5.1 Trial and error method
One alternative would be to start with some arbitrary prescription for the gauge variables, evolve the
solution for a while, stop, look for troublesome regions, and modify the gauge prescription there. One would
do that only a finite number of times and choose smooth prescriptions and smooth transitions between
them, so no problem of well posedness or numerical stability would be created by that procedure. With
some experience, and luck, this procedure could work.
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 [14], and
more recently new indications have been found in [5
]. 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 [1] 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 [51],
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 [29], as said in Section 4.2, 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.
