The Cauchy problem in General Relativity

In this chapter we shall give an outline of the Cauchy problem in General Relativity. We shall show that, given certain data on a spacelike three-surface there is a unique maximal future Cauchy development D+[£f) and that the metric on a subset W of D+(Sf) depends only on the initial data on n if. We shall also show that this dependence is continuous if has a compact closure in .

This discussion is included here because of its intrinsic interest, because it uses some of the results of the previous chapter, and because it demonstrates that the Einstein field equations do indeed satisfy postulate (a) of § 3.2 that signals can only be sent between points that can be joined by a non-spacelike curve. However it is not really needed for the remaining three chapters, and so could be skipped by the reader more interested in singularities.

In § 7.1, we discuss the various difficulties and give a precise formulation of the problem. In §7.2 we introduce a global background metric § to generalize the relation which holds between the Ricci tensor and the metric in each coordinate patch to a single relation which holds over the whole manifold. We impose four gauge conditions on the covariant derivatives of the physical metric g w,ith respect to the background metric These remove the four degrees of freedom to make diffeomorphisms of a solution of Einstein's equations, and lead to the second order hyperbolic reduced Einstein equations for g in the background metric Because of the conservation equations, these gauge conditions hold at all times if they and their first derivatives hold initially.

In § 7.3 we show that the essential part of the initial data for g on the three-dimensional manifold if can be expressed as two three-dimensional tensor fields hab, %ab on £f. The three-dimensional manifold if is then imbedded in a four-dimensional manifold and a metric g is defined on £f such that hab and xab become respectively the first and second fundamental forms ofSf in g. This can be done in such a way that the gauge conditions hold on if. In § 7.4 we establish some

basic inequalities for second order hyperbolic equations. These relate integrals of squared derivatives of solutions of such equations to their initial values. These inequalities are used to prove the existence and uniqueness of solutions of second order hyperbolic equations. In § 7.5 the existence and uniqueness of solutions of the reduced empty space Einstein equations is proved for small perturbations of an empty space solution. The local existence and uniqueness of empty space solutions for arbitrary initial data is then proved by dividing the initial surface up into small regions which are nearly flat, and then joining the resulting solutions together. In § 7.6 we show there is a unique maximal empty space solution for given initial data and that in a certain sense this solution depends continuously on the initial data. Finally in §7.7 we indicate how these results may be extended to solutions with matter.

7.1 The nature of the problem

The Cauchy problem for the gravitational field differs in several important respects from that for other physical fields.

(1) The Einstein equations are non-linear. Actually in this respect they are not so different from other fields, for while the electromagnetic field, the scalar field, etc., by themselves obey linear equations in agiven space-time, they form a non-linear system when their mutual interactions are taken into account. The distinctive feature of the gravitational field is that it is self-interacting: it is non-linear even in the absence of other fields. This is because it defines the space-time over which it propagates. To obtain a solution of the non-linear equations one employs an iterative method on approximate linear equations whose solutions are shown to converge in a certain neighbourhood of the initial surface.

(2) Two metrics and g2°n a manifold are physically equivalent if there is a diffeomorphism Jt which takes gj into g2

* = 62)1 and clearly satisfies the field equations if and only if g2 does. Thus the solutions of the field equations can be unique only up to a diffeomorphism. In order to obtain a definite member of the equivalence class of metrics which represents a space-time, one introduces a fixed 'background' metric and imposes four 'gauge conditions' on the covariant derivatives of the physical metric with respect to the background metric. These conditions remove the four degrees of freedom to make diffeomorphisms and lead to a unique solution for the metric components. They are analogous to the Lorentz condition which is imposed to remove the gauge freedom for the electromagnetic field.

(3) Since the metric defines the space-time structure, one does not know in advance what the domain of dependence of the initial surface is and hence what the region is on which the solution is to be determined. One is simply given a three-dimensional manifold if with certain initial data co on it, and is required to find a four-dimensional manifold Jt, an imbedding 6: and a metric g on Jt which satisfies the Einstein equations, agrees with the initial values on 6(if) and is such that 6(if) is a Cauchy surface for JK. We shall say that (Jt, 6, g), or simply is a development of (if, co). Another development (JC, 6\ g') of (if, co) will be called an extension of if there is a diffeomorphism ctoiJC into which leaves the image of if point-wise fixed and takes g' into g (i.e. 6~yarW = id on ^ and a, g' = g). We shall show that provided the initial data co satisfies certain constraint equations on if, there will exist developments of (if, to) and further, there will be a development which is maximal in the sense that it is an extension of any development of (if, co). Note that by formulating the Cauchy problem in these terms we have included the freedom to make diffeomorphisms, since any development is an extension of any diffeomorphism of itself which leaves the image of if pointwise fixed.

7-2 The reduced Einstein equations

In chapter 2, the Ricci tensor was obtained in terms of coordinate partial derivatives of the components of the metric tensor. For the purposes of this chapter it will be convenient to obtain an expression that applies to the whole manifold Ji and not just to each coordinate neighbourhood separately. To this end we introduce a background metric § as well as the physical metric g. With two metrics one has to be careful to maintain the distinction between covariant and contra-variant indices. (To avoid confusion, we shall suspend the usual conventions for raising and lowering indices.) The covariant and contra-variant forms of g and § are related by

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