6.1 The harmonic formulation
Here, we discuss the IBVP formulated in [264
] for the Einstein vacuum equations in generalized
harmonic coordinates. The starting point is a manifold of the form
, with
a
three-dimensional compact manifold with
-boundary
, and a given, fixed smooth background
metric
with corresponding Levi-Civita connection
, as in Section 4.1. We assume that the time
slices
are space-like and that the boundary surface
is time-like with
respect to
.
In order to formulate the boundary conditions, we first construct a null tetrad
,
which is adapted to the boundary. This null tetrad is based on the choice of a future-directed time-like
vector field
tangent to
, which is normalized such that
. One possible choice is to
tie
to the foliation
, and then define it in the direction orthogonal to the cross sections
of the boundary surface. A more geometric choice has been proposed in [186
], where
instead
is chosen as a distinguished future-directed time-like eigenvector of the second fundamental
form of
, as embedded in
. Next, we denote by
the unit outward normal to
with respect to the metric
and complete
and
to an orthonormal basis
of
at each point
. Then, we define the complex null tetrad by
where
. Notice that the construction of these vectors is implicit, since it depends on the
dynamical metric
, which is yet unknown. However, the dependency is algebraic, and does not involve
any derivatives of
. We also note that the complex null vector
is not unique since
it can be rotated by an angle
,
. Finally, we define a radial function
on
as the areal radius of the cross sections
with respect to the background
metric.
Then, the boundary conditions, which were proposed in [264
] for the harmonic system (4.5), are:
where
,
,
, etc., and where
and
are real-valued given smooth functions on
and
and
are complex-valued given smooth
functions on
. Since
is complex, these constitute ten real boundary conditions for the metric
coefficients
. The content of the boundary conditions (6.2, 6.3, 6.4, 6.5) can be clarified by
considering linearized gravitational waves on a Minkowski background with a spherical boundary.
The analysis in [264
] shows that in this context the four real conditions (6.2),(6.3, 6.4) are
related to the gauge freedom; and the two conditions (6.5) control the gravitational radiation.
The remaining conditions (6.6, 6.7, 6.8) enforce the constraint
on the boundary,
see Eq. (4.6), and so together with the constraint propagation system (4.14) and the initial
constraints (4.15) they guarantee that the constraints are correctly propagated. Based on these
observations, it is expected that these boundary conditions yield small spurious reflections
in the case of a nearly-spherical boundary in the wave zone of an asymptotically-flat curved
spacetime.
6.1.1 Well-posedness of the IBVP
The IBVP consisting of the harmonic Einstein equations (4.5), initial data (4.7) and the boundary
conditions (6.2 – 6.8) can be shown to be well posed as an application of Theorem 8. For this, we first
notice that the evolution equations (4.5) have the required form of Eq. (5.113), where
is
the vector bundle of symmetric, covariant tensor fields
on
. Next, the boundary
conditions can be written in the form of Eq. (5.115) with
. In order to compute the matrix
coefficients
, it is convenient to decompose
in terms of the basis vectors
with
,
,
,
,
,
,
. With respect to this basis, the only nonzero matrix coefficients are
which has the required upper triangular form with zeros in the diagonal. Therefore, the hypothesis of
Theorem 8 are verified and one obtains a well-posed IBVP for Einstein’s equations in harmonic
coordinates.
This result also applies the the modified system (4.16), since the constraint damping terms, which are
added, do not modify the principal part of the main evolution system nor the one of the constraint
propagation system.