ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaAsymptoti ally Simple Solutionsof the Va uum Einstein Equationsin Even DimensionsMi hael T. AndersonPiotr T. Chru�s iel

Vienna, Preprint ESI 1555 (2004) De ember 7, 2004Supported by the Austrian Federal Ministry of Edu ation, S ien e and CultureAvailable via http://www.esi.a .at

ASYMPTOTICALLY SIMPLE SOLUTIONS OF THEVACUUM EINSTEIN EQUATIONS IN EVENDIMENSIONSMICHAEL T. ANDERSON AND PIOTR T. CHRU�SCIELAbstra t. We show that a set of onformally invariant equationsderived from the Fe�erman-Graham tensor an be used to onstru tglobal solutions of va uum Einstein equations, in all even dimen-sions. This gives, in parti ular, a new, simple proof of Friedri h'sresult on the future hyperboloidal stability of Minkowski spa e-time, and extends its validity to even dimensions.1. Introdu tionConsider the lass of va uum solutions to the Einstein equations(M ; g) in n + 1 dimensions, whi h are future asymptoti ally simple,i.e. onformally ompa t, in the sense of Penrose, to the future of a omplete Cau hy surfa e (S ; ). A natural method to try to onstru tsu h spa e-times is to solve a Cau hy problem for the ompa ti�ed,unphysi al spa e-time (M ; �g), and then re over the asso iated physi alspa e-time via a onformal transformation. However, a dire t approa halong these lines leads to severe diÆ ulties, sin e the onformally trans-formed va uum Einstein equations form, at best, a degenerate systemof hyperboli evolution equations, for whi h it is very diÆ ult to proveexisten e and uniqueness of solutions.Friedri h [20,23℄ has developed a method to over ome this diÆ ultyin 3 + 1 dimensions, by introdu ing a system of \ onformal Einsteinequations" whose solutions in lude the va uum Einstein metri s andwhi h transforms naturally under onformal hanges. A variation uponFriedri h's approa h, again in 3 + 1 dimensions, has been presentedin [6℄.In this paper, we develop a di�erent approa h to this issue whi h, be-sides its simpli ity, has the advantage of working in all even dimensions.Date: De ember 21, 2004.Partially supported by NSF grant DMS 0305865 (MA), and the Erwin S hr�odingerInstitute, the Vienna City Coun il, and the Polish Resear h Coun il grant KBN 2P03B 073 24 (PTC). 1

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 2The method, arried out for va uum spa e-times with � > 0 in [2℄, isbased on use of the Fe�erman-Graham (ambient obstru tion) tensor H,introdu ed in [16℄. The tensor H is a symmetri bilinear form, depend-ing on a metri g and its derivatives up to order n + 1, f. Se tion 2for further dis ussion. It is onformally ovariant, (of weight n� 1) andmetri s onformal to Einstein metri s satisfy the system(1.1) H = 0 :When n = 3, i.e. in spa e-time dimension 4, the Fe�erman-Grahamtensor is the well-known Ba h tensor.The main result of the paper, Theorem 4.1, is the proof of the well-posedness of the Cau hy problem for the equation (1.1), for Lorentzmetri s. This leads to a new proof of Friedri h's result on the future \hy-perboloidal" stability of Minkowski spa e-time [21℄ (see Theorem 6.1),and extends the validity of this result to all even dimensions.1 As a orollary we additionally obtain existen e of a large lass of non-trivial,va uum, even dimensional spa e-times whi h are asymptoti ally simplein the sense of Penrose, see Theorem 6.2.We further note that in [2℄ existen e of solutions of the Cau hy prob-lem for (1.1) is obtained by pseudo-di�erential te hniques. Here weshow that the Anderson-Fe�erman-Graham (AFG) equations (1.1) anbe solved using an auxiliary, �rst order, symmetrisable hyperboli sys-tem of equations. This shows that (1.1) is a well posed evolutionarysystem, dire tly amenable to numeri al treatment. Thus, in spa e-timedimension four we provide an alternative to Friedri h's onformal equa-tions for the numeri al onstru tion of global spa e-times [18,19,28℄.Our methods do not apply in odd spa e-time dimensions, where thesituation is rather di�erent in any ase, as one generi ally expe ts poly-homogeneous expansions with half-integer powers of 1=r, where r is,say, the luminosity distan e, ompare [11,26,27,30℄.2. The Anderson-Fefferman-Graham equationsLet, as before, n + 1 denote spa e-time dimension, with n odd. TheFe�erman-Graham tensor H is a onformally ovariant tensor, built out1On e most of the work on this paper was ompleted we have been informed ofthe work by S himming [32℄, who has done a lo al analysis, related to ours, of theCau hy problem for the Ba h equations in dimension four. The appli ation of hiswork to global issues, for instan e on erning the va uum Einstein equations, seemsnot to have been addressed. We are grateful to R. Beig for pointing out that referen eto us.

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 3of the metri g and its derivatives up to order n + 1, of the form(2.1) H = (r�r)n+12 �2[r�r(P ) +r2(trP )℄ + Fn ;where(2.2) P = Ri g �Rg2ng ;and where Fn is a tensor built out of lower order derivatives of themetri (see, e.g., [25℄, where the notation O is used in pla e of H). Itturns out that Fn involves only derivatives of the metri up to ordern � 1: this is an easy onsequen e of Equation (2.4) in [25℄, using thefa t that odd-power oeÆ ients of the expansion of the metri gx in[25, Equation (2.3)℄ vanish. (For n = 3; 5 this an also be veri�ed byinspe tion of the expli it formulae for F3 and F5 given in [25℄.)The system of equations(2.3) H = 0will be alled the Anderson-Fe�erman-Graham (AFG) equations. It hasthe following properties [25℄:(1) The system (2.3) is onformally invariant: if g is a solution, sois '2g, for any positive fun tion '.(2) If g is onformal to an Einstein metri , then (2.3) holds.(3) H is tra e-free.(4) H is divergen e-free.Re all that H was originally dis overed by Fe�erman and Graham [16℄as an obstru tion to the existen e of a formal power series expansion for onformally ompa ti�able Einstein metri s, with onformal boundaryequipped with the onformal equivalen e lass [g℄ of g. This geomet-ri interpretation is irrelevant from our point of view, as here we areinterested in (2.3) as an equation on its own.3. Redu tion to a symmetrisable hyperboli systemLet g be a Lorentzian metri , letr be a onne tion (not ne essarily g- ompatible), and let � denote an operator with prin ipal part g��r�r�(a ting perhaps on tensors). Let u be a tensor �eld, and let r(k)u denoteany tensor formed from the k-th order ovariant derivatives of u. Fork � 0 onsider the system of equations(3.1) �k+1u = F (x; u;ru;r(2)u; : : : ;r(2k+1)u) ;for some smooth F . Here we allow the oeÆ ients of � as well asthe onne tion oeÆ ients to depend smoothly upon x as well as uponthe olle tion of �elds (u;ru;r(2)u; : : : ;r(2k+1)u), in parti ular the

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 4metri g is allowed to depend (smoothly) upon those �elds. We willassume that (3.1) is invariant under di�eomorphisms, although this isnot ne essary for some of our results below, su h as lo al existen e andlo al uniqueness of solutions.We want to show that solutions of (3.1) an be found by solving a�rst order symmetri hyperboli system of PDEs. The idea of the proof an be illustrated by the following example. Consider the equation(3.2) �2u = 0 :Introdu ing (0) = u ; (1) = �u ;it is easily seen that solutions of (3.2) are in one-to-one orresponden ewith solutions of the system(3.3) �� (0) (1) � = � (1)0 � :It is then standard to write a symmetrisable-hyperboli �rst order sys-tem so that solutions of (3.3) are in one-to-one orresponden e withsolutions of the �rst order system with appropriate initial data ( om-pare the al ulations in the proof below).Some work is needed when we want to allow lower order derivativesas in the right-hand-side of (3.1):Proposition 3.1. There exists a symmetrisable hyperboli �rst ordersystem(3.4) P� = H(�) ;where P is a linear �rst order operator and H does not involve deriva-tives of �, su h that every solution of (3.1), with (M ; g) time orientable,satis�es (3.4).Proof. Let feaga=0;:::;n = fe0; eigi=1;:::;n be an orthonormal frame for g,with e0 a globally de�ned unit timelike ve tor; (su h ve tor �elds alwaysexist on time orientable manifolds). We set'(j) = f'(j)a1:::aig1�i�2(k�j) ; where '(j)a1:::ai = ea1 � � �eai�ju ;(3.5) ' = f'(j)g0�j�k ;(3.6) = f (j)g0�j�k ; where (j) = �ju :(3.7)Let us derive a onvenient system of equations for '. First,(3.8) for 1 � i � 2(k� j)� 1 e0'(j)a1:::ai = '(j)0a1:::ai = L('(j)) ;where we use a generi symbol L to denote a linear map whi h may hange from line to line. This gives evolution equations for those

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 5'(j)a1:::ai 's whi h have a number of indi es stri tly smaller than the max-imum number allowed.We ontinue by noting that, again for 1 � i + 2j � 2k � 1, we haveon the one hand�'(j)a1:::ai = �e0'(j)0a1:::ai + nX̀=1 e`'(j)`a1:::ai + L('(j)) ;(3.9)and on the other�'(j)a1:::ai = ea1 � � �eai� (j) + [�; ea1 � � �eai ℄ (j)(3.10) = '(j+1)a1:::ai + L('(j)) :Combining those two equations we obtain(3.11) e0'(j)0a1:::ai = nX̀=1 e`'(j)`a1:::ai + L('(j); '(j+1)) :Note that the ondition i+2j � 2k� 1 implies j < k so that (3.11) analso be rewritten as(3.12) e0'(j)0a1:::ai = nX̀=1 e`'(j)`a1:::ai + L(') :Next, for i+ 2j = 2k � 1 and for ` running from 1 to n we writee0'(j)`a1:::ai = e`'(j)0a1:::ai + [e0; e`℄'(j)a1:::ai(3.13) = e`'(j)0a1:::ai + L('(j)) :The rewriting of (3.12)-(3.13) in the form(3.14) 0BBBB� e0'(j)0a1:::ai �e1'(j)1a1:::ai � � � �en'(j)na1:::ai�e1'(j)0a1:::ai +e0'(j)1a1:::ai +0 +0... +0 . .. ...�en'(j)0a1:::ai +0 � � � +e0'(j)na1:::ai 1CCCCA = L(')makes expli it the symmetri hara ter of (3.12)-(3.13). It is well knownthat this system is symmetrisable hyperboli in the sense of [34, Vol-ume III℄ when e0 is a nowhere vanishing ve tor �eld.2 This provides thedesired system of evolution equations for those '(j)a0a1:::ai 's whi h have2In fa t, (3.14) is symmetri hyperboli in a oordinate system with e0 = �t andeit = 0. However, when g depends upon u and its derivatives it is not useful touse su h oordinates, as the onstru tion of the Gauss oordinate system leads todi�erentiability loss. In any ase Gauss oordinates are not well adapted to the proofof existen e of solutions when g depends upon u.

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 6the maximum number of indi es. (One ould also use (3.14) for anynumber of indi es, but (3.8) is obviously simpler.)If we write (3.14) as �P' = 0 ;where �P is a linear �rst order operator, then the derivatives ea' satisfya �rst order symmetrisable-hyperboli system of equations(3.15) �Pea' = L(';r') ; L(';r') := [�P; ea℄' :The evolution equations for are simply� (i) = (i+1) ; 0 � i � k � 1 ;(3.16) � (k) = F (x; (0); '(0);r'(0)) ;(3.17)where in (3.17) we have expressed the derivatives of u appearing in (3.1)in terms of '(0) and r'(0) using (3.5). By obvious modi� ations of the al ulation starting at (3.9) and ending at (3.14) one an rewrite the left-hand-side of (3.16)-(3.17) as a �rst order symmetrisable hyperboli op-erator a ting on the olle tion of �elds ( ;r ) := f( (i);r (i))g0�i�k.Setting(3.18) � = ( ;r ;';r') ;and letting P be the linear part of the system of equations just des ribed,the proposition follows. �The interest of Proposition 3.1 relies in the fa t, that it is standardto prove existen e and uniqueness of solutions of (3.4) when the initialdata for � are in Hs, s 2 N, for s > n=2 + 1, provided that (M ; g) isglobally hyperboli . If g does not depend on r2k+1u, then the threshold an be lowered3 to s > n=2.Now, not every solution of (3.4) will be a solution of (3.1). Letus show that appropriate initial data for (3.4) will provide the desiredsolutions. When the spa e-time metri g is independent of u, let S bea spa elike hypersurfa e in the spa e-time (M ; g). We hoose e0 to be aunit time-like ve tor �eld normal to S , so that the ei's are tangential atS , and we extend e0 o� S in some onvenient way, whi h might varya ording to the ontext. Sin e (3.1) is an equation of order 2k+2, theasso iated Cau hy data onsist of a set of tensor �elds ff(i)gi=0;:::;2k+13For s > n=2 + 1 the result follows from [34, Volume III, Theorem 2.3, p. 375℄.However, when the symmetri hyperboli system has the stru ture onsidered here,with g not depending upon r2k+1u, the proof in [34℄ applies for s > n=2.

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 7de�ned on S whi h provide initial data for (e0)iu on S :(3.19) (e0)iujS := e0 � � �e0| {z }i times ujS = f(i) ; 0 � i � 2k + 1 :For any i � 0 and ` � 0 we an use (3.19) to al ulate formally (i)jSand (i)0 jS by repla ing ea h o urren e of (e0)ju by f(j), e.g., (0)jS =f(0), e0 (0)jS = f(1), (1)jS = (�u)jS = �� e0e0u+ nXi=1 eieiu+ ��e�u+ �u����S= �f(2) + nXi=1 eieif(0) + �0f(1) + nXi=1 �ieif(0) + �f(0) ;for some linear maps ��, � arising from the detailed stru ture of �, andso on. We will write g(i)(0) for the resulting fun tions (i)jS and g(i)(1) forthe resulting fun tions e0 (i)jS , so that (i)jS = g(i)(0) ; e0 (i)jS = g(i)(1) :Similarly we an al ulate'(i)01:::0` jS := '(i)0 : : : 0|{z}` fa torsjSwhere we repla e ea h o urren e of (e0)ju by f(j), e.g. '(0)01:::0` jS = f(`).We will write h(i)(`) for the resulting fun tions, so that'(i)jS = h(i)(0) = g(i)(0) ; '(i)(1)jS = h(i)(1) = g(i)(1) ; '(i)01:::0` jS = h(i)(`) :When g does depend upon u, then the spa e-time will be built in thepro ess of solving the equations. In the simplest ase of g depending onlyupon u, the pro edure just des ribed should be understood as follows:the initial metri gjS is determined by the initial data f0. We hoosean orthonormal basis feigi=0;:::;n for gjS , and interpret e0 as the unitnormal to S in the spa e-time that will arise out of the initial data.Thus, f(1) will be interpreted as the value of the normal derivative of uat S , and so on, and the above onsiderations remain un hanged whenthis interpretation is used.Proposition 3.2. Let � 2 C(I;Hs(O)), s > n=2 + 2k+ 2, s 2 N, be asolution of (3.4) on a globally hyperboli region I �O with initial data onstru ted as des ribed above. Then u := (0) is a solution of (3.1)and (3.19).

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 8Proof. From (3.16) one has (i) = �i (0) for 0 � i � k. It remains toshow that if '(0)a1:::ai = ea1 � � �eai (0), then (3.17) will oin ide with (3.1).This an be proved by a standard al ulation. One sets�(j)a1:::ai = '(j)a1:::ai � ea1 � � �eai (j) ;and using (3.4) one derives a system of equations whi h show that�(j)a1:::ai = 0 for the initial data under onsideration.However, the omputations involved are avoided by the following ar-gument. Suppose, �rst, that g, F and r are analyti fun tions of alltheir variables. Let us denote by f = ff(j)g0�j�2k+1 the initial data for(3.1); by an abuse of notation we will write f 2 Hs if f(j) 2 Hs�j for0 � j � 2k + 1. We note, �rst, that by using an exhaustion of I � Oby ompa t subsets thereof it suÆ es to prove the result when I �O isa onditionally ompa t subset of the domain of de�nition of the solu-tion. Let fn be any sequen e of analyti initial data whi h onverges inHs(O) to f . Let un be the orresponding solution of (3.4); by stabilityall un's will be de�ned on I �O for n large enough. Similarly, the sta-bility estimates4 for symmetri hyperboli systems [29℄ prove that un isCau hy in C(I;Hs(O))\ C1(I;Hs�1(O)). The results in [1℄ show thatun is analyti throughout I � O .Let bun be a solution of (3.1) on an open neighborhood Un of S inI � O obtained by the Cau hy-Kowalevska theorem. (Note that Un ould in prin iple shrink as n tends to in�nity, but it is neverthelessopen and nonempty for ea h n.) Passing to a subset of Un if ne essarywe an without loss of generality assume that Un is globally hyperboli .Now, uniqueness of the solutions of the Cau hy problem for (3.4)shows that un oin ides with bun on Un. Thus un satis�es (3.1) thereand thus, by analyti ity, everywhere. This shows that maximal globallyhyperboli solutions of (3.1) with analyti initial data are in one-to-one orresponden e with maximal globally hyperboli solutions of (3.4)with the initial data onstru ted as above. Then, for Hs initial data,Proposition 3.2 follows from ontinuity of solutions upon initial data for(3.4).Finally, if the �elds g, r and F are smooth fun tions of their argu-ments, they an be approximated by a sequen e of �elds g(n), r(n) andF (n) whi h are analyti in their arguments. The estimates for (3.4) justdes ribed an similarly be used to show that solutions of the approx-imate problem onverge to solutions of the problem at hand both forequation (3.4) and (3.1), whi h �nishes the proof. �4Note that I � �O is non-timelike by global hyperboli ity, so that integration byparts gives harmless ontributions as far as energy estimates are on erned.

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 9From what has been said so far we obtainTheorem 3.3. Let s > n=2 + 2k+ 2, s 2 N. For any �eldsf(i) 2 Hs�ilo (S ) ; i = 0; : : : ; 2k+ 1 ;there exists a unique solution of (3.1) satisfying (3.19). If the metri gdoes not depend upon r2k+1u, then s > n=2 + 2k + 1 suÆ es.We note that in lo al oordinate systems (t; xi) on an open neighbor-hood U of O � S of the form U = I �O , with S \U = ft = 0g andO- ompa t, the solutions are inu 2 \2k+1i=0 Ci(I;Hs�i(O)) :As usual, the lower bound for the lo al time of existen e of the solu-tion does not depend upon the di�erentiability lass s, so in parti ularsmooth initial data provide smooth solutions. In addition the Cau hyproblem for (3.1) is well-posed, in that given a pair of initial data f1(i),f2(i) whi h are lose in Hs�ilo (S ), then the solutions u1, u2 are also losein \2k+1i=0 Ci(I;Hs�ilo (S )).When g does not depend upon u, there exists a unique maximalglobally hyperboli subset O of M , with S being Cau hy for O , onwhi h the solution exists. This is proved by the usual methods.In the quasi-linear ase one also has the existen e of a maximal glob-ally hyperboli development of the Cau hy data, giving a spa e-time(M ; g). This follows from the fa t that the domains of dependen e forthe system onstru ted above are determined by the light- ones of themetri g, so that a proof along the lines of [5℄, ( ompare [7,8℄), applies.4. The Cau hy problem for the AFG equationsThe Cau hy problem for (2.3) has a similar form to that for theEinstein equations. Sin e the system (2.3) is of order n + 1, the initialdata onsist of an n-dimensional Riemannian manifold (S ; ), n =2k + 1 � 3, with n symmetri two-tensors K(i) pres ribed on S . Thetensor �elds K(i) represent the i-th time derivative of the metri g in aGauss oordinate system around S . Thus, in a neighborhood of S , (ormore pre isely a neighborhood of a bounded domain in S ), one maywrite(4.1) g = �dt2 + (t);where (t) is a urve of metri s on S . Setting e0 = �rt, one has(4.2) K(i) = 12Lie0gjt=0 = 12�it (t)jt=0:

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 10In parti ular K = K(1) is the extrinsi urvature tensor of S in the�nal spa e-time (M ; g).The set ( ;K(1); : : : ; K(n)) is not arbitrary, sin e the equations(4.3) H(e0; �) = 0;only involve t-derivatives of g up to order n, and so indu e (n+1) equa-tions on ( ;K(1); : : : ; K(n)). Be ause (4.3) is di�eomorphism invariant,this is most easily seen in the oordinates (4.1) where g0� = �Æ0�, sothat (4.3) only involves t-derivatives of gab, a; b � 1, up to order n.The fa t that the onstraint equations (4.3) are preserved under theevolution follows in the usual way from the equation ÆH = 0.To des ribe the system of n + 1 onstraint equations (4.3) in moredetail5, the Gauss-Codazzi equations for the embedding S � (M ; g)are:(4.4) R � jKj2 +H2 = R+ 2Ri (e0; e0);(4.5) ÆK � dH = Ri (e0; �);where H = trK. In addition, in a Gauss oordinate system (t; xi) nearS , the Ray haudhuri equation gives(4.6) �tH + jKj2 = �Ri (e0; e0):We �rst point out that the urvature s alar R = Rg is determineddire tly by the initial data; this is in onstrast to the situation with theEinstein equations, where R is determined by the evolution equationsfor the metri . Namely, the left side of (4.4) is determined by the initialdata, as is Ri (e0; e0), by (4.6). Thus R is determined by and K(i),for i = 1; 2.To des ribe the form of the \s alar onstraint equation" H��n�n� =0, n = e0, note that from (2.2) one has trP = (n� 1)Rg=2n. Togetherwith (4.4) and (4.6), this leads to�P00 �r0r0trP = �1� nn � (� +r0r0) �tH+ 12n (� + (1� n)r0r0)R + ::: ;(4.7)where \:::" stands for terms whi h ontain less derivatives of the spa e-time metri . The t-derivatives of the metri of order 4 an el out, as5An expli it form of (4.3) in spa e dimension 3 an be found in [32℄. The param-eterisation of the initial data there is rather di�erent from ours.

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 11expe ted, and one easily �nds that the equation H��n�n� = 0 takes theform(4.8) �n�12 R = �n ;where, as before, R is the urvature s alar of , with � = DkDk theLapla e operator of . Finally, �n is a fun tional of ( ;K(1); : : : ; K(n))whi h does not involve derivatives of initial data of order n+1, while theleft-hand-side does. This shows in parti ular that (4.3) is a non-trivialrestri tion on the initial data.It also follows from (4.7) that �n will ontain terms of the form(4.9) 1n�(trK(n�1))�DkDlK(n�1)kl ;with other o urren es of K(n�1) there, if any, being also linear withat most one spa e-derivative. Thus, the s alar onstraint equationH��n�n� = 0 an be viewed either as a non-linear equation of ordern+ 1 whi h puts restri tions on in terms of the remaining data, or asa se ond order linear PDE for the tra e of K(n�1).One an similarly he k that the equation H0i = 0 takes the form ofa linear �rst order PDE for K(n), with prin ipal part(4.10) Di(K(n)ij � 1ntrK(n)gij) ;where D denotes the Levi-Civita onne tion of .A set ( ;K(1); : : : ; K(n)) will be alled an initial data set if the �elds( ;K(1); : : : ; K(n)) satisfy the onstraint equations (4.3). The olle -tion of initial data sets is not empty, as every solution of the generalrelativisti onstraint equations solves (4.3).The equations (4.3) are preserved by the following family of transfor-mations, related to the onformal invarian e of (2.3). Suppose that thedata set ( ;K(1); : : : ; K(n)) arises from a spa e-time (M ; g) satisfying(2.3). Then ( ;K(1); : : : ; K(n)) satis�es (4.3), and if is any stri tlypositive fun tion onM , then the set (~ ; ~K(1); : : : ; ~K(n)) obtained on Sfrom the metri 2g also satis�es (4.3). For example, if we set(4.11) ! := jS ; !(j) := e0 � � �e0| {z }j fa tors ()jS ;where e0 is, e.g., a geodesi extension of e0 o� S , then it holds that(4.12) ~ = !2 ; ~K(1) = !K(1) + !(1) :Similar but more ompli ated transformation formulae hold for ~K(i), i �2, see Appendix A. This leads to a family of transformations preserving

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 12(4.3) whi h are parameterized by n + 1 fun tions !, !(j), j = 1; : : : ; n,on S , whi h are arbitrary ex ept for the requirement that ! > 0.Re all that under a onformal transformation of the spa e-time metri we have ~gij = 4n�1 gij =) ~R = � 4n�1 �R� 4n(n� 1)�g� :(4.13)It follows from this formula that for any given ! = jS > 0 and !(1) =e0()jS one an hoose !(2) so that solves the linear wave equation(4.14) 4n(n� 1)�g = Rwhen restri ted to S . (This equation is globally solvable in globallyhyperboli spa e-times, but this is irrelevant for the urrent dis ussion.Note that solutions of (4.14) might sometimes develop zeros; these areessential in the analysis of the va uum Einstein equations). Note asremarked above that the urvature s alar R is determined by the order2 part of the initial data set. Taking further e0-derivatives shows thatthe remaining !(j)'s may be hosen so that the onformally transformed urvature s alar ~R, together with its normal derivatives up to order n�2,vanish at S .Now the onformal and di�eomorphism invarian e of (2.3) requiresa suitable hoi e of gauge in order to obtain a well-posed system. Asin [2℄, we use onstant s alar urvature for the onformal gauge andharmoni oordinates for the di�eomorphism gauge; the treatment ofthe onformal gauge is somewhat di�erent here than in [2℄.Thus, we require �rst that(4.15) R = 0 :If (4.15) holds, then (2.3) takes the form(4.16) (r�r)n�12 Ri = �Fn ;In harmoni oordinates fy�g = f(�; yi)g with respe t to the onformalgauge (4.15), one hasRi ab = �12g������gab +Qab(g; �g) ;where Q is quadrati in g and �g. Applying (r�r)(n�1)=2 to this and ommuting (r�r)(n�1)=2 with the �g terms in Q shows that (4.16) hasthe form(4.17) �n+12g g�� = � bFn�� ;where bFn�� still has the form (3.1) and �g = g������ a ts on s alars.

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 13Finally, as initial data for the (gauge-dependent) variables g0�, 0 �� � n, we hoose g0� = �Æ0�; on S :The data gab and ��gab, a; b � 1, are determined by the initial data ,K(1). The derivatives ��g0� are then �xed by the requirement that the oordinates fy�g are harmoni when restri ted to S , i.e.(4.18) �y� = ��g�� + 12g��g����g�� = 0 on S :The higher derivatives �i�g��, i � n, on S are then determined bythe given initial data ( ;K(1); : : : ; K(n)) and by setting to zero the � -derivatives of (4.18) up to order n� 1.These hoi es lead to the following:Theorem 4.1. Consider any lass (S ; [ ;K(1); : : : ; K(n)℄) satisfying the onstraint equations (4.3), with( ;K(1); : : : ; K(n)) 2 Hslo (S )�Hs�1lo (S )� � � � �Hs�nlo (S ) ;s > n=2+n+1, s 2 N, where (S ; ) is a Riemannian metri and wherethe K(i)'s are symmetri two- ovariant tensors; the equivalen e lass istaken with respe t to the transformations of the data dis ussed above.Then there exists a unique maximal globally-hyperboli onformalspa e-time (M ; [g℄) satisfying (2.3), where [�℄ denotes the onformal lass, and an embedding i : S !M ;for whi h is the metri indu ed on S by g, with K(i) given by (4.2).One an always hoose lo al representatives of [g℄ by imposing (4.15).Moreover, the Cau hy problem with su h initial data is well-posed inHslo (S )�Hs�1lo (S )� � � � �Hs�nlo (S ).Remark 4.2. The inequality s > n=2+n suÆ es for existen e of uniquesolutions in lo al oordinate pat hes in Theorem 4.1. One expe ts thatthe use of lo al foliations with pres ribed mean urvature and spa e-harmoni oordinates as in [4℄ should allow one to lower the thresholds > n=2 + n+ 1 to s > n=2 + n in this result.Remark 4.3. The onformal spa e-time (M ; [g℄) is smooth if the ini-tial data are. Similarly, real-analyti initial data lead to real-analyti solutions.Proof. Given any initial data set [( ;K(1); : : : ; K(n))℄ satisfying the on-straint equations, by using the fun tions !(j), j = 2; : : : ; n from (4.11),one an adjust the tensor �elds K(i), i = 2; : : : ; n, so that Rg, togetherwith its transverse derivatives up to order n� 2, vanish on S , (see the

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 14dis ussion following (4.14)). Note that this holds for any 2 [ ℄, sothat ! > 0 and !(1), while �xed, are otherwise freely spe i�able. Solv-ing (4.17) with this initial data, as des ribed in Se tion 3, one obtainsa olle tion of spa e-time oordinate pat hes with a solution of (4.17)there when s > n=2 + n. We re all again that the Cau hy problem for(4.17) is well-posed in the Hslo spa es above.The argument that these gauge hoi es are preserved, so that (4.15)holds and the oordinates y� remain harmoni in these lo al spa e-time oordinate pat hes generated by (4.17), is rather similar to the one forthe Einstein equations [17℄; we give details be ause of some di�eren esin the analyti al tools used.First re all that (4.17) an be written in the form(4.19) �n�12g (R�� � r��� � r���) = �Fn�� ;where �� := �12�gy� :As g solves (4.19), its obstru tion tensor equals(4.20)H�� = (�1)n�12 ��n�12g (r��� +r��� � 12nRgg��)� n� 12n �n�32g r�r�Rg� :The divergen e identity r�H�� = 0 gives then an equation involving �and Rg:(4.21)�n�12g ���� +r�(r��� � 12Rg)� = lower order ommutator terms :Sin e H is tra e-free, we further have from (4.20)(4.22) �n�12g (r��� � 12Rg) = 0 :Be ause Rg vanishes to order n�2 at S , and �j vanishes to order n�1atS by the dis ussion following (4.18), the initial data for this equationvanish. It follows, for instan e from the work in Se tion 3, that(4.23) Rg = 2r��� :This an be used to rewrite (4.21) as(4.24) �n+12g �� = lower order ommutator terms :In this last equation all ommutator terms in (4.21) that involve gradi-ents of Rg have been repla ed by derivatives of � using (4.23).Sin e g satis�es the onstraint equation (4.3) at S , (4.23) and (4.20)imply that �j vanishes to order n at S , so that (4.24) has vanishing

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 15initial data. This system has the form onsidered in Se tion 3, so we on lude that � vanishes. Hen e, the oordinates y� remain harmoni ,Rg = 0 by (4.23), and so H = 0 as well, as desired.The usual pro edure, as used in the ontext of the Cau hy problemfor Einstein's equations, then allows one to pat h the solutions togetherprovided s > n=2 + n + 1. An argument as in [5℄ leads to a unique,(up to di�eomorphism) maximal globally hyperboli manifold (M ; [g℄),with [g℄ satisfying (4.16), with an embedding i : S ! M , with thedesired initial data on i(S ).The well-posedness statement follows immediately from the dis us-sion following Theorem 3.3. �Remark 4.4. In general, there will not be a global smooth gauge for(M ; [g℄) in whi h Rg = 0. The lo al oordinate pat hes where Rg = 0need not pat h together smoothly, preserving Rg = 0, sin e the initialdata ! > 0, !(1) on lo al spa e-like sli es S are freely hosen, and sonot uniquely determined.To see this in more detail, onsider for example the de Sitter spa e-time M = R� Sn, with metri gdS = �dt2 + osh2 t gSn(1):This is a geodesi ally omplete solution of the Einstein equations withRg = n(n + 1), and so satis�es (2.3). The linear wave equation (4.14)with initial data = 1, �t = 2 on S = ft = 0g has a global solutionon M given by = �n� 14 sinh t+ d2 Z0 1 oshn t dt+ d1;for suitable d1; d2. One sees that there are no values of 1; 2 for whi h > 0 everywhere onM , so that there is no natural global R = 0 gaugefor (M ; [gdS℄) with su h initial data.Consider for example the solution = �n�14 sinh t, giving an R = 0gauge in the region t < 0, whi h does not extend to ft = 0g. For t = �"small, the indu ed metri on S�" = ft = �"g is the round metri Æon Sn of small radius Æ = Æ("). To obtain an R = 0 gauge starting atS�" whi h extends up to and beyond ft = 0g, one must hoose ! to bea large onstant. This auses a dis ontinity in the hoi e of gauge forthe metri , but not in the stru ture of the onformal lass.Remark 4.5. One may also onstru t the maximal solution (M ; [g℄) bymeans of lo al gauges satisfying(4.25) Rg = 0

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 16in pla e of the s alar- at gauge (4.15), for any 0 2 R. The proof of thisis the same as before, noting from the form of (4.13) that given ! and!(1) on S , one an �nd !(2) on S su h that ~R = 0 on S .For example, the de Sitter spa e-time is a geodesi ally omplete so-lution in the gauge Rg = n(n + 1). However, it is well-known that(M ; gdS) onformally ompa ti�es to the bounded domain in the Ein-stein stati ylinder gE = �dT 2+gSn(1) where T 2 (��2 ; �2 ). The metri gE is of ourse also a solution of (2.3) with RgE = n(n � 1), whi h isthus a globally hyperboli extension of (M ; gdS), sin e T 2 (�1;1);it is easily seen that this is the maximal solution.Thus, the hoi e RgE = n(n� 1) gives a global gauge for [gdS ℄. Thede Sitter metri itself, with gauge Rg = n(n + 1), has onformal fa tor relative to gE blowing up to 1 as T ! ��2 . To obtain an extensionof [gdS ℄ past the range (��2 ; �2 ) of gdS requires a res aling of the large fa tor to a fa tor of unit size.5. AFG equations vs. Einstein equationsConsider an initial data set (S ; ;K) for the va uum Einstein equa-tions in n+1 dimensions, n odd. Thus (S ; ) is a Riemannian manifold,and the pair ( ;K) satis�es the va uum onstraint equations with os-mologi al onstant � 2 R. Using Einstein's equations one an formally al ulate the derivatives(5.1) K(i)kl := 12�it kljt=0 ; 1 � i � nin a hypotheti al Gauss oordinate system near S in whi h the spa e-time metri g takes the form �dt2 + (t), as in (4.1). This givesProposition 5.1. The initial data set for (2.3) so obtained solves the onstraints (4.3), and any su h globally hyperboli solution of (2.3)given by Theorem 4.1 is onformally Einstein.Proof. Let g be the asso iated maximal globally hyperboli solution ofthe va uum Einstein equations. Then g also solves (2.3), and the resultfollows from the uniqueness part of Theorem 4.1. �A spa e-time with boundary (M ; �g) is said to be a onformal omple-tion at in�nity of a spa e-time (M ; g) if the usual de�nition of Penroseis satis�ed; that is, there exists a di�eomorphism � from M to theinterior of M and a fun tion : M ! R+ [ f0g, whi h is a de�ningfun tion for the boundary I := �M ;

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 17su h that g = ��(�2�g) :A onformal ompletion (M ; �g) isHs smooth if �g 2 Hs(M ), (in suitablelo al oordinates for M ).A set (S ; � ; �K;!; !(1); : : : ; !(n)) is said to be a smooth onformal ompletion at in�nity of a general relativisti initial data set (S ; ;K)if (S ; � ) is a Riemannian manifold with boundary, with S = S [�S ,and with !; !(j) : S ! Rbeing smooth-up-to-boundary fun tions su h that ! is a de�ning fun -tion for the boundary _S := �S ;with(5.2) � = !2 on S . Finally �K is a smooth-up-to-boundary symmetri tensor �eld onS su h that the equation(5.3) �K = !K + !(1)!2 � holds on S ( ompare (4.12)). We further assume that the fun tions!(i), 2 � i � n, are su h that the �elds �K(i), al ulated using the K(i)'sas in (5.1), and the fun tions !(i), de�ned before (4.12), an be extendedby ontinuity to smooth tensor �elds on S .A onformal ompletion will be said to be Hs if � 2 Hs(S ) and�K(i) 2 Hs�i(S ), i = 1; : : : ; n.The above onditions are learly ne essary for the existen e of asmooth onformal ompletion �a la Penrose of the maximal globally hy-perboli development of (S ; ;K); we will see shortly that they arealso suÆ ient. We emphasise, however, that Equations (5.2) and (5.3)alone, together with the requirement of smoothness of � and �K are notsuÆ ient for the existen e of su h spa e-time ompletions. Indeed, itfollows from the results in [3℄ that, in dimension 3+ 1, the requirementof smoothness up-to-boundary of the �K(i)'s, i � 2, imposes further on-straints on � and �K. It would be of interest to work out the expli itform of those last onditions, analogously to [3℄, in all dimensions.We have the following, onformal version of Proposition 5.1; it allowsone to repeat several onstru tions of Friedri h (see [19℄ and referen estherein) for va uum spa e-times with vanishing osmologi al onstantin all even spa e-time dimensions:

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 18Theorem 5.2. Let (S ; ;K) be a general relativisti va uum initialdata set, � = 0, whi h admits an Hs onformal ompletion at in�nity,s > n=2 + n + 1, with s 2 N and n odd. Then there exists an Hsspa e-time with boundary (M ; �g), equal to the onformal ompletion atin�nity of the unique maximal development (M ; g) of (S ; ;K), so thatg = �2�g on M , and I � _S :Proof. The proof is essentially identi al to that of Proposition 5.1. Con-sider the initial data ( ;K(1); : : : ; K(n)) as onstru ted at the beginningof this se tion. One an onformally transform them to initial data�K(i)kl := 12�is� kljt=0 ; 1 � i � nin a hypotheti al Gauss oordinate system near S in whi h the spa e-time metri �g takes the form �g = �ds2 + � (s). Here we use a nor-malisation as in the proof of Theorem 4.1 or Remark 4.5, requiring thevanishing (or onstan y) of R�g. Theorem 4.1 provides a solution ofthis Cau hy problem, while the fa t that this solution is onformallyEinstein follows from Proposition 5.1.The hoi e of the onformal fa tor transforming (M ; [�g℄) to the va -uum Einstein solution (M ; g) of ourse depends on the hoi e of gaugefor [�g℄; we des ribe here how is determined at least in the naturalsettings orresponding to (4.25). Suppose that � = !2 is a geodesi ompa ti� ation of (S ; ), so that for x near _S , !(x) = dist� ( _S ; x).Su h a ompa ti� ation is uniquely determined by the hoi e of a bound-ary metri on the boundary _S . Now the value of !(1), at the zero levelset of ! is determined by the initial data, ( ompare Eq. (3.13) of [3℄in dimension 3 + 1; an obvious modi� ation of that equation holds inall dimensions). Changing time-orientation if ne essary, one will have!(1) = �1 at _S and we extend !(1) to a neighborhood of _S in S tohave the same value. These data determine the ompa ti� ation of theinitial data set (S ; ;K).Hen e, the proof of Theorem 4.1 gives a unique lo al solution �g of(2.3) in the onformal lass [�g℄, with R�g = 0, for any given 0, satisfyingthe initial onditions. Let ' = (n�1)=2, and set g = �2�g = '�4=(n�1)�g.Then ', (and so ), is uniquely determined in a hart for ~M ontaininga portion of _I by the requirement that ' solves the linear wave equation4nn�1��g' � R�g' = 0, with initial data ' = !, �s' = !(1) = �1 on Snear _S , (where s is the Gaussian oordinate).Given su h a solution , let I be the onne ted omponent of theset f = 0 ; d 6= 0g interse ting _S . Sin e, by onstru tion, Ri g = 0,

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 19and �g is smooth up to I , standard formulas for the Ri i urvatureunder onformal hanges show that has the usual stru ture on I , inthat r is null, and r2 is pure tra e, on I . �Remark 5.3. A version of Theorem 5.2, and its proof, also holds for deSitter-type va uum solutions of the Einstein equations, where � > 0. Inthis ase, the ompletion is at future or past spa e-like in�nity I+ orI�; the Cau hy data for (M ; � ) atI+, onsist of the two undeterminedterms g(0), g(n) in the formal Fe�erman-Graham expansion for va uumEinstein solutions with � > 0. This gives an alternate proof of one ofthe results of [2℄.The ase � < 0 leads to initial-boundary value problems. While it is lear that a generalization of Friedri h's analysis of this ase [22℄ shouldexist, pre ise statements require further investigation.6. Appli ations to semi-global and global stability ofgeneral relativisti initial value problems in all evenspa e-time dimensionsLet (S ; 0) be the Poin ar�e metri on the (n + 1)-dimensional ball,with n � 3 and n odd. SettingK0 = 0;the set (S ; 0; K0) is an initial data set for the va uum Einstein equa-tions, denoted standard hyperboloidal initial data. The maximal globallyhyperboli development (M ; g0) of (S ; 0; K0) is given by(6.1) g0 = �d�2 + �2 0;for � 2 R+, with S = f� = 1g. This spa e-time is the interior of thefuture light one about a point in Minkowski spa e-time (the \Milneuniverse"). With respe t to the standard smooth onformal ompa t-i� ation of Minkowski spa e-time as a bounded domain in the stati Einstein ylinder,(6.2) �g0 = �dT 2 + dR2 + sin2R gSn�1(1);one has �K(i) = 0, 1 � i � n; the hypersurfa es f� = onstg orrespondto the level-sets fT = onstg, and so are totally geodesi . As noted inRemark 4.5, this hoi e of gauge is global and satis�es Rg0 = n(n� 1).As an example of appli ation of the results of the previous se tion,one now easily obtains:Theorem 6.1. Let S be an n-dimensional open ball, n odd, and onsider a general relativisti initial data set (S ; ;K) whi h admitsan Hs onformal ompletion at in�nity, s > n=2 + n + 1, s 2 N.

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 20Then there exists " = "(n) > 0 su h that if the asso iated data(S ; [(� ; �K(1); : : : ; �K(n))℄) are "- lose in Hs � : : : � Hs�n to the data(S ; [(� 0; 0; : : : ; 0)℄) asso iated to standard hyperboloidal initial data,then the maximal globally hyperboli development (M ; g) of (S ; ;K)is ausally geodesi ally omplete to the future.The Hs onformal ompa ti� ation (M ; �g) is Hs lose to (M ; �g0),and extends to a larger Hs spa e-time ontaining a regular future time-like in�nity �+ for (M ; �g).Proof. The standard spa e-time (M ; g0) has a onformal ompa ti� a-tion to a bounded domain D in the stati Einstein ylinder (6.2), whereD orresponds to the range of parameters T +R 2 [0; �℄, T �R 2 [0; �℄,R � 0. Future null in�nity I + is given by I + = fT + R = �g; T 2(�2 ; �), with future time-like in�nity �+ = fT = �;R = 0g. The futuredevelopment of (S ; 0; K0) orresponds to the domain D+ = D \ fT ��2g.Clearly, the ompa ti� ation (D; �g0) extends smoothly to a neighbor-hood ~D of D as a globally hyperboli solution of (2.3). The Cau hy datafor su h an extension are an extension of the standard Cau hy data(S ; [(� 0; 0; : : : ; 0)℄) past the boundary _S . Similarly, the initial data(S ; [(� ; �K(1); : : : ; �K(n))℄) for (M ; �g) extend in Hs past a neighbhorhoodof the boundary _S and generate a maximal globally hyperboli spa e-time ( ~M ; ~g), satisfying (2.3). By the Cau hy stability asso iated withTheorem 5.2, for " small, the solution ( ~M ; ~g) is lose in Hs to ( ~D; �g0),and in parti ular is an Hs extension of (M ; �g), where M = f > 0g in~M . This shows that, to the future of S , (M ; g) has an Hs onformal ompletion, whi h extends in Hs to a neighborhood of I and �+. Thisgives the result. �Note that in dimension 3 + 1 the mere requirement[(� ; �K(1); : : : ; �K(n))℄ 2 Hs � : : :�Hs�n ; s > n=2 + n+ 1 ;regardless of any smallness ondition, forbids solutions whi h have loga-rithmi terms with small powers of 1=r in polyhomogeneous expansions.Thus, (similarly to the results of Friedri h), the above theorem appliesfor non-generi initial data sets only.Using Corvino-S hoen type onstru tions together with the abovestability result, as in [9℄, one obtains:Theorem 6.2. There exists an in�nite dimensional spa e of omplete,asymptoti ally simple globally hyperboli solutions of the Einstein va -uum equations in all even dimensions n + 1 = 2(k + 1), n � 3. Thus,

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 21su h solutions are geodesi ally omplete both to the future and past, andhave a smooth onformal ompletion at in�nity.Proof. The Corvino-S hoen gluing te hnique [12, 13℄ an be used, asin [9, 10℄, to onstru t stati , parity-symmetri initial data on Rn, forany n � 3, whi h are S hwarzs hild with m 6= 0 outside a ompa t set,and whi h are as lose to the Minkowskian data as desired. The result-ing maximal globally hyperboli spa e-time then ontains smooth hy-perboloids, lose to standard hyperboli initial data, as in Theorem 6.1,both in the future and in the past. In even spa e-time dimensions theresult follows as in the proof of Theorem 6.1. �We note that all the spa e-times onstru ted in the proof of Theo-rem 6.2 possess a \ omplete I "; this should be understood as om-pleteness of generators of I in the zero-shear gauge, ompare [24℄.6One expe ts the above onstru tion to generalise to initial data whi hare stationary, asymptoti ally at outside of a spatially ompa t set(rather than exa tly S hwarzs hild there). This would require provingthat the resulting spa e-times have smooth onformal ompa ti� ationsnear �0, (in spa e-time dimension four this follows from [14,15,33℄), andworking out \referen e families of metri s" needed for the argumentsin [10℄. Those results are very likely to hold, but need detailed he king;note that one step of [33℄ requires dimension four, and that the familyof asymptoti ally at stationary metri s in higher dimensions might beri her than that in dimension 3 + 1, [31℄.Appendix A. The infinitesimal invarian e group of the onstraint equations.In this appendix we study the Lie algebra asso iated to the groupof onformal transformations preserving the onstraint equations (4.3).This allows one to derive identities whi h shed light on the stru ture ofthose equations.Suppose that H = 0 and that the spa e-time metri g is res aled by2, where, in a Gauss oordinate system (t; xi), so that S = ft = 0g,we have = 1+ " (xi)j! tj ;for some j � 0, and " > 0 small. In the new Gauss oordinates (�t; �xi)we thus have2(�dt2 + gijdxidxj) = �d�t2 + �gk`d�xkd�x` ;6We are grateful to H. Friedri h for useful dis ussions on erning this point.

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 22leading to (�t�t)2 � �gk`�t�xk�t�x` = 2 ;(A.1) �t�t�i�t = �gk`�i�xk�t�x` ;(A.2) gij = �2��gk`�i�xk�j �x` � �i�t�j�t� :(A.3)By de�nition of Gauss oordinates we have (�t; �xi) = (O(t); xi + O(t)),and also (�t; �xi) = (t + O("); xi + O(")). Inserting this in the equationsabove, mat hing powers in Taylor expansions we �nd�t = t+" (j + 1)!tj+1+O("2t2j+3) ; �xi = xi+" Di (j + 2)!tj+2+O("2t2j+2) :At the right-hand-side of (A.3) we have variations related to the fa tthat all the quantities there are evaluated at the point �x = x+"�(�)+: : :;to �rst order, this produ es a Lie derivative-type ontribution. Next,there are variations related to the fa t that �t = t + " � (�) + : : : Ea hterm �ti �K(i)=i! in (twi e) the Taylor expansion of �gij at t = 0 gives thena ontribution to the right-hand-side of (A.3) equal to�2(t+" (j + 1)!tj+1)i �K(i)i! +O("2) = �ti + i� 2j � 2(j + 1)! " ti+j� �K(i)i! +O("2) :>From this we an al ulate the oeÆ ients of an expansion in powersof t of the right-hand-side of (A.3); inverting those relations, for j = 0this leads to(A.4) �K(i) = � (1� "(i� 2) )K(i)+ O("2); i 6= 2;K(i) � "4LD g + O("2); i = 2,where L denotes a Lie derivative; note that LD g is twi e the Hessianof .For j > 0 we obtain instead�K(i) = K(i) ; 0 � i < j ;(A.5) �K(j) = K(j) + " g ;(A.6) �K(j+i) = K(j+i) � (i� 2j � 2)(j + i)!2(j + 1)!i! " K(i)(A.7) +� O("2); 1 � i 6= 2;� "4LD g + O("2); i = 2.The terms proportional to " in those equations des ribe the desiredin�nitesimal a tion.We an view H0� as fun tions of the metri , its derivatives, and ofthe symmetri derivatives D(`1:::`s)K(i)k` , with i = 1; : : : ; n. As H�� is adi�erential operator in the spa e-time metri of order n+1, the possibly

ASYMPTOTICALLY SIMPLE EVEN-DIMENSIONAL SPACE-TIMES 23non-trivial ontributions arise from those D(`1:::`s)K(i)k` for whi h s+ i �n + 1. Di�erentiating the equations 0 = H0� with respe t to ", at" = t = 0 for j > 0 one obtains(A.8)0 = n+1�jXs=0 �H0��D(`1:::`s)K(j)k` gk`D`1:::`s �12 n�jXi=1 (i� 2j � 2)(j + i)!(j + 1)!i! n+1�i�jXs=0 �H0��D(`1:::`s)K(j+i)k` D`1:::`s( K(i)k` )�12 n�1�jXs=0 �H0��D(`1:::`s)K(j+2)k` D`1:::`sk` :At any point x the derivatives D`1:::`s'(x) = D(`1:::`s)'(x) an be ho-sen independently, whi h leads to various identities. The simplest oneis obtained for j = n, then the last two lines give a vanishing ontribu-tion. We parameterise the K(i)'s by their tra e-free part and by tra e,obtaining(A.9) 0 = �H0��D`1:::`strK(n) ; s � 0 ;this gives a he k of the general form of (4.10).When j = n � 1 the last line in (A.8) is zero again, and we obtain0 = 2Xs=0 �H0��D(`1:::`s)K(n�1)k` gk`D`1:::`s +12(2n� 1) 1Xs=0 �H0��D`1:::`sK(n)k` D`1:::`s( K(1)k` )= 2Xs=0 �H0��D`1:::`strK(n�1)D`1:::`s +12(2n� 1)� �H0��K(n)k` K(1)k` + �H0��D`K(n)k` (K(1)k` D` + D`K(1)k` )� :The vanishing of the oeÆ ients in front of the se ond derivatives of leads to the identity(A.10) 0 = �H0��D`1`2trK(n�1) ; onsistently with (4.9).

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