In order to keep general covariance, we follow [232] and choose a fixed smooth background
metric with corresponding Levi-Civita connection
, Christoffel symbols
, and
curvature tensor
. Then, the generalized harmonic gauge condition can be rewritten
as11
Einstein’s field equations in the gauge are equivalent to the wave system
For any given smooth stress-energy tensor , the equations (4.5
) constitute a quasilinear system of ten
coupled wave equations for the ten coefficients of the difference metric
(or equivalently, for the ten
components of the dynamical metric
) and, therefore, we can apply the results of Section 3 to
formulate a (local in time) well-posed Cauchy problem for the wave system (4.5
) with initial conditions
An alternative way of establishing the hyperbolicity of the system (4.5) is to cast it into first-order
symmetric hyperbolic form [164
, 18
, 286
]. There are several ways of constructing such a system; the
simplest one is obtained [164
] by introducing the first partial derivatives of
as new variables,
However, as indicated above, the first-order symmetric hyperbolic reduction (4.9, 4.10
, 4.11
)
is not unique. A different reduction is based on the variables
, where
is the derivative of
in the direction of the future-directed unit normal
to the
time-slices
, and
. This yields a first-order system, which is symmetric
hyperbolic as long as the
slices are space-like, independent of whether or not
is
time-like [18
, 286
].
The hyperbolicity results described above guarantee that unique solutions of the nonlinear wave
system (4.5) exist, at least for short times, and that they depend continuously on the initial data
,
. However, in order to obtain a solution of Einstein’s field equations one has to ensure that the
harmonic constraint (4.3
) is identically satisfied.
The system (4.5) is equivalent to the modified Einstein equations
However, in numerical calculations, one cannot assume that the initial constraints (4.15) are satisfied
exactly, due to truncation and roundoff errors. The propagation of these errors is described by the
constraint propagation system (4.14
), and hyperbolicity guarantees that for each fixed time
of
existence, these errors converge to zero if the initial constraint violation converges to zero, which is
usually the case when resolution is increased. On the other hand, due to limited computer
resources, one cannot reach the limit of infinite resolution, and from a practical point of view one
does not want the constraint errors to grow rapidly in time for fixed resolution. Therefore, one
would like to design an evolution scheme in which the constraint violations are damped in time,
such that the constraint hypersurface is an attractor set in phase space. A general method for
damping constraints violations in the context of first-order symmetric hyperbolic formulations of
Einstein’s field equations was given in [74
]. This method was then adapted to the harmonic
formulation in [224
]. The procedure proposed in [224
] consists in adding lower-order friction
terms in Eq. (4.13
), which damp constraint violations. Explicitly, the modified system reads
With this modification the constraint propagation system reads
A mode analysis for linear vacuum perturbations of the Minkowski metric reveals [224] that forFor a discussion on possible effects due to nonlinearities in the constraint propagation system; see [185].
The results described so far guarantee the local-in-time unique existence of solutions to Einstein’s equations
in harmonic coordinates, given a sufficiently-smooth initial data set . However, since general
relativity is a diffeomorphism invariant theory, some questions remain. The first issue is whether or not the
harmonic gauge is sufficiently general such that any solution of the field equations can be obtained by this
method, at least for short enough time. The answer is affirmative [169
, 164
]. Namely, let
,
, be a smooth spacetime satisfying Einstein’s field equations such that the initial surface
is spacelike with respect to
. Then, we can find a diffeomorphism
in
a neighborhood of the initial surface, which leaves it invariant and casts the metric into the
harmonic gauge. For this, one solves the harmonic wave map equation (4.2
) with initial data
The next issue is the question of geometric uniqueness. Let and
be two solutions of Einstein’s
equations with the same initial data on
, i.e.,
,
.
Are these solutions related, at least for small time, by a diffeomorphism? Again, the answer
is affirmative [169, 164] because one can transform both solutions to harmonic coordinates
using the above diffeomorphism
without changing their initial data. It then follows by
the uniqueness property of the nonlinear wave system (4.5
) that the transformed solutions
must be identical, at least on some sufficiently- small time interval. Note that this geometric
uniqueness property also implies that the solutions are, at least locally, independent of the
background metric. For further results on geometric uniqueness involving only the first and second
fundamental forms of the initial surface; see [127
], where it is shown that every such initial-data set
satisfying the Hamiltonian and momentum constraints possesses a unique maximal Cauchy
development.
Finally, we mention that results about the nonlinear stability of Minkowski spacetime with respect to vacuum and vacuum-scalar perturbations have been established based on the harmonic system [283, 284], offering an alternative proof to the one of [129].
http://www.livingreviews.org/lrr-2012-9 |
Living Rev. Relativity 15, (2012), 9
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