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SPACELIKE HYPERSURFACES WITH NEGATIVE TOTAL ENERGY IN DE SITTER SPACETIME ZHUOBINLIANGANDXIAOZHANG 2 1 Abstract. DeSitterspacetimecanbeseparatedintotwopartsalongtwokinds 0 ofhypersurfacesandthehalf-deSitterspacetimesarecoveredbytheplanarand 2 hyperboliccoordinatesrespectively. Twopositiveenergytheoremswereproved n previously forcertain -asymptoticallydeSitterand -asymptoticallydeSit- a P H terinitialdatasetsbythesecondauthorandcollaborators.Theseinitialdatasets J areasymptotictotimeslicesofthetwokindsofhalf-deSitterspacetimesrespec- 9 tively,andtheirmeancurvaturesareboundedfromabovebycertainconstants. 1 Whilethemeancurvaturesviolatetheseconditions,thespacelikehypersurfaces withnegativetotalenergyinthetwokindsofhalf-deSitterspacetimesarecon- ] c structedinthisshortpaper. q - r g [ 2 1. Introduction v 3 1 Ingeneralrelativity,thepositiveenergytheoremplaysafundamentalrolewhich 2 serves as a consistent verification of the theory. If the positive energy theorem 1 holds,thespacetimewithvanishingtotalenergycanbeviewedasthegroundstate. . 5 Inthe caseofzero cosmological constant, thepositive energy theorem forasymp- 0 toticallyflatspacetimeswasfirstlyprovedbySchoenandYau[9,10,11],andthen 1 byWitten [13]using adifferent method. Recently, Witten’s method wasextended 1 successfully to asymptotically Anti-de Sitter spacetimes and the positive energy : v theorem wasproved completely and rigorously inthecase ofnegativecosmologi- Xi calconstant [12,5,16,8,15]. r Recentcosmologicalobservationsindicatedthatouruniversehasapositivecos- a mological constant. It is therefore important to study whether the positive energy theorem for asymptotically de Sitter spacetimes holds. De Sitter spacetime with cosmological constant Λ = 3 > 0, (λ > 0) is a hypersurface embedded into λ2 5-dimensional Minkowskispacetime R1,4 2 2 2 2 2 3 X0 + X1 + X2 + X3 + X4 = (1.1) − Λ (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) with the induced metric (cf. [6]). It is covered by global coordinates where each time slice is a 3-sphere with constant curvature and has no spatial infinity. As thecorresponding KillingvectorfieldstotheLorentziangenerators aretimelikein someregion ofdeSitterspacetime andspacelike insome otherregion, there isno positive conserved energy in de Sitter spacetime [14, 1]. So the issue of the posi- tiveenergytheoremforasymptoticallydeSitterspacetimesbecomessophisticated. 1 2 ZHUOBINLIANGANDXIAOZHANG However, de Sitter spacetime can be separated into two parts along the hypersurface X0 = X4 andthehalf-de Sitterspacetimeiscoveredbytheplanarcoordinates withthedeSittermetric g˜PdS = −dt2+e2λt (cid:18)(cid:16)dx1(cid:17)2+(cid:16)dx2(cid:17)2+(cid:16)dx3(cid:17)2(cid:19). (1.2) Similarly, de Sitter spacetime can be separated into two parts along the hypersurface X4 = λ and the half-de Sitter spacetime is covered by the hyperbolic − coordinates withthedeSittermetric T R g˜ = dT2+sinh2 dR2+λ2sinh2 dθ2 +sin2θdψ2 . (1.3) HdS − λ (cid:18) λ (cid:16) (cid:17)(cid:19) In [7], two positive energy theorems were proved for certain -asymptotically de P Sitter and -asymptotically de Sitter initial data sets. These initial data sets are H asymptotic to time slices of the two kinds of half-de Sitter spacetimes covered by the planar coordinates and hyperbolic coordinates respectively. And their mean curvatures satisfy (2.1) for -asymptotically de Sitter initial data sets and satisfy P (2.2)for -asymptotically deSitterinitialdatasets. H In the case of zero or negative cosmological constants, the positive energy theorem holds for any asymptotically flat or asymptotically anti-de Sitter spacelike hypersurfaces whose mean curvatures have no restriction as (2.1) or (2.2). This motivatesustostudyfurtherthespecialfeatureoftheconditions (2.1)and(2.2)in the case of positive cosmological constant. In this short paper, we shall construct spacelikehypersurfaceswithnegativetotalenergyinthetwokindsofhalf-deSitter spacetimes. The constructed spacelike hypersurfaces are either -asymptotically P de Sitter or -asymptotically de Sitter. However, their mean curvatures violate H (2.1)inthe -asymptoticallydeSittercaseandviolate(2.2)inthe -asymptotically P H de Sitter case. As the two kinds of half-de Sitter spacetimes satisfy the vacuum Einstein field equations with positive cosmological constant, the dominant energy condition holds automatically. Sothese examples provide counterexamples ofthe positive energy theorem for general asymptotically de Sitter spacelike hypersurfaces. Thus it indicates that the positive energy theorem is no longer the general featureintheoryofgravitywhenthecosmological constant ispositive. We would like to point out that de Sitter spacetime can be also separated into two parts along the hypersurface X3 = 0. The time slices in this half-de Sitter spacetime are hemispheres. The counterexample to Min-Oo’s conjecture related to the rigidity of hemispheres, constructed by Brendle, Marques and Neves, can beunderstood astheanotherfailure ofthepositive energy theorem forspacetimes with positive cosmological constant [3]. We refer to the survey paper [4] for the recentworksonMin-Oo’sconjecture. 2. Positiveenergytheorems In this section, we review two positive energy theorems proved in [7]. Let (N1,3,g˜) be a spacetime satisfying the Einstein equations with a positive cosmological constant Λ = 3 . Let (M,g,K) be a spacelike hypersurface with induced λ2 Riemannian metric g and second fundamental form K. It is -asymptotically de P NEGATIVEENERGY 3 Sitterof order τ > 1 if there is acompact set M such that M M isthe disjoint 2 c − c union of a finite number of subsets M , , M - called the “ends” of M - each 1 k diffeomorphic toR3 B where B isthe·c·l·osed ballofradius r withcenteratthe r r − coordinate origin. Andgandh= K Λgsatisfy − q3 g = 2g¯, h = h¯, g¯ g˘ C2,α(M), h¯ C1,α (M) P P − ∈ −τ ∈ −τ−1 forcertain τ > 1,0 < α < 1,where iscertainpositiveconstant, g˘ isthestandard 2 P metricofR3,Ck,α areweightedHo¨lderspacesdefinedin[2]. Moreover,thescalar τ curvature −L1(M), T L1(M). Let the metric g˘ = 2g˘. Let xi be natural g 0i coordinateRs of∈R3,g¯ = g¯ ∂∈,∂ ,h¯ = h¯ ∂,∂ . ThePtotalPenergy E{ a}ndthetotal ij i j ij i j l linearmomentum P of M(cid:16) for (cid:17)-asympto(cid:16)tically(cid:17) deSitterinitialdataset(M,g,K) lk l P are E = P lim ∂ g¯ ∂g¯ dxi, l j ij i jj 16π r→∞ZSr,l (cid:16) − (cid:17)∗ 2 P = P lim h¯ g¯ tr h¯ dxi. lk ki ki g¯ 8π r Z − ∗ →∞ Sr,l(cid:16) (cid:16) (cid:17)(cid:17) Aninitialdataset(M,g,K)is -asymptotically deSitteroforder τ> 3 ifthere H 2 is a compact set M such that M M is the disjoint union of a finite number of c c subsets M , , M - called the−“ends” of M - each diffeomorphic to R3 B 1 k r ··· − where B is the closed ball of radius r withcenter at the coordinate origin. And g r cothT andh= K λgsatisfy − λ g = H2g¯, h = Hh¯, aij = O e−λτr , ∇˘Hk aij = O e−λτr , ∇˘Hl ∇˘Hk aij = O e−λτr , h¯ij = O(cid:16) e−λτ(cid:17)r , ∇˘Hk h¯ij = O(cid:16) e−λτ(cid:17)r (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) on each end for certain constant T , 0, where = sinh T, a = g¯ e˘ ,e˘ H λ ij i j − g˘ e˘ ,e˘ ,h¯ = h¯ e˘ ,e˘ . Andg˘ isthehyperbolic metric (cid:16) (cid:17) i j ij i j H H (cid:16) (cid:17) (cid:16) (cid:17) R g˘ = dR2+λ2sinh2 dθ2 +sin2θdψ2 . H λ (cid:16) (cid:17) e˘ = ∂ , e˘ = ∂θ , e˘ = ∂ψ is the frame, e˘i is the coframe. ˘ is the 1 R 2 λsinhR 3 λsinhRsinθ { } ∇H λ λ Levi-Civitaconnection ofg˘ . Let ¯, ¯,ρ bethescalarcurvature, theLevi-Civita z connection of g¯ and the distHance fuRnc∇tion with respect to z M respectively. We furtherrequire R¯ + λ62 eρλz ∈ L1(M), ∇¯jh¯ij−∇¯itrg¯ h¯ eρλz ∈∈L1(M). Denote (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) 1 = ˘ ,jg¯ ˘ tr (g¯)+ (a +a )+2 h¯ +h¯ . El ∇H 1j−∇H1 g˘ λ 22 33 22 33 H (cid:16) (cid:17) Thetotalenergy-momentum oftheend M are l 2 ElHν = 1H6π Rl→im∞ZSR,lElnνeRλe˘2∧e˘3 4 ZHUOBINLIANGANDXIAOZHANG where 0 ν 3, (nν) = (1,sinθcosψ,sinθsinψ,cosθ), S is the coordinate R,l ≤ ≤ sphereofradius Rin M. l Thefollowingtwopositiveenergytheoremsareprovedin[7]. Theorem 2.1. Let (M,g,K) be an -asymptotically de Sitter initial data set of order1 τ> 1 whichhaspossiblyaPfinitenumberofapparenthorizons inspace- ≥ 2 time (N1,3,g˜) with positive cosmological constant Λ > 0. Suppose N1,3 satisfies the dominant energy condition. If the trace of the second fundamental form of M satisfies tr (K) √3Λ, (2.1) g ≤ thenforeachend M, l E P . l lg˘ ≥ | | P IfE = 0forsomeend M ,then l l0 Λ (M,g,K) R3, 2g˘, 2g˘ . ≡  P r3P  Moreover, the mean curvature achieves the equalityin (2.1) and the spacetime (N1,3,g˜) is de Sitter along M. In particular, (N1,3,g˜) is globally de Sitter in the planarcoordinates ifitisglobally hyperbolic. Theorem 2.2. Let (M,g,K) be an -asymptotically de Sitter initial data set of order τ > 3 whichhas possibly afinHite number ofapparent horizons inspacetime 2 (N1,3,g˜) with positive cosmological constant Λ > 0. Suppose N1,3 satisfies the dominant energy condition. If the trace of the second fundamental form of M satisfies T T tr (K)sinh √3Λcosh , (2.2) g λ ≤ λ thenforeachend M, l 2 2 2 E E + E + E . lH0 ≥ r lH1 lH2 lH3 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) IfE = 0forsomeend M ,then lH0 l0 T Λ T T (M,g,K) H3,sinh2 g˘ , sinh cosh g˘ . ≡  λ H r3 λ λ H Moreover, the mean curvature achieves the equality in (2.2) and the spacetime (N1,3,g˜) is de Sitter along M. In particular, (N1,3,g˜) is globally de Sitter in the hyperbolic coordinates ifitisgloballyhyperbolic. In general, the conformal factors and are functions on 3-manifolds. The P H above positive energy theorems hold only when and are constants on ends. P H Lete betimelikeunitnormalto M. Itiswell-knownthat 0 1 K = g˜ proj 2 Le0 (cid:0) (cid:1) NEGATIVEENERGY 5 where istheLiederivativeand“proj”meanstheprojectiontothetangentbundle L of M. Fort-slicewithe = ∂ inthemetric(1.2), 0 t tr (K) = √3Λ, g andforT-slice(T , 0)withe = ∂ inthemetric(1.3), 0 T T tr (K) = √3Λcoth . g λ Thus the mean curvatures conditions (2.1) and (2.2)indicate that, in certain sense ofaverage,the3-spaceevolution inspacetime N shouldnotbetoorapidtoexceed thestandard deSitterspacetime inordertokeeppositivity ofthetotalenergy. 3. Negativetotalenergy In this section, weconstruct certain spacelike hypersurfaces withnegative total energy. The mean curvatures of these hypersurfaces violate the condition (2.1) or (2.2). These hypersurfaces are constructed by slightly perturbing the t or T-slices inhalf-deSitterspacetime. Firstofall,weprovide theformula ofthesecond fundamental form ofagraph. Let (Mn,gˆ) be an n-dimensional Riemannian manifold, ˆ be its Levi-Civita con- R ∇ nection, I beanintervalof . Weequip I M withthewrappedproduct × g˜ = dt2+ρ2gˆ, − where ρ = ρ(t) is a positive smooth function. Suppose that Σ is a smooth graph whichisgivenby F :M I M −→ × x (f(x),x). 7−→ Choosingaframe eˆ on M,thetangentspaceofthegraph Σisspannedby i { } F eˆ = ˆ f∂ +eˆ , i = 1,...,n. i i t i ∗ ∇ As∂ +ρ 2ˆ f isorthogonal toΣ,thegraphisspacelike ifandonlyif t − ∇ ρ2 > ˆ f 2. (3.1) |∇ |gˆ Then, ν = ∂t+ρ−2∇ˆf is a timelike unit normal vector field of the graph. Let g be 1 ρ 2 ˆf2 the inducedqm−et−ri|c∇o|gˆn Σ. The components of the second fundamental form with respecttothenormalvector νare K =K(F eˆ ,F eˆ ) = µ ˆ2 f 2ρ ρ 1ˆ f ˆ f +ρ ρgˆ , ij ∗ i ∗ j ∇i,j − ′ − ∇i ∇j ′ ij (cid:16) (cid:17) tr (K) =µρ 2 gîj+µ2ρ 2îf ˆjf ˆ2 f µ3ρ ρ 1+(n+1)µρ ρ 1, g − − ∇ ∇ ∇i,j − ′ − ′ − (cid:16) (cid:17) 1 whereµ = 1 ρ 2 ˆ f 2 −2. − − |∇ |gˆ (cid:16) (cid:17) (i)The -asymptotically deSittercase: Considerthegraphwhere P 1 f(x) = t +ε 1+r2 −2 0 (cid:16) (cid:17) 6 ZHUOBINLIANGANDXIAOZHANG forcertainconstantt inthehalf-deSitterspacetimeequippedwiththemetric(1.2) 0 intheplanarcoordinates. As 1 ρ−2|∇˘ f|2g˘ = ε2 1+r2 −3r2e−λ2(cid:18)t0+ε(1+r2)−2(cid:19), (cid:16) (cid:17) (3.1) holds and the graph is an embedding spacelike hypersurface for ε is suffi- cientlysmall. Notethat 3 ∂ f = ε 1+r2 −2 x, i i − (cid:16) (cid:17) 5 3 ∂2 f =3ε 1+r2 −2 x x ε 1+r2 −2 δ , i,j i j− ij (cid:16) (cid:17) (cid:16) (cid:17) thustheinducedmetricandthesecondfundamental formsofthegraphare gij =e2λfδij ε2 1+r2 −3xixj, − (cid:16) (cid:17) Kij =µ 1e2λfδij ε 1+r2 −32 δij 2ε2 1+r2 −3xixj+3ε 1+r2 −25 xixj , λ − − λ ! (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) 1 where µ = (cid:18)1−ε2r2(1+r2)−3e−2λf(cid:19)−2 = 1 + 12ε2r2(1 + r2)−3e−2λf + O(r−8). Set P= etλ0,g¯ = P−2g,a = g¯ −g˘ andh¯ = P−1 K − λ1g . Wehave, (cid:16) (cid:17) aij = e2(fλ−t0) 1 δij −2ε2 1+r2 −3xixj = e2(fλ−t0) 1 δij+O r−4 , (cid:18) − (cid:19) −P (cid:16) (cid:17) (cid:18) − (cid:19) (cid:16) (cid:17) ∂ a =O r 2 , ∂ ∂a = O r 3 , k ij − k l ij − (cid:16) (cid:17) (cid:16) (cid:17) h¯ij = −1 1e2λfδij+O r−3 1 e2λfδij +O r−4 = O r−3 . P −λ (cid:16) (cid:17)!− λ(cid:18) (cid:16) (cid:17)(cid:19)! (cid:16) (cid:17) Usingtheformulaonthedifferenceofthescalarcurvaturesoftwometrics(c.f. [3], p187), = + ∞ ( 1)s Ric ,as + 1g¯ikg¯jlg¯pq 2˘ a ˘ a 4˘ a ˘ a g¯ g˘ g˘ i lp j kq i kp j lq R R − g˘ 4 − ∇ ∇ − ∇ ∇ Xs=1 D E (cid:16) +4˘ a ˘ a +3˘ a ˘ a ˘ a ˘ a g¯ikg¯jl ˘2 a ˘2a , ∇i kl∇j pq ∇i lq∇k jp −∇i jl∇k pq − ∇i,k jl −∇i,l jk (cid:17) (cid:16) (cid:17) weobtain = g¯ikg¯jl ˘2 a ˘2a +O r 4 Rg¯ − ∇i,k jl −∇i,l jk − (cid:16) (cid:17) (cid:16) (cid:17) = ∆ a + ˘2 a +O r 4 − g˘ jj ∇i,j ij − (cid:16) (cid:17) = 2∆g˘e2(fλ−t0) +O r−4 − (cid:16) (cid:17) = 4e2(fλ−t0)∆g˘f +O r−4 = O r−4 . − λ (cid:16) (cid:17) (cid:16) (cid:17) NEGATIVEENERGY 7 This implies that is L1. Thus, (R3,g,K) is -asymptotically de Sitterwith the g R P constant = etλ0. Notethat P trg(K) =4µ 1µ3+µe−2λf µ2e−2λfε2 1+r2 −3xixj+δij λ − λ (cid:18) (cid:16) (cid:17) (cid:19) 3 5 ε 1+r2 −2 δ +3ε 1+r2 −2 x x ij i j · − ! (cid:16) (cid:17) (cid:16) (cid:17) =3 + 1 ε2r2 1+r2 −3e−2λf +O r−5 . λ 2λ (cid:16) (cid:17) (cid:16) (cid:17) Thisshows that ifε , 0, then tr (K) > 3 forlarge r, which violates the condition g λ (2.1). Nowthetotalenergy is E = P lim ∂ g¯ ∂g¯ dxi j ij i jj 16π r Z − ∗ →∞ Sr(cid:16) (cid:17) = P lim 4εr 1+r2 −32 e2(fλ−t0) 3 −2ε2r 1+r2 −3 dσr 16π r Z λ − P ! →∞ Sr (cid:16) (cid:17) (cid:16) (cid:17) = εetλ0. λ SoE < 0ifε < 0. Furthermore, thetotallinearmomentum 2 2 P = P lim h¯ g¯ tr h¯ dxi = P lim O r 3 dxi = 0. k ki ki g¯ − 8π r Z − ∗ 8π r Z ∗ →∞ Sr(cid:16) (cid:16) (cid:17)(cid:17) →∞ Sr(cid:16) (cid:16) (cid:17)(cid:17) (ii)The -asymptotically deSittercase: Considerthegraphwhere H 3R f(x) = T0+εe−λ forcertainconstant T , 0inthehalf-deSitterspacetimeequippedwiththemetric 0 (1.3)inthehyperbolic coordinates. (Notethatthehyperbolic coordinates areonly validforT , 0,T = 0cannotbechosenhere.) As 0 9ε2 f ρ−2|∇˘Hf|2g˘ = λ2 e−λ6R sinh−2 λ, H (3.1) holds and the graph is an embedding spacelike hypersurface for ε is suffi- cientlysmall. Recall that the standard hyperbolic metric g˘ with the g˘ -orthonormal frame e˘ , e˘ , e˘ . By Cartan’s structure equations de˘iH= ωi e˘j Hand Γj = ωi(e˘ ), we 1 2 3 − j ∧ ki j k obtain R R ω2 = ω1 = cosh dθ, ω3 = ω1 = cosh sinθdψ, ω3 = ω2 = cosθdψ, 1 − 2 λ 1 − 3 λ 2 − 3 1 R 1 R cotθ Γ2 = Γ1 = coth , Γ3 = Γ1 = coth , Γ3 = Γ2 = , 21 − 22 λ λ 31 − 33 λ λ 32 − 33 λsinh R λ andotherωi,Γk vanish. Clearly,thegradient of f is j ij ∇˘Hf = −3λεe−λ3R,0,0,0 . (cid:16) (cid:17) 8 ZHUOBINLIANGANDXIAOZHANG Since 9ε ∇˘H1,1f =e˘1e˘1f −e˘i(f)Γi11 = ∂R∂Rf = λ2e−λ3R, 3ε R ∇˘H2,2f =e˘2e˘2f −e˘i(f)Γi22 = −e˘1(f)Γ122 = −λ2e−λ3R coth λ, 3ε R ∇˘H3,3f =e˘3e˘3f −e˘i(f)Γi33 = −e˘1(f)Γ133 = −λ2e−λ3R coth λ, ˘ f =e˘ e˘ f e˘ (f)Γi = 0, ∇H1,2 1 2 − i 12 ˘ f =e˘ e˘ f e˘ (f)Γi = 0, ∇H1,3 1 3 − i 13 ˘ f =e˘ e˘ f e˘ (f)Γi = 0, ∇H2,3 2 3 − i 23 wecanfindtheHessianof f 3ε R R (cid:16)∇˘Hi,jf(cid:17) = λ2e−λ3Rdiag(cid:18)3,−coth λ,−coth λ(cid:19). Thentheinduced metricandthesecond fundamental formsofthegraphare f 9ε2 f R g = sinh2 e−λ6R dR2+λ2sinh2 sinh2 dθ2 +sin2θψ2 , λ − λ2 ! λ λ (cid:16) (cid:17) 1 f f 9ε 18ε2 f 3R 6R K11 =µ λ cosh λ sinh λ + λ2e−λ − λ3 coth λe−λ !, 1 T T ε T T = sinh 0 cosh 0 + 9+sinh2 0 +cosh2 0 e−λ3R +O e−λ6R , λ λ λ λ2 (cid:18) λ λ (cid:19) (cid:18) (cid:19) 1 f f 3ε R 3R K22 =K33 = µ λcosh λ sinh λ − λ2e−λ coth λ! 1 T T ε T T = sinh 0 cosh 0 + 3+sinh2 0 +cosh2 0 e−λ3R +O e−λ5R , λ λ λ λ2 (cid:18)− λ λ (cid:19) (cid:18) (cid:19) 1 and other Kij = 0, where µ = 1− 9λε22e−λ6R sinh−2 λf −2 = 1+O e−λ6R . Set H = sinh T0,g¯ = 2g,a= g¯ g˘ (cid:16)andh¯ = 1 K co(cid:17)thTλ0g ,weob(cid:16)tain(cid:17) λ H− − H H− − λ ! 2ε T aij = coth 0e−λ3Rdiag 1+O(e−λ3R),1+O(e−λ3R),1+O(e−λ3R) , λ λ (cid:16) (cid:17) (cid:18) (cid:19) 4ε T h¯ij = sinh−1 0e−λ3Rdiag 2+O(e−λ3R), 1+O(e−λ2R), 1+O(e−λ2R) , (cid:16) (cid:17) λ2 λ (cid:18) − − (cid:19) ∇˘Hk aij =O(cid:18)e−λ3R(cid:19), ∇˘Hk,laij = O(cid:18)e−λ3R(cid:19), ∇˘Hk h¯ij = O(cid:18)e−λ3R(cid:19). NEGATIVEENERGY 9 Nowwecomputethescalarcurvature. Notethat 6ε T trg˘ a =a11 +a22+a33 = coth 0e−3λR +O(e−λ6R), λ λ H 6ε T ∇˘Hi,iajj = λ coth λ0∆g˘He−3λR +O(cid:18)e−λ6R(cid:19), 2ε T 2ε T ∇˘Hi,jaij =∇˘Hi,j λ coth λ0e−3λRg˘Hij +O(e−λ6R)! = λ coth λ0∆g˘He−3λR +O(cid:18)e−λ6R(cid:19), 3 3R 3R 5R ∆g˘ e−λ = e−λ +O e−λ , λ2 H (cid:18) (cid:19) Wehave Rg¯ =Rg˘H −DRicg˘H,aEg˘H −∇˘Hi,iajj+∇˘Hi,jaij +O(cid:18)e−λ6R(cid:19) 6 12ε T 18ε T 6ε T = + coth 0e−3λR coth 0e−3λR + coth 0e−3λR +O e−λ5R − λ2 λ3 λ − λ3 λ λ3 λ (cid:18) (cid:19) 6 5R = +O e−λ . − λ2 (cid:18) (cid:19) So Rg¯ + λ62 eρλz is L1 with respect to the metric g¯. Next, we shall show that ¯jh¯(cid:16)ij ¯itr(cid:17)g¯ h¯ eRλ isL1 R3,g˘ . As ∇ −∇ H (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) ∇˘Hi trg˘H(h¯) =O(cid:18)e−λ5R(cid:19), ˘ h¯ =e˘ h¯ h¯ Γk h¯ Γk ∇Hj 1j j 1j − kj j1− 1k jj Xj (cid:16) (cid:17) =e˘ h¯ h¯ Γ2 h¯ Γ3 h¯ Γ1 +Γ1 1 11 − 22 21− 33 31− 11 22 33 (cid:16) (cid:17) (cid:16) (cid:17) 8ε T 4ε T 2 R =∂R sinh−1 0e−λ3R + sinh−1 0e−λ3R coth λ2 λ ! λ2 λ λ λ! 8ε T 2 R sinh−1 0e−λ3R coth +O e−λ6R = O e−λ5R , − λ2 λ −λ λ! (cid:18) (cid:19) (cid:18) (cid:19) ˘ h¯ =e˘ h¯ h¯ Γk h¯ Γk ∇Hj 2j j 2j − kj j2− 2k jj Xj (cid:16) (cid:17) =−h¯jjΓjj2−h¯22Γ2jj+O(cid:18)e−λ5R(cid:19) = O(cid:18)e−λ5R(cid:19), ˘ h¯ =e˘ h¯ h¯ Γk h¯ Γk ∇Hj 3j j 3j − kj j3− 3k jj Xj (cid:16) (cid:17) =−h¯jjΓjj3−h¯33Γ3jj+O(cid:18)e−λ5R(cid:19) = O(cid:18)e−λ5R(cid:19), weobtainthat(cid:16)∇˘Hjh¯ij−∇˘Hi trg˘H (cid:16)h¯(cid:17)(cid:17)eRλ isL1(cid:16)R3,g˘H(cid:17). Denotesy˘mametric2-tensor Γ(X,Y)= ∇¯XY−∇˘HX Y. ThereisconstantC1suchthat|Γ|g˘H ≤C1(cid:12)(cid:12)(cid:12)1∇−H|a|(cid:12)(cid:12)(cid:12)g˘g˘HH = O(cid:16)e−λ3R(cid:17) 10 ZHUOBINLIANGANDXIAOZHANG (c.f. [3],p188). Therefore, (cid:12)∇¯h¯ −∇˘Hh¯(cid:12)g˘H ≤C2(cid:12)h¯(cid:12)g˘H |Γ|g˘H = O(cid:18)e−λ6R(cid:19), and we conclude(cid:12)(cid:12)that ¯jh¯ij(cid:12)(cid:12) ¯itrg¯ h¯(cid:12)(cid:12) (cid:12)(cid:12) eRλ is L1 R3,g˘ . Thus (R3,g,K) is - ∇ −∇ H H asymptotically deSitte(cid:16)rwiththecon(cid:16)sta(cid:17)n(cid:17)t = sin(cid:16)h T0. N(cid:17)ow H λ f f tr (K)=µsinh 2 δ +µ2sinh 2 ˘ f ˘ f ˘ f g − λ ij − λ∇Hi ∇Hj !∇Hi,j 1 f 4 f µ3coth + µcoth − λ λ λ λ 3 T 12ε T = coth 0 sinh−2 0e−λ5R +O e−λ6R . λ λ − λ2 λ (cid:18) (cid:19) Therefore, T 3 T 12ε trg(K)sinh λ0 − λcosh λ0 = −λ2sinh T0e−λ5R +O(cid:18)e−λ6R(cid:19). λ Nowwecalculate thetotalenergy-momentum. As 1 =˘ ,jg¯ ˘ tr (g¯)+ (a +a )+2(h¯ +h¯ ) E ∇H 1j −∇H1 g˘ λ 22 33 22 33 H 6ε T 18ε T = coth 0e−3λR + coth 0e−3λR − λ2 λ λ2 λ 4ε T 16ε T + coth 0e−3λR sinh−1 0e−3λR +O e−λ5R λ2 λ − λ2 λ (cid:18) (cid:19) 16ε T = tanh 0e−3λR +O e−λ5R , λ2 2λ (cid:18) (cid:19) wehave 2 EνH =1H6π Rl→im∞ZSREnνeRλe˘2∧e˘3 sinh2 T0 16ε T R = λ lim tanh 0e−3λRnνeRλλ2sinh2 sinθdθ dψ 16π R→∞ZSR λ2 2λ λ ∧ ε T T = tanh 0 sinh2 0 nνsinθdθ dψ. 4π 2λ λ Z ∧ S1 Since(nν)= (1,sinθcosψ,sinθsinψ,cosθ),weobtain T T εtanh 0 sinh2 0, if ν = 0, EνH =  2λ λ  0, if ν = 1,2,3. SoE0H < 0ifεT0 < 0andinthiscase(2.2)doesnotholdforlarge R. WeremarkthatthedeSittermetric(1.3)collapsesatT = 0whichcanbeviewed asthebigbangoftheuniverse. Itisinteresting tofind,atthebigbang, that lim E = 0. 0H T0 0 →