SRab hgabR gaigbSRii WagbAj gbA l8g SgVt g6Ri

= g\ - = (det g)"1 ((det g)^)le = (det g (7.6)

The plan is now as follows. We choose some suitable background metric § and express the Einstein equations in the form

Rab-\Rgab = 8{Rab-\Rgab)+ &<*>- = 8nTab. (7.7)

One regards this as a second order non-linear set of differential equations to determine g in terms of the values of it and its first derivatives on some initial surface. Of course to complete the system one has to specify the equations governing the physical fields which make up the energy-momentum tensor Tab. However even when this is done one does not have a system of equations which uniquely determines the time development in terms of the initial values and first derivatives. The reason for this is, as was mentioned above, that a solution of the Einstein equations can be unique only up to a diffeomorphism. In order to obtain a definite solution one removes this freedom to make diffeomorphisms by imposing four gauge conditions on the covariant derivatives of g with respect to the background metric We shall use the so-called 'harmonic' conditions ft" = 4>bcic = o which are analogous to the Lorentz gauge conditions Ai.i — 0 in electrodynamics. With this condition one obtains the reduced Einstein equations gVQ^W + (terms in 4>cdu and 006) = l6nTab- 2fiab + Q^M. (7.8)

We shall denote the left-hand side of (7.8) by E^j(^cd), where E^ is the Einstein operator. For suitable forms of the energy-momentum tensor T"6 these are second order hyperbolic equations for which we shall demonstrate the existence and uniqueness of solutions in §7.5. We still have to check that the harmonic conditions are consistent with the Einstein equations. That is to say: we derived (7.8) from the Einstein equations by assuming that <fibcic was zero. We now have to verify that the solution that (7.8) gives rise to does indeed have this property. To do this, differentiate (7.8) and contract. This gives an equation of the form gi}ftbHj + Bbi + Cbrfr* = 1 6ttT0». a, (7.9)

where a semi-colon denotes differentiation with respect to g, and the tensors Bcbi and Cb depend on §ab, g0* and gr°®|c. Equations (7.9) may be regarded as second order linear hyperbolic equations for ijfb. Since the right-hand side vanishes, one can use the uniqueness theorem for such equations (proposition 7.4.5) to show that will vanish everywhere if it and its first derivatives are zero on the initial surface. We shall see in the next section that this can be arranged by a suitable diffeomorphism.

We still have to show that the unique solution obtained by imposing the harmonic gauge condition is related by a diffeomorphism to any other solution of the Einstein equations with the same initial data. This will be done in § 7.4 by making a special choice of the background metric.

7.3 The initial data

As (7.8) is a second order hyperbolic system it seems that to determine the solution one should prescribe the values of gab and g^^W on the initial surface S(£f), where ue is some vector field which is not tangent to 6(Sf). However not all these twenty components are significant or independent: some can be given arbitrary initial values without changing the solution by more than a diffeomorphism, and others have to obey certain consistency conditions.

Consider a diffeomorphism which leaves 6(£f) pointwise fixed. This will induce a map which takes g0* &tpe 6{£f) into a new tensor /i*gab at p. If naeT*p is orthogonal to 6(Sf) (i.e. na Va = 0 for any VaeTp tangent to 6[Sf)) and normalized so that = -1

then, by suitable choice of ¡i, na//,lttgat' can be made equal to any vector at p which is not tangent to 6{Sf). Thus the components nagob are not significant. On the other hand as /i leaves 6{£f) pointwise fixed, the induced metric hab = 6*gab on if will remain unchanged. It is therefore only this part of g which lies in 6(6?) which need be given to determine the solution. The other components T^gr"6 can be prescribed arbitrarily without changing the solution by more than a diffeomorphism. Another way of seeing this is to recall that we formulated the Cauchy problem in terms of certain data on a disembodied three-manifold Sf and then looked for an imbedding into some four-manifold Now on £f itself one cannot define a four-dimensional tensor field like g but only a three-dimensional metric h, which we shall take to be positive definite. The contravariant and covariant forms of h are related by h°»hbc = 6ac, (7.10)

where now d°c is a three-dimensional tensor in if. The imbedding 6 will carry h^ into a contravariant tensor field 6^,hab on 6(£f) which has the property na6*h°»=0. (7.11)

As nagaX' is arbitrary, one may now define g on 6(Sf) by gab = 61fhab-uaub, (7.12)

where ua is any vector field on 6{£f) which is nowhere zero or tangent to 6{£f). Defining g^ by (7.1), one has:

¿06 = 0*006. nag°» =-nauaub, g^Wu" = -1. (7.13) Thus hgj, is the metric induced on Sf by g and ua is the unit vector orthogonal to S(if) in the metric g.

The situation with the first derivatives gabicuc is similar: nagabicuc can be given any value by suitable diffeomorphisms. However there is now an additional complication in that g^^ depends not only on g but also on the background metric § on In order to give a description of the significant part of the first derivative of g in terms only of tensor fields defined on if, we proceed as follows. We prescribe a symmetric contravariant tensor field xab on ^■ Under the imbedding xab is mapped into a tensor field G^x^ on 6(£f). We require that this is equal to the second fundamental form (see § 2.7) of the submanifold 0(5") in the metric g. This gives

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