One approach has been taken in [45,
46], and [47] where the equation
4.1
has been modified to (in the notation of [39]):
With this new evolution equation for the lapse function, they analyze the principal part of the equations and see that if f > 0 then it has imaginary eigenvalues and a complete set of eigenvectors. Thus, up to the smoothness requirement on the eigenvalues with respect to the wave vector, the system seems to be at least strongly hyperbolic and so well posed. This prescription enlarges the system a bit; as one also has to correctly include in it the corresponding equations for the first space derivatives of the lapse function, which become now a dynamical variable.
Although the system is well posed in the sense of the theory
of partial differential equations, it has some instabilities from
the point of view of the ordinary differential equations. A quick
look at the toy model in [33] shows that if we take constant initial data for (in that
paper's notation)
,
, and
, and null data for
A, and
D, then the resulting system is just a coupled set of ordinary
equations. One can see that
, and so
If
f
=1, the harmonic time gauge in this notation, nothing happens at
first sight. See nevertheless [33]. If
f
>1 and
then we have a singularity in a finite time. The same happens if
f
<1 and initial data is taken so that
is negative. Thus we see that this gauge prescription can
generate singularities which do not have much to do with the
propagation modes, and so with the physics of the problem. In [33
] and [54
] numerical simulations have been carried out to study this
problem. Needless to say, these instabilities would initially
manifest themselves in numerical calculations via the forming of
large gradients on the various fields coupled to the above
fields, and the time at which they appear depends on the size of
the trace of the momentum variable. In [54] a proposal to deal with this problem is made which consists of
smoothing out the lapse via a parabolic term. In view of the fact
that this problem already arises for constant data, it is
doubtful that such a prescription can cure it.
Note that the above prescription for the evolution of the
lapse for
is identical to the one considered in [49], namely equation
4.1
. It is most probably the case then that the same sort of
instability would be present there, although the equations
considered there are different, due to the inclusion of terms
proportional to the scalar constraint in order to render the
system symmetric hyperbolic.
In [33] there is also a study of another type of singularity which is not ruled out with the choice of the harmonic gauge, f =1. This singularity seems to be of a different nature, and is probably related to the instability of the harmonic gauge already mentioned. It clearly has to do with the non-linearities of the theory.
It should be mentioned that there are a wide variety of possibilities for making bigger hyperbolic systems out of those which are hyperbolic for a prescribed lapse-shift pair, or for the generalized harmonic gauge variant. In that respect, perhaps the systems which are more amenable to a methodological and direct study are the ones in the frame or in Ashtekar's representations, for there, as discussed in the previous section for the Ashtekar's representation systems, § 4.3, the possibilities to enlarge the system and keep it symmetric hyperbolic are quite clear and limited.
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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 |