6*Xab = W^B^H - gcigd}gi}ikuk + gbiu% + gc{u<ib). (7.15)

This may be inverted to give gablc uc in terms of6+xab'-

where Wb is some vector field on 8(£f). It can be given any required value by a suitable diffeomorphism ¡i.

The tensor fields hP* and xab cannot be prescribed completely independently on if. For multiplying the Einstein equations (7.7) by na, one obtains four equations which do not contain the second derivatives of g out of if. Thus there must be four relations between gab, g^^u* and naT'a'. Using (2.36) and (2.35), they can be expressed as equations in the three-manifold if\

where a double stroke | denotes covariant differentiation in if with respect to the metric h, and R' is the curvature scalar of h.

The data to on Sf that is required to determine the solution therefore consists of the initial data for the matter fields (in the case of a scalar field for example, this would consist of two functions on if representing the value of 4> and its normal derivative) and two tensor fields hab and y^ on£f which obey the constraint equations (7.17—18). These contraint equations are elliptic equations on the surface if which impose four constraints on the twelve independent components of (tf*,X*). In such situations, one can show one can prescribe eight of these components independently and then solve the constraint equations to find the other four, see e.g. Bruhat (1962). We shall call a pair (y, to) satisfying these conditions, an initial data set. We then imbed y in some suitable four-manifold J( with metric g and define gab on 8(y) by (7.12) for some suitable choice of ua. We shall take ua to be gabnb. Thus it will be the unit vector orthogonal to 8(y) in both the metric g and We shall also exploit our freedom of choice of Wa in the definition of gab^cuc by (7.16) to make \¡rb zero on 6(y). This requires wb = -glKid9ce6*h<*+

(Note that all the derivatives in (7.19) are tangent to 6(y) as is required by the fact that the fields involved have been defined only on 6(Sf).) To ensure that r¡rb vanishes everywhere one also needs to be zero on 6(y). However this now follows from the constraint equations providing the reduced Einstein equations (7.8) hold on 6(y). One may therefore proceed to solve (7.8) as a second order non-linear hyperbolic system on the manifold with metric

(Note that there are 10 such equations for the 0's; in proving the existence of solutions of these 10 equations we do not split them into a set of constraint equations and a set of evolution equations, and so the question as to whether the constraint equations are conserved does not arise.)

In this section we shall reproduce some results on second order hyperbolic equations given in Dionne (1962). They will be generalized to apply to a whole manifold, not just one coordinate neighbourhood. These results will be used in the following sections to prove the existence and uniqueness of developments for an initial data set (y, to).

We first introduce a number of definitions. We use Latin letters to denote multiple contravariant or covariant indices; thus a tensor of type (r, s) will be written as RTj, and we denote by |/| —r the number of indices that the multiple index I represents. We introduce a positive definite metric eab on Jl and define eu = eabecd-- ePQ, = eabecd■ ■ ■ ep9>

r times r times where |J| = | J| = r. We then define the magnitude \K*j\ (or simply, |K|) as [KTjKlmeILeJM)l where repeated multiple indices imply contraction over all the indices they represent. We define | DmKIJ | (or simply, |i>mK|) to be \KIJ{L\ where \L\ = m and as before, | indicates covariant differentiation with respect to

Let Jf be an imbedded submanifold of Jl with compact closure in J(. Then H^y.^Hm is defined to be where dcr is the volume element on Jf induced by e. We also define ||m to be the same expression where the derivatives are taken only in directions tangent to Jf. Clearly, HK,./^^ 3s ¡K,^T||m.

The Sobolev spaces Wm(r, s,Jf) (or simply Wm(Jr)) are then defined to be the vector spaces of tensor fields Kxj of type (r, s) whose values and derivatives (in the sense of distributions) are defined almost everywhere on Jf (i.e. except, possibly, on a set of measure zero; for the rest of this section 'almost everywhere' is to be understood almost everywhere) and for which \KIJ,Jf|m is finite. With the norms

|| , ||m the Sobolev spaces are Banach spaces in which the Cm tensor fields of type (r, s) form dense subsets. If e' is another continuous positive definite metric on ^ then there will be positive constants Cx and C2 such that

CAR1 A < I^I'^CjI^I on Jf, and Cx\K'Jtjr\m < Wj,^ < C2||^,^i|m.

Thus || will be an equivalent norm. Similarly another Cm background metric will give an equivalent norm. In fact it follows from two lemmas given below that if Wm(J^) and 2m is greater than the dimension of Jf, then the norm obtained using the covariant derivatives defined by is again equivalent.

We now quote three fundamental results on Sobolev spaces. The proofs can be derived from results given in Sobolev (1963). They require a mild restriction on the shape oiJf. A sufficient condition will be that for each point p of the boundary dJf it should be possible to imbed an w-dimensional half cone in Jf with vertex at p, where n is the dimension of Jf. In particular this condition will be satisfied if the boundary dJf is smooth.

Lemma 7.4.1

There is a positive constant Pt (depending on Jf, e and g) such that for any field KTj e Wm(jV) with 2m > n, where n is the dimension of./f,

From this and the fact that the vector space of all continuous fields KTj on Jf is a Banach space with norm sup |K|, it follows that if

K!j e Wm(jV) where 2m > n, then Kzj is continuous on Jf. Similarly if KIJ e Wm+r(jV), then KTj is C on ¿V.

Lemma 7.4.2

There is a positive constant P2 (depending on jV, e and §) such that for any fields KTj, LFq e Wm(J/~) with 4m > n,

From this and the previous lemma it follows that if n < 4 and 2m > n, then for any two fields RTj, LpQeWm{^V), the product RTjLpq is also in

IfjV' is an (n~ l)-dimensional submanifold smoothly imbedded in^/f", there is a positive constant P3 (depending on JT, jV\ e and §) such that for any field KTj e

We shall prove the existence and uniqueness of developments for (y.to) when h^e and e TP+a(y) where a is any non negative integer. (If £f is non-compact, we mean by h0* e Wm[6f) that habg Wm(J/") for any open subset jV of ¿f with compact closure.) A sufficient condition for this is that hab be Ci+a and xab he C3+a on by lemma 7.4.1, a necessary condition is that hai be C2+a and xab be Cl+a. The solution obtained for g0* will belong to Wi+a(Jf) for each smooth spacelike surface JC and so the (2 + o)th derivatives will be bounded, i.e. g06 will be C(2+a)~ on uT

These differentiability conditions can be weakened to cases such as shock waves where the solution departs from W* behaviour on well-behaved hypersurfaces; see Choquet-Bruhat (1968), Papapetrou and Hamoui (1967), Israel (1966), and Penrose (1972a). However no proof is known for cases in which such departures occur generally. The W1 condition for the existence and uniqueness of developments is an improvement on previous work (Choquet-Bruhat (1968)) but it is somewhat stronger than one would like since the Einstein equations can be defined in a distributional sense if the metric is continuous and its generalized derivatives are locally square integrable (i.e. if g is C° and W1). On the other hand any Wp conditions for p less than 4 would

not guarantee the uniqueness of geodesies, or, for p less than 3, their existence. Our own view is that these differences of differentiability conditions are not important since as explained in § 3.1, the model for space-time may as well be taken to be

In order to prove the existence and uniqueness of developments wd now establish some fundamental inequalities (lemmas 7.4.4 and 7.4.6) for second order hyperbolic equations, in a manner similar to that of the conservation theorem in §4.3.

Consider a manifold Jl of the form JifxR1 where is a three-dimensional manifold. Let be an open set of J( with compact closure which has boundary and which intersects ^f(O), where ^f(i) denotes the surface x {t}, teR1. Let and <%(t') denote the parts of °U for which t 3s 0 and t' 3s t 3s 0 respectively (figure 48). On let § be a C2~ background metric and let e be a C1- positive definite metric. We shall consider tensor fields KTj which obey second order hyperbolic equations of the form

L(K) = A"»K'Jioh + RQpia+C"QJR°P = F',, (7.20)

where A is a Lorentz metric on (i.e. a symmetric tensor field of signature + 2), B, C and F are tensor fields of type indicated by their indices, and | denotes covariant differentiation with respect to the metric g.

Lemma 7.4.4

(2) there exists some Qt > 0 such that on and A<*Wa Wb > Qt eabWa Wb for any form W which satisfies A^t^W/, = 0,

(3) there exists some Q2 such that on

|A| < Q2, |DA| < g2, |B| |C|^g2, then there exists some positive constant P4 (depending on e, g, Q1 and Q2) such that for all solutions RTj of (7.20),

One forms the 'energy tensor' S^ for the field RTj in analogy to the energy-momentum tensor of a scalar field of unit mass (§3.2):

gab = {(¿«¿w - IA°»A«1) RiJlcRPm _ RPQ}e^eIP. (7.21)

The tensor S06 obeys the dominant energy condition (§4.3) with respect to the metric A (i.e. if Wa is timelike with respect to A then S°*>Wa Wb ^ 0 and jS"6^ is non-spacelike with respect to A). Moreover by conditions (2) and (3) there will be positive constants Q3 and Qt such that

<23(|K|'+ |DK|2) < < <24(|K|* + |DK|2). (7.22)

We now apply lemma 4.3.1 to S^, taking as the compact region 3F

and using the volume element d© and covariant differentiation defined by the metric f _ S^tad&b < f _ S"%adab

where P is a positive constant independent of S^. (The sign has been changed in the first term on the right-hand side since the surface element d&b of the surface Jif(t) is taken to have the same orientation as i.e. d&b = tlb da where dcr is a positive definite measure on 3?(t).)

Since e and § are continuous there will be positive constants <26 and Qt such that on ~ , ,„

where dcr is the area element on 3^(t) induced by e. Thus by (7.22) and (7.23) there is some Q-, such that

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