Perhaps the mayor drawback of this approach is the fact that
the strict harmonic gauge, the most conspicuous choice, does not
behave well under evolution. This has been known for a long time
[32], and more recently new indications have been found in [33]. These findings are of course not valid for the generalized
harmonic time gauge because one can trivially take any solution,
draw a well behaved foliation on it, and identify the generalized
gauge that
works
for that solution. Some attempts have been made in order to get
loose from the generalized harmonic time condition, presumably
with the intention of later imposing equations on the free
variables, otherwise independent fields. In [56
] a non-strictly hyperbolic system is found by taking yet another
time derivative of the evolution equation for the momentum
variable. In that way, they are able to prescribe in a completely
free way the lapse-shift pair. In doing this, they obtain a
non-strictly hyperbolic system, in the sense of Leray-Ohya, [27], which I presume in the language of first order systems means
that it is a weakly hyperbolic, but with certain other properties
which imply that the system is well posed in Gevrey classes of
functions
. The resulting system, once brought to first order, has a rather
big number of variables. It is not clear that one can stablish
numerical stability and convergence for these types of systems,
for at least in the continuum estimates an infinite number of
derivatives are involved.
In [35], as said in the previous section, a system in the frame representation is made where the corresponding gauge variables can also be given arbitrarily. The importance of this freedom is that in this case one can prescribe directly the lapse. This is in contrast to the case of the generalized harmonic time condition, in which one prescribes the lapse up to the square root of the metric and so finds out what the lapse really was only after solving the problem.
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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 |