1.2 Main subject and plan of the review

In the nineties, researchers have realized that other or more general forms of gauge conditions are needed to address some pressing problems, in particular, the numerical studies of regions of high curvature – like black hole formation. One should also mention the study of slow solutions, that is, solutions which weakly depend on the value of the velocity of light – in particular those which have a well behaved Newtonian limit. For the first problem, the issue has been whether the coordinate system chosen, that is, whether the gauge condition, stays well behaved during evolution and does not cause unphysical singularities, singularities which, for the numerical problem, are as bad as real ones. For the second problem, the issue has been whether the gauge is well behaved and does not prevent uniform continuity of slow solutions as the speed of light goes to infinity.

In both cases it is clear that one can choose bad gauges which would make the problem intractable and which nevertheless are genuine smooth coordinate conditions in the whole region of interest.

The rest of this review describes the efforts triggered by the need to deal with the above mentioned problems. An attempt to compare these efforts in all respects appears to be a hopeless task. The variables they use are different. Thus we shall concentrate on revealing the points these theories hold in common and discuss the key properties of each. These properties can vary a lot from one system to another. The properties can be seen as virtues or defects of the system according to the uses to which people put them. For instance, a gauge condition can be incorporated into the systems as:

1. a solution to a hyperbolic equation, and so be incorporated as a part of the hyperbolic system;
2. a solution to an elliptic equation, given a mixed hyperbolic-elliptic system;
3. a solution to a parabolic equation, given a mixed hyperbolic-parabolic system;
4. or it can be given as a fixed function in space-time, chosen by a rule of thumb by looking at the initial data.

Each one of these possibilities can be implemented in a mathematically rigorous manner in most of the general schema proposed. And probably each of them would be of relevance in some specific implementation. The point, in other words, is that it is very difficult to go beyond a superficial or general description of the methods before some particular, extended, and fruitful use legitimizes the job.

In Section 2, I give a short summary of hyperbolic theory, describing the case which is completely understood, namely the constant coefficient case, and mention what of it can be extended to the case of interest here, namely quasi-linear systems. For physicists, this discussion should be a complement to Geroch’s lecture notes on symmetric hyperbolic systems [34]. This discussion follows very closely Chapter II of [48], see also Chapter IV of [36].

In Section 3, I first describe the general problem of adapting the theory of hyperbolic systems to general relativity and the gauge issue. A companion to this section is, besides Geroch’s lectures, the paper of Friedrich in Hyperbolic Reductions [29]. In particular we discuss the standard approach, that is, the harmonic gauge.

In Section 4, I present the more recent approaches, and divide them into four classes, according to the type of variables used.

In Section 5, I compare the different implementations that have been made of the approaches introduced in the previous sections, and discuss the impact these approaches have had on the problems where they have been applied.

In Section 6, I consider the role the constraint equations play in these new systems. In the harmonic gauge, constraint equations become evolution equations. The consistency of the gauge is all that is needed to ensure equivalence between Einstein’s equations and the harmonic system. In the new systems, one is not incorporating the constraints, and so one should make sure that, if the constraint equations are satisfied initially, they are satisfied during evolution. I claim that, in the initial value formulation, this follows from uniqueness of solutions to the equations and from the fact that the modified evolution vector fields proposed are tangent to the constraint sub-manifold. I also mention the difficulties that appear when considering an initial-boundary value problem.