Gravity, Stability and Energy Conservation on the Randall-Sundrum Brane-World

10 Pages·0.2 MB·English

by  Misao Sasaki

DAMTP February 1, 2008
University of Cambridge
Gravity, Stability and Energy Conservation
on the Randall-Sundrum Brane-World
Misao Sasaki1,4, Tetsuya Shiromizu2,4,5 and Kei-ichi Maeda3,6
1Department of Earth and Space Science, Graduate School of Science,
Osaka University, Toyonaka 560-0043,Japan
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2DAMTP, University of Cambridge
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2 Silver Street, Cambridge CB3 9EW, United Kingdom

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a 3Isaac Newton Institute, University of Cambridge,
J 20 ClarksonRoad, Cambridge CB3 0EH, United Kingdom

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1 4Department of Physics, The University of Tokyo, Tokyo 113-0033,Japan

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v 5Research Centre for the Early Universe(RESCEU),
3 The University of Tokyo, Tokyo 113-0033,Japan
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6Department of Physics, Waseda University, Shinjuku, Tokyo 169-8555,Japan
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WecarefullyinvestigatethegravitationalperturbationoftheRandall-Sundrum(RS)singlebrane-
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worldsolution[hep-th/9906064],basedonacovariantcurvaturetensorformalismrecentlydeveloped
/
h byus. Usingthiscurvatureformalism, itisknownthatthe‘electric’partofthe5-dimensionalWeyl
-t tensor, denoted by Eµν, gives the leading order correction to the conventional Einstein equations
p on thebrane. Weconsider thegeneral solution of theperturbation equationsfor the5-dimensional
e Weyltensorcausedbythematterfluctuationsonthebrane. Byanalyzingitsasymptoticbehaviour
h in the direction of the 5th dimension, we find the curvature invariant diverges as we approach the
:
v Cauchy horizon. However, in the limit of asymptotic future in the vicinity of the Cauchy horizon,
i the curvature invariant falls off fast enough to render the divergence harmless to the brane-world.
X WealsoobtaintheasymptoticbehaviorofEµν onthebraneatspatialinfinity,assumingthematter
r perturbation is localized. We find it falls off sufficiently fast and will not affect the conserved
a
quantities at spatial infinity. This indicates strongly that the usual conservation law, such as the
ADMenergyconservation,holdsonthebraneasfarasasymptoticallyflatspacetimesareconcerned.
OUTAP-109;DAMTP-1999-173;UTAP-356; RESCEU-49/99
I. INTRODUCTION
A recent discovery by Randall and Sundrum of the exact solution that describes Minkowski branes in the 5-
dimensional anti-de Sitter space [1,2] has attracted much attention. In particular, in their second paper [2], they
showed that a single Minkowski brane solution is possible if the brane has positive tension, and opened up the
possibility of dimensional reduction without compactifying extra dimensions. Subsequently, a various aspects of, and
variants of the RS solution have been discussed by many authors. Among them are work on non-linear plane-waves
[3], on black strings and black cigars [4,5], on the AdS/CFT correspondence [6,7], on cosmological solutions [8–16],
on the stability of the RS solution [17] and on the quantum creation of the brane world [18](see also [19,20])
Meanwhile, we formulated a covariant set of equations that describes both the 5-dimensional gravity and the 4-
dimensional gravity on the brane [21]. With Z -symmetry, which is expected to hold from an M-theoretic point of
2
view[22,23],wefoundthenegativetensionbrane-worldwouldnotbeallowedsinceitwouldbeaworldofanti-gravity.
Then Garriga and Tanaka showed that the negative tension brane is a world of a negative Brans-Dicke parameter at
the linear perturbation order [17].
On the other hand, we found a positive tension brane has the correct sign of gravity, and the equations reduce to
the conventionalEinstein equations in the low energy limit, provided that the extra term due to the ‘electric’ part of
the 5-dimensional Weyl tensor, E , is negligible [21]. Thus it is urgently important to clarify the effect of E to
µν µν
the brane-world. Part of this program was first done by Randall and Sundrum themselves [2] and more rigourously
by Garriga and Tanaka [17], and they showed that the effect is small at large distances from the source. However,
this conclusion was obtained only for static perturbations. Furthermore, since they adopted the metric perturbation
formalism,theasymptoticbehaviorofthecurvatureperturbationsin5dimensionsastheCauchyhorizonisapproached
was not straightforwardto see.
Inthispaper,wecarefullyinvestigatethefirstorderperturbationoftheRSbrane-worldintermsofthe5-dimensional
Weyl tensor. We focus on the single brane model. We first briefly review our curvature tensor formalism. Then we
derive the evolution equation for E , and give an expression for the retarded Green function with appropriate
µν
boundaryconditiononthe brane. We recovertheresultobtainedin[2,17]onthe braneforstaticsources. Forgeneral
spacetime dependent sources, we evaluate the asymptotic behavior of the Weyl tensor. We find the Weyl curvature
invariant diverges as we approach the Cauchy horizon. Nevertheless, an inspection of the asymptotic behavior as we
approachinfinitefuturealongneartheCauchyhorizon,thisinstabilitydoesnotaffectthebrane-worldsinceitinfinitely
redshifts away. This infinite redshift effect was suggested by Chamblin and Gibbons [3]. Then we discuss the energy
conservation on the brane. Local energy-momentum conservation is guaranteed by the 4-dimensional covariance.
We find globally conserved energy exists as well, just as in the conventional Einstein gravity, for asymptotically flat
spacetimes.
II. THE EFFECTIVE EINSTEIN EQUATIONS ON THE BRANE
In this section, we briefly review the effective gravitational equations on the brane and the equations for the
5-dimensional Weyl tensor which were derived in [21].
We consider a 5-dimensional spacetime with negative vacuum energy but otherwise vacuum,
G =κ2T ; T =−Λg , (1)
µν 5 µν µν µν
and a brane in this spacetime as a fixed point of the Z -symmetry. κ is the 5-dimensional gravitational coupling
2 5
constant (κ2 =8πG ). We assume the metric of the form,
5 5
ds2 =(n n +q )dxµdxν =dχ2+q dxµdxν, (2)
µ ν µν µν
wheren dxµ =dχisunitnormaltotheχ=constanthypersurfaces,oneofwhichcorrespondstothebrane,andq is
µ µν
the induced metric on the χ=constant hypersurfaces. Then, thanks to the Z -symmetry, the effective 4-dimensional
2
gravitationalequations on the brane take a form that resembles the conventionalEinstein equations:
(4)G =−Λ q +8πG τ +κ4π −E , (3)
µν 4 µν N µν 5 µν µν
where
1 1
Λ = κ2 Λ+ κ2λ2 , (4)
4 2 5(cid:18) 6 5 (cid:19)
κ4λ
G = 5 , (5)
N
48π
1 1 1 1
π =− τ τ α+ ττ + q τ ταβ − q τ2, (6)
µν 4 µα ν 12 µν 8 µν αβ 24 µν
E =C nαnβ (7)
µν µανβ
and C is the 5-dimensional Weyl tensor. λ is the vacuum energy on the brane and gives the brane tension. τ
µναβ µν
is the energy-momentum tensor of the matter on the brane. It should be noted that E here is not the one exactly
µν
on the brane (which is proportionalto the delta function), but the one that is evaluated by taking the limiting value
to the brane. But for simplicity, we call it E on the brane in the rest of the paper. It can be evaluated from either
µν
side of the brane due to Z -symmetry.
2
As G , which corresponds to the 4-dimensional Newton constant, has the same sign as λ, we assume a positive
N
tension brane to obtain the conventional gravity on the brane. In the case of a two brane model with positive and
2
negative tensions, it has been arguedthat it is possible to recoverthe normal gravityif the distance between the two
branes is fine-tuned [17] and if some mechanism to stabilize the distance is introduced [15]. In this paper, however,
we focus on a single brane model and assume λ>0.
The difference of Eq. (3) from the Einstein gravity is the presence of π and E on the right-handside. With κ
µν µν 5
and λ of a very high energy scale, it is easy to see that π can be safely neglected in the low energy limit. On the
µν
otherhand,sinceE comesfromthe 5-dimensionalWeyltensor,thereis noa priori reasontoexpectthatitis small
µν
even in the low energy limit. In fact, in the two-brane model, it is this part that contributes dominantly to recover
Einstein gravityonthe negativetension brane when the distance between the branes is fine-tuned [17]. We note that
because of the contracted Bianchi identities, we have
DµE =κ4Dµπ , (8)
µν 5 µν
where D is the covariant differentiation with respect to q . Hence, in the low energy limit when we can neglect
µ µν
π , E is transverse-tracelesswith respect to q .
µν µν µν
The effective gravitational equations on the brane (3) are not closed but one must solve the gravitational field in
the bulk at the same time. The 5-dimensional equations in the bulk we have to solve are
1
£ E =DµB + κ2Λ(K −q K)+Kµν(4)R
n αβ µ(αβ) 6 5 αβ αβ µανβ
+3Kµ E −KE +(K K −K K )Kµν, (9)
(α β)µ αβ αµ βν αβ µν
£ B =−2D E +K σB −2B K σ, (10)
n µνα [µ ν]α α µνσ ασ[µ ν]
£ (4)R =−2(4)R Kσ −2D B , (11)
n µναβ µνσ[α β] [µ |αβ|ν]
where K = (1/2)£ q and B = q ρq σC nβ. These equations are derived from the 5-dimensional Bianchi
µν n µν µνα µ ν ρσαβ
identities [21].
The equation for E has an alternative form that will be more convenient for later use;
µν
3 1
£ E =DµB +Kµν(4)C +4Kµ E − KE − q KµνE
n αβ µ(αβ) µανβ (α β)µ 2 αβ 2 αβ µν
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+2K˜µ K˜ K˜ν − K˜ K˜µνK˜ , (12)
α µν β 6 µν αβ
where K˜ is the traceless part of K ,
αβ αβ
1
K˜ =K − q Kµ . (13)
αβ αβ 4 αβ µ
Equations (9) (or (12)), (10) and (11) are to be solved under the boundary condition at the brane,
DµE | =κ4Dµπ (14)
µν brane 5 µν
1
B | =2D K | =−κ2D τ − q τ , (15)
µνα brane [µ ν]α brane 5 [µ ν]α 3 ν]α
(cid:16) (cid:17)
where we used the expression of K which is obtained by the Israel junction condition and Z −symmetry,
µν 2
1 1 1
K | =− κ2λq − κ2 τ − q τ . (16)
µν brane 6 5 µν 2 5 µν 3 µν
(cid:16) (cid:17)
III. THE FIRST ORDER PERTURBATION AROUND THE RS-BRANE WORLD
In this section, we derive the equations for the first order perturbation around the RS solution [2]. As we have
noted in the previous section, we focus on the case of a single brane with positive tension.
3
A. The perturbation equations
The backgroundbulk spacetime is taken to be the anti-de Sitter spacetime whose metric is
ℓ2 1
ds2 =dχ2+e−2χ/ℓη dxidxj = dz2+ η dxidxj , (17)
ij z2 ℓ2 ij
h i
whereℓ= −6/Λκ2,z :=eχ/ℓ,i=0,1,2,3andη isthemetricoftheMinkowskispacetime,η dxidxj =−dt2+dx2.
5 ij ij
χ = ∞ orpz = ∞ is the Cauchy horizon. The RS solution is obtained by putting a brane on a surface z = z∗ and
glueing two identical copies of the region z ≥ z of the AdS [1,2]. Since the coordinate z is scale-free, we may set
∗
z =1 without loss of generality. This solution is obtained from Eq. (3) by putting Λ =0 and τ =E =0.
∗ 4 µν µν
We nowconsiderthefirstorderperturbationofthe RSsolution. We assumeτ is asmallquantityof(ǫ)andsolve
µν
the perturbation linear in ǫ. Hence, we have
(4)G =8πG τ −E +O(ǫ2) (18)
µν N µν µν
on the brane.
In the first order of τ , the extrinsic curvature K can be written as
µν µν
1
K =− q +k , (19)
αβ αβ αβ
ℓ
where k is O(ǫ). The Lie derivative of k with respect to nµ is just E apart from the signature,
αβ µν µν
£ k =k µk −E =−E +O(ǫ2). (20)
n αβ α βµ αβ αβ
Using Eq. (19), Eq. (12) simplifies to
2
£ E =DµB + E +O(ǫ2). (21)
n αβ µ(αβ) ℓ αβ
On the other hand, Eq. (10) reduces to
£ B =−2D E +O(ǫ2). (22)
n µνα [µ ν]α
Combining these two equations, we obtain the wave equation for E in 5 dimensions,
µν
4 4
£ £ +K +D2+ E = 2 + E =0, (23)
n n 0 ℓ2 αβ 5 ℓ2 αβ
h (cid:16) (cid:17) i h i
where 2 is the 5-dimensional d’Alembertian. In the coordinates (z,xµ), this becomes
5
3 4
∂2− ∂ +ℓ22 + E =0, (24)
z z z 4 z2 αβ
h i
where 2 is the 4-dimensional d’Alembertian. Substituting Eq. (15) to Eq. (21), we obtain the boundary condition
4
of E on the brane,
µν
E 1 1 1
∂ µν =− κ2ℓ D2τ + D D τ − q D2τ . (25)
z(cid:16) z2 (cid:17)z=1 2 5 h µν 3 µ ν 3 µν i
In addition to the above, since we have DµE = O(ǫ2) on the brane, and £ DµE = O(ǫ2), E is transverse-
µν n µν µν
traceless with respect to the 4-metric q everywhere in the bulk.
µν
Once E is solved, we can integrate Eqs.(10) and (11) to obtain B and (4)C . To the first order, these are
µν αβµ αβµν
expressed in the coordinates (z,xi) as
2ℓ
∂ B =− ∂ E (26)
z µνα z [µ ν]α
∂ (z2(4)C )=ℓz[∂ρB η −∂ρB η ]−2ℓz∂ B (27)
z µναβ ρ(µ[α) β]ν ρ(ν[α) β]µ [µ |αβ|ν]
The above equationswill be usedwhen we evaluate the asymptotic behaviour ofthe Weylcurvature nearthe Cauchy
horizon. To discuss the behaviour near the Cauchy horizon, we consider an invariant quantity defined by the 5-
dimensional Weyl tensor,
C Cµναβ =(4)C (4)Cµναβ +6E Eµν +4B Bµνα+O(ǫ3). (28)
µναβ µναβ µν µνα
4
B. The retarded Green function
The general solution of E can be expressed in terms of the Green function satisfying
µν
(2 +4/ℓ2)G(z,x)=−δ5(z,x). (29)
5
As usual, we assume there is no incoming waves from past Cauchy horizon. Hence we consider the retarded Green
function. Inadditionto this causalboundary condition,wehaveone moreboundaryconditionatthe brane. Because
of the boundary condition for E , Eq. (25), we must impose the corresponding condition on the Green function,
µν
G
∂ =0. (30)
z(cid:16)z2(cid:17)z=1
With the Green function satisfying this condition, E is given by
µν
Eµν(z,x)= d4x′(−g(x′))1/2gzz(x′) ∂z′G(z,x; z′,x′)Eµν(x′)−G(z,x; z′,x′)∂z′Eµν(x′)
Zz′=1 h i
E (x′)
=−Zz′=1d4x′(−g(x′))1/2gzz(x′)G(z,x; z′,x′)∂z′(cid:18) µzν′2 (cid:19)
κ2 1 1
= 5 d4x′G(z,x; z′,x′) 2 τ + ∂ ∂ τ − q 2 τ(x′) . (31)
4 µν µ ν µν 4
2 Zz′=1 (cid:16) 3 3 (cid:17)
For bounded sources, which we are mainly interested in, the last line of the above equation can be re-expressed as
κ2 1
E (z,x)= 5 δαδβ2 + ηαβ(∂ ∂ −η 2 ) d4x′G(z,x; z′,x′)τ (x′). (32)
µν 2 (cid:20) µ ν 4 3 µ ν µν 4 (cid:21)xZz′=1 αβ
Now let us construct the retarded Green function. The general form of a mode function satisfying Eq. (23) is
e−iωpmt+ip·x
u (z,x)=N z2(J (mz)+b N (mz)) , (33)
m,p m 0 m 0
(2π)3/2 2ω
pm
p
whereω = p2+m2/ℓ2 andN isanormalizationconstant. Theboundarycondition(25)requiresthatthemode
pm m
functions shoupld satisfy (30), which determines bm as
J (m)
1
b =− . (34)
m
N (m)
1
ThenN isfixedbyrequiringthatthemodefunctionsshouldbeproperlynormalizedwithrespecttotheKlein-Gordon
m
norm. This gives
m1/2
N = . (35)
m
(1+b2 )ℓ
m
p
Using the mode functions obtained above, the retarded Green function is constructed as
2z2 ∞ N (m)J (mz)−J (m)N (mz)
GR(z,x; z′,x′)|z′=1 =− πℓ Z dm 1 N0(m)2+J1(m)20 gm(x,x′) (36)
0 1 1
where g (x,x′) is the 4-dimensional Green function given by
m
dωd3p e−iω(t−t′)+ip·(x−x′)
g (x,x′)= . (37)
m Z (2π)4 (m/ℓ)2+p2−(ω+iǫ)2
5
C. Static case
First let us consider the static case. In this case, the Green function is given by the integration of the retarded
Greenfunctionoverthetime. Thenthe4-dimensionalpartgivesrisetothefactore−mr/ℓ. Hencetheintegrationwith
respect to m is dominated by the contribution of small m, and we obtain
∞ ℓz2
G(z,x;1,x′)= dtG (z,t,x;1,0,x′)≃ , (38)
Z R 4π[r2+(ℓz)2]3/2
−∞
where r =|x−x′|.
On the brane the far field approximation gives
ℓ
G(1,x;1,x′)≃ . (39)
4πr3
From Eq. (32), we obtain the leading order of E as
µν
4G ℓ2 3
E ≃ N d3x ρ+ P ,
00 r5 Z (cid:16) 2 (cid:17)
1
E ≃ δ E , E ≃0, (40)
ij ij 00 0i
3
where weassumedthe perfect fluidform, τ =ρu u +P(q +u u ), forthe energymomentumtensor. The result
µν µ ν µν µ ν
is consistent with those obtained in Refs. [2,17].
For z ≫1, that is, near the Cauchy horizon, the static Green function is approximately given by
1 3 r 2
G(z,x;1,x′)≃ 1− . (41)
4πℓ2z 2 ℓz
h (cid:16) (cid:17) i
Hence we obtain the asymptotic behaviour as
E =O(∂2G)=O(z−3), ∂ E =O(z−5). (42)
µν r µ αβ
With the help of Eqs. (26) and (27), the above gives
B =O(z−5), (4)C =O(z−7). (43)
µνα µναβ
Therefore we have
E Eµν =O(z−2), B Bµνα =O(z−4), (4)C (4)Cµναβ =O(z−6), (44)
µν µνα µναβ
from which we obtain the estimate,
C Cµναβ =O(z−2). (45)
µναβ
This means that the perturbation remains regular at the Cauchy horizon for the static case.
D. General case
We now turn to the general case of time-dependent sources. We first note that the 4-dimensional Green function,
g (x,x′), can be evaluated exactly as
m
1 ∂ |s|
g (x,x′)= θ(t−t′) θ(s2)J m (46)
m 2π ∂s2 0 ℓ
h (cid:16) (cid:17)i
where s2 = (t−t′)2 −r2 and r = |x−x′|. Inserting this expression to Eq. (36), we find the part that contains
the derivative of θ(s2) (hence is proportional to δ(s2)) will not contribute. This is because this part of g becomes
m
independentofm,hencethe integrationwithrespecttomgivesδ(z−1),butz =1isoutsidethedomainofdefinition
of E (recall that E on the brane is actually defined by taking the limit to the brane). Thus we have
µν µν
6
z2 ∂ ∞ N (m)J (mz)−J (m)N (mz) |s|
GR(z,x; z′,x′)|z′=1 =−π2ℓθ(t−t′)θ(s2)∂s2 Z0 dm 1 N01(m)2+J11(m)20 J0(cid:16)m ℓ (cid:17). (47)
ToevaluatetheasymptoticbehaviorofG ,wemayapproximatetheaboveintegralbyassumingm≫1. Physically
R
thisisbecauseonlythehighfrequencymodes(inthe5-dimensionalsense)canreachthenullinfinity,whichcorresponds
to the Cauchy horizon in the present case. Since z > 1 we can then use the asymptotic form of the Bessel functions
except the function J (m|s|); we keep it as it is since we do not want to restrict the range of s2. Thus, we obtain
0
θ(t−t′)θ(s2)z3/2 ∂ ∞ |s|
GR(z,x; z′,x′)|z′=1 ≃− π2ℓ ∂s2 Z0 dmcos[m(z−1)]J0(cid:16)m ℓ (cid:17)
z3/2
=θ(t−t′)θ(s2)θ(s2−ℓ2(z−1)2) . (48)
2π2[s2−ℓ2(z−1)2]3/2
On the brane (z =1+0), the Green function becomes
θ(s2)θ(t−t′)
G (1,x;1,x′)≃ , (49)
R 2π2s3
which is in accordance with a naive expectation. This implies the behaviour of E in the far-field region as
µν
E ∼O(∂2G )=O(s−5). (50)
µν s R
This should be compared with the energy momentum tensor of a radiative field, τ ∼ s−2 at null infinity. Thus we
µν
conclude that E cannot carry away the energy momentum from a system to infinity. We will come back to this
µν
point in the next section.
To investigate the asymptotic behavior near the Cauchy horizon, it is more convenient to work in the null coordi-
nates, u=s−ℓ(z−1) and v =s+ℓ(z+1). Then with the help of Eqs. (26) and (27), we find
(v−u)3/2
E =O(∂2G )=O ,
µν s R (cid:16) (vu)5/2 (cid:17)
(v+u)(v−u)3/2
B =O(∂ E )=O(∂3G )=O ,
µνα α µν s R (cid:16) (vu)7/2 (cid:17)
(v−u)3/2
(4)C =O(∂ B )=O(∂4G )=O . (51)
µναβ α µνβ s R (vu)7/2
(cid:16) (cid:17)
We take the limit v →∞ along u=constant to approach the Cauchy horizon. Then E =O(v−1), B =O(v−1)
µν µνα
and (4)C =O(v−2). Therefore, the curvature invariant (28) diverges in this limit as
µναβ
C Cµναβ =O(v4). (52)
µναβ
ThismeansthattheCauchyhorizonisunstabletothe perturbation. Thebadbehaviourofthe Weyltensor,however,
does not necessarily imply an instability of the brane-world. It will not affect the brane if the divergence disappears
in the limit of distant future; u = αv → ∞ (α =const.< 1). This is the infinite redshift effect. In the present case,
if we take this limit, we find C Cµναβ = O(u−3). Hence the brane-world is unaffected by the instability of the
µναβ
Cauchy horizon.
IV. THE ENERGY CONSERVATION ON THE BRANE
Inthe conventionalEinsteingravity,the ADMenergyisconservedifthe energy-momentumtensorτ decaysfaster
µν
than ∼ r−3 towards spatial infinity. For the matter source of compact support, this condition is trivially satisfied
[24–26]. In the present case, we have E which is a part of the 5-dimensional Weyl curvature, in addition to the
µν
matter energy-momentum tensor. However,as we have seen in the previous section, although the 4-geometry on the
brane is affected by E , the effect seems to be quite subtle: First, E itself is locally conservedat the linear order,
µν µν
independent of τ . Second, both the static and dynamic perturbations fall off sufficiently rapidly at large distances
µν
from the source. So we expect that the conservation of the total energy holds also in the brane-world, provided the
brane geometry is asymptotically flat.
7
To re-confirm this expectation, in this section we present a more detailed discussion of the energy conservation. A
brief review of the asymptotic structure at spatial infinity is given in Appendix A. The following argument is based
on a recently developed formalism of the conformal infinity [25,26], which is much easier to deal with than the old
formalism [24].
First we express the asymptotic behavior of the 4-dimensional Ricci tensor as
1 1
L =(4)R − q (4)R≃L(0)+L(1)r−1+L(2)r−2+L(3)r−3+O . (53)
µν µν 6 µν µν µν µν µν r4
(cid:16) (cid:17)
The total energy of a system is naturally defined by the electric part of the 4-dimensional Weyl tensor, (4)E =
µν
(4)C rαrβ, because it is the one that describes the tidal force. rα is the unit vector of the radial direction. By
µανβ
a conformal transformation, one can expand the spatial infinity to introduce the structure of a unit 3-dimensional
timelike hyperboloid. We denote the metric and the covariant derivative of it by p and D , respectively. Then,
µν µ
assuming L(0) =L(1) =L(2) =0, the leading order behavior of (4)E for r →∞ can be expressed as
µν µν µν µν
1 1 1 1
(4)E ≃ (D D +p )F(1)− (L(3)+p L(3)rαrβ) +O , (54)
µν r 2 µ ν µν 2 µν µν αβ r2
h i (cid:16) (cid:17)
where F(1) is a function on the unit 3-dimensional timelike hyperboloid. In the present case, we have contributions
from not only τ but also E to the Ricci tensor. From the analysis in the previous section at the spatial infinity,
µν µν
E =O(r−5). Hence, if the matter source is localized, we have L(n) =0 for n=0, 1, 2, 3. Thus,
µν µν
1 1
(4)E ≃ (D D +p )F(1)+O , (55)
µν 2r µ ν µν r2
(cid:16) (cid:17)
Since (4)E is traceless, (1)F satisfies
µν
(D2+3)F(1) =0. (56)
Using this and introducing (4)E˜ =r(4)E , we can show
µν µν
Dµ(4)E˜ =0. (57)
µν
Now we define the ADM energy by
1
E :=− dSµνǫ E˜αβt (58)
ADM µνα β
8π Z
C
where where tµ is the asymptotic conformal Killing vector associated with the global time-translation symmetry
inducedatspatialinfinityandC isa2-dimensionalclosedsurfaceinthe 3-dimensionalunittimelikehyperboloid. The
aboveintegrationis invariantunder the changeofthe closedsurfaceC because Eq.(57)givesD ((4)E˜µνt )=0. This
µ ν
means that E is conserved.
ADM
V. SUMMARY AND DISCUSSION
We have carefully investigatedthe asymptotic behavior of the linear perturbation aroundthe single brane solution
of the RS brane-world scenario [2]. It is known that the effect of the 5-dimensional curvature appears in the form
of E , which is the ‘electric’ part of the 5-dimensional Weyl tensor, on the right-hand side of the effective Einstein
µν
equationsonthebrane[21]. TodealwiththispartoftheWeyltensordirectly,wehaveemployedthecurvaturetensor
perturbation formalism developed in [21] instead of the metric perturbation formalism.
For static perturbations, we have found that E on the brane falls off as r−5 from the source and the Weyl
µν
curvature in the bulk behaves regularly near the Cauchy horizon (z = ∞) as C Cµναβ = O(z−2), in accordance
µναβ
with the previous result obtained in the metric perturbation formalism [2,17].
For generic perturbations, we have found the perturbation diverges as C Cµναβ = O(v4) near the Cauchy
µναβ
horizon (v = ∞, u =const.). This implies instability of the Cauchy horizon. However, we have found that this
instabilitydoesnotaffectthebrane-worldbecauseC Cµναβ =O(u−3)foru=αv →∞(α<1)duetotheinfinite
µναβ
redshift effect. This supports the conjecture by Chamblin and Gibbons [3] that the Cauchy horizoninstability would
not affect the brane-world.
8
We have also found that E decays rapidly as s−5 on the brane for generic spacetime-dependent perturbations,
µν
where s2 = t2−r2. This is to be contrasted with the case of a radiative field (like electromagnetic radiation) that
behaveslikes−2(Toavoidtheconfusion,aradiationfieldbehaveslike∼r−4 atspatialinfinity.). ThisimpliesE does
µν
notcarryenergyawayfromthesystemtoinfinity. TogetherwiththefactthatE =O(r−5)forstaticperturbations,
µν
we have concluded that the ADM energy is well-defined also in the brane-worldand it is conserved.
One remaining issue is the positivity of the ADM energy. In the linear perturbation, the right-hand side of the
effectiveEinsteinequationshastheadditionalterm,−E . If−E uµuν >0,whereuµisanarbitrarytimelikevector,
µν µν
thebranewillbestableinthesemi-classicallevel. Inthisconnection,recentworkoncosmologicalsolutionsthat−E
00
is proportionalto the mass of the 5-dimensionalSchwarzshild-AdSspacetime [10–12,18]implies it is positive. On the
contrary, our result of the perturbation analysis given in Eq. (40) has the opposite sign. One might suspect that our
result is incorrect. However, there is a good reason to believe that our result is indeed correct. Remember that E
00
is the tidal force in the 5th direction. When there is a mass in the bulk, this means we have a force that tends to
‘tearapart’the matter onthe brane inthe 5thdirection,hence givesa repulsivecontribution. This means the energy
is positive, in accordance with cosmological results obtained in [10–12,18]. On the other hand, because of the very
nature of the tidal force, a mass on the brane exerts a force that tends to squeeze the matter in the 5th direction
at least in the far zone, meaning it works as an attractive force. This means it is a negative energy contribution,
just like the Newtonianeffective gravitationalenergy. It is therefore impossible to judge whether the positivity of the
ADM energy is guaranteed or not within the present perturbation analysis. Some global non-linear analysis will be
necessary to resolve this issue.
ACKNOWLEDGEMENTS
We wouldliketothank J.Garriga,G.W.Gibbons,H.Kodama,A.Ishibashi,H.Ishihara,T.TanakaandN.Turok
for discussions. Partof this work was done while MS and KM were participating the program,“Structure Formation
inthe Universe”,atthe NewtonInstitute, UniversityofCambridge. We aregratefulto the NewtonInstitute fortheir
hospitality. MS’s work is supported in part by Monbusho Grant-in-Aid for Scientific Research No. 09640355. TS’s
workissupportedbyJSPSPostdoctalFellowshipforResearchAbroad. KM’sworkissupportedinpartbyMonbusho
Grant-in Aid for Specially Promoted Research No. 08102010.
APPENDIX A: ASYMPTOTIC FLATNESS AT SPATIAL INFINITY
We briefly describe the definition of asymptotic flatness and some useful results obtained in 4-dimensional asymp-
totically flat spacetimes (see Refs. [25,26] for the details and exact descriptions ). In this appendix the notation
basicallyfollowsRef. [26],whichis slightly differentfromthe maintext ofthis paper. Belowthe suffix a,b,...denotes
the abstract index [27].
Definition: Aspacetime(M,q )willbesaidtobeasymptoticallyflatatspacelikeinfinityI ifthereexistsasmooth
ab
function Ω satisfying the following features (i), (ii) and the energy momentum tensor satisfies the fall off condition
(iii);
(i) Ω| =0 and dΩ| 6=0.
I I
(ii) The following quantities have smooth limits on I.
ra =Ω−4qabD Ω, (A1)
b
pˆ =Ω2(q −F−1Ω−4D ΩD Ω)=Ω2p , (A2)
ab ab a b ab
where F =£ Ω.
r
(iii) Tµν :=Tab(eµ)a(eν)b =O(Ω4) near I, where {(eµ)a}µ=0,1,2,3 is a tetrad of the metric qab.
In this formalism, Ω∼r−1. For example, the extrinsic curvature of the Ω=cosnt. hypersurface is written as
1 1
κ := £ p =Ω−1F1/2pˆ − F−1/2£ pˆ , (A3)
ab rˆ ab ab r ab
2 2
where rˆa :=ra/ rbr . From the 4-dimensional Einstein equations under the condition (i), (ii) and (iii), we find
b
p
9
F=ˆ1 and (3)Rˆ =ˆ2pˆ , (A4)
ab ab
where the hatted equality, =ˆ, denotes the evaluation on I. This gives the 3-dimensional Riemann tensor as
(3)Rˆ =ˆ2pˆ pˆ , (A5)
abcd a[c d]b
which means that the 3-dimensional Ω=0 surface is locally a unit 3-dimensional timelike hyperboloid.
A part of the gravitationalfield is described by the electric part of the 4-dimensional Weyl tensor,
1
(4)E =κ κc−£ κ +D a −a a − (pcpd+p rˆcrˆd)L , (A6)
ab ac b rˆ ab (a b) a b 2 a b ab cd
where aa =rˆbD rˆa and L =(4)R −(1/6)q (4)R. Around Ω=0, (4)E is expanded as (4)E = (4)E(ℓ)Ωℓ.
b ab ab ab ab ab ℓ=0 ab
It can be shown that (4)E(0) =0 and P
1 1
(4)E(1) = (Dˆ Dˆ +pˆ )F(1)− (L(3)+p L(3)rˆcrˆd), (A7)
ab 2 a b ab 2 ab ab cd
where F =1+ F(ℓ)Ωℓ.
ℓ=1
FromaratherPlengthyargument,wefindthereexistsanasymptoticconformalkillingvector,ξˆa,suchthatξˆa=ˆDˆaα
and£ pˆ =ˆ2αpˆ orDˆ Dˆ α+pˆ α=ˆ0. TheconformalKillingvectorinducesthetime-translationsymmetryatspatial
ξˆ ab ab a b ab
infinity. The asymptotic conformal Killing vector tµ appeared in Sec. IV is just Dˆµα.
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