The standard approach, or the 4-D covariant approach

3.1 The standard approach, or the 4-D covariant approach

The standard approach to overcome this problem has been to “fix the gauge”, that is, by imposing some extra condition on the metric coordinate components which would select one and only one representative from each equivalent class of Einstein’s geometries. With a clever choice of gauge fixing, commonly the so-called harmonic gauge, Einstein’s equations can be “reduced” to a hyperbolic system by removing from them the parts which the gauge condition would make vanish. This reduced system is equivalent to the full Einstein equations if one can prove that, possibly after imposing further initial data set constraints, the solutions of the reduced system satisfy the gauge conditions previously imposed to the system. Alternatively, one can think of this method as fixing some components of the tensor obtained by taking the difference between two connections, the one associated with the metric tensor at which we are looking and the one associated with some other arbitrary background metric [23 , 38

A convenient way to describe this scheme is by introducing a background metric, $&tidle;gab$ , thus the gauge is not a coordinate condition, but rather a condition which links the physical metric with the background one. In this approach, see [38 ], the basic variable is a densitized symmetric tensor, $ab ab Φ := gδg$ , where $g$ is the metric determinant with respect to the background one,⁷ and $ab ab ab δg := g − &tidle;g$ .

In these variables Einstein’s equations become,

1 1 -gcd&tidle;∇c &tidle;∇dΦab − gc(a&tidle;∇c Ψb) + -gab&tidle;∇c Ψc (2 ) 2 2 + (terms in &tidle;∇cδgde and δgde ) (3 ) = 8πgT ab − g &tidle;Gab, (4 )

where $&tidle;∇c$ is the covariant derivative associated with $&tidle;gab$ , $G&tidle;ab$ its Einstein tensor, and $∘ -- Ψa := &tidle;∇bΦab = &tidle;∇b ggab ˆg$ .

In that gauge, Einstein’s equations are “reduced” to a hyperbolic system by removing from them all terms containing $Ψa$ , for this quantity is assumed to vanish in this gauge. By doing this one gets a set of coupled wave equations, one for each metric component. Thus by prescribing at an initial (space-like) hypersurface values for $ab Φ$ and of its normal derivatives one gets unique solutions to the reduced system. When are such solutions solutions to Einstein’s equations? That is, under what conditions does $Ψa$ vanish everywhere? It turns out that the Bianchi identity grants that when $Φab$ satisfies the reduced equations, then $Ψa$ satisfies a linear homogeneous second order hyperbolic equation. Standard uniqueness results for such systems implies that if initially $ab Φ$ is chosen so that $a Ψ$ and its normal derivative vanish at the initial surface, then they vanish everywhere on the domain of dependence of that surface. Thus the question is now posed on the initial data, that is, on whether it is possible to choose appropriate initial data for the reduced system, $(Φab,Φ˙ab )$ , in such a way that $(Ψa, ˙Ψa)$ vanish initially. It turns out, using that the reduced equations are satisfied at the initial surface, that one can indeed express $a Ψ$ and its normal derivative at the initial surface, in terms of $ab Φ$ and its derivatives (both normal and tangential to the initial surface). Thus one finds there are plenty of initial data sets for which solutions to the reduced system coincide with solutions to the full Einstein system. Are they all possible solutions to Einstein’s equations, or are we loosing some of them by imposing this scheme? The answer to the first part of the question is affirmative (subject so some asymptotic and smoothness conditions), for one can prove that given “any” solution to the Einstein equations, there exists a diffeomorphism which makes it satisfy the above harmonic gauge condition.

It is important to realize that it is not necessary to set $a Ψ$ to zero to render the Einstein equations hyperbolic; it just suffices to set it equal to some given vector field on the manifold, or any given vector function of the space-time points and on the metric, but not its derivatives. So there are actually many ways to hyperbolize Einsteins’s equations via the above scheme. We shall call all of them harmonic gauge conditions, and reserve the name full harmonic condition to the one where $a Ψ ≡ 0$ .

An important advantage of this method is that some gauge conditions, like the full harmonic gauge, are four-dimensional covariant – although a background metric is fixed – a condition which can be very useful for some considerations.

One drawback of this method, at least in the simplest version of the harmonic gauge, i.e. the full harmonic gauge, was recognized early [14 ]. The drawback is the fact that this gauge condition can be imposed only locally, and generically breaks down in a finite evolution time. A related problem has been discussed recently in [5 ] in the context of the hyperbolizations of the ADM variables with the harmonic gauge along the temporal direction. The above disadvantage can be considered just a manifestation of another: the lack of ductility of the method, that is the fact that one has been able do very little besides imposing the full harmonic gauge condition, and that for each new harmonic gauge condition one would like to use, a whole study of the properties of the reduced equations would have to be undertaken. Although there are many other gauge conditions besides the harmonic one, the issue of the possibility of their global validity, or the search for other properties of potential use, do not seem to have been considered. For a detailed discussion of this topic, see [28 , 29 ] and also [38 ].

One can summarize the situation by noticing that in this setting one needs to prescribe a four vector as a harmonic gauge condition. Since the theory keeps its four dimensional covariance, then it is hard to choose any other vector but zero, that is the full harmonic gauge. Since recently there have been no advances in this area, I do not elaborate on it.