The asymptotic of static isolated systems and a generalized uniqueness for Schwarzschild

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1 Home Search Collection Journal About Contact u My IOPcience The aymptotic of tatic iolated ytem and a generalized uniquene for Schwarzchild Thi content ha been downloaded from IOPcience. Pleae croll down to ee the full text. View the table of content for thi iue, or go to the ournal homepage for more Download detail: IP Addre: Thi content wa downloaded on /7/6 at 4:47 Pleae note that term and condition apply.

2 Claical and Quantum Gravity Cla. Quantum Grav. (5 95 (6pp doi:.88/64-98//9/95 The aymptotic of tatic iolated ytem and a generalized uniquene for Schwarzchild Martin Reiri Max Planck Intitute für Gravitationphyik Am Mühlenberg D-4476 Golm, Germany martin@aei.mpg.de Received 7 February 5, revied June 5 Accepted for publication July 5 Publihed 4 September 5 Abtract It ha been proven that any tatic ytem that i pacetime-geodeically complete at infinity, and whoe pacelike-topology outide a compact et i that of minu a ball, i aymptotically flat. The matter i aumed to be compactly upported and no energy condition i required. A imilar (though tronger reult alo applie to black hole. Thi allow u to tate a large generalization concerning the uniquene of the Schwarzchild olution in not requiring aymptotic flatne. The Korotkin icolai tatic black-hole how that for the given generalization, no further flexibility in the hypothei i poible. Keyword: iolated ytem, tatic olution, aymptotic. Introduction Aymptotic flatne i the baic notion ued in general relativity (GR to model ytem that can be thought of a iolated from the ret of the univere. It wa ued by Eintein himelf, at leat in heuritic form, and i now a tandard piece of differential geometry and of gravitational and theoretical phyic. The notion of aymptotic flatne i alo epitemologically linked to the ewtonian theory of gravitation. In the 96 manucript The Foundation of the Generalized Theory of Relativity, Eintein addreed what he called an epitemological defect (but not mitake of claical mechanic, whoe origin he linked to Mach. He imagined two bodie, A and B, made of the ame fluid material and ufficiently eparated from each other that none of the propertie of one could be attributed to the exitence of the other. Oberver at ret in one body ee the other body rotating at a contant angular velocity, yet thee ame oberver The following paage i baed on a text I prepared for a highlight in CQG /5/95+6$. 5 IOP Publihing Ltd Printed in the UK

3 Cla. Quantum Grav. (5 95 M Reiri Figure. Repreentation of an AF end and a non-af end. meaure a perfectly round urface in one cae and an ellipoid of rotation in the other cae. He then aked: Why i thi difference between the two bodie?. ecearily, he continue, the anwer cannot be found inide the ytem A + B only; it mut lie exterior to the ytem: the outer empty pace. Eintein found that the ource of the peculiar diparity wa omitting that the empty pace hould alo obey phyical law. Thee law, which treat part A and B of the ytem A + B + EXTERIOR EMPTY SPACE on an equal footing, are the Eintein equation of GR. There i one point in Eintein elegant concluion that remain lightly inconcluive. It can be argued on the bai of GR that the abolute pace of the 8th and 9th centurie wa an inevitable concept, a correction to the ewtonian gravity are imply too mall. Although thi i unquetionable, it can alo be demanded of GR to explain why thi background olution, repreenting the EXTERIOR EMPTY SPACE of the ytem decribed earlier, i o ditinguihed in a theory that treat the geometry and the aymptotic of pace eentially a a variable. Depite their importance, mathematical analyi of thee quetion wa only recently initiated. The firt general reult came in with Anderon uniquene of the Minkowki pacetime []. The reult ay that a geodeically complete tatic pacetime with no material ource (i.e. vacuum i flat, and therefore covered by the Minkowki pacetime. Thu, the only geodeically complete and imply connected olution of the tatic Eintein equation empty of matter i Minkowki. Thi nicely illutrate the ditinguihed place that the Minkowki pacetime ha among the phyically relevant olution. Around the ame time, Anderon [] tarted a ytematic analyi of the global geometry of geodeically incomplete pacetime including, for intance, pacetime with boundary or ingularitie, and ufficient condition for AF were given. The concluion of [] apply directly to pacetime with compact ource a one can alway excie a region containing matter and retrict the attention to the reulting pace. More recently, neceary and ufficient condition for aymptotic The reult in [] apply to trictly tationary olution a well. In thi article, we will refer only to tatic olution.

4 Cla. Quantum Grav. (5 95 M Reiri flatne were invetigated again in [8, 9] (ee alo []. We explain thee reult in ome detail a they will be relevant for the ret of the paper. We begin with a formal definition of the vacuum tatic data et that will be ueful later. Definition.. A tatic vacuum data et ( Σ; g, conit of a mooth three-manifold Σ, poibly with boundary, a mooth Riemannian metric g, and a mooth function, uch that, (i ( Σ; g i metrically complete, (ii i trictly poitive in the interior Σ of Σ, and (iii (g, atifie the vacuum tatic Eintein equation, Ric =, Δ =. (. ote that if vanihe omewhere, then it doe o only in point at the boundary of Σ (though could alo be trictly poitive there. For intance, Σ can be part of a larger pace after removing a region containing the ource (if any. The reult in [8] and [9] concern the aymptotic of the end of Σ and are independent of the geometry of the tate in the bulk of the manifold (including the boundary. They are tated a follow. Suppoe that a cloed region Σ in the interior Σ of Σ i diffeomorphic to minu an open three-ball. If the pace ( Σ, g = g i metrically complete, then the pace ( Σ ; g, i aymptotically flat with Schwarzchildian fall off. The analyi in [8, 9] i made uing the conformal metric g = g becaue of the remarkable propertie that thi metric ha. The Ricci curvature i Ricg = ln ln and, in particular, i non-negative. Moreover, a alo hown in [], the pace ( Σ, g ha quadratic curvature decay (from Σ provided that it i metrically complete. If ( Σ, g i metrically complete and >, then ( Σ, g i metrically complete, but in principle there i no reaon to aume the completene of the econd pace without any aumption on. In thi article, we prove that the completene of ( Σ, g (in a ituation a decribed above follow from the uitable and phyically natural geodeic completene (until the boundary of the aociated pacetime. The reult i ummarized in theorem. and ay that iolated ytem, a defined below, are indeed alway aymptotically flat. A hould be clear to the reader, the definition below i intended to capture the intuitive notion of a phyical iolated ytem, but of coure without making any reference to the aymptotic. The definition i a bit formal but it will give a good mathematical frame to be ued later. Definition.. A globally hyperbolic tatic pace-time ( M, g i called a tatic iolated ytem if there i an open et K of M containing (if any the material ource, uch that, (i the region M K i diffeomorphic to (, (. and on thi region the pace-time metric i of the form g = d t + g, (. where the lape > and the patial metric g are t-independent ( t i the tatic Killing, and

5 Cla. Quantum Grav. (5 95 M Reiri (ii ( M K, g i geodeically complete until it boundary, namely, geodeic (of any pacetime character at either end of it boundary or defined for infinite parametric time. To illutrate thi definition, the implet example of a tatic iolated ytem that one can imagine i a tatic pherically ymmetric tar. In thi cae, the pace-time M i diffeomorphic to and the material ource (the tar i contained inide the open pacetime region K = { p M: r( p < r*, where r i the areal coordinate and r* i the radiu of the tar. Thi region i repreented chematically in the left of figure. Outide thi region K, the pacetime i Schwarzchild and clearly atifie (i and (ii. We comment on everal apect of the definition. The topological condition (. in (i of the definition above i the mot natural if one i decribing an atrophyically realitic ytem like a neutron tar. On the other hand, the exitence of a tatic Killing field doe not automatically imply the exitence of global coordinate where the pacetime metric take the form of (. (though locally thi i alway the cae where t. The problem of the global exitence of uch coordinate i difficult and will not be conidered here. In thi ene, (. ha to be conidered a an aumption and not a a conequence of taticity. The geodeic completene until the boundary in (ii i a neceary condition to enure that (roughly peaking the phyical boundary of M K i ut K. Geodeic are either infinite or they reach K. The information that will be crucial for proving the metric completene of ( Σ =, g = g (from which AF will follow a explained earlier i that concerning the completene of the geodeic that move further and further away from the boundary. Thi hould become clear during the proof later. Once more, we tre that a M K i free of matter, the data et (, g on Σ = { t = ( M K atifie the vacuum tatic equation Ric =, Δ =. (.4 From now on we will call (ii imply geodeic completene at infinity: Thi terminology i utified by the following fact: geodeic completene until the boundary hold if every pacetime geodeic, whoe proection into leave any compact et, i complete. In thi etup we prove the following. Theorem.. Static iolated ytem are aymptotically flat with Schwarzchildian fall off. Thi theorem i an expreion of the remarkable conitency of GR a a phyical theory and how the inevitability of aymptotic flatne in certain context. The definition of Schwarzchildian fall off that we ue in thi theorem (and alo above i the implet one and refer to the decay of g and on a (uitable coordinate patch. Concretely, i i i ( g gs + ( S = ( r 4, where gs = ( + m r (dr + r d Ω and S = ( mr ( + m r are, repectively, the uual metric and lape of the Schwarzchild olution. The Schwarzchildian fall off doe not play any pecial technical role in thi article but it i important to tate, a we did in theorem., the type of decay that tatic iolated ytem have. Along the ame line a in theorem., we can generalize the celebrated uniquene of the Schwarzchild olution (Irael [6], Robinon [], Bunting Maood-Ul Alam [] to a uniquene tatement among a (a priori much larger cla of tatic olution than thoe AF. Accordingly, we conider tatic olution given by vacuum tatic data ( Σ ; g,, i.e. with The Irael breakthough in 967 wa the firt uniquene theorem for Schwarchild and required that could be choen a a global radial coordinate. 4

6 Cla. Quantum Grav. (5 95 M Reiri Ric =, Δ =, (.5 and with a compact but not necearily connected horizon Σ = { =. A tated earlier, the olution are aid to be geodeically compete at infinity if pacetime geodeic, of any pacetime character, either end at the horizon (i.e. the boundary or are defined for infinite parametric time. The uniquene theorem i the following. Theorem.4. Let ( Σ; g, be the data et of a tatic vacuum pacetime with a compact horizon and which i geodeically complete at infinity. Then, the pacetime i Schwarzchild if a connected component of the complement of an open et of Σ containing the boundary i diffeomorphic to. Eentially, the AF hypothei that wa required in earlier verion of the uniquene theorem can be replaced by a neceary and ufficient topological condition. Oberve that in thi tatement nothing i aid about the other connected component (if any of the complement of the compact et. In principle, there could be many other unbounded connected component. That thi cannot happen mut be dicerned after ome analyi. Thi i imilar in pirit to topological cenorhip the ame type of theorem a in [4], although our technique i different a we cannot rely on any given tructure at infinity. To undertand the importance and cope of thi theorem, let u conider two purely relativitic example. The firt i of coure the Schwarzchild black hole. It i a tatic vacuum olution with a topological-pherical hole, it curvature decay to zero at infinity, and the pacetime i geodeically complete at infinity. However (though not alway properly emphaized, Schwarzchild i not the only tatic vacuum black hole olution in + dimenion enoying thee attribute. The other olution we are referring to i the Korotkin icolai tatic black hole [7]. It repreent a topologically pherical hole that i not inide an open (infinite three-ball a in Schwarzchild, but inide an open (infinite olid-toru. It i axially ymmetric and ha the aymptotic of a tatic Kaner [7] pacetime. It pace i not imply connected; for thi reaon, the horizon i prolate, a it feel the influence of itelf along an axi of ymmetry of finite length. The particular Kaner aymptotic i the imultaneou reult of the preence of the hole on one ide and of the non-trivial global topology on the other. Finite cover of the olution yield tatic pacetime with a finite number of black hole in equilibrium. From the point of view of the general theory of relativity, the Korotkin icolai and Schwarzchild olution are perfectly acceptable, although one i AF and the other i not. Thi how that the topological aumption in theorem.4 cannot be eliminated altogether. In parallel to the dicuion given at the beginning of the introduction, it i worth noting that theorem. can be interpreted a a reult of aymptotic uniquene (here aymptotic flatne, and that, in thi ene, it i a cloe relative of the uniquene of the flat Minkowki pacetime among complete (imply connected vacuum tatic pacetime a proved by Anderon in []. Anderon reult i a direct conequence of a curvature decay that we will explain in ection.. We tre, however, that uch decay i not nearly ufficient to deduce aymptotic flatne. The Korotkin icolai olution atifie thi curvature decay and i not AF. The ret of the article i roughly organized a follow. Section.,. and. deal with ome important fact about the global tructure of the vacuum tatic olution. Section contain the proof of theorem. and.4. Propoition. how the exitence of a natural partition of tatic end of the form. Propoition. then prove that the lape can 5

7 Cla. Quantum Grav. (5 95 M Reiri have only three type of behaviour at infinity and propoition. prove the completene of g on the end. The proof of theorem. and.4 are given afterward.. Background material A mooth Riemannian metric g on a mooth connected manifold Σ (with or without boundary, compact or not induce the metric { ( γpq γpq dit( p, q = inf length : mooth curve oining pto q. (. The pace ( Σ; g i aid to be metrically complete if ( Σ; dit i complete. If Σ ha a compact boundary, then metric completene i equivalent to the geodeic completene until the boundary of ( Σ ; g (by Hopf Rinow. On the other hand, geodeic in ( Σ; g lift to geodeic perpendicular to the tatic Killing field in the aociated pacetime, i.e. in M = Σ, g = d t + g. (. Hence, if Σ i compact, then geodeic completene until the boundary of ( M; g implie the metric completene of ( Σ ; g. Thi i ued in propoition.. Geodeic completene until the boundary of ( M; g i a baic aumption in the two main theorem in thi article. However, regarding poible mathematical application, it i important to aume only the metric completene of the data whenever poible. We will make ome remark in thi repect. If Σ, we define the metric annulu ( a, b of radii < a < b by ( a, b = { p Σ: a < dit( p, Σ < b, (. where dit( p, Σ = inf{dit( p, q: q Σ... Anderon curvature decay Anderon curvature decay [] i an important property of tatic olution. It ay that there i a univeral contant η > uch that for any tatic data ( Σ ; g,, we have Ric ( p η p dit ( p,, and ( η dit ( p,. (.4 Σ Σ g The optimal contant η can be een to be greater than or equal to one, but it i not know if it i one. Upper bound can be given, but they are far from one. A an application of the curvature decay, let u prove here the propoition that will be ued in the proof of theorem.4 to rule multiple end when it i known that there i one that i AF. In the tatement, we ue Σ δ to denote the manifold reulting from removing from Σ the tubular neighbourhood of Σ and radiu δ, i.e. Σδ = Σ { p: dit g( p, Σ < δ. We aume that δ < δ with δ mall enough that Σ δ i alway mooth. Propoition.. Let ( Σ; g, be a tatic vacuum initial data et with a compact horizon ( Σ = { = and ( Σ; g be metrically compete. Then there i < ϵ < uch that for every ϵ < ϵ there i δ < δ, uch that ϵ ( Σ δ ; g i metrically complete and Σ δ i trictly convex (with repect to the outward normal. g 6

8 Cla. Quantum Grav. (5 95 M Reiri Proof. Given < ϵ <, the convexity of Σ δ for mall enough δ δ i direct (and we leave it to the reader a the factor ϵ blow up the boundary Σ uniformly (oberve, however, that a ϵ <, Σ remain at a finite ditance from the bulk of Σ. So let u prove that if we choe mall enough ϵ, the pace ϵ ( Σ δ, g i metrically complete. A we aume δ < δ, it i ufficient to prove that if ϵ i mall enough, then ϵ ( Σ δ, g i metrically complete 4. We will do that below, and the argument i thu independent of δ. It i enough to prove that if ϵ i mall enough, then the following hold: for any equence of point p i whoe g ditance to Σ δ diverge, the ϵ ( g ditance to Σ δ alo diverge. Equivalently, it i ufficient to prove that for any equence of curve γ i tarting at Σ δ and ending at p i, we have i ϵ ( d, (.5 ( γi where i the g arc length of γ i tarting from Σ δ. ow, a we will how below, the curvature decay (.4 immediately implie the etimate ( g( Σ η ( p c + dit p, δ (.6 for any p Σ, where η > i univeral but c depend on ( Σ, g and δ. dit ( γ(, Σ, then we have g i δ ( i Thu, if ϵ < η, then i γ ( c( + η. (.7 i d = ( γi c ( + c ( ϵ η ( d ϵ ϵ ϵ η ϵ c ϵ ( ϵ η ( g( pi Σδ ϵ η (( + i A (.8 ϵ η + dit, (.9 a wihed. ow we briefly comment on the derivation of (.6. Let γ ( be a geodeic oining a point p to Σ δ and realizing the ditance between p and Σ δ. Let p be the point of interection between γ and Σ δ. Then ( p ln ( p ( p Σδ γ dit g( p, Σδ dit g, ( η = d d (. dit g, δ ( p Σδ ( = δ ( p Σ δ dit, ln δ η, (. 4 A Σδ = Σδ ( Σδ Σδ ϵ and ( Σδ Σδ i a mooth compact manifold, the manifold ( Σ δ, g i metrically complete if ϵ (, g i metrically complete. Σ δ 7

9 Cla. Quantum Grav. (5 95 M Reiri and hence ( p Σ δ dit, ( p ( p δ η. (. One then ue the general etimate dit g( p, Σδ dit g( p, Σ δ δ + and ( p max{ ( q: q Σ δ to how eaily (.6 for a uitable c big enough. There are two propertie about the pace ϵ ( Σ δ, that will be central in the proof of theorem. and that are worth highlighting here. Firt, the manifold ϵ ( Σ δ, g i geodeically convex, that i, any two point in Σ δ can be oined by a length-minimizing geodeic contained inide Σ δ. Thi i indeed a direct conequence of the trict convexity and compactne of the boundary Σ 5 δ. In particular, if Σ ha two end, then o doe Σ δ, and one can guarantee the exitence of a line diverging along the ϵ two end. Second, but not le important, the Ricci curvature of the metric g = g ha the expreion 6 Ric = f + f f, (. c where f and c depend on ϵ and are given by ( ϵ ϵ f = ( + ϵln, and = c ( + ϵ. (.4 In particular, if < ϵ <, then c > and the c-bakry Emery Ricci tenor Ric which i defined by c f, Ric c f = Ric + f f f, (.5 c i zero. In the proof of theorem., we will ue thee two obervation together to make ome imple mean-curvature comparion (a la Bakry Emery... The ball covering property A oberved in [], Liu ball covering property hold for (metrically complete tatic olution ( Σ; g with compact boundarie. amely, for any < a < b, there i r and n uch that for any r r there i a et of ball { B ( pi, ar, pi ( ar, br, i =,, nr n covering ( ar, br. Here and below i the cloure of. A a direct corollary, we ee that for any < a < b and r r, a in the ball covering property, any two point in the ame connected component of ( ar, br can be oined by a curve of length le than or equal to nar entirely contained in ( ar, br. 5 Oberve that a length-minimizing equence of curve (with fixed end-point mut remain at a definite ditance away from the boundary, a otherwie their length could be reduced in a definite amount (due to the trict convexity. With thi property granted, the limit of the equence (or of a ubequence if neceary mut be a geodeic in Σ δ by tandard argument. 6 For thi, if g e = ϕ g, then Ric = Ric ( ϕ ϕ ϕ ( Δϕ + ϕ g and V = V k ( V iϕ + Vi ϕ ( V k ϕ g i. i i 8

10 Cla. Quantum Grav. (5 95 M Reiri Let c ( ar, br be a connected component of ( ar, br. By the curvature decay (.4, we have η ar all over c ( ar, br. By integrating thi inequality along curve a in the previou paragraph, we obtain 7 { p p c ar br { p p c ar br max ( : (, min ( : (, Thi i a type of Harnack inequality for and i fundamental. C( a, b. (.6 Remark.. It i not known at the moment if a imilar ball covering property hold for trictly tationary olution. Thi i a main obtacle to extending theorem. to tationary iolated ytem... Spacetime geodeic in tatic pacetime Let ( Σ; g, be a tatic vacuum data and let ( M, g be it aociated pacetime. We recall here a ueful way to decribe pacetime geodeic Γ( τ in term of certain metric conformal to g in Σ. Thi goe back at leat to the work of Weyl [] from 97. Let γ = Π( Γ be the proection of Γ into Σ. Then it i eay to ee that γ atifie the equation a γ γ =, (.7 where γ = dγ dτ and a i the contant a = g( Γ,. Moreover, we have a γ = ε +, (.8 where the norm on the left-hand ide i with repect to g and ε =,, according to the character type of the geodeic. Then define e ϕ by t a ϕ e = ε +, (.9 wherever the right-hand ide i poitive (thi include the proection of the geodeic. Finally, we conider the conformal metric ϕ ϕ gˆ = e g, gˇ = e g, (. and ue d, dˆ = ed ϕ, and dˇ = e ϕ d to denote the element of length of γ with repect to g, gˆ and ǧ, repectively. In thi etup, we have the following characterization: if Γ( τ i a pacetime geodeic, then γ (ˆ i a geodeic of ĝ and dτ = dˇ. Converely, if γ (ˆ i a geodeic of ĝ, then the curve Γ ( ˇ = š a ( γ ( ˇ dˇ, γ( ˇ Σ = M (. i a pacetime geodeic with g( Γ, Γ = ε, and hence with τ = š. 7 If γ ( i a curve of length le than or equal to nar oining the point p and p, then ln ( p ( p = ( γ d ( η ar nar = nη. From thi we deduce e n η ( p ( p e n η and (.6 follow (note n = n( a, b. 9

11 Cla. Quantum Grav. (5 95 M Reiri Two point are particularly important about thi characterization of pacetime geodeic: (i pacetime geodeic can be contructed out of the proected curve which in turn can be eaily found through length-minimization, and (ii a the affine parameter of pacetime geodeic i the ǧ-arc length of the proected curve, there i a way to link pacetime geodeic completene at infinity to the metric completene of gˇ = g. We will exploit thee two obervation during the proof of propoition.. We will only ue the characterization of null geodeic, i.e. ϵ =, although other type of geodeic can be ueful in imilar context.. The proof Every mooth, connected, compact, boundaryle and orientable urface F embedded in divide into two connected component. Below we will work with uch urface F embedded in and will denote by M(F the cloure of the bounded connected component of ( F. Two fact are imple to check. Firt, for any dioint F and F uch that M( F i for i =,, either F M ( F or F M ( F (here = interior. Second, if a et { Fi, i =,, n of uch urface i uch that belong to a bounded component of Σ i = Fi then there i at leat one F i uch that M( F i. We will ue thee i= n fact in the proof of the following propoition. Propoition.. Let ( Σ; g, be a metrically complete vacuum tatic data et with Σ. Then, there i a et of (mooth, connected, compact, boundaryle and orientable urface { S ; =,,,,, uch that the following hold for every :. S i embedded in ( +, +,. Σ M( S, and. M( S M( S +. The urface S i will be ued only a reference inide the manifold Σ; their geometrie = play no role. Oberve that Σ M( Sk = = km( S+ M( S with the union dioint and that S + S = ( M( S+ M ( S. Thi lat obervation will be ued when we apply the maximum principle to on M( S M ( S. + Proof. In the argument that follow, we treat Σ and inditinctly. The contruction of the urface S, =,,, i a follow. Let f : Σ [, be a (any mooth function uch that f on { p: dit( p, Σ + and f on { p: dit( p, Σ +. Let x be any regular value of f in (,. Then we can write f ( x = F Fn, where each F i i a (connected, compact, boundaryle and orientable urface embedded in ( +, +. ow, a Σ i the dioint union of the et f (( x,, f ( x i i = = = F i and f ((, x, and a { p : dit( p, + f Σ ((, x, we conclude that Σ, which lie inide f (( x,, mut belong to a bounded component of i= n Σ i F = i. Hence Σ M( F i * for ome F i*, (ee the beginning of thi ection. We et S = F i *. We now verify that the urface S atify propertie. By contruction the S already atify and. ow, either M( S M ( S+ or M( S+ M ( S. If M( S+ M ( S, then + S + { p: dit( p, Σ <, which i impoible becaue + 4 S + (, +. Thu, M( S M ( S+, howing property.

12 Cla. Quantum Grav. (5 95 M Reiri We claim that for any, the urface S + and S lie in the ame connected component of the annuli 4 ( +, +. To ee thi, conider a ray γ (,, tarting at Σ at =, (i.e. dit( γ(, Σ = for all ; i arc-length. Let be the lat time that γ ( S and let + be the firt time that γ ( S +. Then, + becaue S ( +, + and 4+ + becaue + 4 S + (, +. Hence the arc { γ (: [, + ] mut lie inide 4 ( +, + becaue dit( γ(, Σ = for all. We then conclude that S and S + mut lie in the ame connected component of 4 ( +, +. Thi claim and propoition. will be ued in the proof of the following propoition. Propoition.. Let ( Σ; g, be a metrically complete vacuum tatic data et with Σ and >. Then, one of the following hold:. converge uniformly to zero over the end of Σ,. converge uniformly to infinity over the end of Σ,. C < < C for contant < C < C <. Proof. To horten notation, we will write max{ ; Ω: max{ ( p: p Ω, where Ω are compact et (ame notation for min{ ; Ω. Suppoe that there i a divergent equence p i for which ( pi a i. We claim that, in thi cae, tend uniformly to zero over the end. For every i let i be uch that p M( S M ( S. Suppoe firt that { S i i i i max ;. (. Then, for any i > i, the maximum principle give { M( S M( S { { S { S i i i i max ; max max ;, max ;. (. Letting i and uing (., we obtain { Σ i ( { i up ( p: p M S max ; S, (. where the right-hand ide tend to zero a i tend to infinity. Thi prove that tend uniformly to zero a claimed. To prove (., we recall, a wa pointed out earlier, that S i and S lie in the ame i connected component of (, +. Oberve too that the annuli (, + can be written a ( aribri with a =, b = 4 and ri = i. Therefore, we can ue the dicuion of ection. to deduce that { S c { S S i i i max ; min ;, (.4 where the contant c i independent of i. On the other hand, by the maximum principle, we have { { ( ( min ; S S min ; M S M S ( pi. (.5 i i i i

13 Cla. Quantum Grav. (5 95 M Reiri Combining (.4 and (.5, we obtain { Si max ; ( p, (.6 i where the right-hand ide tend to zero. Thi implie (. a deired. In the ame manner, one prove that if there i a divergent equence p i uch that ( p i a i, then tend uniformly to infinity over the end. If none of the ituation conidered above occur, then < C < < C for the contant C, C. To how the aymptotic flatne for iolated ytem uing [8, 9], we need only to prove the completene of g uing the aumption that the tatic pacetime i geodeically complete at infinity. Thi i done in the next propoition. Propoition.. Let ( Σ; g, be a tatic vacuum data et, with Σ and > on Σ. Aume that the aociated pacetime M = Σ, g = d t + g (.7 i geodeically complete at infinity. Then the pace ( Σ; g i metrically complete. Proof. The proof i made by contradiction. So let u aume that ( Σ; g i not metrically complete. We will explain later how thi contradict the geodeic completene at infinity. During the proof, we ue the ame notation a in propoition.. We will alo aume, a wa explained in ection, that under the hypothei of the propoition, the pace ( Σ; g i metrically complete. We begin by proving that = { S max ; <. (.8 = Let β:[, + ] M( S+ M ( S be any curve with β ( S and β ( + S+. We claim that then { + ( ( d c max ; S β, (.9 where the contant c i independent of. To ee thi, we write and note that { ( + ( + ( β( d min ; M S M S length( β (. +. length( ( + β = 6, becaue it i S ( +,, and + 4 S + (,,. min{ ; M( S+ M ( S max{ ; S, becaue { + ( ( { + min ; M S M S min ; S S (. by the maximum principle, and becaue

14 Cla. Quantum Grav. (5 95 { S+ S c { S min ; max ;, (. where c i independent of, a wa explained in ection., 8. The formula (.9 i then obtained making c = 6c. ow, if ( Σ; g i not metrically complete, then one can find a equence of point p i, with dit g( pi, Σ but with dit g ( pi, Σ uniformly bounded. From the definition of dit, thi implie that there i a equence of curve αi(; [, i] tarting at Σ and ending at p i, for which = i ( α ( d K <, (. = where K i independent of. For every i let i be the greatet uch that pi M( S. Then, for every i one can find an interval [ i,, +, i] uch that the curve β defined by β ( = αi (, [ i,, +, i], ha range in M( S+ M ( S and moreover with β (, S and β ( +, S +. Uing (.9 we write i = i = i ( β { K ( αd d c max ; S. (.4 = i = i = +, i i, Taking the limit i give (.8 a wihed. We proceed now with the proof. By propoition. we know that mut go uniformly to zero at infinity otherwie would be bounded from below away from zero and the metric g would be automatically complete. If uniformly at infinity, then ( Σ; g i metrically complete. A wa explained in ection., null-pacetime geodeic proect into ( g geodeic and the affine parameter i the ( g arc length. We will ee below that if ( Σ; g i not metrically complete, then there i an infinite ( g geodeic whoe ( g length i finite. Thi would be againt the hypothei that the pacetime i geodeically complete at infinity and the proof will be finihed. Let Γ (, be a ray for the metric to g tarting at Σ. For each, let be the lat time that Γ ( S. Let Γ be the retriction of Γ to [, + ]. Then Γ ( Σ M ( S and Γ i the concatenation of the curve Γ,. ow, = = = + = = ( { ( ( Γ( d = Γ ( d max ; S length Γ, (.5 = = where to obtain the inequality we ue { Γ { Σ + ( ( { up ( : [, ] up ( p: p M S max ; S, (.6 which i obtained from the incluion Γ ( Σ M ( S (for the firt inequality and the maximum principle (for the econd. Thu, if we prove that for a contant c independent of we have ( Γ c length, (.7 M Reiri 8 A in the proof of propoition., recall that S and S + lie in the ame connected component of ( +, + (ee remark after the proof of propoition., and oberve too that we can write ( +, + = ( ar, br with a =, b = 6 and r =.

15 Cla. Quantum Grav. (5 95 M Reiri then we can ue (.8 in conunction to (.5 to conclude that ( Γ ( d < (.8 which would imply that there i an incomplete null geodeic in the pacetime. Let u prove then the inequality (.7. We will play with the fact that Γ i a ray for g. Firt, note + ( d length max ; ( Γ ( Γ length( Γ { Γ max { ; S, (.9 where the econd inequality i obtained from the incluion Γ Σ M ( S and becaue max{ ; Σ M( S max{ ; S by the maximum principle. Then recall from the dicuion after propoition. that S and S + lie in the ame 4 connected component c ( +, + 4 of ( +, +. Hence, Γ ( ( S and Γ ( + ( S+ alo lie in 4 c ( +, +. Then, a in ection., we can oin Γ ( to Γ ( + through a curve Γ of length le than or equal to c,(c i a contant independent of, entirely contained in a connected component 4 c ( +, + of 4 ( +, +. Thi curve Γ mut have ( g length greater than or equal to the ( g length of Γ becaue Γ, (being a ray, minimize the ( g length between any two of it point. Thu, we can write c ( ( d ( d Γ min ;, + + Together with (.9, we obtain length ( Γ But from (.6 we have { Γ max ; { + 4+ c( min ;, ( Γ { c( { Γ max ; c min ;, { + 4+ c( where c i independent of. Thu, (.7 follow. { c( + 4+ { c( (.. (. max ;, c, (. min ;, Proof of theorem.. From the ame definition of a tatic iolated ytem, we know that the pacetime outide a et (invariant under the Killing field i ( g t g M =, = d +, (. which i decribed by the data ( ; g,. A the pacetime i geodeically complete at infinity, we can ue propoition. to deduce that the metric g i complete on. Theorem. in [8] then apple and aymptotic flatne follow. (Remark: the concept of an iolated ytem ued in [8] i the ame a in thi paper but with the extra aumption that i bounded from below away from zero outide a compact 4

16 Cla. Quantum Grav. (5 95 M Reiri et. A noted in [8], theorem. till hold if thi hypothei on i replaced by the metric completene of g. Remark.4. If the matter model (which i alway aumed to be compactly upported, atifie the weak energy condition, then the concluion of theorem. can be een to follow only from the metric completene of the tatic data. The geodeic completene at infinity i unneceary. We can now prove theorem.4. Proof of theorem.4. Suppoe that a connected component of the complement of a compact et in Σ i diffeomorphic to minu a cloed ball. Then, a in the proof of theorem., thi component ha to be an AF end of Σ. If we prove that Σ ha only one end, then the main theorem in [5] how that Σ i diffeomorphic to minu a finite et of open ball. The Irael [6] Robinon [] Bunting Maood-ul Alam [] uniquene theorem then applie and the olution i Schwarzchild. Let u prove then that Σ mut have only one end. We will proceed by contradiction. Aume then that Σ ha more than one end. From now on, we work in a pace ϵ ( Σ δ, g a in propoition. but with ϵ <. The end that wa AF (and had Schwarzchildian fall off for g i alo AF for ϵ g. On thi end conider large ( almot round embedded phere S. On thee phere we have ϵg area( S, while for the mean curvature θ S (with repect to the outward unit normal n we have θs 4π area( S. Hence, one can clearly take an embedded phere S ufficiently far away that θs n ( ( + ϵ > (.4 at every point of S. We work with uch S below. The particular combination (.4 will be relevant. The phere S divide Σ δ into two connected component. Denote by Σ δ the cloure of the connected component of Σ δ S containing Σ. We have Σδ = Σ S and, more importantly, Σ δ contain at leat one more end. Since Σ δ i trictly convex, we can contruct a geodeic ray γ (,, in Σ Σ and with the following propertie:. γ ( tart at S and perpendicular to it,. γ ( diverge through and end in Σ δ a,. dit ϵ ( γ (, S = for all. g δ Thee propertie imply that the expanion θ (, along the geodeic γ (, of the congruence of geodeic emanating perpendicularly to S mut remain finite for all (i.e. θ ( > for all. If not then there i a focal point on γ after which property fail. We will prove now that indeed θ ( = for ome >, thu reaching a contradiction. Let m ( = θ ( + ( ( + ϵ (, (.5 where ( = ( γ ( and ( = d ( γ (d. At =, m i equal to minu the left-hand ide of (.4, and i therefore negative (note that γ ( = n. On the other hand, a we explained in ection., if ϵ <, then the Bakry Emery Ricci tenor 5

17 Cla. Quantum Grav. (5 95 c Ric f = Ric + f f f (.6 c i zero, where f = ( + ϵln and c = ( ϵ ϵ ( + ϵ. ow, it i hown in [] (appendix A that m( atifie the differential inequality m M Reiri m. (.7 + c Thu, if m ( <, then there i > uch that m ( =. But a ( ( i finite for all, then we mut have θ ( =. Remark.5. If the complement of a compact et in Σ i diffeomorphic to and ( Σ; g i metrically complete, then the olution i alo Schwarzchild (i.e. the geodeic completene of the pacetime at infinity i unneceary. To ee thi, oberve firt that cannot go uniformly to zero on the end of Σ becaue thi would violate the maximum principle ( i harmonic and i zero only on Σ. By propoition. i then bounded away from zero on the end and aymptotic flatne follow. Remark.6. It i eay to how that propoition.,. and. hold true when Σ with being a compact two-urface of arbitrary genu (propoition. correpond to =. Thi could be of interet in further tudie. Acknowledgment I am grateful to Marc Mar for intereting dicuion on related topic and for carefully reading part of the manucript. Reference [] Anderon M T On tationary vacuum olution to the Eintein equation Ann. Henri Poincaré [] Anderon M T On the tructure of olution to the tatic vacuum Eintein equation Ann. Henri Poincaré [] Bunting G L and Maood-ul Alam A K M 987 onexitence of multiple black hole in aymptotically Euclidean tatic vacuum pace-time Gen. Relativ. Gravit [4] Galloway G J, Schleich K, Witt D M and Woolgar E 999 Topological cenorhip and higher genu black hole Phy. Rev. D 6 49 [5] Galloway G J 99 On the topology of black hole Commun. Math. Phy [6] Irael W 967 Event horizon in tatic vacuum pace-time Phy. Rev [7] Korotkin D and icolai H 994 The Ernt equation on a Riemann urface ucl. Phy., B [8] Reiri M 4 Stationary olution and aymptotic flatne: I. Cla. Quantum Grav. 55 [9] Reiri M 4 Stationary olution and aymptotic flatne: II. Cla. Quantum Grav. 55 [] Reiri M 5 On Ricci curvature and volume growth in dimenion three J. Differ. Geom [] Robinon D C 977 A imple proof of the generalization of Iraelʼ theorem Gen. Relativ. Gravit [] Wei G and Wylie W 9 Comparion geometry for the Bakry Emery Ricci tenor J. Differ. Geom [] Weyl H 97 Zur gravitationtheorie Ann. Phy., Berlin