A first step towards formulating a well-posed IBVP for the BSSN system was performed in [52], where the evolution equations (4.52, 4.53, 4.56 – 4.59) with a fixed shift and the relation were reduced to a first-order symmetric hyperbolic system. Then, a set of six boundary conditions consistent with this system could be formulated based on the theory of maximal dissipative boundary conditions. Although this gives rise to a well-posed IBVP, the boundary conditions specified in [52] are not compatible with the constraints, and therefore, one does not necessarily obtain a solution to the full set of Einstein’s equations beyond the domain of dependence of the initial data surface. In a second step, constraint-preserving boundary conditions for BSSN with a fixed shift were formulated in [220], and cast into maximal dissipative form for the linearized system (see also [15]). However, even at the linearized level, these boundary conditions are too restrictive because they constitute a combination of Dirichlet and Neumann boundary conditions on the metric components, and in this sense they are totally reflecting instead of absorbing. More general constraint-preserving boundary conditions were also considered in [220] and, based on the Laplace method, they were shown to satisfy the Lopatinsky condition (5.27).

Radiative-type constraint-preserving boundary conditions for the BSSN system (4.52 – 4.59) with dynamical lapse and shift were formulated in [315] and shown to yield a well-posed IBVP in the linearized case. The assumptions on the parameters in this formulation are , , , , which guarantee that the BSSN system is strongly hyperbolic, and as long as , they allow for the gauge conditions (4.62, 4.63) used in recent numerical calculations, where and ; see Section 4.3.1. In the following, we describe this IBVP in more detail. First, we notice that the analysis in Section 4.3.1 reveals that for the standard choice the characteristic speeds with respect to the unit outward normal to the boundary are

where is the normal component of the shift. According to the theory described in Section 5 it is the sign of these speeds, which determines the number of incoming fields and boundary conditions that must be specified. Namely, the number of boundary conditions is equal to the number of characteristic fields with positive speed. Assuming is small enough such that , which is satisfied asymptotically if and , it is the sign of the normal component of the shift, which determines the number of boundary conditions. Therefore, in order to keep the number of boundary conditions fixed throughout evolution- Boundary conditions on the gauge variables

There are four conditions that must be imposed on the gauge functions, namely the lapse and shift. These conditions are motivated by the linearized analysis, where the gauge propagation system, consisting of the evolution equations for lapse and shift obtained from the BSSN equations (4.52 – 4.55, 4.59), decouples from the remaining evolution equations. Surprisingly, this gauge propagation system can be cast into symmetric hyperbolic form [315], for which maximal dissipative boundary conditions can be specified, as described in Section 5.2. It is remarkable that the gauge propagation system has such a nice mathematical structure, since the equations (4.52, 4.54, 4.55) have been specified by hand and mostly motivated by numerical experiments instead of mathematical analysis.In terms of the operator projecting onto vectors tangential to the boundary, the four conditions on the gauge variables can be written as

Eq. (6.10) is a Neumann boundary condition on the lapse, and Eq. (6.11) sets the normal component of the shift to zero, as explained above. Geometrically, this implies that the boundary surface is orthogonal to the time slices . The other two conditions in Eq. (6.12) are Sommerfeld-like boundary conditions involving the tangential components of the shift and the tangential derivatives of the lapse; they arise from the analysis of the characteristic structure of the gauge propagation system. An alternative to Eq. (6.12) also described in [315] is to set the tangential components of the shift to zero, which, together with Eq. (6.11) is equivalent to setting at the boundary. This alternative may be better suited for IBVP with non-smooth boundaries, such as cubes, where additional compatibility conditions must be enforced at the edges. - Constraint-preserving boundary conditions

Next, there are three conditions requiring that the momentum constraint be satisfied at the boundary. In terms of the BSSN variables this implies As shown in [315], Eqs. (6.13) yields homogeneous maximal dissipative boundary conditions for a symmetric hyperbolic first-order reduction of the constraint propagation system (4.74, 4.75, 4.76). Since this system is also linear and its boundary matrix has constant rank if , it follows from Theorem 7 that the propagation of constraint violations is governed by a well-posed IBVP. This implies, in particular, that solutions whose initial data satisfy the constraints exactly automatically satisfy the constraints on each time slice . Furthermore, small initial constraint violations, which are usually present in numerical applications yield solutions for which the growth of the constraint violations can be bounded in terms of the initial violations. - Radiation controlling boundary conditions

Finally, the last two boundary conditions are intended to control the incoming gravitational radiation, at least approximately, and specify the complex Weyl scalar , cf. Example 32. In order to describe this boundary condition we first define the quantities and , which determine the electric and magnetic parts of the Weyl tensor through and , respectively. Here, denotes the volume form with respect to the three metric . In terms of the operator projecting onto symmetric trace-less tangential tensors to the boundary, the boundary condition reads with a given smooth tensor field on the boundary surface . The relation between and is the following: if denotes the future-directed unit normal to the time slices, we may construct an adapted Newman-Penrose null tetrad at the boundary by defining , , and by choosing to be a complex null vector orthogonal to and , normalized such that . Then, we have . For typical applications involving the modeling of isolated systems one may set to zero. However, this in general is not compatible with the initial data (see the discussion in Section 10.3), an alternative is then to freeze the value of to the one computed from the initial data.The boundary condition (6.14) can be partially motivated by considering an isolated system, which, globally, is described by an asymptotically-flat spacetime. Therefore, if the outer boundary is placed far enough away from the strong field region, one may linearize the field equations on a Minkowski background to a first approximation. In this case, one is in the same situation as in Example 32, where the Weyl scalar is an outgoing characteristic field when constructed from the adapted null tetrad. Furthermore, one can also appeal to the peeling behavior of the Weyl tensor [328], in which is the fastest decaying component along an outgoing null geodesics and describes the incoming radiation at past null infinity. While can only be defined in an unambiguous way at null infinity, where a preferred null tetrad exists, the boundary condition (6.14) has been successfully numerically implemented and tested for truncated domains with artificial boundaries in the context of the harmonic formulation; see, for example, [366]. Estimates on the amount of spurious reflection introduced by this condition have also been derived in [88, 89]; see also [135].

Living Rev. Relativity 15, (2012), 9
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