By lemma 7.4.2, f |DB|« |DK|2dcr < ij2||B, Jf(t')n ^+||22||K, JT(t')n ^+||22, where, by condition (4) and lemma 7.4.3, ||B,^f(i')n <^+||2 < P3Q3. The term involving CPIQJ[d can be bounded similarly. Thus by lemma, 4.3.1 there is some constant Qs such that f (|D2K| + |DK|2)dcr<<26(f (|D2K|2+|DK|2)dor

+ Ç\\K, 3t(t') (] <^+||22d<'+ f |DF|2dori. (7.34) Jo J vu) )

< 2P42{||K, 3f(0) n WIS + IIF, ®(t)U). (7.35) Adding this to (7.34), one obtains

+Jj|K, JT(Í') n ®+|s,di' + |F,<8r(i)|1«}. (7.36)

where <2e = <26+ 2P4. By a similar argument to that in lemma 7.4.4, there is some constant Q7 such that

From lemma 7.4.1 it now follows that on

Using this one may proceed in a similar way to establish an inequality for || K, Jf(t)n The divergence of the 'energy' tensor now gives a term of the form qJ (|D8K|2+ |D2B|2|DK|2) dcr. (7.39)

By lemma 7.4.2 the second term above is bounded by

Qb fy ||B, Jf(n n l|K, mn n ^+||22, where by condition (4), ||B, ^'(É) n ^+03 is defined for almost all values of f and is square integrable with respect to f. Thus one can obtain an inequality for ||K, Jf?(t) n in the same manner as for || K, ^+¡2- The procedure for higher order derivatives is similar. □


There exist constants P%a and P7 a such that ||K, Jf(l)n ^+||4+a < Pe.jK, jT(0)n ®\i+a

and ||K,^+||4+a<P7 „{ditto}, where ua is some C8+a vector field on 34?(0) which is nowhere tangent to 3t(0).

By (7.20), the second and higher derivatives of K out of the surface ^f(O) may be expressed in tferms of F and its derivatives out of 3t(0), KJj,aua and derivatives of K in the surface 3f(0). By lemma 7.4.3,

The second result follows immediately, since t is bounded on □

We can now proceed to prove the existence of solutions of linear equations of the form (7.20). We first suppose that the components of A, B, C, F, u and g are analytic functions of the local coordinates x1, xxs and x4 (x* — t) on a coordinate neighbourhood V and take the initial data R*j = 0KIJ and Klj\av? = 1KIJ to be analytic functions of the coordinates x\ x2 and a;3 on 3^(0) n "V". Then from (7.20) one can calculate the partial derivatives ^(K^dt2, B^K^St* dx1, d^K^^dfi, etc. of the components of K out of the surface (0) in terms of derivatives of 0Kand xKin 3?(0). One can then express K*j as a formal power series in x1, x2, xz and t about the origin of coordinates p. By the Cauchy-Kowaleski theorem (Courant and Hilbert (1962), p. 39) this series will converge in some ball V(r) of coordinate radius r to give a solution of (7.20) with the given initial conditions. One now selects an analytic atlas from the C00 atlas of covers 0) n with coordinate neighbourhoods of the form "V"(r) from this atlas, and in each coordinate neighbourhood constructs a solution as above. One thus obtains a solution on a region ^(£4) for some f2 > 0. One then repeats the process using (¿2)- By the Cauchy-Kowaleski theorem, the ratio of successive intervals of t for which the power series converges is independent of the initial data and so the solution can be extended fco the whole of in a finite number of steps. This proves the existence of solutions of linear equations of the form (7.20) when the coefficients, the source term and the initial data are all analytic. We shall now remove the requirement of analyticity.

¡A,^(0)nn>+«< |B,jr(0)n*| i+a<P3Q> |C, Jt(0) n <^||2+a < PsQt

Thus there will be some constant Qt such that

Proposition 7.4.7

(6) 0Ke Wi+a(J^(0) n W), xKe TP+°(.?f (0) n then there exists a unique solution Ke Wi+aC%+) of the linear equation (7.20) such that on Jf(0), K*j = 0KTj and KTJiaua = Ji1^

We prove this result by approximating the coefficients and initial data by analytic fields and showing that the analytic solutions obtained converge to a field which is a solution of the given equations with the given initial conditions. Let A„ (n — 1,2,3,...) be a sequence of analytic fields on which converge strongly to A in Wi+a(<%+). (An is said to converge strongly to A in Wm if ||An —Aj|m converges to zero.) Let Bn, Cn and Fn be analytic fields on which converge strongly to B, C and F respectively in T73+a(<2f+), and let 0K„ and 1Krt be analytic fields on Stif (0) n <25f which converge strongly to 0K and XK in Wi+a(3f(0) n and n respectively. For each value of n there will be an analytic solution Kn to (7.20) with the initial values Knxj = 0KnTj, KnTJ]aua = Jt^j. By the corollary to lemma 7.4.6, ||K„, <%+\\1+a will be bounded as w->oo. Therefore by a theorem of Riesz (1955) there will be a field Ke Wi+a(%+) and a subsequence K„. of the K„ such that for each b, 0 < b < 4 + a, D6K„. converges weakly to DbK. (A sequence of fields In*j on jV is said to converge weakly to Pj if for each C® field JTj, f V^d^f I'jJ'idv)

Since A„, Bn and Cn converge strongly to A, B and C in W3(<%+), sup|A —A„|, supjB —Bn| and sup |C —Cn| will converge to zero. Thus i/n.(Kn.)will converge weakly to £(K). But Ln.(Kn.) is equal to Fn. which converges strongly to F. Therefore £(K) = F. On Jff(0) fl & Kn.Tj and Kn.TJlaua will converge weakly to Krj and KTJiaua which must therefore be equal to 0KIJ and 1KIJ respectively. Thus K is a solution of the given equation with the given initial conditions. By proposition 7.4.5 it is unique. Since each K„ satisfies the inequality in lemma 7.4.6, K will satisfy it also. □

7.5 The existence and uniqueness of developments for the empty space Einstein equations We shall now apply the results of the previous section to the Cauchy problem in General Relativity. We shall first deal with the Einstein equations for empty space (Tab = 0), and shall discuss the effect of matter in §7.7.

The reduced Einstein equations

are quasi-linear second order hyperbolic equations. That is, they have the form (7.20) where the coefficients A, B and C are functions of K and DK (actually, in this case A^ = gab is a function of <fiab and not of ^>abtc). To prove the existence of solutions of these equations we proceed as follows. We take some suitable trial field <p,ab and use this to determine the values of the coefficients A, B and C in the operator E. Using these values we then solve (7.42) as a linear equation with the prescribed initial data and obtain a -new field <pab. We thus have a map a which takes <f>' into <f>", and we show that under suitable conditions this map has a fixed point (i.e. there is some $ such that a(<f>) = <f>). This fixed point will be the desired solution of the quasilinear equation.

We shall take the background metric § to be a solution of the empty space Einstein equations and choose the surfaces (t) n and n <25r+ to be spacelike in Then by lemma 7.4.1 there will be some positive constants Qa such that if for some value of a ^ 0

then the coefficients A', B' and C' determined by <p' satisfy conditions (1), (2) and (4) of lemma 7.4.6 for given values of and Qs. From (7.41) one then has

Thus the map a: Wl+a{<%+) will take the closed ball W(r)

of radius r (r < Qa) in Wi+a(1t+) into itself provided that

We shall show that a has a fixed point if (7.44) holds and if r is sufficiently small.

Suppose ipi and <f>2 are in W(r). The fields = a(4>i) and <f>2" = a(<f>2) satisfy = 0, E2'(<f>2") = 0 where is the

Einstein operator with coefficients A/, B/ and C/ determined by <f>y.

Since the coefficients A^, Bx' and C/ depend differentiably on <f>y' and D</>i for tpi in W(r), there will be some constant Qi such that on

|B'X-B'2| < QM'i-W + Wi-WJh (7-46) IC^-C',! < «.(l^-^l + ID^'i-D^',!)..

Therefore by lemmas 7.4.1 and 7.4.6,

|(E\-E'2)(4>\)\ < Srg.A^A.ail^'i-^l + ID^-D^I). We now apply lemma 7.4.4 to (7.45) to obtain the result

- 4>\ <*+lli < rQsWi ~ <*>'*> ^+111. (7-47)

where Qs is some constant independent of r. Thus for sufficiently small r, the map a will be contracting in the |[ norm (i.e. ||a(0i)- a(02)|| i < ~ ^ii) the sequence ctn(<p\) will converge strongly in Wl(%+) to some field <f>. But by the theorem of Riesz some subsequence of the an(<f>\) will converge weakly to some field $ e W(r). Thus <p must equal and so be in W(r). Therefore a(<J>) will be defined. Now

As n ->oo, the right-hand side tends to zero. This implies that ||a($) — </>, = 0 and so that a(<f>) = </>. Since the map a is contracting the fixed point is unique in W(r). We have therefore proved:

Proposition 7.5.1

If § is a solution of the empty space Einstein equations, the reduced empty space Einstein equations have a solution <f>e Wi+a(%+) if f0<£, jr(0)n^f4+a and 11^,^(0)n<^j[3+a are sufficiently small. 0) n wiU be bounded and so <p will be at least C(2+a)-. □

This solution will be locally unique even among solutions which are not in Wl(<%+).

Proposition 7.5.2

Let $ be a C1- solution of the reduced empty space Einstein equations with the same initial data on an open set "f c jf(0) f| Then <J> = <f> on a neighbourhood of "V in

Since $ is continuous one can find a neighbourhood W of "V in ^ such that the conditions of lemma 7.4.4 hold for A, B and C. As before one has E($-<f>) = -(i!-E)(<t>). (7.48)

Similarly there will be some Q6 such that

Applying lemma 7.4.4 to (7.48) one obtains an inequality of the form da;/df $ QiX, where * = - <f>, 3f(t') n «f+^di'.

Proposition 7.5.1 shows that if one makes a sufficiently small perturbation in the initial data of an empty space solution of the Einstein equations one obtains a solution in a region What one wants however is to prove the existence of developments for any initial data h^ and xab which satisfy the constraint equations on a three-manifold Sf. To do this we proceed as follows. We take to be Ri, e to be the Euclidean metric and § to be the flat, Minkowski metric (this is a solution of the empty space Einstein equations). In the usual Minkowski coordinates x1, x2, a? and x* (x* = t) we take to be such that d%(\ is spacelike and Jf?(0) n consists of the points for which (a;1)2-!- (z2)2 + (x3)2 ^ 1, x* = 0. The idea now is that any metric appears nearly flat if looked at on a fine enough scale. Therefore if one maps a sufficiently small region of onto 3^(0) n one can use proposition 7.5.1 and obtain a solution on We then repeat this for other portions of if and join up the resulting solutions to form a manifold Jt with metric g which is a development of to).

Let "fy be a coordinate neighbourhood in £f with coordinates y1, y2 and y3 such that at p, the origin of the coordinates, the coordinate components of hai equal dab. Let ^(/J be the open ball of coordinate radius about p. Define an imbedding ^(fi) -> by xi = /1_1i/i (i = 1,2,3), x* = 0. By the usual law of transformation of a basis, the components of 6ithab and 6*Xab with respect to the coordinates {a;} are /i-2 times the components of hab and with respect to the coordinates {y}. We define new fields h,ab and x>ab on by h,ab = f12hab and x>ab = fiZXab- Then since h is continuous (in fact C2+a) on one can make g,ab-{jab and g>abicuc arbitrarily small on Jf(0) n by taking fx sufficiently small, where g>ab and g'ab[cuc are defined from h>ab and x>ab in the manner of §7.3. The derivatives ofg>ab and g,abicuc in the surface J4?(0) will also become smaller as fx is made smaller. Thus *%]\i+a and J^', .?f(0)n ^¡[3+a can be made small enough that proposition 7.5.1 can be applied and a solution for <f>' obtained on Then gf6 = f1~2g'ab will be a solution of the reduced Einstein equations with the initial data determined by A06 and xab-Similarly one can obtain a solution on the part ofon which t < 0.

One can now cover if by coordinate neighbourhoods "f~a(fa) of the form ^(/x), map them by imbeddings 6a to neighbourhoods of the form and obtain solutions gj^ on °Ua. The problem now is to identify suitable points in the overlaps to make the collection of the °lla into a manifold with a metric g. To do this we make use of the harmonic gauge condition

By the definition (7.3) of ¿r°6c, this is equivalent to g^ST"^ = 0. Therefore for any function z,

If the background metric is the Minkowski metric and z is one of the Minkowski coordinates x1, x2, a;3 and xthe right-hand side of (7.50) will vanish. Suppose now one has an arbitrary Wi+a Lorentz metric g on a manifold Jl. In some neighbourhood <=■ Jt one can find four solutions z1, z2, z3 and zi of the linear equation z;ab9ab = 0 (7.51)

which are such that their gradients are linearly independent at each point of W. We may then define a diffeomorphism by a^ = z° (a = 1,2, 3,4). This diffeomorphism will have the property that the metric fi-^g^ on will satisfy the harmonic gauge condition with respect to the Minkowski metric § on ^t. Thus if the metric g is a solution of the Einstein equations on the metric ¡x^ g will be a solution of the reduced Einstein equations on ^with the background metric

The procedure to identify points in the overlap between two neighbourhoods °lia and is therefore to solve (7.51) on °lla for the coordinates zf, Zp3 and Zpl using the initial values for z^a and Xpalbub determined by the overlap of the coordinate neighbourhoods "Ka and ■Vp on Sf. In fact xfi\aua =0 (i = 1, 2, 3) and xp\aua = 1 where ua = 8l8xaa is the unit vector in orthogonal to Stf (0) in the metric Thus Zp1 = xa* though x^ will not in general be equal to xj. By proposition 7.4.7. the coordinates xfia will be C(2+a)_ functions on °Ua. (In proposition 7.4.7 the background metric with respect to which the covariant derivatives are taken has to be C(6+a)~. Thus it cannot be applied directly to (7.51), since the covariant derivatives are taken with respect to g, which is only T74+a. However one can introduce a C6+a background metric g and express (7.51) in the form

Was this article helpful?

0 0

Post a comment