036 (2004).[30] R. Arnowitt, S. Deser and C. W. Misner, Gravitation: an introduction to current research , ed. L. Witten (Wiley,New York, 1962).[31] L. E. Kidder, M. A Scheel and S. A. Teuklsky,
Phys. Rev. D , 064017 (2001).[32] G. Nagy, O. E. Ortiz and O. A. Reula, “Strongly hyperbolic second order Einstein’s evolution equations”, Phys. Rev. D , 044012 (2004).[33] O. Sarbach, G. Calabrese, J. Pullin and M. Tiglio, “Hyperbolicity of the BSSN system of Einstein evolution equations”, Phys. Rev. D , 064002 (2002).[34] C. Gundlach and J. M. Martin-Garcia, “Well-posedness of formulations of the Einstein equations with dynamical lapseand shift conditions”, Phys. Rev. D , 024016 (2006).[35] C. Gundlach and J. M. Martin-Garcia, “Symmetric hyperbolic and consistent boundary conditions for second order Einsteinequations”, Phys. Rev. D , 044032 (2004).[36] H. Beyer and O. Sarbach, “On the well-posedness of the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein’sfield equations”, Phys. Rev. D , 104004 (2004).[37] S. Frittelli and R. O. G´omez, “Einstein boundary conditions for rhe Einstein equations in the conformal-traceless decom-position”, Phys. Rev. D70