CC BY
32
Kanno S.,' Soda J.2
A UNIFIED VIEW OF RS BRANEWORLDS
Department of Physics, Kyoto University, Kyoto 606-8501, Japan PACS numbers: 98.80.Cc), 98.80.Hw, 04.5Q.-fh
I, Introduction
Nowadays, the most promising and seemingly a unique candidate for quantum theory of gravity is the string theory. Remarkably, it can be consistently formulated only in 10 dimensions [1]. This fact requires a mechanism to fill the gap between our real world and the higher dimensions. Conventionally, the extra dimensions are considered to be compactified to a small compact space of the Planck scale. However, recent developments of superstring theory invented a new idea, the so-called braneworld. The brane world scenario has been the subject of intensive investigation for the past few years. Although there are many braneworld models, there are similarities in those models. Hence, we study a simple toy model constructed by Randall and Sundrum as a representative {'2,3].
There are various views of RS braneworlds depending on the approach one uses. The purpose of this paper is to unify the various views using the low energy gradient expansion method [4,5] and give insights into the physics of the braneworld.
The organization of this paper is as follows: In Sec.
II, we summarize various views of the braneworlds obtained by different methods. In. Sec. Ill, key questions are presented. In See. IV, we explain the gradient expansion method. In Sec. V and VI, the single-brane model and the two-brane model are analyzed separately. The final section is devoted to the answers to the key questions.
II. Various Views of Braneworlds
Randall and Sundrum proposed a simple model where the four dimensional brane with tension o is embedded in the five-dimensional asymptotically anti-de Sitter (AdS) bulk with a curvature scale £. This single-brane model is described by the action [3]
(1)
where K and k2 are the scalar curvature and gravitational constant in five-dimensions, respectively. We impose Z2 symmetry on this spacetime, with the brane at the fixed point. The matter £mma is confined to the brane. Throughout this paper, represents the
induced metric on the brane. Remarkably, the internal dimension is non-compact in this model. Hence, we do not have to care about the stability problem.
Originally, they proposed the two-brane model as a possible solution of the hierarchy problem {2]. The action reads
v J ¡~®,e ¡pi
¿=®,e
where © and © represent the positive and the
t]€.pjtiw fpns'joti brainy f^nftrtivffv,
Be h 111'"J. re >"!'.' i ■ we lis. 1 ' • ' , II 1 '
prototypes.
', ... iC>>' Sf i""'J<.-.
I*n »••>'»*ijr *t o - ip »»;*'< v of the single-brane 'HhM 's 'he dn i'e:.i cas- tt : studied. It is easy to
„-duce the eil-H. >\e i ned îh> u equation as
//2=—p+kv+4-
eK at
(3)
where H , a0 and p are, respectively, the Hubble parameter, the scale factor and the total energy density of each brane. The Newton's constant can be identified as BîtGy = k2 / Î . Here, C is a constant of integration associated with the mass of a black hole in the bulk. This constant (.' is referred to as the dark radiation in which the effect of the bulk is encoded.
There are two kinds of corrections, the high energy correction p2 and the bulk correction C which exists even in the low energy regime. Thus, the deviation
1 E-mail: sugumi@tap.scphys.kyoto-u.ao.jp
2 E-mail: jiroitap.scphys.kyoto-u.ac.jp
from the conventional Einstein theory is expected even in the low energy regime.
As to the two-brane model, the same effective Friedmann equation can be expected because this equation can be deduced without referring to the bulk equations of motion. In the two-brane case, however, the meaning of C is obscure.
ear Perturbation Approach
The other useful method to investigate the braneworld is the linearized analysis.
In the case of the single-brane model, it was shown that the gravity is localized on the biar o in spire of the noncompact extra dimension. Consetjiunllv it turned out that the conventional linearized 1 insu m equation approximately holds at scales large o>ri[> t id with the curvature scale £. It should be stater., dithis result can be attained by imposing the outgoing boundary conditions.
In the case of the two-brane model, Garriga and Tanaka analyzed linear»wi wpvify pp4 yho'vn
that the gravity o,> ib. hr.*» • Hij..,. ■ »•. hr-"* Dicke theory at lc . *„.•••• t. ¡. 7r • >- >• ¡r >•
linp'.sri'jad i< met'"1 '•..... ,, ..... .... ■ *,...... .. ,. j ■- t i >■ ¡« -.■■■■
In the case of the braneworld, the brane acts as the cutoff. Therefore, there is no divergences in the above expressions. Hence, we can freely rearrange the terms as follows
S* + = + SCfT + [R1 terms], (7)
This tells us that the brane models can be described as the conventional Einstein theory with the cutoff CFT and higher order curvature terms. In terms of the equations of motion, the AdS/CFT correspondence reads
(4, K2
Gpv =^(7;, +T^) + [R2 terms],
(8)
where the R2 terms represent the higher order
curvature terms and T^ denotes the energy-
momentum tensor of the cutoff version of conformai field theory.
D. Geometrical Approach
Here, let us review the geometrical approach [8]. In the Gaussian normal coordinate system: ds2=dy2 + g^(y,j?)cb»dx\ (9)
¡5}
we can write the 5-dimensional Einstein tensor iijiv in
terms of the 4-dimensional Einstein tensor Guv and the
.:I..l'...!.,•.:>.____________! i_iit.,u.m
Hilbeit action SS4, we obtain the effective action for
. The statement of the
the boundary field ij) = i>c]
AdS/CFT correspondence is that the resultant effective action can be equated with the partition functional of some conformally invariant field theory (CFT), namely
exp[fSH[Ocl]]« (expr J<t»^)m , (4)
where 0 is the field in CFT. This action must be defined at the infinity where the conformal symmetry exists as the asymptotic symmetry. Hence, there exist infrared divergences which must be subtracted by the counter terms. Thus, the correct formula becomes
expliS5d[«J)d] + iSa] = (expi, (5)
where we added the counter terms
Sa=Siaae-S4i-[R2 terms],
(6)
extrinsic curvature as !S> (*«)
G(W = Guv + Km - gm, K - KK + 2K XK
yX v
6
y * " P" a/ fl
where we have introduced the extrinsic curvature
HO)
(11)
K =~~g
"ijv 2 6ilv->
and the last equality comes from the 5-dimensional
Einstein equations. Taking into account the Z2 symmetry, we also obtain the junction condition
-sï^=y(-os; +rv). (i2)
Here, T^ represents the energy-momentum tensor
of the matter. Evaluating Eq. (10) at the brane and substituting the junction condition into it, we have the "effective" equations of motion
where SbmK and S4i are the brane action and the 4-dimensional Einstein-Hilbert action, respectively.
Here, the higher curvature terms [R2 terms] should be understood symbolically.
(4) K2 4
Gfjv = ~TfV + k Knv - EMV where
V ~ ^v +12
(13)
F = C I
^'¡av Wv ly~0 '
Here Cym is the Weyl tensor. We assumed the relation ? 6
k a = - • (14)
I
so that the effective cosmological constant vanishes.
The geometrical approach is useful to classify possible corrections to the conventional Einstein equations. One defect of this approach is the fact that the projected Weyl tensor can not be determined without solving the equations in the bulk,
III. Key Questions
As a landmark, we set a sequence of questions. We consider the single-brane model and two-brane model, separately.
A. Single-brane model
Is the Einstein theory recovered even in the nonlinear regime?
In the case of the linear theory, it is known that the conventional Einstein theory is recovered at low
energy.
On the other hand, the cosmological consideration suggests the deviation from the conventional Friecliriann equation even in the low energy regime. This is due to the dark radiation term.
Therefore, we need to clarify this discrepancy.
How does the AdS/CFT come into the braneworld?
It was argued that the cutoff CFT comes into the braneworld. However, no one knows what is the cutoff CFT. It is a vague concept at h'-nt fr~m die point of view of the classical gravity. Moid»- n, it should be noted that the AdS/CFT corre»|Mi'dei..c is a specific conjecture. Indeed, originally, Maldacena conjectured that the supergravity on AdSsxSs is dual to the four-dimensional /1/ = 4 super Yang-Mills theory [9]. Nevertheless, the AdS/CFT correspondence seems to be related to the brane ' world model as has been demonstrated by several people [10].
Hence, it is important to reveal the role of the AdS/CFT correspondence starting from the 5-dimensional general relativity.
How are the AdS/CFT and geometrical approach
related?
The geometrical approach gives
(4) K2 ^
G(tv = —j'l+K 7Epv — E^.
On the other hand, the AdS/CFT correspondence yields
(4) yA
G,v =—(7T +T^) + [R2 terms].
An apparent difference is remarkable. It is an interesting issue to clarify how these two descriptions are related. Shiromizu and Ida tried to understand the AdS/CFT correspondence from the geometrical point of view {.11]. They argued that itfl
corresponds to the trace anomaly of the cutoff CFT on the brane. However, this result is rather paradoxical because there exists no trace anomaly in an odd dimensional brane although я1^ exists even in that
case. Thus, the more precise relation between the geometrical and the AdS/CFT approaches remains to be understood.
Since both the geometrical and AdS/CFT approaches seem to have their own merit, it would be beneficial to understand the mutual relationship.
B. Two-brane model
How is the geometrical approach consistent with the Brans-Dicke picture?
Irrespective of the existence of other branes, the geometric approach gives the effective equations (13). The effect of the bulk geometry comes intothe brane • l\ 4 Vdy •!' , Vi" С '<" . I, • '
V _ ■ ■!<•.< 'Лчг I ' I I1
T • !■ ,i ,
\>. ' « <"• ■ . !'■ 1 ■'■, J.'i - O1' , t I. >■ i '4 " ! J", ' M-.
'vvhiie, нош ше lineai analysis, и itiius out tuai the
system can be described by the Brans-Dicke theory where the extra scalar field is nothing but the radion. How can we reconcile these seemingly incompatible pictures?
«thai «чрЬк'л ti e AdS/CFT correspondence in the "vm brane Mirdt E.
In the single-brane model, there are continuum Kaluza-Klein (KK)-spectrum around the zero mode. They induce the CFT matter in the 4-dimensional effective action.
In the two-brane system, the spectrum become discrete. Hence, we can not expect CFT matter on the brane. Nevertheless, KK-modes exist and affect the physics on the brane.
So, it is still interesting to know what kind of 4-dimensional theory mimics the effect of the KK-modes.
IV. Gradient Expansion Method
Our claim in this paper is that the gradient expansion method gives the answers to all of the
questions presented m the previous section. Here, we give the formalism developtu jy us [4].
We use the Gaussian not ual coordinate system (9) to describe the geometry of the brane world. Note that the brane is located at >• = 0 in this coordinate system. Decomposing the extrinsic curvature into the txaceless part and the trace part
= 2„¥ +jhllvK, K = -|-log ,fg, (15)
we obtain 'the basic equations which hold in the bulk;
' (4) 1 (4)" 06)
4 ya yP _ i. ¡jZ, a - "'(4)" R 12 (17)
K,--K2 > 4
- Za|iZ„R = ■
>
V^-I^JC-O,
(18) (19)
<4(
where R\, is the curvature on the brane and V
denotes the covariant derivative with respect to the metric g . One also have the junction condition
\ -t r > () ■ . V
, pa'''"'" >
(20)
a k2cf 11, <8Cl
energy. Here, we have used- the relation (14) and Einstein's equation on the brane, R~ g^/L2 ~ K2p/£..
Thus, the anti-Newtonian or gradient expansion method used in the cosmological context is applicable to our problem [12].
At zeroth order, we can neglect the curvature term. Then we have
10) (0)(0)
■x (0) (0) (0) 4
= 0,
y a y fi _ ^ a~ „1 '
12
i2
m i «)) (0) to) A 4 t
(0) a (0)
Equation. (22) can be readily integrated into (7=0,
(22)
(23)
(24)
(25)
(26)
condition wtu t>e imposed at the brane. as it is the condition for the induced metric ftpv, it is naturally
interpreted as the effective equations of motion for gravity on the brane.
Along the normal coordinate y, the metric varies
with a characteristic length scale £; g ~ gv J i .
Denote the characteristic length scale of the curvature fluctuation on the brane as L; then we have R ~ gpv / L2. For the reader's reference, let us take £ -1 nun, for example. Then, the relation (14) give the scale, K2 = (108 GeV)"3 and ct = ITc\'4. In this paper, we will consider the low energy regime in the sense that the energy density of matter, p, on the brane is smaller than the brane tension, i.e.,p/o«l. In this regime, a simple dimensional analysis
P
where €gv is the constant of integration. Equation (25) also requires = 0. If it could exist, it would
represent a radiation like f5n;:> or- »b- br-r" r>r"l " strongly anisotropic ur'- a • ■> • >>
srm must vanish V • <>> i ■ .' iion. Therefore, we simply put CMV =0, hereafter,
it is easy to solve the remaining equations. The is 4 1*
using the definition of the extrinsic curvature
(27)
(0) J g (0)
K 2dy 8 (28)
we get the zeroth order metric as
ds2=dy2+a2(y)hm(xliWdx'/, a(y) = e\ (29) where the tensor is the induced metric on the brane.
From the zeroth order solution, we obtain
(0) (fi) K\-$tK
y=0
3 k2 = = -—08;;.
£ 2 v
(30)
(21.)
implies that the curvature on the brane can be neglected compared with the extrinsic curvature at low
Then we get the well known relation k1g-6/£. Here, we will assume that this relation holds exactly. It is apparent that C% is not allowed to exist.
The iteration scheme consists in writing the metric as a sum of local tensors built out of the induced
metric on the brane, the number of gradients increasing with the order. Hence, we will seek the metric as a perturbative series
«
(y,*) = a (y)i^iv(*") + * )№(y,Jt")
(2)
(31)
« 41» Ep -—E'1 =
(4) ) (4)
4
«SU
£
(i)
(1) 7 CO
=-K = 0,
t
m
' it ix
"5 0)
(34)
(35)
(36)
(37)
where the superscript (1) represents the order of the derivative expansion and | denotes the covariant derivative with respect to the metric k^. Here,
(4)
R\
(l)
means that the curvature is approximated by
(4)
(4)
U> 0
K=--TR(hy 6 a
(38)
where a2(y) is extracted and we put the Dirichlet
boundary condition <0
^v(y = 0,^) = 0, (32)
so that gvx(y = 0,x) = hiiv(x) holds at the brane. Other quantities can be also expanded as
1 U) (2)
1 (33)
(1) (2)
In our scheme, in contrast to the AdS/CFT correspondence where the Dirichlet boundary condition is imposed at infinity, we impose it at the finite point >• = 0, the location of the brane. Furthermore, we carefully consider the constants of integration, i.e., homogeneous solutions. These homogeneous solutions are ignored in the calculation of AdS/CFT correspondence. However, they play the important role in the braneworld.
V. Single brane model (RS2)
stein Gravity at Lowest Order
The next order solutions are obtained by taking into account the terms neglected at zeroth order. At first order, Eqs. (16)-(19) become
Hereafter, we omit the argument of the curvature for simplicity.
Simple integration of Eq. (34) also gives the traceless part of the extrinsic curvature as
(39)
where the homogeneous solution satisfies the constraints
lpv=0, 3^=0. (40)
As we see later, this term corresponds to dark radiation at this order.
The metric can be obtained as
8 ¡iv ™ —T
1
(41)
where we have imposed the boundary condition,
CO
Let us focus on the role of in this part. As this order, the junction condition can be written as
(42)
"(l) (1)"
fv 8v K
-f ^--SStfl + y^—r"v.
2 v 2 ' j 2 %
(44)
(45)
taking the Ricci tensor of a /jpv in place of . It is also convenient to write it in terms of the Ricci tensor of , denoted R"v(h).
Substituting the zeroth order metric into R, we obtain
Using the solutions Eqs. (38), (39) and the formula „ 1 (43)
+6
we calculate the projective Weyl tensor as (t) p
Then we obtain the effective Einstein equation
1 If2 C)
2 l
At this order, we do not have the conventional Einstein equations. Recall that the dark radiation exists even in the low energy regime. Indeed, the low energy effective Priedmann equation becomes
+ (46)
3.r a0(t)
This equation can be obtained from Eq. (45) by imposing the maximal symmetry on the spatial part of the brane world and the conditions (40).
Hence, we observe that %,lv is the generalization of the dark radiation found in the cosmological context.
The nonlocal tensor %[i¥ must be determined by the boundary conditions in the bulk. The natural choice is
asymptotically AdS boundary condition. For this boundary condition, % =0- h is this boundary
condition that leads to the conventional Einstein theory in linearized gravity. Assuming this, we have
= (47)
Thus, Einstein theory is recovered at the leading
order!
1 * -jS/CFT Emerges
In this subsection, we do not include the % field because we have adopted the AdS boundary condition. Of course, we have calculated the second order solutions with the contribution of the % field. It merely
adds extra terms such as , etc.
At second order, the basic equations can be easily deduced. Substituting the solution up to first order into the Ricci tensor and picking up the second order quantities, we obtain the solutions at second order. The trace part is deduced algebraically as
<2,1 /3 / ?
So I 15 <x 9
r ?
"]2«U
R" R>'
("{ v lur
(48)
part.
(49)
where ¿H'v is defined by bjd*x>Ph-
R^—R2
(50)
= jd'xiftiS^Sg^.
The tensor Svv is transverse and traceless, <5^ = 0, «5»; =0. ' (51)
lite homogeneous solution fv must be traceless.
Moreover, it must satisfy the momentum constraint. To be more precise, we must solve the constraint equation
(52)
8 = ¡d^xyFh^bg^
S¡d4x^h~R2 ---- /¿W^M^.
(54)
Notice that = Wv - fcMv / 3. Hence, only <SPV and ICPV are independent. Thanks to the Gauss-Bonnet
topological invariant, we do not need to consider the Riemann squared term. In addition to these local tensors, we have to take into account the nonlocal tensor tMv with the property = 0. Thus, we get
f J?y?Pa-Ij?2 32 vl p 3
1
+— 24'
RRt
1 |+if„
(55)
+| a + — 4
where the constants a and P represents the freedom of the gravitational wave in the bulk. The condition fv = 0 leads to
(56)
Dression is the reminiscent of the trace •. the CPT. It is possible to use the result of . point. For example, we can choose the ,V" = 4 super Yang-Mills theory as the confonnal matter. In that case, we simply put ¡3 = 0, This is the way the AdS/CFT correspondence comes into the brane world scenario.
Hp to the second order, the junction condition gives
id' .....-WR + 2e
2
If we define
tcft = _2—— t
""pv " 2 pv '
we can write
,4) K2
t4' + od?>
3 '
_ K" 7 M v
G (iv + TfJ1 ) - 2i2aSyy
2 tl
(57;
(58)
(59)
As one can see immediately, there are ambiguities
in integrating this equation. Indeed, there are two types of covariant local tensor whose divergences vanish;
1 2
(53)
Let us try to arrange the terms so as to reveal the
geometrical meaning of the above equation. We can calculate the projective Weyl tensor as
E \= f |Vv + 2#v + 2a^v +1
where
4 6
8 16
Substituting this expression into Eq. (57) yields our main result
(60)
(61)
(4) K2
G = —T + el
nv 2 v
(2) P^-E
iiv
(62)
Notice that contains the nonlocal part and the free parameters a and p. On the other hand, P\ is
determined locally. One can see the relationship in a more transparent way. Within the accuracy we are considering, we can get P* = using the lowest
order equation R% = (k2U)(T\ -H2WJ). Hence, we
can rewrite Eq. (62) as
(4) K2 ^ (2)
^ |1V~ "T ^v + K V ~ ^ ¡IV
(63)
Now, the similarity between Eqs. (13) and (63) is apparent. Thus we get an explicit relation between the geometrical approach and the AdS/CFT approach, However, we note that our Eq. (63) is a closed system of equations provided that the specific conformal field theory is chosen.
Now we can read off the effective action as
o?i2
JL
2k1
^dAx4~hK + Smma + SCFT
?r2
/¿V-aI^-I*2
(64)
6 k:
j {d4x\f-hR2
tensor contributions which describe the non-local 5-dimensional effect. Although Eqs. (65) and (66) are non-local individually, with undetermined % • one can combine both equations to reduce them to local equations for each brane. Since appears only
algebraically, one can easily eliminate from Eqs.
(65) and (66). Defining a new field lF = l-Q2, we find
k2(1 - ¥)2
if
(67)
J-fVl¥k, -Ig^T,,
2»F(1
1
(68)
2(1-¥)
where | denotes the covariant derivative with respect to the metric hvv. Since O (or equivalently ¥ ) contains the in . . between the two brane.s.
we can 'r I- •, ' x <h- . • -i
W
îten
by eliminating CMV
where we have used the relations Eqs. (50), (53) and (54) and we denoted the nonlocal effective action constructed from t4\, as S™ .
VI.
A. Scalar-Tensor Theory Emerges
We consider the two-brane system in this section. Without matter on the branes, we have the relation 'e-^'g^*"* »Qy-*"» where d is the
distance between the two branes. Although Q is constant for vacuum branes, it becomes the function of the 4-dimensional coordinates if we put the matter on the brane.
Adding the energy momentum tensor to each of the two branes, and allowing deviations from the pure AdSs bulk, the effective (non-local) Einstein equations on the branes at low energies take the form [4,8],
= (65)
W)=-yrvf|f- (66)
where %¥=g^bnm\ and the
terms proportional to %MV are 5-dimensional Weyl
from Eqs. (65) and (66). Then,
ZMv=-
2¥ I
......
i
3 f,„„,„, I
(69)
-I I**,,
2(1* 2
Note that the index of F|lv is to be raised or lowered by the induced metric on the © -brane, / .
The effective action for the ® -brane which gives Eqs. (67) and (68) is
3
H>R
|a
(70)
2(1 -T)
+ jd*x*niC* + ld4xsTh(l -¥)2£e.
It should be stressed that the radion has the conformal coupling.
i. AdS/CFT in two-brane system?
In the two-brane case, it is difficult to proceed to the next order calculations. Hence, we need to 'invent a new method [13]. For this purpose, we Shall start with the effective Einstein equation obtained by Shromizu, Maeda, and Sasaki
= T 4» tr _ p
|IV * p.v pv }i>
(71)
where jc is the quadratic of energy momentum tensor T^ and E(1V represents the effect of the bulk geometry.
Here we have set 8%G = 1. This geometrical projection approach can not give a concrete prediction, because we do not know E without solving the equations of motion in the bulk. Fortunately, in the case of the homogeneous cosmology, the property E^ = 0 determines the dynamics as
+ (72)
3 a0
This reflects the interplay between the bulk and the brane dynamics on the brane.
What we want to seek is an effective theory which contains the information of the bulk as finite number of constant parameters like C in the homogeneous universe. When we succeed to obtain it, the •»•v»,<J£H-;'t c.rrtf'rr>a«i« ircciv crn be coi-siioctet, in - - >",: vAlih.mgh tin i on re if pre < * ction can no: otch',. qe.dii/ith e undi •• tanjing of tK; ">h,,;oji of
• u 1Í M '
i., * J .. i >. i i i • ' • 1 >t.' i
\ i oc. 'i Y* his h;it¡,iiaS'i t í aiviíüV íri- cr ui,■;<;,n • is valid even in the non-linear regime. 11ms, at low energy, the action can be expanded by the local terms with increasing orders of derivatives of the metric #
and the radion ¥ [4].
Let us illustrate our method using the following action truncated at the second order derivatives:
=^Jd4x^m- 2A0F)
2-
com, ¥
(74)
For the vacuum brane, we can put Tm, + Kgv = -Xg^. Hence, the geometrical effective equation reduces to
=~2v -Xg^. (75)
First, we must find E^. The above action (74) gives the equations of motion for the metric as
+_L(V V V
Y V
(76)
r = _A
H» ij/
The right hand side of this Eq. (76) should be identified with -E^ - Xgm. Hence, the condition
Ji'l =0 becomes
ÍA 4
= V^V^—(A-
3¥ p 3
■IT).
(77)
o¥ =
|¥a¥Va¥-—R+—A', 2(0 J 2© to
(78)
This is the equation for the radion ¥ . However, we also have the equation for ¥ from the action (74) as
„2¥
where the prime denotes the derivative with respect to ¥ . In order for these two Eqs, (77) and (78) to be compatible, A and co must, satisfy
cti 1 co'
■ — - — - -
(79)
¿00
-rA-X¥)= —(2X-A'), (80)
3 co
where we used I? = 4X which comes from the trace part of Eq. (75). Eqs. (79) and (80) can be integrated as
¥
À(¥) = A. + %f(l - ¥) ,
(81)
21.-Y
where the constant of integration y represents the ratio of the cosmological constant on the negative tension brane to that on the positive tension brane. Here, one of constants of integration is absorbed by rescaling of *-F . In doing so, we have assumed the constant of integration is positive. We can also describe the negative tension brane if we take the negative signature-
Thus, we get the effective action
Sea = |d*XyPg x
which is nothing but the scalar-tensor theory with coupling function oXT) and the potential function Aff). Note that this is the most general local action which contains up to the second order derivatives and has the general coordinate invarianee. It should be stressed that the scalar-tensor theory is, in general, riot related to the braneworld. However, we know a special type of scalar-tensor theory corresponds to the low energy braneworld [4,5,14]. Here, we will present a simple derivation of this known fact.
-VR-
2
-f--V^FV Y •
4(1-¥) "
X-Xy(l-Wf
(82)
Surprisingly, this completely agrees with the previous result (70). Our simple symmetry principle E^ = 0 has determined the action completely.
As we have shown in [15], if y<~l there exists a static deSitter two-brane solution which turns out to be unstable. In particular, two inflating branes can collide at *P = 0. This process is completely smooth for, the observer on the brane. This fact led us to the born-
again scenario. The similar process occurs also in the ekpyrotic (cyclic) model [16] where the moduli approximation is used. It can be shown that the moduli approximation is nothing but the lowest order truncation of the low energy gradient expansion method developed by us [4]. Hence, it is of great interest to see the leading order corrections due to KK modes to this process.
Let us apply the procedure explained above to the higher order case:
xj^ft - 2A(¥) -iffivyV^
+C(Y)Vil¥V(1lFo¥ + DÇ¥)Rti¥ +G(W)R2 +H(W)RvxRilv
(83)
Now we impose the conformai symmetry on the fourth order derivative terms in the action (83) as we did in the previous example. Starting from the action (83), one can read off the equation for the metric from which E can be identified. The compatibility between the equations of motion for '*¥ and the equation £"',, =0 determines the coefficient functional»
in the action (83).
Thus, we find the 4-dimeasional effective action with KK corrections as
5eff = ¡d\xj~,
+£2 ¡d\x^'
1
2
—d—yt^w w 4(1 *
+
(1-*F)2 3
(oWy +
RdV+-
_1_
4(1-¥)4 1
(1-T)3 1
(VfWu¥)2 RV'fVM'
(84)
VII. Conclusion
We have developed the low energy gradient expansion scheme to give insights into the physics of the braneworld such as the black hole physics and the cosmology. In particular, we have concentrated on the specific questions in this paper. Here, we summarize our answers obtained by the gradient expansion method. 'Our understanding of RS braneworlds would be useful for other brane models.
A Siiigle-brane model
Is tie Einstein theory .recovered even lit the nonlinear regime?
We have obtained the effective theory at the lowest
order as
14) 9
GV—n-fxV (85)
Here we have the correction % which can be
interpreted as the dark radiation in the cosmological situation.
On 'the other hand, j« 1 • c-r»'-.'>1 v, •• .
,".i> > i iO -c
v. t , '11 ;
-1 i Hi
2(1 - 40 " 3(1-Y)2 +JR2
where constants j and k can be interpreted as the variety of the effects of the bulk gravitational waves. It should be noted that this action becomes non-local after integrating out the radion field. This fits the fact that KK effects are non-local usually. In principle, we can continue this calculation to any order of derivatives.
holds.
In this sense, the answer is yes.
How does the AdS/CFT come into the braneworld?
The CFT emerges as the constant of integration which satisfies the trace anomaly relation
(«6)
This constant can not be determined a priori. Here, the AdS/CFT correspondence could come into the braneworld. Namely, if we identify some CI;7! with t, then we can determine the boundary condition.
How are the AdS/CFT and geometrical approach "related?
The key quantity in the geometric approach is obtained as (2)
Pl+lx^+laS^+pfC^
(87)
The above expression contains T which can be interpreted as the CFT matter. Hence, once we know Em, no enigma remains. In particular, Pm = it is
independent of the x . In odd dimensions, there exists
no trace anomaly, but Ppv exists. In ^dimensions, it^
accidentally coincides with the trace anomaly in CI-T.
It is interesting to note that the high energy and the Weyl term corrections found in the geometrical approach merge into the CPT matter correction found in the AdS/CFT approach.
iv;o-brane model
»"V is hi goo. iftrica ;.pp>oacJh consistent with the
j.^-lMcHpicture'
In the geometrical approach, no radion seems to appear. On the other hand, the' linear theory predicts the radion as the crucial quantity. The resolution can be attained by obtaining Epv (%pv in our notation). The resultant expression
2¥
[(n^en)
2(1-V;
conformally coupled scalar-tensor theory where the radion plays the role of the scalar field.
What, replaces the AdS/CFT correspondence in the two-brane model?
In the case of the single-brane model, the out-going boundary condition at the Cauchy horizon is assumed. This conforms to AdS/CFT correspondence. Indeed, the continuum KK-spectrum are projected on the brane as CFT matter.
On the other hand, the boundary condition in the two-brane system allows only the discrete KK-spectrum. Hence, we can not expect CFT matter on the brane. Instead, the radion controls the bulk/brane correspondence in two-brane model. In fact, the higher derivative terms of the radion mimics the effect of the bulk geometry (KK-effect) as we have shown explicitly.
Hence, the AdS/CFT correspondence does not exist. Instead, the AdS/radion correspondence exists.
V.M-fO»itei»C9ft,fitt$
I • - supported in part by Grant-in-Aid
for Scientific Research Fund of the Ministry of Education, Science and Culture of Japan No. 155476 (SK) and No. 14-540258 (IS) and also by a Grant-in-Aid for the 2 J si Century COE "Center for Diversity and Universality in Physics".
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