Metric Fluctuations in Brane Worlds

6 Pages·0.11 MB·English

by  M. G. Ivanov

hep-th/9912242
Metric Fluctuations in Brane Worlds
M. G. Ivanov∗
Department of Physics; University of California; 94720-7300, Berkeley, CA , USA
Moscow Institute of Physics and Technology;
Institutsky Per.9, Dolgoprudny, Moscow Reg., Russia
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0 and
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2 I. V. Volovich†

n Steklov Mathematical Institute; Gubkin St.8, 117966, Moscow, Russia
a
J
January 19, 2000
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v
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2 Abstract
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9 Recently, a realization of the four-dimensional gravity on a brane in five-
9
dimensional spacetime has been discussed. Randall and Sundrum have shown that
/
h
theequationforthelongitudinalcomponentsofthemetricfluctuationsadmitsanor-
t
- malizable zero mode solution, which has been interpreted as the localized gravity
p
e on the brane. We findthatequations, which includealso thetransverse components
h
of the metric fluctuations, have a zero-mode solution, which is not localized on the
:
v brane. This indicates that probably the effective theory is unstable or, in other
i
X words, actually it is not four-dimensional but five-dimensional. Perhaps a modifi-
r cation of the proposal by using matter fields can lead to the trapping of gravity to
a
the brane.
∗[email protected], [email protected]
†[email protected]
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1 Introduction
Recently arealizationofthefour-dimensionalgravityonabraneinfive-dimensionalspace-
time has been discussed [1]-[20]. Randall and Sundrum [1, 2] have shown that the longitu-
dinal components of the metric fluctuations satisfy to the quantum mechanical equation
with the potential, which includes an attractive delta-function. As a result one has a
normalizable zero mode, which has been interpreted as the localized gravity on the brane.
However in order to speak about the localized gravity one has to demonstrate that
not only longitudinal but also the transverse components of the metric are confined to the
brane. In this note we point out that the transverse components of the metric fluctuations
satisfy to the equation with the potential, which includes the repulsive delta-function.
There is a zero mode solution, which is not localized on the brane and this indicates
that there is unstability or, in other words, that the effective theory actually is not four-
dimensional but five-dimensional. Therefore it seems the originalproposalfrom[1,2]does
not lead to the realization of the four-dimensional gravity on the brane in five-dimensional
spacetime. Perhaps a modification of this proposal by using matter fields can lead to the
trapping of gravity to the brane.
2 Metric perturbation
The action has the form
1
I = dDx |g|(R+2Λ)+I , (1)
Brane
2
Z q
The RS solution is [1-3]
η dxMdxN
ds2 = MN , (2)
(k |zi|+1)2
i
M,N = 0,...,3+n, i = 1,...,n, zi = xi+3, µ,ν = 0,1,2,3,
P
where D = n+4 and ηMN is a Minkowski metric with the signature (+,−,−,−,...). The
metric (2) has singularities at zi = 0 for any i = 1,...,n, which correspond to intersect-
ing n+2-branes. Their intersection at zi = 0 for all i = 1,...,n is 3-brane, which bears
standard model fields and corresponds to observable 4-dimensional universe. The prop-
agation of the longitudinal (µ,ν)-components of metric perturbation in the background
(2) was studied in the papers [2, 3]. These perturbations are bounded in the vicinity of
the 3-brane. In this note we consider the perturbation of the transverse ((i,j) and (i,µ))
components.
We write the solution in the form [1-3]
g0 = H−2η , (3)
MN MN
n
H = k |zi|+1, (4)
i=1
X
where
−2
k2 = Λ, (5)
n(n+2)(n+3)
2
and Λ is the bulk cosmological constant.
Let us consider perturbations, which are parametrized by h in the following way
MN
g = H−2(η +h ) = H−2g˜ (6)
MN MN MN MN
and fix the gauge
h ηMN = 0, ηKM∂ h = 0. (7)
MN K MN
The Einstein tensor G = R − 1g R is
MN MN 2 MN
∇˜ ∇˜ H ∇˜ ∇˜ H ∇˜ H∇˜ H
G = G˜ +(D−2) M N +g˜ g˜KL − K L +(D −1) K L , (8)
MN MN " H MN ( H 2H2 )#
where all objects, which bear tilde are calculated by using the metric g˜ ,
MN
g˜MN = ηMN −hMN = ηMN −ηMKh ηLN, (9)
KL
We compute all objects up to the first order in h . We obtain
MN
∂ ∂ H −Γ˜K ∂ H
G = G˜ +(D−2) M N MN K (10)
MN MN
H
∂ ∂ H −Γ˜I ∂ H ∂ H∂ H
+ (D −2)g˜ g˜KL − K L KL I +(D −1) K L ,
MN ( H 2H2 )
where
ηKL
Γ˜K = (∂ h +∂ h −∂ h ). (11)
MN 2 N LM M LN L MN
Using the gauge conditions (7)
ηKLΓ˜I = 0 (12)
KL
we get
1 ∂ ∂ H −Γ˜K ∂ H ∂ ∂ H ∂ H∂ H
G = − 2h +(D−2) M N MN K +g˜ g˜KL − K L +(D −1) K L ,
MN 2 MN " H MN ( H 2H2 )#
(13)
where 2 = ηMN∂ ∂ . Now by using Einstein equations
M N
G = Λg +Tbranes (14)
MN MN MN
we can identify ∂∂H-terms with Tbranes, because they contribute only on the brane sur-
faces, and using (4), (5) we identify the (∂H)2-term with Λg . Therefore one gets
MN
1 ∂ H ∂ H∂ H
− 2h −(D −2) K Γ˜K −(D −2)(D−1)η hKL K L = 0. (15)
2 MN H MN MN 2H2
Finally we obtain the wave equation for metric perturbation in the form
∂ H ∂ H∂ H
2h +(D−2) L ηKL(∂ h +∂ h −∂ h )+(D−2)(D−1)η hKL K L = 0.
MN H N KM M KN K MN MN H2
(16)
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3 Propagation of metric perturbations
Function H does not depend on xµ. It allows us to rewrite the equation (16) in the
following form
∂ H ∂ H∂ H
2h +(D−2) m ηmn(∂ h +∂ h −∂ h )+(D −2)(D−1)η hmn m n = 0,(17)
ij H j ni i nj n ij ij H2
∂ H
2h +(D −2) m ηmn(∂ h +∂ h −∂ h ) = 0,(18)
iµ µ ni i nµ n iµ
H
∂ H ∂ H∂ H
2h +(D −2) m ηmn(∂ h +∂ h −∂ h )+(D −2)(D −1)η hmn m n = 0.(19)
µν H ν nµ µ nν n µν µν H2
To solve the system one can solve equation (17) to find h , then substitute h into
ij ij
equation (18) to find h and finally substitute h into equation (19) to find h .
iµ iµ µν
If h = 0, then equation (19) coincides with the wave equation derived in [2, 3] for
iµ
the longitudinal polarization of the perturbation
∂ H
2−(D −2) m ηmn∂ h = 0. (20)
n µν
H !
This equation can be transformed into the wave equation with attractive delta-function
2
+V(−)(z) hˆ = 0, (21)
µν
2
(cid:18) (cid:19)
n(n+2)(n+4)k2 (n+2)k
V(−)(z) = − δ(zj),
8H2 2H
j
X
where hˆ = H−(n+2)/2h. There is a bound state, which corresponds to the localized four-
dimensional gravity [2, 3]. The zero-mass state corresponds to hˆ = cH−(n+2)/2eipx, p pµ =
µ
0, so
h = c eipx, (22)
µν µν
where c is a constant polarization tensor.
µν
4 Non-longitudinal polarization in 5 dimensions
In the simplest case of one extra dimension equations (17)-(19) acquire the following form
2
∂ H ∂ H
2−3 5 ∂ −12 5 h = 0, (23)
5 55
 H H ! 
 ∂ H 
2h −3 5 ∂ h = 0, (24)
5µ µ 55
H
∂ H ∂ H ∂ H
2+3 5 ∂ h −3 5 (∂ h +∂ h )+12η h 5 = 0. (25)
5 µν µ 5ν ν 5µ µν 55
H ! H H !
The equation (23) can be transformed into the wave equation with the repulsive delta-
function
2
+V(+)(z) hˆ = 0, (26)
55
2
(cid:18) (cid:19)
169k2 3k
V(+)(z) = − + δ(z),
8H2 2
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where hˆ = H3/2h and we denote xM = (xµ,z).
Zero-mass state corresponds to the solution h ∼ eipx, p pµ = 0. Let us set
µ
h = 0. (27)
55
After the substitution of h (27) into equation (24) we have
55
2h = 0. (28)
5µ
Let us set
h = c (z)eipx, h = ψ (z)eipx. (29)
5µ µ µν µν
Then, from (28) and (25) one gets
′′
c = 0, (30)
µ
′′ ′
−ψ +3f ψ −i(c p +c p ) = 0, (31)
µν µν µ ν ν µ
(cid:16) (cid:17)
where we denote f(z) = ∂ H/H. As a simple explicite solution we take
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ψ = c +i(c p +c p )z, c = c = const, c = const. (32)
µν µν µ ν ν µ µν νµ µ
To satisfy the gauge conditions (7) we have also to set c pµ = 0, c ηµν = 0, c pν = 0.
µ µν µν
Let us summarize our solution
h = 0,
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h = c eipx,
5µ µ
h = (c +i(c p +c p )z)eipx, (33)
µν µν µ ν ν µ
px = p xµ, p pµ = 0, c pµ = 0, c ηµν = 0, c pν = 0,
µ µ µ µν µν
where c , c and p are constants. We assume, of course, that one takes the real (or
µ µν µ
imaginary) part of the above expressions.
To conclude, we obtain the zero mode solution (33) of the equations for the metric
perturbation with non-longitudinal polarization. One has a massless vector field h on
5µ
the brane. The perturbation (33) is not localized on the brane because h depends
µν
linearly on z. This indicates that probably the effective theory is unstable or, in other
words, actually it is not four-dimensional but five-dimensional. Perhaps the coupling with
matter fields [5] could lead to the trapping of gravity to the brane.
We are grateful to I.Ya. Arefeva and B. Dragovich for useful discussions. I.V.V. is
supported in part by INTAS grant 96-0698 and RFFI-99-01-00105.
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