6.2 Boundary conditions for BSSN
Here we discuss boundary conditions for the BSSN system (4.52 – 4.59), 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 [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
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 [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:
- 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].