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 [1012] and [11], where Equation 4.1 has been modified to (in the notation of [18]):

˙ N-2 N = 2 f(N )P,
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 [5Jump To The Next Citation Point] shows that if we take constant initial data for (in that paper’s notation) α = α0, g = g0, and K0, and null data for A, and D, then the resulting system is just a coupled set of ordinary equations. One can see that 2 g = g0(K ∕K0 ), and so

2 α˙-= (-α-)2K-0(1 − f). K K g0
If f = 1, the harmonic time gauge in this notation, nothing happens at first sight. See nevertheless [5Jump To The Next Citation Point]. If f > 1 and α0- K0 > 0 then we have a singularity in a finite time. The same happens if f < 1 and initial data is taken so that Kα00 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 [5Jump To The Next Citation Point] and [6Jump To The Next Citation Point] 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 f = f0 > 0 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, 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 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.

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