Spacelike hypersurfaces with negative total energy in de Sitter spacetime

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by  Zhuobin Liang

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 hypersur-
face 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 hyper-
surface 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 the-
orem 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 hypersur-
faces. 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 cosmo-
logical 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 gˆij+µ2ρ 2ˆif 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
→