These geometric existence and uniqueness problems have been solved in the context of the Cauchy
problem without boundaries; see [127] and Section 4.1.3. However, when boundaries are present, several
new difficulties appear as pointed out in [186]; see also [187
, 184]:
Although a complete answer to these questions remains a difficult task, there has been some recent
progress towards their understanding. In [186] a method was proposed to geometrically single out a
preferred time direction
at the boundary surface
. This is done by considering the trace-free part of
the second fundamental form, and proving that under certain conditions, which are stable under
perturbations, the corresponding linear map on the tangent space possesses a unique time-like eigenvector.
Together with the unit outward normal vector
, the vector field
defines a distinguished adapted
null tetrad at the boundary, from which geometrically meaningful boundary data could be
defined. For instance, the complex Weyl scalar
can then be defined as the contraction
of the Weyl tensor
associated to the metric
along the null vectors
and
, and the definition is unique up to the usual spin rotational freedom
, and
therefore, the Weyl scalar
is a good candidate for forming part of the boundary data
.
In [355] it was suggested that the unique specification of a vector field
may not be a fundamental
problem, but rather the manifestation of the inability to specify a non-incoming radiation condition
correctly. In the linearized case, for example, setting the Weyl scalar
to zero computed from the
boundary-adapted tetrad is transparent to gravitational plane waves traveling along the specific null
direction
, see Example 32, but it induces spurious reflections for outgoing plane waves
traveling in other null direction. Therefore, a genuine non-incoming radiation condition should be, in fact,
independent of any specific null or time-like direction at the boundary, and can only depend on
the normal vector
. This is indeed the case for much simpler systems like the scalar wave
equation on a Minkowski background [153], where perfectly absorbing boundary conditions are
formulated as a nonlocal condition, which is independent of a preferred time direction at the
boundary.
Aside from controlling the incoming gravitational degrees of freedom, the boundary data should also
comprise information related to the geometric evolution of the boundary surface. In [187
] this was achieved
by specifying the mean curvature of
as part of the boundary data. In the harmonic formulation
described in Section 6.1 this information is presumably contained in the functions
,
and
, but
their geometric interpretation is not clear.
In order to illustrate some of the issues related to the geometric existence and uniqueness problem in a
simpler context, in what follows we analyze the IBVP for linearized gravitational waves propagating on a
Minkowski background. Before analyzing this case, however, we make two remarks. First, it should be
noted [186] that the geometric uniqueness problem, especially an understanding of point (iii), also has
practical interest, since in long term evolutions it is possible that the gauge threatens to break down at
some point, requiring a redefinition. The second remark concerns the formulation of the Einstein IBVP in
generalized harmonic coordinates, described in Sections 4.1 and 6.1, where general covariance was
maintained by introducing a background metric on the manifold
. IBVPs based on this approach
have been formulated in [369
] and [264
] and further developed in [434] and [433]. However, one
has to emphasize that this approach does not automatically solve the geometric existence and
uniqueness problems described here: although it is true that the IBVP is invariant with respect to
any diffeomorphism
, which acts on the dynamical and the background metric
at the same time, the question of the dependency of the solution on the background metric
remains.
Here we analyze some of the geometric existence and uniqueness issues of the IBVP for Einstein’s field equations in the much simpler setting of linearized gravity on Minkowski space, where the vacuum field equations reduce to
where Let us consider the linearized Cauchy problem without boundaries first, where initial data is specified at
the initial surface . The initial data is specified geometrically by the first and second
fundamental forms of
, which, in the linearized case, are represented by a pair
of covariant
symmetric tensor fields on
. We assume
to be smooth and to satisfy the linearized
Hamiltonian and momentum constraints
Theorem 9. The initial-value problem (6.15, 6.18
) possesses a smooth solution
, which is
unique up to an infinitesimal coordinate transformation
generated by a vector
field
.
Proof. We first show the existence
of a solution in the linearized harmonic gauge , for which Eq. (6.15
)
reduces to the system of wave equations
. The initial data,
,
for this system is chosen such that
,
and
,
, where
satisfy the
constraint equations (6.17
) and where the initial data for
and
is chosen smooth but
otherwise arbitrary. This choice implies the satisfaction of Eq. (6.18
) with
and
and the initial conditions
and
on the constraint fields
. Therefore,
solving the wave equation
with such data, we obtain a solution of the linearized
Einstein equations (6.15
) in the harmonic gauge with initial data satisfying (6.18
) with
and
. This shows geometric existence for the linearized harmonic formulation.
As for uniqueness, suppose we had two smooth solutions of Eqs. (6.15, 6.18
). Then, since the equations
are linear, the difference
between these two solutions also satisfies Eqs. (6.15
, 6.18
) with trivial data
,
. We show that
can be transformed away by means of an infinitesimal gauge
transformation (6.16
). For this, define
where
is required to satisfy the
inhomogeneous wave equation
It follows from the existence part of the proof that the quantities and
,
corresponding to linearized lapse and shift, parametrize pure gauge modes in the linearized harmonic
formulation.
Next, we turn to the IBVP on the manifold . Let us first look at the boundary
conditions (6.2
– 6.5
), which, in the linearized case, reduce to
Theorem 10. [355] The IBVP (6.15, 6.18
, 6.21
) possesses a smooth solution
, which is unique
up to an infinitesimal coordinate transformation
generated by a vector field
.
In conclusion, we can say that, in the simple case of linear gravitational waves propagating on a
Minkowksi background, we have resolved the issues (i–iii). Correct boundary data is given to
the linearized Weyl scalar computed from the boundary-adapted tetrad. To linear order,
is invariant with respect to coordinate transformations, and the time-like vector field
appearing in its definition can be defined geometrically by taking the future-directed unit
normal to the initial surface
and parallel transport it along the geodesics orthogonal to
.
Whether or not this result can be generalized to the full nonlinear case is not immediately clear. In our
linearized analysis we have imposed no restrictions on the normal component of the vector field
generating the infinitesimal coordinate transformation. However, such a restriction is necessary in
order to keep the boundary surface fixed under a diffeomorphism. Unfortunately, it does not
seem possible to restrict
in a natural way with the boundary conditions constructed so
far.
http://www.livingreviews.org/lrr-2012-9 |
Living Rev. Relativity 15, (2012), 9
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