### 5.2 Hyperbolic extensions

One would also like to have recipes which could be used automatically during evolution, that is,
algebraic or differential equations, which would not only fix uniquely the evolution of the gauge variables,
but which would also result in a well posed evolutionary problem.
One approach has been taken in [10, 12] and [11], where Equation 4.1 has been modified to (in the
notation of [18]):

With this new evolution equation for the lapse function, they analyze the principal part of
the equations and see that if 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 [5] shows that if we take constant initial data for (in that paper’s
notation) , , and , and null data for , and , then the resulting
system is just a coupled set of ordinary equations. One can see that , and
so

If
, the harmonic time gauge in this notation, nothing happens at first sight. See nevertheless [5]. If
and then we have a singularity in a finite time. The same happens if 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 [5]
and [6] 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 [6] 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 [32], 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 [5], there is also a study of another type of singularity which is not ruled out with the choice of the
harmonic gauge, . 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 Section 4.3
for the Ashtekar’s representation systems, the possibilities to enlarge the system and keep it symmetric
hyperbolic are quite clear and limited.