A convenient way to describe this scheme is by introducing a
background metric,
, thus the gauge is not a coordinate condition, but rather a
condition which links the physical metric with the background
one. In this approach, see [5
], the basic variable is a densitized symmetric tensor,
, where
g
is the metric determinant with respect to the background
one,
and
.
In these variables Einstein's equations become,
where
is the covariant derivative associated with
,
its Einstein tensor, and
.
In that gauge, Einstein's equations are ``reduced'' to a
hyperbolic system by removing from them all terms containing
, for this quantity is assumed to vanish in this gauge. By doing
this one gets a set of coupled wave equations, one for each
metric component. Thus by prescribing at an initial (space-like)
hypersurface values for
and of its normal derivatives one gets unique solutions to the
reduced system. When are such solutions solutions to Einstein's
equations? That is, under what conditions does
vanish everywhere? It turns out that the Bianchi identity grants
that when
satisfies the reduced equations, then
satisfies a linear homogeneous second order hyperbolic equation.
Standard uniqueness results for such systems implies that if
initially
is chosen so that
and its normal derivative vanish at the initial surface, then
they vanish everywhere on the domain of dependence of that
surface. Thus the question is now posed on the initial data, that
is, on whether it is possible to choose appropriate initial data
for the reduced system,
, in such a way that
vanish initially. It turns out, using that the reduced equations
are satisfied at the initial surface, that one can indeed express
and its normal derivative at the initial surface, in terms of
and its derivatives (both normal and tangential to the initial
surface). Thus one finds there are plenty of initial data sets
for which solutions to the reduced system coincide with solutions
to the full Einstein system. Are they all possible solutions to
Einstein's equations, or are we loosing some of them by imposing
this scheme? The answer to the first part of the question is
affirmative (subject so some asymptotic and smoothness
conditions), for one can prove that given ``any'' solution to the
Einstein equations, there exists a diffeomorphism which makes it
satisfy the above harmonic gauge condition.
It is important to realize that it is not necessary to set
to zero to render the Einstein equations hyperbolic; it just
suffices to set it equal to some given vector field on the
manifold, or any given vector function of the space-time points
and on the metric, but not its derivatives. So there are actually
many ways to hyperbolize Einsteins's equations via the above
scheme. We shall call all of them
harmonic gauge conditions, and reserve the name
full harmonic condition
to the one where
.
An important advantage of this method is that some gauge conditions, like the full harmonic gauge, are four-dimensional covariant -although a background metric is fixed- a condition which can be very useful for some considerations.
One drawback of this method, at least in the simplest version
of the harmonic gauge, i.e. the full harmonic gauge, was
recognized early, [32]. The drawback is the fact that this gauge condition can be
imposed only locally, and generically breaks down in a finite
evolution time. A related problem has been discussed recently in
[33
] in the context of the hyperbolizations of the ADM variables
with the harmonic gauge along the temporal direction. The above
disadvantage can be considered just a manifestation of another:
the lack of ductility of the method, that is the fact that one
has been able do very little besides imposing the full harmonic
gauge condition, and that for each new harmonic gauge condition
one would like to use, a whole study of the properties of the
reduced equations would have to be undertaken. Although there are
many other gauge conditions besides the harmonic one, the issue
of the possibility of their global validity, or the search for
other properties of potential use, do not seem to have been
considered. For a detailed discussion of this topic, see [34
,
35
], and also [5].
One can summarize the situation by noticing that in this setting one needs to prescribe a four vector as a harmonic gauge condition. Since the theory keeps its four dimensional covariance, then it is hard to choose any other vector but zero, that is the full harmonic gauge. Since recently there have been no advances in this area, I do not elaborate on it.
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Hyperbolic methods for Einstein's Equations
Oscar A. Reula http://www.livingreviews.org/lrr-1998-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |