An existence theorem for the Cauchy problem on the light-cone for the vacuum Einstein equations with near-round analytic data Текст научной статьи по специальности «Математика»

Том 153, кн. 3

УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА

Физико-математические пауки

2011

UDK 530.12

AN EXISTENCE THEOREM FOR THE CAUCHY PROBLEM ON THE LIGHT-CONE FOR THE VACUUM EINSTEIN EQUATIONS WITH NEAR-ROUND ANALYTIC DATA

Y. Choquet-Bruhat, P.T. ChruscAel, J.M. Martin-Garcia

Abstract

A class of characteristic general rolat.ivist.ic initial data satisfying a near-roundness condition at. tlio tip of a light.-coue is introduced. It. is shown that, for any such analytic data there exists a corresponding solution of the vacuum Einstein equations defined in a future neighborhood of the vertex.

Key words: characteristic Caucliy problem on a light-cone, vacuum Einstein equations.

Introduction

In the paper fl] we studied the Cauchy problem for the Einstein equations with data on a characteristic cone CO • We used the tensorial splitting of the Ricci tensor of a Lorentzian metric g on a manifold V as the sum of a quasidiagonal hyperbolic system acting on g and a linear first order operator acting on a vector H, called the wave-gauge vector. The vector H vanishes if g is in wave gauge; that is, if the identity map is a wave map from (V, g) onto (V, g), with g being some given metric, which we have chosen to be Minkowski. The data needed for the reduced PDEs is the trace, which we denote by g, of g on CO • However, because of the constraints, the intrinsic, geometric, data is a degenerate quadratic form g on Co • Given g, the trace g is determined through a hierarchical system of ordinary differential equations1 along the rays of CO, deduced from the contraction of the Einstein tensor with a tangent to the rays, which we have written explicitly and solved. We have called these equations the wave map gauge constraints and shown that they are necessary and sufficient conditions for the

solutions of the hyperbolic system to satisfy the full Einstein equations. We have also

g

inducing a given g (for details see fl]). Further references to previous works on the problem at hand can be found in fl].

Existence theorems known for qnasilinear wave equations with data on a characteristic cone give also existence theorems for the Einstein equations, if the initial data is Minkowski in a neighbourhood of the vertex. For more general data problems arise duo to the apparent discrepancy between the functional requirements on the characteristic data of the hyperbolic system and the properties of the solutions of the constraints, due to the singularity of the cone CO at to vertex O. The aim of this work is to make progress towards resolving this issue, and provide a sufficient condition for the validity

O

fast-decay conditions of [4]. More precisely, we prove that analytic initial data arising

1For previous writing of these equations in the case of two intersecting surfaces in four-dimensional spacetime, see Rendall [2] and Damour Schmidt [3].

from a metric satisfying (3.3). (3.4) together with the "noar-roundnoss" condition of Definition 6.1 lead to a solution of the vacuum Einstein equations to the future of the light-cone.

1. Cauchy problem on a characteristic cone for quasilinear wave equations

The reduced Einstein equations in wave-map gauge and Minkowski target are a quasi-diagonal, quasi-linear second order system for a set v of scalar functions v1, I = 1,..., N, on Rn+1 of the form

A^(y,v)D^v + f (y, v, Dv) = 0, y = (yA) e Rn+1, n > 2, f = (f1). (1.1)

If the target is the Minkowski metric and takes in the coordinates ya the canonical form

n

n = -(dy0)2 + ^>yi)2, (1.2)

i= 1

then.

dv1 02vI

dv=d^> a, m = 0,1 ,...,n. (1.3)

We will underline components in these ya coordinates.

In the case of the Einstein equations the functions AxM = gxM do not depend directly on y, they are analytic in v in an open set W C RN. For v e W the quadratic form gAM

is of Lorentzian signature. The functions f1 are analytic in v e W and Dv e R(n+1)N,

y

The characteristic cone CO of vertex O for a Lorentzian metric g is the set covered by future directed null geodesies issued from O. We choose coordinates ya such that the coordinates of O are ya = 0 and the components aam(0, 0) take the diagonal Minkowskian values, (-1,1,..., 1) .If v is C1,1 in a neighbour hood U of O and takes its values in W there is an eventually smaller neighbourhood of O, still denoted U, such that CO n U is an n dimensional manifold, differentiable except at O, and there exist in U coordinates y := (ya) = (y0, yi, i = 1,n) in which CO is represented

O

co := {r - y0 = 0}, r := {^(yi)^1/2 , (1.4)

CO

tangent to the vector (' with components = 1, = . Inspired by this result and following previous authors we will set the Cauchy problem for the equations (1.1) on a characteristic cone as the search of a solution which takes given values on a manifold

v

v = (1.5)

where overlining means restriction to Co • The function ip takes its values in W and is such that (' is a null vector for A, i.e. when A = g

= loo + Vloi + '' : = 0. (1.6)

We use the following notations:

CO := co n {0 < t := y0 < T}, YO := {y0 > r} , ^^e toerior of CO, YO := yo n {0 < y0 < T}.

arid wo sot

£T := CO n {y0 = t}, diffeomorphic to Sn-1 ,

ST := YO n {y0 = t}, diffeomorphic to the ball Bn-1 .

We recall the following theorem, which applies in particular to the reduced Einstein equations.

Theorem 1.1. Consider the problem, (1.1), (1.5). Suppose that:

1. There is an open set U x W C Rn+1 x RN, Y^ c U where the functions gxM are smooth in y and v. The function f is smooth2 in y G U and v G W and in Dv G R(n+1)N.

2. For (y, v) G U x W the quadratic form gxM has Lorentzian signature; it takes the Minkowskian values for y = 0 and v = 0. It holds that <(O) = 0.

3a. The function < takes its values in W. The cone C^ is null for the metric

gv(y,<)-

3b. < is the trace on CO of a smooth function in U.

Then there is a number 0 < T0 < T < +<» such that the problem (1.1), (1.5) has v yT0 which can be extended by continuity to a smooth function defined on a neighbourhood of the origin in Rn+1.

< T0 = T

2. Null adapted coordinates

It has been shown (see fl] and references therein) that the constraints are easier to solve in coordinates xa adapted to the null structure of CO, defined by

x0 = r - y0, x1 = r andxA = ^A(r-1yi), (2.1)

where A = 2,..., n, local coordinates on the sphere Sn-1, or angular polar coordinates. Conversely

n

y0 = x1 - x0, yi = r©i(xA) with J3©i(xA)2 = 1.

i=1

x

n = -(dx0)2 + 2dx0dx1 + (x1)2sn-1, (2.2)

with

sn-1 := sAB dxAdxB , the metric of the round sphere Sn-1.

Recall that in these coordinates the non zero Christoflel symbols of the Minkowski metric are, with SAC, the Christoflel symbols of the metric s,

ff4 = —J;® , YBic = 5'fc. , r'^ = —x1sab , Tab = ~x1sab ■ (2-3) x1

O x0 = 0

Q

xA = constant, so that (' := ——¡- is tangent to those geodesies. The trace g on Co of

axL _ _

the spacetime metric g that we are going to construct is such that gn = 0 and g1A = 0;

we use the notation

g = g00(dx°)2 + 2fodx°dx1 + 2vAdx°dxA -\-gABdxAdxB, (2.4)

means Cm, with m being some integer depending on the problem at hand and the considered function. In particular C^ and Cw (real analytic functions) are smooth.

g

for a metric g to have such a trace on a null cone x0 = 0.

The Lorentzian metric g induces on CO a degenerate quadratic form g which reads in coordinates x1, xA

g = gAB dxA dxB, (2-5)

i.e. gn = giA = 0 while gABdxAdxB = gABdxAdxB is an x1 -dependent Riemannian metric on Sn-1 induced on each by g, we denote it by g^. While g is intrinsically-defined. it is not so for g00, z/q • va , they are gange-dependent quantities.

gg yi CO

dy0 yi

inclusion mapping of Co in the coordinates ya is y° = r, hence —- = — .it holds

dyi r

that

:I l/i'dn'. with gi:j = r^y'y^g^ + r^1(y3g0i + y'g0j) + g^. (2.6)

For Theorem 1.1. to apply to the wavc-gangc reduced Einstein equations, the corii-

y CO

spacetime functions. The solution of the reduced equations satisfy the full Einstein equations if and only if these initial data satisfy the wave-map gange constraints. We have

x

are admissible coordinates for Rn+1 only for r > 0. The change of coordinates from x to y, smooth for r > 0, is recalled below; the components of a spacetime tensor T in the coordinates x are denoted by , while in the coordinates y they are denoted by Ta,3.

Lemma 2.1. It holds that:

,y m , ^

Too = Too, Tn = Too + 2—Tm + ^oi = —{Too +Ty¿€>1),

dO* dO* dO* d©j

Conversely, if T1A = Tn = 0, then

dxA

Too = Tío, T0i = -(Too + Toi)»'~ V - Tqa—— , --— dy>

Zk - + 2T01)r-W + ToAr-^ + y^) +

Hereinafter, we shall often abbreviate partial derivatives as follows

d „ d d

= T7> ' = TT' = 0 4 , dxu dx1 dxA

an = A q. = A

dy°' — dy'

3. Characteristic data

3.1. Basic (intrinsic) characteristic data. The basic data on a characteristic

cone for the Einstein equation is a degenerate quadratic form. We will define this data

Cg

C CO

x0 = r — y0 = 0. If we take the yi as coordinates on CO, it holds that, see (2.6),

C = r.;,/,, ,/,,< With Cy = Cij + r" VIk + y'CjO) + r-2ylyi~Coo , (3.1)

where Caß are the components in the y coordinates of the trace C of C on Co (not to be mistaken with the induced quadratic form C). Assuming that CO is a null cone

for C with degenerate.

for C with generators £_ = 1. = —. (1.6) implies that the quadratic form C is

r

y l * ~ — y l — y l ¡J * —

— — Cij = Con + 2— Cpj + Cij = 0. (3.2)

Lemma 3.1. Two spacetime metrics C and, C' with components linked by Ci/ := Cjj_ + r"1 (aiyj + aji/i) + r~2ay'yj, Ci0' := Cj_o - oi; Coo' := Coo - a,

with arbitrary a. and a induce on Co the same quadratic form C, i.e. C.j = Cij. Proof. Elementary calculation using the identity written above. □

In what follows, to simplify computations we will make the restrictive condition that Coi = 0, Coo = —1, i.e. (3-3)

The set {y° = ?•} is then a null cone for the metric C, with generator ^ = 1, P_ = —,

r

if and only if

yiCii = yj (3.4)

(compare [op.

y

dinates x\ xA gives

C = Cab dxAdxB with CAB = CAB, C\A = Cn = 0,

with CAB ^^^^g ^^e components of a x1 = r-dependent Riemannian metric on the sphere Sn-1

f yl \ _ _ dyl _

— I Cqq + ~Coi J = Cn = 1, Coa = ~-g^x Coj = 0, Coo = Coo = —1.

C

C00 = C0A = C1A = 0, C01 = C11 = 1,

_AB

while C are the elements of the inverse of the positive definite quadratic form with CAB

3.2. Full characteristic data. We have seen in [1] that the trace g of a Lorentzian metric g satisfying the reduced Einstein equations is a solution of the full Einstein equations if and only if it satisfies the wave map gauge constraints. These constraints Ca =0 are deduced in vacuum from the identity satisfied by the Einstein tensor S : ^

1 S aft = Ca + La

where Ca is linear and homogeneous in the wave gauge vector H while Ca depends only on g and its derivatives among Co and the given target g. Given g, i.e. "gab = Cab , glA = gn = 0. the remaining components z/q = goi' VA = 9oa< 9oo aro determined by the constraints and limit conditions at the vertex O which can always be satisfied

by choice of coordinates (see [1]). The Cagnac Dossa theorem applies to components in the y coordinates. Lemma 2.1 gives

dxA

goo = goo> goi = -(goo + v0)»'" V - Vi, with Ui_ := , (3.5)

_ dxA dxB i + vjy±) + 9ae

while, for the chosen metric C and gAB = Cab

9ij_ = (goo + 2i/o)r W + r 1 [ylVj_ + y3 Ui) + gAB - ,

-pr ,• _ dxA dxB

L:, r ~yly° + gAB-

Therefore.

- T woo T -^u - i ji yy T < U^TI/^

ffii = Cij + (ffoo + 2^0 " l)r-2yV + + (3.6)

4. Null second fundamental form

We have defined in I the null second fundamental form of (CO, g) as the tensor x on CO defined by the Lie derivative3 with respect to the vector I of the degenerate quadratic form g, namely in the coordinates x1 , xA :

Xab ■= \{c-eg)ab = \d{gAB, (4.1)

Xai := \(Ceg)ai = 0, xn := \(^g)n = 0. (4.2)

In view of the application of the Cagnac Dossa theorem, we look for smooth extensions. We define a smooth spacetime vector field L, vanishing at O and with trace d

colinear with i = ——- on Co. by its components in the xa and ya coordinates dx1

respectively:

L := yx^~r = x°—^ + x1—^-r, hence L = x1( = rl. dyA dxu dx1

We assume that the metric C is smooth in U, a neighbourhood of O in Rn+1, i.e. its components Cj are of class Cm, with m as large as necessary in the considered context, functions of the ya. We define a symmetric Cm-1 2-tensor X, identically zero in the case where C = n, the Minkowski metric, by:

X:=±CLC-C. (4.3)

In y coordinates one has4, using y®Cjj = yj

Xoo = = 0, Xa = -!// 'IC:; + !,'n,( (4.4)

1

2

with

y'dqCjj = 0,

3Recall that in arbitrary coordinates x1 the Lie derivative reads

(LeC)HK = I1 di Chk + CHI dK I1 + CKI dn I1.

4 Recall that to underline components in the y coordinates and overline restrictions to Co .

and, using dhyl = Slh

.'/ //' 'h <'; = yhMy'Cii) ~ yhCi1dhyi = 0, (4.5)

which imply

/■ .V.; = // -V:; = 0. (4.6)

In x coordinates we find, using the values of the components C0a and C1a of the metric C, that the tensor X obeys the key properties

XM0 = 0, XM1 = 0, (4.7)

while

Xab = ^(x°d0CAB + x1d1CAB) - Cab ■ (4.8) XAB Co

Xab = \x1diCab ~ Cab = x1xab ~ gab- (4-9)

By X we still denote the mixed Cm-1 tensor on spacetime obtained from X by lifting an index with the contravariant associate of C; its y components are the Cm-1 functions

2 Xl = C2l {y0d0Ca(3 + g:":<\ ,j.

C

Xj = \c^{y°d0Cih + g1/>,(■:, = = (4.10)

V'/-; »• (4.11)

where the index of L has been lowered with the metric C, so that this is equivalent to (4.6). In x coordinates X^ are the only non-vanishing components of X. Their traces Co

_c i

XA = -xlgBCd{gAB - 5°A = x1^ -

hence

1_Rr'rs _ 1 —C

Xa '■=

and

XCa ■= ^9bcdi9äb = -k + <*a), (4-12)

2 x1

1 _ 4 D „ _ tl- X n—i , ^ "» \

r := -gABdigAB = — + -j-- (4.13)

The trace of the tensor X is the Cm-1 function

tvX = XS=X£ = CabXab. (4.14)

Co

tvX = X^ = gABXAB = - gAB d{gAB - (n - 1), (4.15)

|xf := XaXC = {Xi X'l- + + « " 1} • (4-16)

l

5. A critérium: admissible series

To show that the integration of the constraints, which appear as ODE in x1, leads to traces on the cone of smooth spacetime functions, we shall nse the following lemma, introduced by Cagnac (unpublished) for formal series, but used here for real analytic functions, a special class of Cœ functions.

Lemma 5.1. A function is the trace f on CO of a spacetime function f analytic in U n Y"O, U is a neighbourhood, of O, if and only if it admits on U n CO a convergent expansion of the form

z^/o+EV' (3-D

p=i

with

7; J,.:....:»' +7'p,il...il>-1©<1 •••©<p-1, (5.2)

where fo, fp,i1...ip and f jil...ip_1 are numbers. Such a series is called an admissible series. A coefficient fp of the form (5.2) is called an admissible coefficient of order p.

Proof. If / is analytic it admits an expansion in Taylor series

^^ 1 dP f

/ = ^2 fai...apyai ■ ■ ■ y°'P, fai...ap ■= -f dy0l ^ ^ ^ QyClp (O). (o.3)

One goes from the formulas (5.3) to (5.1), (5.2) by replacing y® by r6^d yo by r, and conversely, in Q fl Co or in Q. □

Remark 5.1. The identity (5.1) is equivalent to saying that / is of the form f = f 1 + rf2, wit h f1 and f2 analytic fonctions of y®.

We say that an admissible series is of minimal order q if the coefficients fp are identically zero for p < q.

Proposition 5.1. If the metric C is analytic and satisfies the conditions (3.3), (3.4) then the functions trA" and |A"|2 are admissible series of minimal orders 2 and 4 respectively.

The following lemmas will be very nsefnl when integrating the constraints.

Lemma 5.2. If fp and hq are admissible coefficients of order p and q respectively, then fp + hp and fphq are admissible coefficients of order p and p + q respectively.

Proof. Elementary computation of (fp + hp)r^d fphqrp+q replacing rO^y y® and r2 by Sj(y')2. □

Suppose that f and h are admissible series of minimal orders q^d qh. The following are easy-to-check consequences of the lemma:

1) f h is an admissible series of minimal order qf + qh;

2) if qf = qh, the n f + h is an admissible series of the same minimal order;

3) if f (0) = 0 Mid qf = 0, then 1/f is an admissible series of also minimal order 0;

4) rd1f is an admissible series of minimal order qf, unies s qf =0 and then it has a larger minimal order.

Lemma 5.3. If k and h are admissible series with h of minimal order qh > 1 and the constant k0 = k(0) > 0, then the ODE

rd\f + kf = h (5.4)

admits one and only one solution f which is also an admissible series of the same minimal order qh as h. The result extends to qh = 0 if k0 > 0.

Proof. Expand

www

f = £ fPrP, k = £ kprp, h = £ hprp, (5.5)

p=0 p=0 p=qh

honco

w

rdif = ^2 pfprP'

p=i

plug into the ODE (5.4) and proceed to identifications. We obtain by equating to zero the constant term

kofo = ho, (5.6)

h0 = 0 k0 = 0

case where h0 = 0, i.e. qh > 1, and take f0 = 0. We get the successive equalities

h+k0h = hu i.e. /1 = -A_, (5.7)

1 + k0

f0 = 0

p-i

(p + k0)fp kq fp-q = hp . (5.8)

q=l

For p < qh, we have hp = 0 ^^d ^^e recurrence relation gives fp = 0. Therefore, the

fh

U, = (5-9)

qh + k0

We assume the series for k and h converge for all directions ©* and radius cr < 1; that is, we assume that there exists a constant c such that

|kp| < cp, and |hp| < cp. (5.10)

k0 > 0

, „ . hi c

Assume now that

|fp| <cp for P<P0, (5.11)

p

\fP\ < ^ < • (5.12)

p + k0

The bounds on |fp| show that the series for f also converges. It is an admissible series of minimal order qf = qh.

When qh = 0, i.e. h0 = 0 and k0 = 0 we take

, ho

Jo — y ko

and we set

F = f - fo.

It satisfies the equation

rd1F + kF = H, with H := h - kf0. (5.13)

We have

Ho = 0

and we apply the previous result to F. □

Corollary 5.1. If f and h are admissible series related by (5.4) and p + k0 > 0, and r-p h is an admissible series then r-p f is an admissible series of the same minimal order.

Proof. Set f = rp^. If f satisfies (5.4) then ^ satisfies the equation

rdift + (p + k)^ = r-ph.

Remark 5.2. The following example is a case of a differential equation of the form (5.4) with qh = 1, but k0 is a negative integer, which does not admit as a solution an admissible series. Let

— — — 1 — r — r2 — r3 — ■■ ■ , h = --7T = r + r3 + r5 + • • • ,

r — 1 1 — r2

We can solve the ODE explicitly,

f=^T(foc+log^ r — 1 r

with fw an arbitrary which cannot be expanded in powers of r

near 0. However if we ch ange k to r/(r —1), the n k0 changes fr om — 1 to 0, the problem disappears. Remark that the problem also disappears if we change h to r2/(1 — r2), i.e. qh = 2.

In what follows, we will assume the metric C, of the form (3.3) and satisfying (3.4) is analytic, takes Minkowskian values at the vertex O, and is such that the components of its trace on CO satisfy

Cij = Sij + cij, = Sij + tL, (5.14)

where and ç^. have admissible expansions of minimal order 2 while dp ca has an admissible expansion of minimal order 1. The definition (4.10) implies then that

^V' -2LL\r7TTT ■ ,/7^] (5.15)

has an admissible expansion of minimal order 2.

6. The first wave-map gauge constraint

Wo have deduced our first constraint (see [1]) from the identity

tsw = ru = -dit + v°d1v0t - r2t(ti +t)~ xa xb, (6-d

with

ri=Wi + Fi, W^-uog^rsAB. (6.2) Hence for the first wave-map gauge constraint in vacuum we have the equation

ci := -dlt + v°div0t - ir(r - uo gabrsab) ~ xa xb = (6-3)

When is known this equation reads as a first order differential equation for v°

^d^o = r-ldlT + i(r - v0gABrsAB) + r-lxl xb- (6.4)

It can be written as a linear equation for v° — 1,

div0 + a(v° — 1)+ b = 0, (6.5)

with

a-t^drt+^r + t-^xw \x\2=xaxb, (6-6)

b:=a-^gABrsAB- (6.7) In the fiat case ~gAB = i]AB , r = ——-, Xa = ~ ^A equation reduces to:

diV° + \{V°- 1)—=0; 2 r

it has one solution tending to 1 when r tends to zero, v° = 1. In the general case (6.5) reads, with f := v° — 1,

rdif + kf + h = 0, k := ar, h := br = ar - ^ gABr2sAB • (6.8)

Recall that x1 = r and

Hence

i i _r,

xca = - 9b°di9âb = ~ (xa + sca). (6.9)

2 r

2 |A|2 + 2trA + n - 1

M — „9 '

tr X n — 1 -, r

t=-+-, t-1 =-6.10

r r «-1 + trA

1

where tr A" is an admissible series of minimal order 2. The function < 1 H--tr A"

n— 1

is the trace of a Cw function as long as 1 H---tr A" does not vanish, hence always

n— 1

in a neighbourhood of O sinee tr X vanishes there.

It holds that

n-l + trX ditvX

д\Т =--^--1--, (6.11)

(6.12)

r 77,- 1+trX

Also we cari write

т

^ы2^ '"у тт-^ ^ = г1 +-• (o.i3)

|Х |2 + 2tr X + П-1 _ \Х\2 + tr X

г ( п - 1 + tr X ) г ( п - 1 + tr X ) Г

Finally computation gives

п - 1 гдгtrX + trX + IXI2

k = ar=-^- + —- —1 ' • (6.14)

2 n _ 1 + tr X

n_1

We see that in a neighbourhood of О, к---— admits an admissible development

of minimal order 2.

On the other hand, since in the ж coordinates nia = 0> r2sAB = пав and we have assumed that С = С = 0, С = 1, g = С , we have

—AB 2 _ —AB _ ТЧАВ _ 7о

g г sab = g 'пав = с -пав = с i]aß -1. Hence using now the values of Caß

l-gABr2SAB = ^ (1 + C'iSjj — 2) = ~~ТГ~ + (6.15)

where cij Sij has an admissible development of minimal order 2. We conclude that

_rd 1 tr X + tr X + IX12 1 — (n- 1 + trX) 2-

h EE —^-' ' - - C« <Sy (6.16)

admits also such a development. Lemma 5.3 applies, and we have proved the following theorem:

Theorem 6.1. If the basic characteristic data are induced on Co by a Cu (i.e. analytic) metric of the form (3.3), hence satisfying (2.4), then v0 — 1 admits an admissible expansion of minimal order 2, hence is the trace in a neighbourhood of O of a Cu spacetime function. Then v0 = = 1./!îû neighbourhood

of O, it holds that v0 = N0, N0 = (X0)-1.

In the expression of the characteristic initial data in y coordinates there appears r-2(v0 — 1) , which, though being continuous on each ray as r tends to zero, is not an admissible expansion. We introduce the following definition.

Definition 6.1. A metric C satisfying (3.3), (3.4) is said to be near-round at the vertex if there is a neighbourhood of O where r_1Cjj = ciy and r~2cij5^ = D with dkj and D being admissible series.

C Cij has an analytic extension Cij of the form,

with dij being analytic extension of d^ ,

Cij = ^ij + cij 7 cij = y dij ,

therefore,

¡), < ':; EE 0 for ,, II. llOllCO il, C ' EE 0 for ,/ II.

arid

Cij = Sij + y°S1 with cf£_ being some analytic functions. Hence

C^Cij = {:y0f{D+£i.dij}.

Using the definition of we see that if C is near-round at the vertex, then

Xo£ = y°Yo0, with Yjj := 1 {djj + y°d0dij + u'n, >!:;]. Ym=Yqo=0. (6.17) Hence

trX = y° tr V. with tr V ^ + + yhdhdM).

An elementary computation shows that tr Y is of the following form (with Z being an analytic function)

tr Y = y°Z, hence tr X = (y°)2Z.

On the other hand,

x[ = C^Xjh = y^Yjh ■■= r 17,

therefore.

\X\2=XlXi=(y°)2^Yi.

We deduce from these formulas that if C is near-round at the vertex then r~2 trX = Z and r~2 |A"|2 = |y|2 are admissible series.

Theorem 6.2. A sufficient condition for r-2(v° — 1), with v° solution of the first wave-map gauge constraint, to have an admissible expansion is that the Cw metric C given by (3.3) which induces the basic characteristic data be near-round at the vertex.

Proof. Since f := v° — 1 satisfies Eq. (6.8), ^ := r-2(v° — 1) satisfies

rdi^ +(2 + &)<£ + r-2h = 0. (6.18)

The expression (6.16) shows that for a metric C round at the vertex r-2h admits an admissible development, the application of Corollary 5.1 gives the result. □

7. The Ca constraint

We have written in [1] the CA constraint in vacuum as

CA = + TU) + VBXA ~ T/AT + dA QWi + vodiv0^ ,

where is defined as

u := -2v°diVA + I/' /' X m + (V° - ^ uA + gABgCD(SBD - fBGD). (7.1)

Using the first constraint we find

VodlV° + ^W1 = -a, (7.2)

where a is given by (6.14), hence

dA[a+^r) = r-ldAF( trX,|X|2),

where

■ 2 rd itr X + tr X + |X|2 1-

F(trX, IXI2) := —^-!--_' ' + -trX =

V '' 1 7 n-l + trX 2

rd itrX + ±{(n + l)trX + |trX|2} + |X|2 n-l+ trX

O

We have:

CA = {diU + TU) + VbXA ~ r-ldAF{tFX, |X|2) = 0. (7.3)

7.1. Equations for ÇA. We set £1 = £0 = 0 on the cone and we define by

dxa dxA

It holds that

y% = 0, (7.5)

because (recall that x1 = r, yl = r©i (xA ))

= ">< ' = rSA (1 1 ">< . (7.6)

dyi dx1 dyi ' dr dxA r dxA

We have

dxA — dxA dy The equation (7.7) implies that

dy1 _ dy1 dxB _ g _

Si = = a sB = sA- ('■')

( d a , ..-l^A dyi

ii v 01— —j axA

hence

C'4 = I?* {¿1 ' k'' +T)}+ VBXÎ ~ r~1dAF(tvX, |X|2) = 0.

Since tiX~ = trX" is a scalar function and the equation of Co in the x coordinates is x0 = 0 and y0 does not depend on xA, it holds that

d —^ d dy' d dy' d

■ tl- X ee —r tr X ee — tl- X ee — tl- X,

dxA dxA dxA dy' dxA dy'

analogously

a w . SÀIAT, ¿TWXW) - m

dxA dxA dy'-' dxA ' dxA dy'

Wo now computo, with covariant dorivativos V takon in the Riomannian niotric gAB = Cab ,

VbxÍ - VB (il5, + ¿¿f) - ¿ VBXI

The Christoifel symbols CBC of the Riemannian connection V are equal (recall that CB0 = CB1 = C00 = 0) to the trace on CO of the Christoifel symbols with the same

( the covariant derivative

C

VBX% = ¿(c)vBxI

Since the XB are the only non-vanishing components of the tensor X, and due to the

C

(G)VßX.f = (°)vax% = (g)VaXf

and Eqs. (7.3) read

c4= dyj 1 f 1

dxA r | 2

+ (WaX? - trX, |X|2)| = 0. (7.8)

The parentheses constitute a linear diagonal operator on the of the type considered in Lemma 5.3. Equating it to zero gives an equation with solution j an admissible series of minimal order 1. We denote by Sj the extension of & to spacetime. that is we have

= (7.9)

where Ej_ are analytical functions beginning by linear terms.

7.2. Equations for We now consider Eqs. (7.1). which read diVA + - T2Wi) VA - 2/' • \ 1 - \ vogAB9GD(SBD ~ ? CD) + \ "oU = 0 . (7.10)

We set

with v such that

g0i EE -ut + ALu with L¿ = C¿¿ L¿, L¿ = y°, (7.11)

that is. using (7.11). Then (compare (3.5))

Hence

i'. I.' .'/ <>: (7.12)

A = (LjL1) 1 g0i L3 = r V 9oj-

dy1 _ _ dy1

« dy^ —i .

^ ~dxA — ' —

Wo recall that

C1 ^ C

xa = - (x a + da ), lci- rvcxa = "ga a + z/a • r

Therefore, after product by r, the equations can be written as follows: dy I rdiVi - I rWiVi + \ /' - 2F A - \ ru0 E A = 0,

dxA \ — 2 — 2 —) " 2

with

FA := ucX^, EA := gAB g°D(SBD - fBCD).

By definition it holds that

_Q -—

VCXA = gocXA j

and since XA are the only non-vanishing components of the mixed tensor X in the coordinates x

dya dyP

gocX% = g0\X\ = —lg2\Xf3.

Recalling that A"? = 0 we find (using (4.11), X- Lj = 0)

___(• dy1 -7 dy1 -7

FA SS X A JJ^ILAL

We then remark that S^p — CBD is the trace on CO of the difference of the components of the Christoffel symbols, r^ and C^, with these angular x indices of the Minkowski metric n and the metric C :

Ea := gabgCD(SBD - C%D), with Ea ee CabCcd(vbd - CBD).

Using the expressions of ^d C and the vanishing of the Christoffel symbols of n y

dyl ^ -TvTT _ dy*

EA ee - c«m) ee

with analytic functions, components of Christoffel symbols of the metric C in y

yC

<■:,< ',(,, = \ Chk{dhcik + dhCjk - djChk). We recall from (6.15) that

rWi = -is0 gABr2sAB = -vo {n - 1 + cvSjj}, and we find that Eqs. (7.10) can be written as

dy* „

_—Ci = 0

dxA

where £j is the following linear operator on z/j

' n- 1 1 , , , 1 ——; 1 t 1

f i7 _ 1 1 _"I -r 1 1 -—

rdiVi + V0 | + ^chkSh A Vi - 1VjX.f + -mr§i - - n> (•:,( = 0. (7.13)

We extend Lemma 5.3 as follows.

Lemma 7.1. If kj and hi are admissible series of minimal orders 1, and the constant k0 > 0, then the ODE

rd\Ui_ + k0Ui_ + l.-';!*^ = hi (7.14)

admits a solution z/j, which is also an admissible series of the same minimal order h

Recalling that v0 — 1 is an admissible series of minimal order 2 we see that this lemma applies to (7.13). We have proved:

Theorem 7.1. If the basic characteristic data is induced by an analytic metric C satisfying (3.3), (3.4), Eq. (7.13) admits one and only one solution which is an admissible series of minimal order 2. We denote by Nj_ its spacetime extension.

We now prove:

Theorem 7.2. A sufficient condition for r~2Vi_ to have an admissible expansion is that the C metric C given by (3.3) which induces the basic characteristic data be near-round at the vertex.

Proof. Using the relation between X and Y the linearity of F in trX and |X |2 we see that for C round at the vertex the equation satisfied by reads

\ {r-0;k + k(n + r2Z)}=hi>

with _

h, := ^Va((y°)%a) - ^-F((yO)HvY, (y°)4|y|2).

We deduce from the linearity of F in tr X Mid |X |2 that r-1hj admits an admissible expansion, the same holds therefore (see Corollary 5.1) for r-1j. The equation satisfied by Vj reads

rdM + v0 | + r2 Whfcj Vi - 2/';/; Tv = hi, (7.15)

hi = -mr§i + Chk(dhdik + dhdik - djdhk).

An extension of Lemma 7.1 shows that r~2z/j admits an admissible expansion because it is so of hi. □

8. The Co constraint

The last unknown in g, the only unknown in the constraint C0, is

9 on = goo •

The constraint Co has a simpler expression in terms of g11. Since g11 is linked to ~g00 by the identity

g°%0 + 9119w +9A19AO =

we have

g00 = -gnM2 +gABVBVA = -gnM2 ■ Ll^i'r (8.1)

We have seen fl] that the C0 constraint can be written in vacuum as

a1C+(« + r)C+i {diW1+ (k, + t)W1+R-y2gabuzb+gabvazb \= 0, (8.2)

with

C-=(di + K + + (8.3)

k = v-d^o -\{WI+t), WI = v0W°, W° = W1 = -rgABsAB. (8.4)

8.1. Equation for In the flat case it holds that

_ — ]

v o,v = 1, tv = —W liV = ——, kv = 0.

The function Z reduces to

C„ := d^11 + -Tv(gu - 1), (8.5)

and the equation for Z reads, using = 0,

n - 1 , 1 i n - 1 (n - 1)2 ~ )

That is, using the scalar curvature of the S"-1 round sphere of radius r which is5

Rv = r-2(n - 2)(n - 1), (8.6)

the equation

n- 1

diCv + --C„ = 0

r

has the only bounded solution =0. From (8.5) results we have then g^1 = 1. We now study the general case.

Z

rd1 (rZ) +(rZ) k rZ +(rZ) h = 0, (8.7)

(rC)A- := t(Ki + t) - 1 = r jy<Vo + t,(t - WT) J - 1, (8.8)

r2 2

Hence

K)/?.:=!_ ^W' + ^T-W^W' + R+^d^W1 - ^UZb +9AB^AZB J • (8.9)

We have shown that i/° — 1 and rd\vo, hence also ri/°diz/q , admit admissible expansions of minimal order 2, and we have seen that rr and rW\ are admissible series with terms of order zero (n - i^d -(n - 1) respectively. Hence k is an admissible series of the

{r<)h := y jdilF1 + (« + t)W1 +R-\ gABUZB + gABVaZb } •

5See for instance [6, p. 140].

zero-order term n — 2, and we have

(rZ)k = n — 2 + k1;

where k1 is an admissible series of minimal order 1. We study the terms appearing in (rZ)h. The term

rV<9iz/oW1 = rv-diva rW1 (8.10)

has an admissible expansion of minimal order 2 (see Lemma 5.4). The term

r'Ll^ (8.11)

has an admissible expansion of order 4 because & has an admissible expansion of order 1. As for the term

the Christoifel symbols CAC of the Riemannian connection V are equal (recall that CB0 = CB1 = C00 = 0) to the trace on Co of the Christoifel symbols with the same

C

(C)V the covariant derivative

in the metric C _

VA6J = (C')VASB.

Since the SB are the only non-vanishing components of the vector S, and due to the

C

-AB

C c°)V,4Sg = Ca/3 (c)VaS/3. (8.12)

Hence the scalar r2C Va^b has an admissible expansion of minimal order 2. We have seen that in the flat case

= ^ = (8.13)

and

fliFj + rWl, EE ( ^ - 1 = -Rv . (8.14)

In the general case we compute

diw1 + \{t - w^w1.

Recall that

n - 1 tr X

t =-+-;

set

W1 = Wj + F, with F := (;ga/3 - if'3) f */3.

Using the values of the Christoifel symbols f^g and the components of ^d ^ m x coordinates we find

F = -(CAB - nAB)x'sAB EE --(Caß - Vaß)vaß EE

r

1 / -1 ~Pyaß \ 1 / -1 ~pvaß \ 1 / T^J \ ^ —¿i f

EE -{n+l-C 11aß ) EE - ( n + 1 - C 11aß ) = -{n-C IJjj ) = --ÇjL <>ij i

r

—1 _ n- 1 tlßii n TÏ71 - n ~ 1 , - yhdh?j)6ii

hence

TT = -—- - ¿^TT = +

and

Using the value of the scalar curvature Rn of the round sphere S"-1 of radius r we find that

r2 \ d{W + -(r - W )W \ = —r2Rv +

,1 1, —UtttI 2

where $ is an admissible expansion of minimal order 2,

$ := —yhdhC^_ôij -(n- - ± (n - 1 + chkôhk) (tFX + .

To compute r2i?, we use formulas given in [1]. The formulas (10.33) and (10.37) of fl] for a general metric in null adapted coordinates are

gABRAB = 2(a1+rJ1 + r) [(9! + rji + T-)g11 + r1] +R-2gABT\AT\B -2gAB V.4r|B,

Rn = ~diT+ T\1T - XAXB ,

and

S'oi = — vogabRab + Riava ~ -vognRii.

In the case of the metric C, it holds that C01 = 1, C0A = 0, Coo = -1, and C11 = 1. Hence

(c>rh T\A = 0, (c')ri ^ A{cABdlcAB + cABd0CAB),

and the above formulas reduce to (recall that R = (C)R):

-ab

ab

t + C do Cab

+ R,

Rii = -diT - xB xA,

and

-2 ^S0I=9ABRab+RIU

from which we deduce

R = 2d\t + t2 + (dl + T)CABd^ClE + Xa X'b ~ 2 (c)soi.

We have

Since C is an analytic metric in a neighbourhood of O, ^Saß admit admissible expansions and. hence r2 ^S'oi also admits an admissible expansion of minimal order 2. Recall that

n - 1 tr X , l0 IXI2 + 2 tr X + n- 1 r =--1--and |xh = J—!-^-:

hence

r2{2dlT + r2 +XaXb} = (n ~ !)(« " 2) + 2(™ " l)trX + 2rc>itrX + (trX)2 + |X|2.

Finally we remark that d0CAB are the non-vanishing components of the Lie derivative of the metric C with respect to the vector m with x components m0 = 1, m1 = m A = 0. hence with y components mr_ = — 1, ml = 0 that

c oqlab=<~ l-my-aji = l o ot-q/j,

hence

-ab

C d0CAB = ça daCij = ça doc,

0Cij

-AB-

has an admissible expansion of minimal order 1 and r~(di + t)C OqCab has an admissible expansion of minimal order 2. We have proved that

r~

"2R = r2Rn +

where

^ = 2(n - l)tr X + 2r9itrX + (trX)2 + |X|2 + r2(dl + t) Cv dnc,-,-has an admissible expansion of minimal order 2. Hence

r2 jôiWl + \{t -Wl)Wl + R^ =$ +

In conclusion, we have shown that has an admissible expansion of minimal

order 2, the same is therefore true for r(.

8.2. Equation for a := g11 — 1. The definition of ( gives for a := g11 — 1 an equation which reads

rdia + r a+ 7-(2K + T + W1 - 2() = 0. (8.15)

Theorem 8.1. If the ba.iie characteristic data is induced in a neighbourhood of O by an analytic metric C satisfying (3.3), (3.4), then Eq. (8.15) admits in this neighbourhood one and only one solution a which is an admissible series of minimal order 2.

This implies that g00 + 1 is also an admissible series of minimal order 2. Proof. Using the definition of k in (8.4), the equation (8.15) reads

rdia + r + i-r^ a + r(v°d\v0 - Ç) = 0,

that is,

rdxa + (a)ka + (a)h = 0, (8.16)

with

{a)k := r (v°div0 ~ rW^j , {a)h := r (v'd^o - Ç) .

(8.17)

Previous results show that this equation is of a form to which Lemma 5.3 applies with ko = — and ^h of minimal order 2. It has therefore a solution a, admissible series of minimal order 2.

The identity (8.1) shows the property of g00 + 1. □

Theorem 8.2. If in addition to the hypothesis of the previous theorem the given metric C is near-round at the vertex, then r~2(g00 + 1) is an admissible series in a neighbourhood of the vertex.

Proof. The result will follow from the proof that (a)h given in (8.17) is such that r-2 (a) h is an admissible series. By previous results, it remains only to prove that r-2 (rZ)

r We have

~2™h := ^ jr-2($ + *) + W1 - \gABUtB +9ABVAZB } •

(8.18)

C

such that the required condition is satisfied. Indeed,

z/°<Vow1 EE ^r-'dmrrW1 has an admissible expansion because it is so for r~1d-ii/Q and rW1.

9ABUZB = Llkk S^VAZB = C^ (g)VaS/3

have admissible expansions because & does.

The assumptions on and the identity yhdylr = r show that

yhdhcijSij = r2{yhdh7% + 2d?%),

hence r-2^, with $ and ^ given above have admissible expansions.

It remains to show the property for

Since C is an analytic metric in a neighbourhood of O, has an admissible

expansion. Denoting by ^Kij = — — daCij the second fundamental form of C relative to the slicing of R"+1 by y0 = constant, we know that6,

We have

(c)Vj{cMOKth) = {dj(c)Kth-C^Ktk-C*PKjk).

The functions ^Kij are analytic and the Christoffel symbols Cj^ are products by y° of analytic functions, while elementary computations give

-2yiC^dj {c)Kih = y'C^didoCih = C^{djdoWdH) - ¿>Qcjh},

hence using y'cih = 0 and S^Cjh = (y°)2Z. We deduce from these results that r~1yt(°')Soi also has an admissible expansion.

□

6 See for instance [7, Chapter 6].

9. Conclusions

9.1. An existence theorem. We have shown that when the metric C given by (3.3) which induces on Co the basic characteristic data g (i.e. gAB = Cab = Cab) is analytic, then the functions щ, щ, ~g00 have admissible expansions. We have shown that if moreover С is near-round at the vertex (definition 6.2). then the functions г~2(г/о — 1). r_1z/i, and r~2(g00 + 1) have also admissible expansions. These results imply the following theorem (the notations are those of Section 1):

Theorem 9.1. If the metric C given by (3.3), (3.4) which induces the basic characteristic data on the cone CQ is smooth everywhere, and moreover analytic and near-round in a neighbourhood of the vertex, then there exists a number T0 > 0 such that the wave-gauge reduced vacuum Einstein equations with characteristic initial determined by C

induces on CQ0 the same quadratic form as C.

Proof. It results from the formulae

goo = goo, м = -{goo + va)}»'" V - with у'щ = 0, (9.1)

9il ~ ¿a = {goo + 1 + 2(^o - I !'' ' .</ .</' + г"1 {y*^ + yjVi) + c^, (9.2)

and the theorems of previous sections that ~g00 + 1, g0i, and g^ — Sij have admissible expansions, hence are the trace on Co of analytic functions. We apply the Cagnac-Dossa theorem. □

This theorem and the results of fl] lead then to the following sufficient conditions for the existence of a solution of the full Einstein equations.

Theorem 9.2. If the metric C given bу (3.3) which induces the basic characteristic Co

a vacuum Einsteinian spacetime (yQ0 , g) which induces on cQ0 the same quadra,tic C

JMM was supported by the French ANR grant BLAN07-1_201699 entitled "LISA Science", and also in part by the Spanish MICINN project FIS2009-11893. PTC was supported in part by the Polish Ministry of Science and Higher Education grant Nr N N201 372736.

Резюме

И. Шоке-Брюа, П.Т. Хрусьцель, Х.М. Мартин-Гарсиа. Теорема существования решения задачи Коши о световом конусе для вакуумных уравнений Эйнштейна с «почти-закруглеппыми» аналитическими данными.

Представлен класс характеристических релятивистски обобщенных исходных данных. удовлетворяющих условию понппшакруглеииоспт па вершине светового конуса. Показано, что для любых подобных аналитических даппых существует соответствующее решение для вакуумных уравнений Эйнштейна, определенное в окрестности вершины в световом конусе будущего.

Ключевые слова: характеристическая задача Коши о световом конусе, вакуумные уравнения Эйнштейна.

References

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2. Rendall A.D. Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations // Proc. R. Soc. London A. - 1990. -V. 427. - P. 221-239.

3. Damour T., Schmidt B. Reliability of perturbation theory in general relativity //J. Math. Phys. - 1990. - V. 31. - P. 2441-2453.

4. Choquet-Bruhat Y., Chrusciel P.T., Martin-Garcia J.M. An existence theorem for the Cauchy problem on a characteristic cone for the Einstein equations // Contemporary Mathematics. - 2011. - V. 554. - P. 73-81.

5. Chrusciel P.T., Jezierski J. On free general relativistic initial data on the light cone. -arXiv:1010.2098 [gr-qc]. - 2010. - URL: http://arxiv.org/pdf/1010.2098v1.pdf.

6. Choquet-Bruhat Y., DeWitt-Morette C. Analysis, manifolds and physics, Part II. - Amsterdam: North-Holland Publ.g Co., 2000. - 560 p.

7. Choquet-Bruhat Y. General relativity and the Einstein equations. - Oxford: Oxford Univ. Press, 2009. - 840 p.

Поступила в редакцию 28.11.10

Choquet-Bruhat, Yvonne PliD, Full Member of tlio French Academy of Sciences, Paris, France.

Шоке-Врюа, Ивонн доктор паук, полноправный член Французской академии паук, г. Париж, Франция.

E-mail: ycbQihes.fr

Chrusciel, Piotr Т. PliD, Professor, University of Vienna, Vienna, Austria.

Хрусьцель, Петр Т. доктор паук, профессор Венского университета, г. Вена, Австрия.

E-mail: piotr.chrusciclQunivic.ас.at

Martin-Garcia, José Maria PliD, Postdoctoral Research Fellow, Paris Institute of Astrophysics, Paris, France: Universe and Theories Laboratory, Meudon, France.

Мартин-Гарсиа, Хосе Мария доктор паук, научный сотрудник Парижского астрофизического института, г. Париж, Франция: научный сотрудник Лаборатории Вселенной и теорий о пей, Мёдоп (пригород Парижа), Франция.