1 1 On the Center of Mass in General Relativity 0 2 Lan–Hsuan Huang n a J Abstract. The classical notion of center of mass for an isolated system in 3 generalrelativityisderivedfromtheHamiltonianformulationandrepresented byafluxintegralatinfinity. Incontrasttomassandlinearmomentumwhich ] arewell-definedforasymptotically flatmanifolds,center ofmassandangular G momentumseemlesswell-understood,mainlybecausetheyappearasthelower D ordertermsintheexpansionofthedatathanthosewhichdeterminemassand linear momentum. This article summarizes some of the recent developments . h concerning centerofmassanditsgeometricinterpretationusingtheconstant t mean curvature foliationnear infinity. Several equivalent notions ofcenter of a massarealsodiscussed. m [ 1 v 6 1. Introduction 5 4 General relativity is the theory of the Einstein equation. Let (M,g˜) be a four 0 dimensional manifold with a Lorentz metric g˜. The Einstein equation says . 1 1 0 G:=Ric(g˜)− R(g˜)g˜=T, 2 1 1 where Ric(g˜) and R(g˜) are the Ricci tensor and the scalar curvature of g˜ respec- : v tively. The Einstein tensor G corresponds to the gravitational field of spacetime, i while the stress-energy-momentum tensor T is related to the matter content of X spacetime. r a If M is a spacelike hypersurface in (M,g˜), we denote the induced Riemannian metric by g and the induced second fundamental form by K. It follows from the Gauss and Codazzi equations combined with the Einstein equation that g and K must satisfy the following constraint equations: R −|K|2+(tr K)2 =2µ, g g g div (K−tr Kg)=J, g g whereR isthe scalarcurvatureofg,andµandJ arerespectivelythe localenergy g density and the local momentum density of matter defined by T. It is more conve- nient to introduce the momentum tensor π := K −(tr K)g. Then the constraint g 2000 Mathematics Subject Classification. Primary: 53A10,83C05. Theauthor waspartiallysupportedbytheNSFthroughDMS-1005560. 1 2 LAN–HSUANHUANG equations become: 1 R −|π|2+ (tr π)2 =2µ, g g 2 g div π =J. g The triple (M,g,π) satisfying the constraint equations is called an initial data set. An important class of initial data sets which model isolated systems is the class of asymptotically flat manifolds. An initial data set (M,g,π) is called asymptotically flat at the decay rate q greater than 1/2, if there exists a coordinate system {x} outside some compactset, sayB , suchthat g (x)=δ +O(|x|−q) and π (x)= R0 ij ij ij O(|x|−1−q) with the corresponding decay rates on higher derivatives. If µ and J vanish, (M,g,π) is called vacuum. Here we allow nonzero µ and J if they satisfy the decay condition µ=O(|x|−2−2q) and J =O(|x|−2−2q). The firstmajor investigationinto generalrelativity as a dynamicalsystemwas by Arnowitt, Deser, and Misner [1]. Later, various people further studied the correct formulations of total energy (the ADM mass), linear momentum, center of mass, and angular momentum for asymptotically flat manifolds (e.g. [10, 19, 4]) and their well-definedness (e.g. [3, 7, 9, 11]). The mass and linear momentum are defined by the flux integrals at infinity: 1 xj (1.1) m= lim (g −g ) dσ , ij,i ii,j 0 16π r→∞ r Z|x|=r i,j X 1 xi Pj = lim π dσ . ij 0 8π r→∞ r Z|x|=r i X The mass and linear momentum are well-defined because they are independent of the chosen asymptotically flat coordinates as proven by Bartnik [3] and Chru´sciel [5]. The positive masstheorembySchoen–Yau[20, 22]andby Witten[23]states: if (M,g,π) is asymptotically flat with the dominant energy condition µ ≥ |J| , g then m is non-negative. Moreover, m = 0 if and only if (M,g,π) is isometric to a hypersurface in Minkowski spacetime. ReggeandTeitelboim[19]observedthatcenterofmassandangularmomentum are not generally defined for asymptotically flat manifolds unless some parity con- dition at infinity is imposed (see also [2, 6] for other proposed conditions). Given an asymptotically flat coordinate system, we denote by fodd = (f(x)−f(−x))/2 and feven = (f(x)+f(−x))/2 the odd and even parts of a function f. They are defined outside B when the coordinates {x} exist. R0 Definition 1.1. (M,g,π) is asymptotically flat satisfying the RT condition if (M,g,π) is asymptotically flat at a decay rate q > 1/2 and g,π are respectively asymptotically even/odd, i.e. godd =O(|x|−1−q) πeven =O(|x|−2−q), ij ij µodd =O(|x|−3−2q) Jodd =O(|x|−3−2q), andsimilarlyonhigherderivatives. Whenwesaythat(M,g)is asymptoticallyflat satisfyingtheRTcondition,wemean(M,g)satisfyingthecorrespondingconditions by letting π =0. ON THE CENTER OF MASS IN GENERAL RELATIVITY 3 While all known physicalexamples of asymptotically flat manifolds satisfy the RT condition, there exist mathematical solutions to the vacuum constraint equa- tionswhichviolatetheRTcondition,andtheircenterofmassandangularmomen- tum are ill-defined [4, 13]. However, these examples may not be physical in the sensethatthe evolutionofthe initialdatais unknown. Nevertheless,underthe RT condition, the Hamiltonian formulation of center of mass and angular momentum proposed by Regge–Teitelboim [19] and by Beig–O´ Murchadha [4] are defined by 1 xj (1.2) Cl = lim xl (g −g ) dσ ij,i ii,j 0 16πmr→∞ r Z|x|=r i,j X  xi xl − g −g dσ , il ii 0 r r Z|x|=r i (cid:18) (cid:19) # X 1 xj Jp = lim π Yi dσ , 8πmr→∞Z|x|=r i,j ij (p) r 0 X where Y = ∂ ×x (cross product) are the rotation vector fields. (p) ∂xp In addition to the Hamiltonian formulation of center of mass (1.2), there are otherproposeddefinitionssuchas(4.1)and(3.4)below. Ageometricinterpretation ofcenterofmassusestheconstantmeancurvaturefoliationinM nearinfinitypro- posed by Huisken and Yau [15]. They used the volume preserving mean curvature flow to construct the constant mean curvature foliation, under the condition that (M,g) is strongly asymptotically flat, i.e. g = (1+ 2m)δ +O(|x|−2). Similar ij |x| ij results were proven by Ye using the inverse function theorem [24]. Metzger gen- eralized the results to small perturbations of the strongly asymptotically flat data [17]. Thoseauthorsalsoprovedthatthe foliationis unique under someconditions. AmoregeneraluniquenessresultwasobtainedbyQingandTian[18]. Forstrongly asymptotically flat metric which is conformally flat near infinity, Corvino and Wu proved the geometric center of Huisken–Yau’s foliation is equal to center of mass [9], and we later removed the condition of being conformally flat [11]. However,therearevariousinterestingphysicalsolutionstotheconstraintequa- tions which are not strongly asymptotically flat. For example, the family of Kerr solutions is a family of the exact solutions to the vacuum Einstein equation which model the rotating black holes with angular momentum. They are not strongly asymptotically flat, but they do satisfy the RT condition. In [12], we generalized the earlier results of the constant mean curvature fo- liation to asymptotically flat manifolds with the RT condition, which is a natural condition to impose when center of mass is discussed. Furthermore, the foliation that we constructed is unique under some conditions. From our construction, we can show that the foliation is asymptotically concentric, whose geometric center is consistent with the classical notion of center of mass. Anewingredientinconstructingthefoliationisthatforanyasymptoticallyflat manifoldwiththeRTcondition,onecanfindacanonicalfamilyoftheapproximate spheres S(p,R). They are constructed from perturbing the coordinate spheres S (p):={x:|x−p|=R} centered at p of radius R. The approximate spheres are R betteradaptedtotheasymptoticsymmetrythanthecoordinatespheres. Although their meancurvatureis notexactly constant,they sharemany nice propertieswith the constant mean curvature surfaces constructed from them. For example, the 4 LAN–HSUANHUANG leadingordertermsofthelowesttwoeigenvaluesofthestabilityoperatorare−2/R2 and 6m/R3. Moreover, {S(p,R)} form a foliation centered at a fixed p for R large when m>0. In many applications, the approximate spheres should be as good as the constant mean curvature surfaces, except that they are insensitive to center of mass. In order to construct the constant mean curvature surfaces from the approxi- mate spheres, we have to choose the center p of S(p,R), and a suitable choice of p would allow us to find a nearby constant mean curvature surface. The dependence ofthechoiceofponcenterofmassisbasedonthe newobservationofthe following identity (xl−pl)H dσ =8πm(pl−Cl)+O(R1−2q) for l=1,2,3, S 0 Zx∈SR(p) where H is the mean curvature of S (p). This identity can be thought as an S R alternativedefinitionofcenterofmass. Italsoenablesustoshowthatthegeometric center of each constant mean curvature surface is close to center of mass, and the limit of the geometric centers is precisely center of mass. The analytic underpinnings of the above identity rely crucially on the density theorem (Theorem 2.3 below) that we obtained in [11]. The density theorem is a refinement of the density theorem established by Corvino and Schoen [8]. An initialdataset(M,g,π)issaidtohaveharmonic asymptotics if,outsideacompact set, g = u4δ and π = u2(L δ−(divX)δ) for some function u and vector field X. X Because of the constraint equations, u and X are harmonic up to the lower order i terms. Moreover, the quantities m,P,C,J can be read off of the expansion of u and X. The density theorems involve finding suitable spaces so that initial data setswithharmonicasymptoticsformadensesubsetofasymptoticallyflatdatasets in a weighted Sobolev topology, and the physical quantities are continuous in that topology. The method to construct the approximate sequence of the initial data sets in thedensitytheoremscanbeemployedtogeneratemoreinitialdatasets,oftenwith specified asymptotics and physical properties. Several further applications can be found in, for example [13, 14]. The article is organizedasfollows. We presentthe density theorems inSection 2,inwhichaslightlymoregeneralpropositionisproven. InSection3,wesketchthe construction of the constant mean curvature foliation. Several equivalent notions of center of mass are then discussed in Section 4. 2. The density theorems In the previous section, the definition of asymptotically flat manifolds involves the pointwise regularity on g and π. A slightly weaker regularity condition on asymptotically flat manifolds is defined by weighted Sobolev spaces (see more de- tailed discussions in, for example [3]). Definition 2.1 (Weighted Sobolev spaces). For a non-negative integer k, a non- negative real number p, and a real number q, we say f ∈Wk,p(M) if −q 1 p p kfk := Dαf ρ|α|+q ρ−3dvol <∞, W−k,qp(M) ZM|αX|≤k(cid:16)(cid:12) (cid:12) (cid:17)   (cid:12) (cid:12)  ON THE CENTER OF MASS IN GENERAL RELATIVITY 5 where α is a multi-index and ρ is a continuous function with ρ = |x| when the coordinates {x} are defined. When p=∞, kfkW−k,q∞(M) = esssMup|Dαf|ρ|α|+q. |αX|≤k Using the weighted Sobolev norms, we say that (M,g,π) is asymptotically flat atthe decay rate q if (g−δ,π)∈W2,p(M)×W1,p (M), and in addition (M,g,π) −q −1−q satisfies the RT condition if (godd,πeven)∈W2,p (M \B )×W1,p (M \B ). −1−q R0 −2−q R0 The proof of the positive mass theoremby Schoen and Yau [20] was originally stated for strongly asymptotically flat data. Later, they extended the previous proof to allow general asymptotically flat data by a density argument [21]. They observed that, given a scalar-flat metric g, there exists a sequence of scalar-flat metrics with harmonic asymptotics whose mass converges to the mass of g. To generalize their density result, one has to consider not only the scalar curvature equation but the full constraint equations, which is more subtle. The following theorem by Corvino and Schoen is the analogue for the full constraint equations. Theorem 2.2 (Corvino–Schoen [8]). Suppose p > 3/2 and q ∈ (1/2,1). Let (g −δ ,π ) ∈ W2,p(M)×W1,p (M) be a vacuum initial data set. There is a ij ij ij −q −1−q sequence of solutions (g¯ ,π¯ ) with harmonic asymptotics. Given any ǫ > 0, there k k exist k >0 so that 0 kg−g¯kkW−2,qp(M) ≤ǫ, kπ−π¯kkW−1,1p−q(M) ≤ǫ, for all k ≥k0. Moreover, themass and linear momentumof (g¯ ,π¯ ) are within ǫ of those of (g,π). k k The theorem says that the solutions with harmonic asymptotics are dense among general asymptotically flat solutions. Moreover, the mass and linear mo- mentum, which can be explicitly expressed in the expansion of the solutions with harmonic asymptotics, converge to those of the original initial data set in these weightedSobolevspaces. However,intheabovetheoremthecenterofmassandan- gularmomentummaynotconverge,neitheraretheydefinedgenerallyforasymptot- ically flat manifolds. In the following, we show that in some more refined weighted Sobolev spaces, solutions with harmonic asymptotics form a dense subset inside asymptotically flat solutions with the RT condition so that the center of mass and angular momentum are continuous in that topology. Also, we can remove the con- dition that (g,π) is vacuum. Theorem 2.3 ([11]). Suppose p > 3/2 and q ∈ (1/2,1). Let (g −δ ,π ) ∈ ij ij ij W2,p(M)×W1,p (M)beaninitialdataset. Supposethat(godd,πeven)∈W2,p (M\ −q −1−q ij ij −1−q B )×W1,p (M \B ). R0 −2−q R0 There is a sequence of data (g¯ ,π¯ ) with harmonic asymptotics. Given any k k ǫ > 0, (g¯ ,π¯ ) is within an ǫ-neighborhood of (g,π) in W2,p(M)×W1,p (M) for k k −q −1−q k large as in the above theorem. Moreover, there exist R and k =k (R) so that 0 0 kg¯koddkW−2,1p−q(M\BR) ≤C, kπ¯kevenkW−1,2p−q(M\BR) ≤C, for all k ≥k0. Furthermore,mass,linearmomentum,centerofmass,angularmomentumof(g¯ ,π¯ ) k k are within ǫ of those of (g,π). The existence statement of the sequence of solutions with harmonic asymp- totics in the above two theorems involves solving an elliptic system for u and k 6 LAN–HSUANHUANG X by the inverse function theorem. In the proof of Theorem 2.3, we study more k carefully the elliptic systemthat u ,X anduodd,Xodd satisfy. Thenwe apply the k k k k boost trap argument to improve the decay rates of uodd and Xodd and prove that k k {(g¯ ,π¯ )} alsosatisfy the RT condition. The full proofs of the abovetwo theorems k k are rather technical. Some arguments there can prove the following proposition which is slightly more general (in the sense that the sequence {(g¯ ,π¯ )} need not k k have harmonic asymptotics) and is of independent interest. Proposition2.4. Supposethatp≥1andq >1/2. Assumethat(g,π)and(g¯ ,π¯ ) k k are asymptotically flat. If for any ǫ>0, there exits k so that 0 kg−g¯kkW−2,qp(M\R0) ≤ǫ, kπ−π¯kkW−1,1p−q(M\R0) ≤ǫ, for all k ≥k0, then the mass and linear momentum of (g¯ ,π¯ ) are within ǫ of those of (g,π). k k In addition to the above conditions, if (g,π) and {(g¯ ,π¯ )} satisfy the RT con- k k dition and if there is a constant c so that for k large, (2.1) k(g−g¯k)oddkW−2,1p−q(M\BR0) <c, k(π−π¯k)evenkW−1,2p−q(M\BR0) <c, then the mass, linear momentum, center of mass, angular momentum of (g¯ ,π¯ ) k k are within ǫ of those of (g,π). Proof. Bythe definitionofmassandthe divergencetheorem,foranys large, xj 16πm(g¯ )= lim [(g¯ ) −(g¯ ) ] dσ k k ij,i k ii,j 0 r→∞ |x| Z|x|=r i,j X xj = [(g¯ ) −(g¯ ) ] dσ k ij,i k ii,j 0 |x| Z|x|=s i,j X + [(g¯ ) −(g¯ ) ]dx. k ij,ij k ii,jj Z|x|≥s i,j X In the volume integral [(g¯ ) −(g¯ ) ] is the leading order term of the i,j k ij,ij k ii,jj scalar curvature R(g¯ ). Hence, by the constraint equations, the integrand is of the k P order O(|x|−2−2q), so there is a constant s such that 0 ǫ [(g¯ ) −(g¯ ) ]dx≤ , for all s≥s . k ij,ij k ii,jj 0 4 Z|x|≥s i,j X To handle the boundary term, we symbolically denote xj [g −g −((g¯ ) −(g¯ ) )] dσ ≤ |D(g−g¯ )|dσ . ij,i ii,j k ij,i k ii,j 0 k 0 |x| (cid:12)Z|x|=s i,j (cid:12) Z|x|=s (cid:12) X (cid:12) (cid:12) (cid:12) Fir(cid:12)st, if p≥3, by the Sobolev embedding inequality [3(cid:12), (1.9)], Ms\uBpR0|D(g−g¯k)||x|1+q ≤ckD(g−g¯k)kW−1,1p−q(M\BR0) ≤ckg−g¯kkW−2,qp(M\BR0). Fix ǫ and s . There exists k′ so that 0 0 ǫ kg−g¯kkW−2,qp(M\BR0) ≤ 8cπs1−q, for all k ≥k0′. 0 ON THE CENTER OF MASS IN GENERAL RELATIVITY 7 Therefore, ǫ |D(g−g¯ )|dσ ≤ max(|D(g−g¯ )|s1+q)4πs1−q ≤ . Z|x|=s0 k 0 |x|=s0 k 0 0 2 Combining the above estimates, we prove that |m(g)−m(g¯ )| ≤ ǫ for all k ≥ k′. k 0 When 1≤p≤3, by the definition of the weighted Sobolev norm, ∞ (|D(g−g¯ )|r1+q)pr−3dσ r2dr ≤kg−g¯ kp <∞. ZR0 Z|x|=r k 0 k W−2,qp(M\BR0) Therefore, for each k, there exists s with s →∞ so that k k ǫp (|D(g−g¯ )|s1+q)ps−1dσ ≤ s−1/2, k k k 0 2p k Z|x|=sk forotherwisethevolumeintegralwoulddiverge. ThenbytheHo¨lderinequality,for 1 =1− 1, p∗ p 1 |D(g−g¯ )|dσ ≤ (|D(g−g¯ )|s1+q)ps−1dσ p sp1∗−q k 0 k k k 0 k Z|x|=sk Z|x|=sk ! ≤ ǫs−21p+p1∗−q. 2 k Notice that − 1 + 1 −q < 0 for 1 ≤ p ≤ 3 and q > 1/2. Then there exists k′ so 2p p∗ 0 thatthe abovesurfaceintegralislessthanǫ/2forall k≥k′. Combiningthe above 0 estimates, we conclude |m(g)−m(g¯ )|≤ǫ. The proof of the convergence of linear k momentum is similar. If we have additionally condition(2.1), we showthat center of mass converges. By the definition of center of mass and the divergence theorem, 16πm(g¯)Cl(g¯) xj xi xl = xl (g¯ −g¯ ) − g¯ −g¯ dσ ij,i ii,j il ii 0  |x| |x| |x|  Z|x|=s i,j i (cid:18) (cid:19) X X   + xl (g¯ −g¯ )dx ij,ij ii,jj Z|x|≥s i,j X xj xi xl = xl ((g¯odd) −(g¯odd) ) − g¯odd −g¯odd dσ  ij ,i ii ,j |x| il |x| ii |x|  0 Z|x|=s i,j i (cid:18) (cid:19) X X   + xl ((g¯odd) −(g¯odd) )dx. ij ,ij ii ,jj Z|x|≥s i,j X The volume integral is of the order O(s1−2q) by the constraint equations and con- dition(2.1), soitis lessthanǫ/4forall slargeenough. Forthe boundaryterm, we need to estimate the surface integral |x||D(g¯ −g)odd|+|(g¯ −g)odd| dσ . k k 0 Z|x|=s (cid:2) (cid:3) 8 LAN–HSUANHUANG We then proceed as above and show that, given ǫ > 0, there is a sequence {s } k with s →∞ so that for k large k ǫ |x||D(g¯ −g)odd|+|(g¯ −g)odd| dσ ≤ . k k 0 2 Z|x|=sk (cid:2) (cid:3) Wecanthenconcludethat|Cl(g)−Cl(g¯ )|≤ǫ. Theprooffortheangularmomen- k tum is analogous. (cid:3) Thesecondpartofthestatementintheabovepropositionisoptimalinthesense thatcondition(2.1)isnecessary. IntherecentjointworkwithRickSchoenandMu– TaoWang[14],weareabletoconstructperturbeddata(g¯ ,π¯ )closetosomegiven k k asymptoticallyflatdatawiththeRTcondition(g,π)inW2,p(M)×W1,p (M). The −q −1−q energy-momentum vector of the new data sets is equal to that of (g,π), but their centerofmassandangularmomentumcanbe arbitrarilyprescribed. Inparticular, the perturbed data sets violate condition (2.1). 3. The constant mean curvature foliation near infinity In the asymptotically flat region, the mean curvature H of the coordinate S sphere S (p) is close to the constant 2/R, which is the mean curvature of S (p) R R in Euclidean space. The difference H −2/R is measured by h = g−δ. Because S H (x)=div (ν )wherediv isthedivergenceoperatorof(M,g)andν istheunit S g g g g outward normal vector field on S (p) with respect to g. By direct computations, R we have 2 1 (xi−pi)(xj −pj)(xk −pk) H (x)= + h (x) S R 2 ij,k R3 i,j,k X (xi−pi)(xj −pj) xj −pj (3.1) +2 hij(x) R3 − hij,i(x) R i,j i,j X X 1 xj −pj h (x) ii + h (x) − +E (x), ii,j 0 2 R R i,j i X X where E (x)=O(R−1−2q) and Eodd(x)=O(R−2−2q). 0 0 We first examine (3.1) when g is strongly asymptotically flat, i.e. g = (1+ 2m)δ+O(|x|−2). Substituting h = 2mδ +O(|x|−2) into (3.1) and simplifying |x| ij |x| ij the summation terms, we derive 2 4m 6m(x−p)·p 9m2 H (x)= − + + +O(R−3+|p|R−4). S R R2 R4 R3 ThelinearizedmeancurvatureoperatoronS (p)hasanapproximatekernelspanned R by {x1−p1,x2−p2,x3−p3}. It was then observedby Ye [24] that the projection ofthelastthreetermsoftherighthandsideaboveintotheapproximatekernelcan be annihilatedbychoosingsuitable p,whichwelateridentified withcenterofmass [11]. Then the inverse function theorem can be applied to find the constant mean curvaturesurface Σ nearthe coordinatesphere S (p). The two surfaces areclose R R inthe sense that ΣR is the graphofφoverSR(p) andkφkC2,α(SR(p)) is bounded by a constant uniformly in R. Inoursituation,becauseg hasweakerasymptoticsanddecayrates,theinverse function theorem cannot be directly applied because the asymptotics of H −2/R S in (3.1) may be far from being a constant. It simply reflects the fact that the ON THE CENTER OF MASS IN GENERAL RELATIVITY 9 coordinate spheres are defined with respect to some asymptotically flat coordinate chart which is not canonical, so the constant mean curvature surfaces may not be closetothecoordinatespheres. Onemayconsidertochooseabetterasymptotically flat coordinate chart, but it seems hard to handle and simplify all the summation terms in (3.1). Instead, we explicitly construct a family of approximate spheres S(p,R)whichreflectthe asymptoticsbetter thanthe coordinatespheresforeachp andlargeR. Moreover,fromour construction,the approximatespheres also adapt better the RT condition [12]. Lemma 3.1. Let (M,g) be asymptotically flat satisfying the RT condition. There exists a constant c independent of R so that, for each p and for R large, there is an approximate sphere S(p,R)= x+φ(x)ν :φ∈C2,α(S (p)) . g R Here φ∗ (the pull back of φ on(cid:8)S1(0)) satisfies (cid:9) (3.2) kφ∗kC2,α(S1(0)) ≤cR1−q, k(φ∗)oddkC2,α(S1(0)) ≤cR−q. Moreover, the mean curvature of S(p,R) is 2 (3.3) H = +f¯+O(R−1−2q), S R where f¯:=(4πR2)−1 fdσ and f =H −2/R. SR(p) 0 S Remark. Whenq =1R,φisboundedbyaconstant,sotheapproximatespherestays within a constant neighborhood of the coordinate sphere. However, when q < 1, the size of φ may increase as R increases. Nevertheless, S(p,R) has asymptotic antipodal symmetry with respect to the center p as seen in (3.2). After showingthe existence ofthe family ofthe approximatespheres,weprove that we can perturb them to construct constant mean curvature surfaces, if the center p is chosen correctly. The following identity (3.4) plays a key role to locate the center p of the approximate spheres. Lemma 3.2. Let (M,g) be an asymptotically flat manifold satisfying the RT con- dition. For l =1,2,3, 2 (3.4) (xl−pl) H − dσ =8πm(pl−Cl)+O(R1−2q). S 0 R ZSR(p) (cid:18) (cid:19) First, it is simple to see that if the metric g =u4δ, then (3.4) holds as follows. The left hand side above is 2 (xl−pl) H − dσ S 0 R ZSR(p) (cid:18) (cid:19) xj −pj u4−1 = (xl−pl) 4u3u − dσ +O(R1−2q) ,j 0  R R  ZSR(p) j X  xj −pj  =−pl 4u3u dσ ,j 0 R ZSR(p) j X xj −pj xl−pl + 4xlu3u −u4 dσ +O(R1−2q). ,j 0  R R  ZSR(p) j X   10 LAN–HSUANHUANG On the other hand, computing m and Cl when g =u4δ, we have xj −pj 4u3u dσ =−8πm+O(R1−2q), ,j 0 R ZSR(p) j X xj −pj xl−pl 4xlu3u −u4 dσ =−8πmCl+O(R1−2q). ,j 0  R R  ZSR(p) j X We then conclude that (3.4) holds for g =u4δ. For general metric g satisfying the RT condition,one maytend to applyTheorem2.3 atthis stage. However,it seems that we can only prove that (3.4) holds for a sequence of radii {R } up to some i error ǫ. In other words, it only allows us to conclude that 2 lim (xl−pl) H − dσ =8πm(pl−Cl). S 0 R→∞ZSR(p) (cid:18) R(cid:19) However,forourpurposeto constructthe constantmeancurvaturesurfaceateach radius R, we need to show that (3.4) holds for each R. Therefore, we study the summationterms in(3.1)separatelyandobservesomecancellationsinthe integral in (3.4) after applying the divergence theorem. In particular, our argument shows that (3.4) holds when the asymptotically flat coordinates {x} are global, without usingthedensityargument. Whenthecoordinates{x}aredefinedonlyoutsidethe compact set, the (outer) boundary term is then handled by the density theorem (Theorem 2.3). The proof is computationally involved, so we omit the detailed computations and present only the sketch of the proof below. Sketch of proof. We define, for l =1,2,3, 1 (xi−pi)(xj −pj)(xk−pk) Il(R)= (xl−pl) h (x) dσ . g 2 ij,k R3  0 ZSR(p) i,j,k X   Because the asymptotically flat coordinates may not be defined in the interior, we use the divergence theorem in the annulus A = {R ≤ |x−p| ≤ R }. Using 1 integrationbypartsandsimplifyingtheexpression,weobtainanidentitycontaining purely the boundary terms (3.5) Il(R )−Il(R)=Bl(R )−Bl(R) for all R ≥R, g 1 g g 1 g 1 where Bl(R) equals the boundary integrals: g 1 xj −pj (xi−pi)(xj −pj) (xl−pl) h (x) −2h (x) dσ 2 ij,i R ij R3 0 ZSR(p) i,j (cid:20) (cid:21) X 1 xl −pl xi−pi + h (x) +h (x) dσ . ii il 0 2 R R ZSR(p) i (cid:20) (cid:21) X We would like to show that Il(R) = Bl(R) for each R large and for l = 1,2,3. It g g suffices to prove that lim Il(R )= lim Bl(R ). g 1 g 1 R1→∞ R1→∞ Firstnoticethatifg =u4δ outsideacompactset,thenbydirectcomputations,for any R large (so that g =u4δ on B (p)), 1 R1 Il(R )=Bl(R ). g 1 g 1