A first step towards formulating a well-posed IBVP for the BSSN system was performed in , 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  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 , and cast into maximal dissipative form for the linearized system (see also ). 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  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  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 arepositive 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 evolution26 one has to ensure that either or at the boundary surface. If the condition is imposed asymptotically, the most natural choice is to set the normal component of the shift to zero at the boundary, at . The analysis in  then reveals that there are precisely nine incoming characteristic fields at the boundary, and thus, nine conditions have to be imposed at the boundary. These nine boundary conditions are as follows:
In terms of the operator projecting onto vectors tangential to the boundary, the four conditions on the gauge variables can be written as6.10) 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  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.
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 , 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, . Estimates on the amount of spurious reflection introduced by this condition have also been derived in [88, 89]; see also .
Living Rev. Relativity 15, (2012), 9
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