One approach has been taken in [10, 12] and , where Equation 4.1 has been modified to (in the notation of ):
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  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
Note that the above prescription for the evolution of the lapse for is identical to the one considered in , 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 , 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.
© Max Planck Society and the author(s)