W ZllaB

where || indicates covariant differentiation with respect to Proposition 7.4.7 can then be applied to this equation.)

Since the gradients of Zpa are linearly independent on^f (0) n <%a, they will be linearly independent on some neighbourhood °li*a of ^f(O) in The metric will be at least C1" on in typ. Since it will obey the reduced empty space Einstein equations on p in the background metric § and since it has the same initial data on 6p("fa n "fp), it must coincide with g^ on some neighbourhood fyp of 6p(-fa fl in Wp. This shows that one may join together <%"a and <%'p to obtain a development of the region U "Vp of Taking the covering of Sf to be locally finite, one may proceed in a similar fashion to join together the subsets of the other neighbourhoods {<%a} to obtain a development of £f, i.e. a manifold with a metric g and an imbedding B\£f->J( such that g satisfies the empty space Einstein equations and agrees with the prescribed initial data to on which is a Cauchy surface for If (JC, g') is another development of (£f, to) one can by a similar procedure establish a diffeomorphism /i between some neighbourhood of 8'(Sf") in and some neighbourhood of 6(Sf) in such that fi*gfab = gab. We have therefore proved:

The local Cauchy development theorem

If hab e W'and e Ws+a(Sf) satisfy the empty space constraint equations there exist developments g) for the empty space Einstein equations such that g eW^Jt) and geW*+a(jr) for any smooth spacelike surface Stf. These developments are locally unique in that if (J!', g') is another W4+° development of (SP, to) then (J(, g) and (JC, g') are both extensions of some common development of

That g follows from lemma 7.4.6 since the surfaces of constant f can be chosen arbitrarily. □

7.6 The maximal development and stability

We have shown that if the initial data satisfied the empty space constraint equations one can find a development, i.e. one can construct a solution some distance into the future and past of the initial surface. In general, this development can be extended further into the future and past to give a larger development of (SP, to). However we shall show by an argument similar to that of Choquet-Bruhat and Geroch (1969) that there is a unique (up to a diffeomorphism) development (JK, g) of (SP, to) which is an extension of any other development of(S, to).

Recall that (uflf gx) is an extension of (JK2, g2) if there is an imbedding /i: such that n*g2 = gi, and such that 8^~i/jUd2 is the identity map on SP. Given a point qeSP, and a distance 5 one can uniquely determine points pieJfi and p2eJ?2 by going a distance s along the geodesies orthogonal to 6t(SP) and 82(SP) through 8x(q) and 62(q) respectively. Since fi(p2) must equal pu the imbedding ¡1 must be unique. One can therefore partially order the set of all developments of (SP, to), writing (Jf2, g2) < g1) if gx) is an extension of 62)- If now {(J!a, gj} is a totally ordered set (a set sf is said to be totally ordered if for every pair a, b of distinct elements of si, either a ^ b or b < a) of developments of (SP, to), one can form the manifold JC as the union of all the where for (J!a, gj s£ (^5, gp) each pasJ!a is identified with /J-ap(pa) e where is the imbedding. The manifold Jt' will have an induced metric g' equal to ¡iaif ga on each fia(J?a) where fia: is the natural imbedding. Clearly (.Jt', g') will also be a development of (SP, to); therefore every totally ordered set has an upper bound, and so by Zorn's lemma (see, for example, Kelley (1965), p. 33) there is a maximal development (jfH, g) of (SP, to) whose only extension is itself.

We shall now show that (¿f(, §) is an extension of every development of (SP, to). Suppose (JC, g') is another development of (SP, to). By the local Cauchy theorem, there exist developments of (SP, to) of which (jfH, g) and (-JC, g') are both extensions. The set of all such common developments is likewise partially ordered and so again by Zom's lemma there will be a maximal development (JC, g") with the imbed-dings p,\JC-+Ji and fi': Jt"-+Jt', etc. Let JK+ be the union of Jl, Jl' and JK", where each p" eJ?" is identified with ft,(p")eJl and ¡x'(p") bJK'. If one can show that the manifold JK+ is Hausdorff, the pair (J?+, g+) will be a development of (S", to). It will be an extension of both (Jl, g) and (JC, g'). However the only extension of (Jl, g) is (Jl, g) itself, and so (Jl, g) must equal (J(+, g+) and be an extension

Suppose that JK+ were not Hausdorff. Then there exist points p e (p,(JC))' <=■ Jl and p e (¡i'(.J("))' <=■ J(' such that every neighbourhood % of p has the property that /¿'(p.*1^)) contains p'. Now since (JK", g") is a development, it will be globally hyperbolic as will its image p,(JC) in Jl. Therefore the boundary of ¡L(JC) in Jl must be achronal. Let y be a timelike curve in Jl with future endpoint at p. Then p' must be a limit point in J(' of the curve ¡i'prx(y). In fact it must be a future endpoint, since strong causality holds in (Ji', g'). Thus the point p is unique, given p. ¡Further, by continuity vectors at p' can be uniquely associated with vectors at p. Thus one can find normal coordinate neighbourhoods ofp mJK and of p' in J(' such that under the map fi'pr1 points of ^ fl ¡L(JC) are mapped into points of fl /i'(JK") with the same coordinate values. This shows that the set^" of all 'non-Hausdorff' points of (fi(JK"))' is open in (fi(JK"))'. We shall suppose that^ is non-empty, and so obtain a contradiction.

If A is a past-directed null geodesic in J( through p e ¡F, then since one can associate directions at^ with directions atp', one can construct a past-directed null geodesic A' through p' in J(' in the corresponding direction. To each point of A n (p-(J("))' there will correspond a point of A' fl and so every point of X fl (fL(JK"))' will be in^". Since

B(Sf) is a Cauchy surface for J/, A must leave (fi(J("))' at some point q. There will be some point f e ¿F in a neighbourhood of q such that there is a spacelike surface through f which has the property that (¿P — f) e ¡L(JC). There will be a corresponding spacelike surface — (fi'prx(M — f)) u r' in Jl' through the corresponding point r'. The surfaces and may be regarded as images of a three-dimensional manifold df under imbeddings 3V-+JK and \jr': df-^Jl' such that is the identity map on 3V—

The induced metrics and ^'*(g') on 3/f will agree since and —p' are isometric. By the local Cauchy theorem, they will be in Wi+a(3#>). Similarly the second fundamental forms will agree and be in W3+a(J^). Neighbourhoods of J? in J( and in JC would be W4+a developments of . By the local Cauchy theorem they must be extensions of the same common development (JK*, g*). Joining (.Jl*, g*) to (JH", g") one would obtain a larger development of (5^, to), of which (Ji, g) and (Jf, g') would be extensions. This is impossible, since , g") was the largest such common development. This shows that must be Hausdorff, and so that (yK, g) must be an extension

We have therefore proved: The global Cauchy development theorem

If hab e Wl+a(£f) and xab e W^Sf) satisfy the empty space constraint equations, there exists a maximal development g) of the empty space Einstein equations with g e W4+a(J?) and g e W4+a(J^) for any smooth spacelike surface ¿F. This development is an extension of any other such development.

We have so far only proved that this development is maximal among Wi+a developments. If a is greater than zero, there will also be jp+a-i; jp+a-2^ developments which are extensions of the development. However, Choquet-Bruhat (1971) has pointed out that these developments must all coincide with the W4 development. This is because one can differentiate the reduced Einstein equations and then regard them as linear equations on the W* development, for the first derivatives ofgab. Then using proposition 7.4.7 one can show that gab is W6 on the TP development, if the initial data is Wh. By continuing in this way, one can show that if the initial data is C®, there will be a C® development which will in fact coincide with the W4 development.

We have proved the existence and uniqueness of maximal developments only for W* or higher metrics. In fact, it is possible to prove the existence of developments for Ws initial data, but we have not been able to prove the uniqueness in this case. It may be possible to extend the W* maximal development either so that the metric does not remain in W4, or so that 6(£P) does not remain a Cauchy surface. In the latter case, a Cauchy horizon occurs; examples of this were given in chapter 6. On the other hand it may be that some sort of singularity occurs, in which case the development cannot be extended with a metric which is sufficiently differentiable to be interpreted physically. In fact, theorem 4 of the next chapter will show that if Sf is compact and Xabhab is negative everywhere on Sf, then the development cannot be extended to be geodesically complete with a C2~ metric, i.e. with locally bounded curvature.

We have shown there is a map from the space of pairs of tensors (hab, x"*) on £f which satisfy the constraint equations to the space of equivalence classes of metrics g on a manifold Jl, which, by proposition 6.6.8, is diffeomorphic to if xE>. If two pairs (hab, x"*) and (h'ab,x'ab) are equivalent under a diffeomorphism A: Sf -+SP (i.e. A, A"6 — h,ab andA^x"6 = they will produce equivalent metrics g. We thus have a map from equivalence classes of pairs (hab, xab) to equivalence classes of metrics g. Now h°b and xab together have twelve independent components. The constraint equations impose four relations between these, and the equivalence under diffeomorphisms may be regarded as removing a further three arbitrary functions, leaving five independent functions. One of these functions may be regarded as specifying the position of 6(Sf) within the development , g). Therefore maximal developments of the empty space Einstein equations are specified by four functions of three variables.

One would like to show that the map from equivalence classes of (hab, xab) to equivalence classes of g is continuous in some sense. The appropriate topology on the equivalence classes for this is the Wr compact-open topology (cf. § 6.4). Let § be a Cr Lorentz metric on ^ and be an open set with compact closure. Let V be an open set in WT(%) and let 0(°U, V) be the set of all Lorentz metrics on Jl whose restrictions to lie in V. The open sets of the Wr compact open topology on the space of all WT Lorentz metrics on J( are defined to be the unions and finite intersections of sets of the form 0(U, F). The topology of the space of equivalence classes of

WT metrics on ^ is then that induced by the projection n: &r(J() <£*(J()

which assigns a metric to its equivalence class (i.e. the open sets of <£*(.J() are of the form n(Q) where Q is open in Similarly the

WT compact open topology on the space £2,(5^) of all pairs (hab, which satisfy the constraint equations is defined by sets of the form 0(°U, V, V) consisting of the pairs for which fi^eV and ^e V where V and V are open sets in WT(i?) and WT~\y) respectively. The C00 metrics on JK form a subspace Sf^J?) of the space S? (JK) of all Lorentz metrics on .J(. Since a C® metric is WT for any r, one has the WT topology on ^(Ji). One can then define the C® or W® topology on as that given by all the open sets in the Wr topologies on

^(Ji) for every r. The C°° topology on and on QJS) are defined similarly.

One would like to show that the map Ar from the space of equivalence classes of pairs (hab, x°*>) to the space Jifr*(~#) of equivalence classes of metrics is continuous with the Wr compact open topology on both spaces. In other words, suppose one has initial data hab e Wr(£f) and xab e W*'1^) which gives rise to a solution g e Wr(~#) on Jt. Then if "V is a region of Ji with compact closure, and e > 0, one would like to show there was some region'S/ of with compact closure and some 6 > 0 such that || g' - g, Tf ||r < e for all initial data (h'ab, x'ab)

such that ||h'-h,^fr < and ||x'-X.®ir-i < R This result may be true, but we have been unable to prove it. What we can prove is that this result holds if the metric is Cr+1)_. This follows immediately from proposition 7.5.1, taking g to be the background metric and to be some suitable neighbourhood of J~('V) n J+(6(y)). In fact if one examines lemma 7.4.6, one sees that the condition on the background metric can be weakened from C<r+1)~ to JF(r+1), but not to Wr, since the (r— l)th derivatives of the Riemann tensor of the background metric appear. (By the background metric being WT+1 we mean that it is Wr+1 with respect to a further Cr+1 background metric.) Thus the map Ar: from the equivalence classes of initial data to the equivalence classes of metrics will be continuous in the Wr compact open topology at every Wr+1 metric. Although the Wr+1 metrics form a dense set in the Wr metrics, there is a possibility that the map might not be continuous at a Wr metric which was not also a Wr+1 metric. However oo+1 = oo and so the map Am: £2*„(y)J?) will be continuous in the C00 topology on both spaces.

One can express this result as:

The Cauchy stability theorem

Let (JK, g) be the JF5+a (0 < a < oo) maximal development of initial data he W^S?) and xe and let "T be a region of J+((9(S))

with compact closure. Let Z be a neighbourhood of g in JSf 5+„( V) and % be an open neighbourhood in 6(£f) of i) 6(£f) with compact closure. Then there is some neighbourhood Y of (h, x) in Qs+a(&) such that for all initial data (h', x') £ Y satisfying the constraint equations, there is a diffeomorphism ¡x'.JC-^Ji with the properties

(2) M+t'eZ, where W, g') is the maximal development of (h', x')- CD

Roughly speaking what this theorem says is that if the perturbation of initial data on the Cauchy surface 6(Sf) is small on ) D 6(Sf), then one gets a new solution which is near the old solution in "V. In fact the perturbation of the initial data has to be small on a slightly larger region of the Cauchy surface than fl 6(S/')J since the null cones will be slightly different in the new solution and so "V may not lie in the Cauchy development ofJ~(T^) n

7.7 The Einstein equations with matter

For simplicity we have so far considered the Einstein equations only for empty space. However similar results hold when matter is present providing that the equations governing the matter fields xF(j/j obey certain physically reasonable conditions. The idea is to solve the matter equations with the prescribed initial conditions in a given space-time metric g'. One then solves the reduced Einstein equations (7.42) as linear equations with the coefficients determined by g' and with the source term T'ab determined by g' and by the solution for the matter fields. One thus obtains a new metric g" and repeats the procedure with g" in place of g'. To show that this converges to a solution of the combined Einstein and matter equations one has to impose certain conditions on the matter equations. We shall require:

(a) if {o^e W4+a(3t?) and {^¿e W3+a(3f) are the initial data on an achronal spacelike surface in a JF4+a metric g, there exists a unique solution of the matter equations in a neighbourhood of in D+(3f) with {¥«)}e W4+a(Jf") for any smooth spacelike surface Jf", and ¥w = 0*tt>. on A>;

(b) if {¥(J)} is a JF5+° solution in the JF6+a metric g on the set then there exist positive constants Q1 and 02 such that

«> w for any Wi+a solution {W'(i)} in the metric g' such that ||g'-g,^+||4+„<0i and U)

(c) the energy-momentum tensor Tab is polynomial in Va'j, ^u/j-.a and

Condition (a) is the local Cauchy theorem for the matter field in a given space-time metric. Condition (6) is the Cauchy stability theorem for the matter field under a variation of the initial conditions and under a variation of the space-time metric g. If the matter equations are quasi-linear second order hyperbolic equations, these conditions may be established in a similar manner to that for the reduced Einstein equations, providing that the null cones of the matter equations coincide with or lie within the null cone of the spacetime metric g. In the case of the scalar field or the electromagnetic potential which obey linear equations, these conditions follow from proposition 7.4.7. One can also deal with a scalar field coupled to the electromagnetic potential; one fixes the metric and the electromagnetic potential, solves the scalar field as a linear equation in that metric and potential, and then solves the electromagnetic field in the given metric with the scalar field as the source. Iterating this procedure one can show that one converges on a set of the form to a solution of the coupled scalar and electromagnetic equations in the given metric, providing that the initial data are sufficiently small. One then shows, by rescaling the metric and the fields, that for eil+ sufficiently small (as measured by the space-time metric g) one can obtain a solution for any suitable initial data. The same procedure willworkforanyfinite number of coupled quasi-linear second order hyperbolic equations, where the coupling does not involve derivatives higher than the first.

The equations of a perfect fluid are not second order hyperbolic, but form a quasi-linear first otder system. (For the definition of a first order hyperbolic system, see Courant and Hilbert (1962), p. 577.) Similar results can be obtained for such systems providing that the ray cone coincides with or lies within the null cone of the space-time with metric g. The requirement that the matter equations should be second order hyperbolic equations or first order hyperbolic systems with their cones coinciding with or lying within that of the space-time metric g, may be thought of as a more rigorous form of the local causality postulate of chapter 3.

With the conditions (a), (b) and (c) one can establish propositions 7.5.1 and 7.5.2 for the combined reduced Einstein's equations and the matter equations; from these, the local and global Cauchy development theorems and the Cauchy stability theorem follow.

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