- Geometric existence. Let be any smooth solution of Einstein’s vacuum field equations on the manifold corresponding to initial data on and boundary data on , where and represent, respectively, the first and second fundamental forms of the initial surface as embedded in . Is it possible to reproduce this solution with any of the well-posed IBVP mentioned so far, at least on a submanifold with ? That is, does there exist initial data and boundary data for this IBVP and a diffeomorphism , which leaves and invariant, such that the metric constructed from this IBVP is equal to on ?
- Geometric uniqueness. Is the solution uniquely determined by the data ? Given a well-posed IBVP for which geometric existence holds, the question about geometric uniqueness can be reduced to the analysis of this particular IBVP in the following way: let and be two solutions of the IBVP on the manifold with corresponding data and . Suppose the two solutions induce the same data on and on . Does there exist a diffeomorphism , which leaves and invariant, such that the metrics and corresponding to and are related to each other by on ?

These geometric existence and uniqueness problems have been solved in the context of the Cauchy problem without boundaries; see [127] and Section 4.1.3. However, when boundaries are present, several new difficulties appear as pointed out in [186]; see also [187, 184]:

- It is a priori not clear what the boundary data should represent geometrically. Unlike the case of the initial surface, where the data represents the first and second fundamental forms of as a spatial surface embedded in the constructed spacetime , it is less clear what the geometric meaning of should be since it is restricted by the characteristic structure of the evolution equations, as discussed in Section 5.
- The boundary data in the boundary conditions (6.2, 6.3, 6.4, 6.5) for the harmonic formulation and the boundary data in the boundary condition (6.14) for the BSSN formulation ultimately depend on the specific choice of a future-directed time-like vector field at the boundary surface . Together with the unit outward normal to , this vector defines the preferred null directions and , which are used to construct the boundary-adapted null tetrad in the harmonic case and the projection operators and in the BSSN one. Although it is tempting to define as the unit, future-directed time-like vector tangent to , which is orthogonal to the cross sections , this definition would depend on the particular foliation the formulation is based on, and so the resulting vector would be gauge-dependent. A similar issue arises in the tetrad formulation of [187].
- When addressing the geometric uniqueness issue, an interesting question is whether or not it is possible to determine from the data sets and alone if they are equivalent in the sense that their solutions and induce the same geometric data . Therefore, the question is whether or not one can identify equivalent data sets by considering only transformations on the initial and boundary surfaces and , without knowing the solutions and .

Although a complete answer to these questions remains a difficult task, there has been some recent progress towards their understanding. In [186] a method was proposed to geometrically single out a preferred time direction at the boundary surface . This is done by considering the trace-free part of the second fundamental form, and proving that under certain conditions, which are stable under perturbations, the corresponding linear map on the tangent space possesses a unique time-like eigenvector. Together with the unit outward normal vector , the vector field defines a distinguished adapted null tetrad at the boundary, from which geometrically meaningful boundary data could be defined. For instance, the complex Weyl scalar can then be defined as the contraction of the Weyl tensor associated to the metric along the null vectors and , and the definition is unique up to the usual spin rotational freedom , and therefore, the Weyl scalar is a good candidate for forming part of the boundary data .

In [355] it was suggested that the unique specification of a vector field may not be a fundamental problem, but rather the manifestation of the inability to specify a non-incoming radiation condition correctly. In the linearized case, for example, setting the Weyl scalar to zero computed from the boundary-adapted tetrad is transparent to gravitational plane waves traveling along the specific null direction , see Example 32, but it induces spurious reflections for outgoing plane waves traveling in other null direction. Therefore, a genuine non-incoming radiation condition should be, in fact, independent of any specific null or time-like direction at the boundary, and can only depend on the normal vector . This is indeed the case for much simpler systems like the scalar wave equation on a Minkowski background [153], where perfectly absorbing boundary conditions are formulated as a nonlocal condition, which is independent of a preferred time direction at the boundary.

Aside from controlling the incoming gravitational degrees of freedom, the boundary data should also comprise information related to the geometric evolution of the boundary surface. In [187] this was achieved by specifying the mean curvature of as part of the boundary data. In the harmonic formulation described in Section 6.1 this information is presumably contained in the functions , and , but their geometric interpretation is not clear.

In order to illustrate some of the issues related to the geometric existence and uniqueness problem in a simpler context, in what follows we analyze the IBVP for linearized gravitational waves propagating on a Minkowski background. Before analyzing this case, however, we make two remarks. First, it should be noted [186] that the geometric uniqueness problem, especially an understanding of point (iii), also has practical interest, since in long term evolutions it is possible that the gauge threatens to break down at some point, requiring a redefinition. The second remark concerns the formulation of the Einstein IBVP in generalized harmonic coordinates, described in Sections 4.1 and 6.1, where general covariance was maintained by introducing a background metric on the manifold . IBVPs based on this approach have been formulated in [369] and [264] and further developed in [434] and [433]. However, one has to emphasize that this approach does not automatically solve the geometric existence and uniqueness problems described here: although it is true that the IBVP is invariant with respect to any diffeomorphism , which acts on the dynamical and the background metric at the same time, the question of the dependency of the solution on the background metric remains.

Here we analyze some of the geometric existence and uniqueness issues of the IBVP for Einstein’s field equations in the much simpler setting of linearized gravity on Minkowski space, where the vacuum field equations reduce to

where denotes the first variation of the metric, its trace with respect to the Minkowski background metric , and is the covariant derivative with respect to . An infinitesimal coordinate transformation parametrized by a vector field induces the transformation where .Let us consider the linearized Cauchy problem without boundaries first, where initial data is specified at the initial surface . The initial data is specified geometrically by the first and second fundamental forms of , which, in the linearized case, are represented by a pair of covariant symmetric tensor fields on . We assume to be smooth and to satisfy the linearized Hamiltonian and momentum constraints

where . A solution of Eq. (6.15) with the induced data corresponding to up to a gauge transformation (6.16) satisfies where and are smooth and represent the initial gauge freedom. Then, one has:Theorem 9. The initial-value problem (6.15, 6.18) possesses a smooth solution , which is unique up to an infinitesimal coordinate transformation generated by a vector field .

Proof. We first show the existence of a solution in the linearized harmonic gauge , for which Eq. (6.15) reduces to the system of wave equations . The initial data, , for this system is chosen such that , and , , where satisfy the constraint equations (6.17) and where the initial data for and is chosen smooth but otherwise arbitrary. This choice implies the satisfaction of Eq. (6.18) with and and the initial conditions and on the constraint fields . Therefore, solving the wave equation with such data, we obtain a solution of the linearized Einstein equations (6.15) in the harmonic gauge with initial data satisfying (6.18) with and . This shows geometric existence for the linearized harmonic formulation.

As for uniqueness, suppose we had two smooth solutions of Eqs. (6.15, 6.18). Then, since the equations are linear, the difference between these two solutions also satisfies Eqs. (6.15, 6.18) with trivial data , . We show that can be transformed away by means of an infinitesimal gauge transformation (6.16). For this, define where is required to satisfy the inhomogeneous wave equation

with initial data for defined by , , , . Then, by construction, satisfies the harmonic gauge, and it can be verified that . Therefore, is a solution of the wave equation with trivial initial data, and it follows that and that is a pure gauge mode. □It follows from the existence part of the proof that the quantities and , corresponding to linearized lapse and shift, parametrize pure gauge modes in the linearized harmonic formulation.

Next, we turn to the IBVP on the manifold . Let us first look at the boundary conditions (6.2 – 6.5), which, in the linearized case, reduce to

There is no problem in repeating the geometric existence part of the proof on imposing these boundary condition, and using the IBVP described in Section 6.1. However, there is a problem when trying to prove the uniqueness part. This is because a gauge transformation (6.16) induces the following transformations on the boundary data,Theorem 10. [355] The IBVP (6.15, 6.18, 6.21) possesses a smooth solution , which is unique up to an infinitesimal coordinate transformation generated by a vector field .

In conclusion, we can say that, in the simple case of linear gravitational waves propagating on a Minkowksi background, we have resolved the issues (i–iii). Correct boundary data is given to the linearized Weyl scalar computed from the boundary-adapted tetrad. To linear order, is invariant with respect to coordinate transformations, and the time-like vector field appearing in its definition can be defined geometrically by taking the future-directed unit normal to the initial surface and parallel transport it along the geodesics orthogonal to .

Whether or not this result can be generalized to the full nonlinear case is not immediately clear. In our linearized analysis we have imposed no restrictions on the normal component of the vector field generating the infinitesimal coordinate transformation. However, such a restriction is necessary in order to keep the boundary surface fixed under a diffeomorphism. Unfortunately, it does not seem possible to restrict in a natural way with the boundary conditions constructed so far.

Living Rev. Relativity 15, (2012), 9
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