6.2 Boundary conditions for BSSN

Here we discuss boundary conditions for the BSSN system (4.52View Equation4.59View Equation), which is used extensively in numerical calculations of spacetimes describing dynamic black holes and neutron stars. Unfortunately, to date, this system lacks an initial-boundary value formulation for which well-posedness in the full nonlinear case has been proven. Without doubt the reason for this relies on the structure of the evolution equations, which are mixed first/second order in space, and whose principal part is much more complicated than the harmonic case, where one deals with a system of wave equations.

A first step towards formulating a well-posed IBVP for the BSSN system was performed in [52Jump To The Next Citation Point], where the evolution equations (4.52View Equation, 4.53View Equation, 4.56View Equation4.59View Equation) with a fixed shift and the relation f = μ ≡ (4m − 1)∕3 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 [52Jump To The Next Citation Point] 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 [220Jump To The Next Citation Point], 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 (5.27View Equation).

Radiative-type constraint-preserving boundary conditions for the BSSN system (4.52View Equation4.59View Equation) with dynamical lapse and shift were formulated in [315Jump To The Next Citation Point] and shown to yield a well-posed IBVP in the linearized case. The assumptions on the parameters in this formulation are m = 1, f > 0, κ = 4GH ∕3 > 0, f ⁄= κ, which guarantee that the BSSN system is strongly hyperbolic, and as long as e4ϕ ⁄= 2α, they allow for the gauge conditions (4.62View Equation, 4.63View Equation) used in recent numerical calculations, where f = 2∕α and κ = e4ϕ∕α2; 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 m = 1 the characteristic speeds with respect to the unit outward normal si to the boundary are

s s s ∘ -- s √ ---- s √ -- β , β ± α, β ± α f, β ± α GH, β ± α κ, (6.9 )
where βs = βis i 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 |βs| is small enough such that -- √---- |βs∕α | < min {1,√ f , GH, √ κ}, which is satisfied asymptotically if βs → 0 and α → 1, 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 s β > 0 or s β ≤ 0 at the boundary surface. If the condition s β → 0 is imposed asymptotically, the most natural choice is to set the normal component of the shift to zero at the boundary, βs = 0 at 𝒯. The analysis in [52] 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:
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