CC BY
45
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Neves I',1
ON THE STAB! RANDALL-SUNDRUM BRANEWGRLDS WITH CONFORMAL BULK FIELDS
Departamento de Física, Faeuiclacle cíe Ciencias e Tecnología, Universidade do Algarve Campus de Gambeias, 8000-
117 Faro, Portugal
11ntroduction
In the Randall-Sundrum (RS) scenario [1,2] the observable Universe is a 3-brane world of a Z, symmetric 5-dimensional anti-de Sitter (AdS) space. In the RSI model [1] the AdS orbifold has a compactified fifth dimension and two brane boundaries. The gravitational field is bound to the hidden positive tension brane and decays towards the visible negative tension brane. In this model the hierarchy problem is reformulated as an exponential hierarchy between the weak and Planck scales [1]. In the RS2 model [2] the AdS orbifold is non-compact with an infinite fifth dimension and a single positive tension brane. Gravity is localized on the positive tension brane now interpreted as the visible brane.
At low energies the theory of gravity on the observable brane is 4-dimensional general relativity and the cosmology may be Friedmann-Robertson-Walker [1]-[10]. In the RSI model this is only possible if the radion mode is stabilized and this as been achieved using a scalar field in the bulk [3,6,9,10]. The
gravitational collapse of matter was also analyzed in the RS scenario [11]-[16). Using an extended black string solution (first discussed in a different context by Myers and Perry [1?]) Chamblin, Hawking and Reall showed that it was possible to induce on the brane the Schwarzschild black hole metric [11]. However, this solution is divergent at the AdS horizon and at the black string singularity. Consequently, it is expected to be unstable [11,18]. A black cylinder localized near the brane which is free from naked singularities was conjectured to be its stable decay product. This exact 5-dimensional solution has not yet been found. The only known static black holes localized on a brane remain to be those found for a 2-brane in a 4-dimensional AdS space [12]. The problem lies in the simultaneous nonsingular localization of gravity and matter in the vicinity of the brane [11], [13]-[16], This has lead to another conjecture stating that D + l -dimensional black hole solutions localized on a D-1 -brane should correspond to quantum corrected D -dimensional black holes on the brane [15]. This is an extra motivation to look for 5-dimensional collapse solutions localized on
a brane. In addition, the effective covariant Gauss-Codazzi approach [19,20] has permitted the discovery of many braneworld solutions which have not yet been associated with exact 5-dimensional spacetimes [21]-[24],
In tills paper we continue the research on the dynamics of a spherically symmetric RS 3-brane when the bulk is filled with conformal matter fields [16,25] (see also [26]). In our previous work [16,25] we have found a new class of exact 5-dimensional dynamical solutions for which gravity is localized near the brane by the exponential RS warp. These solutions were shown to be associated with conformal bulk fields characterized by a stress-energy tensor f* of weight -4
and by the equation of state T“ - 2T55. 'They were also shown to describe on the brane the dynamics of inhomogeneous dust, generalized dark radiation and homogeneous polytropic matter. However, the density and pressures of the conformal bulk fluid increase with the coordinate of the fifth dimension. In the RS2 model this generates a divergence at the AdS horizon as in the Schwarzschild black string solution. In the RS 1 model this is not a difficulty because the space is cut before the AdS horizon is reached. However, the solutions must be stable under radion field perturbations. In this paper we analyze this problem using a saddle point expansion procedure based on the action [28,29]. In section 2 we present a brief review of the conformal «analysis performed on the 5-dimensional Einstein equations which leads to the set of braneworld backgrounds supported by the class of conformal bulk matter fields with weight -4. in section 3 we consider a dimensional reduction on the RS action and determine the radion effective potential. In section 4 we identify our braneworld solutions as its extrema and show that the exponential RS warp and a radion independent equation of state for the conformal bulk fields lead to unstable solutions. In section 5 we present our conclusions and a new class of stable braneworld solutions which describe on the brane the dynamics of inhomogeneous dust, generalized dark radiation and homogeneous polytropic matter.
2 Einstein Equations and Conformal Bulk Fields
Let (t, r,0,<|), z) be a set of comoving coordinates in the RS orbifold. The most general metric consistent with the Z2 symmetry in z and with 4-dimensional spherical symmetry on the brane may be written as
ds = Q (-e^dt1 + el“drl + RldQ,l2 + dzl ),
When the 5-dimensional space is filled with bulk matter fields characterized by a lagrangian LB the dynamical RS action is given by
S-
I
r[A,S{z ~zl}) + X'b(z - 4 )] + Lg
(2)
•\/<?55 j
The Planck brane with tension "A, is assumed to be located at z = z0 and the visible brane with tension X' at ^ • AB is the negative bulk cosmological constant, A Noether variation on the action (2) gives the Einstein field equations
¿V = -k£{A88* +
1 - 1 (3)
+ ¡u(z - 4)+rscz - z;)](5; - ) - t; ,
V 855 J
where the stress-energy tensor associated with the bulk fields is defined by
SL ,
ua *
■L&
sr
(4)
xnai
and is conserved in the bulk, =0 ,
To find exact solutions of Einstein equations (3) we ,s
assumptions. Let us first cons id cotiformal transformation the bulk
energy tensor has conformal weight s, 7,7 = fi Then Eq. (3) may be re-written as G; = ~6£i~‘ (¥tl£2)gvp¥pQ + 3£ry%¥(1Q -
-3Q~18^£poVp VCQ KjQ2 {A^' +
+ir1 mz - z0)+x'(z - z0 )](8;; - 858’) --qs+27;v).
Similarly under the conformal transformation the conservation equation becomes
(5)
Vjf + £T‘ [(s + UTfdfi - Tf 3VQ] = 0.
(6)
(1)
where the metric functions Q = Q (t,r,z),
A = A(t, r,z), B = B(t, r, z) and R = R(t, r, z) are Z2 symmetric. Q. is the waip factor and R the physical radius of the 2-spheres.
If in addition it is assumed that T* = QT2T* then equation (5) may be separated in the following way
g;=k*7;\ (?)
6Gf2V(l£WpiV'' -3Q-,V11VpQg|W +
+1CF1 vpv0i25 pas; = {A,8; + (8)
+cri m.z - z0)+x'8(z - z;)](s; - 858^) ].
Because of the Bianchi identity we must also have Vfl? =0, (9)
37l!‘d„Q~TdvQ-0. . (10)
Equations (7) and (9) are 5-dimensional Einstein equations with matter fields present in the bulk but without a brane or bulk cosmological constant. They do not depend on the conformal warp factor which is dynamically defined by equations (8) and (10). Consequently, only the warp reflects the existence of the brane or of the bulk cosmological constant. Note that this is only possible for the special class of conformal bulk fields which have a stress-energy tensor with weight -4.
Because the equations still depend non-linearly on the metric functions A, B and R let us further assume that A = A(t, r), B -- Bit, r), R = R(t, r) and fi = Q(z). Then we obtain
= vX=o, (ii)
Gj' = *:jT5s, (12)
6Qr (itQ'f = -k;Q2Ab, (13)
3£r*9iG = -k,Q2 {A „ +
5 / (14)
+Q [,l8(.s - Zq) + X8(z - z'})]}
and (see also [5] and [27])
Iff =TC, (15)
where the iatin indices represent the coordinates t,r,Q and <j). Our branewofid geometries [J 6,2.5] are solutions of equations (11)715) when the stress-tensor is diagonal,
?7 = diag (—p, pr,pT, pr, p,). (16)
where p, pr, p., and. denote the bulk density and pressures, and Q, is the exponen warp,
I
fiRSfe)=l
(17)
radion effective potential. For the calculation it is convinient to work with the coordinate y related to z
by z = ley" for y > 0 . Let us write the most general metric consistent with the Z, symmetry in y and with 4-dimensional spherical symmetry on the brane in the form
ds2 ~a2dsj +b1dy2, ds;=-dt2+e2Bdr2+R2dQl,
where the metric functions a = a(t, r, y), B = B(t. r, y),
R = R(t,r, >*) and b = b(t,r,y) are Z2 symmetric. Now a is the warp factor, R is still the physical radius of Che 2-spheres and b is related to the radion field.
The 5-dimensional dynamical RS action is now given by
r R
(19)
S = ld4xdy^g
[X6(>>) + X'5(y - 7irc.)] + LB
In the new coordinates the Planck brane is located at y - 0 and the visible brane at . Our braneworld backgrounds correspond to the metric functions h - 1, B = B(t,r), R = R(t,r) and a = QK(y) where
= (20) To calculate the radion potential we consider the dimensional reduction of the action (19). Using the
metric (18) we obtain *J~g = a'byj~^4 and
1 /' 6
R =
,,/v
a v
■f VV.a -
U-Zfll+Zo’
where zQ = I and I is the AdS radius given by / = l/y-AgK5 /6 with Kj = 8jtM53 defined by the fundamental 5-dimensional Planck mass Ms. Then the Planck brane is located at z0 = I and the observable brane is located at z'f, = lenr,: n where rc is the RS compactification scale [1]. The former has a positive tension X and the latter a negative tension X' ~ ~X where X - -ABl. They are twin Universes with identical collapse or cosmological dynamics.
3 The Radion Potential
To analyze the behaviour of these solutions under radion field perturbations we apply a saddle point expansion procedure based on the action [28,29]. As a starting point this requires the determination of the
ab
gfVccNdb I-
(21)
/ o N ora 2 ~j2 da
3 + 2-i—
v Cl J a
where gf is the inverse metric associated with the 4-dimensional line element ds2A, RA is the 4-dimensional Ricci scalar and the covariant derivatives are 4-dimensional. Consider the particular metric setting defined by a~£le^ and b = eil where Q = Q(y) and (J = P(i, r). Then in the Einstein, frame the dimensional reduction leads to
5= -V
2’c,'i,S4 ’J’ (22>
where y = p/K4\/2/3 is the canonically normalized radion field. The function V = V(y) is the radion potential and it may be written in the form
A
+Z|rf3i),4(AB-4)+ (23)
+f |<i>i24[X6(j) + r8(>'-Jirc)],
where the field % is defined as % = e~^ll3)Vi. Note that
the integration in the fifth dimension is performed in
the interval [-%rc,Tirc} and that we have chosen
jrfyQ2^. (24)
4 The Radion Reid Instability
To analyze the stability of our braneworld solutions we consider a saddle point expansion of the radion field potential V [28,29]. This procedure requires the determination of its first and second variations. To do so we consider the integral of the radion potential written in the form
V=jdAx^V( T) =
= J'dsxy[-g\-gx
3 nV »VQ _>_| + 2—i—
Q J 12
> +
(25)
+ fd'xJ-g {As - LB + X[^8()0 + ^'5(>■ - ii#; )j}.
Taking into account that an integration by parts and the Z2 symmetry leads to
=.... |,£iyil3a"Q(26)
we find that the first variation of the integral of the radion potential is given by
^ jdx5^Pg%[l&(y) + X'8(y - nrc)] -
5y H 3
+7l
7?-27?
12 2d2yQ
~TX ~
iq Q
For our solutions Q is taken to be the exponential RS warp factor Ors and satisfies the following warp equation in the y coordinate 6 32Q
Kj Q
- = AB +2[X,8(y) + A,'5(y-Jtrc)].
On the other hand the bulk matter fields which must have a stress-energy tensor of conformal weight -4 must also obey the equation of state (15). Integrating in the fifth dimension with Q = we find
f - = ^ Ag/(1 - ^ " )X(4z-l-3x2).
87 Vfo
The critical extrema of the radion potential are the nonzero finite roots of the polynomial equation
lOX1 -4% + l) = 0. They are & =1 and %, =1/3 .
Our braneworld solutions correspond to the first root X, = 1. The same happens if the bulk matter is absent as in the RS vaccum solutions. The other extremum is not a solution of the Einstein equations for the RSI model. Indeed, a rescaling of the coordinates shows that the point %2 =1/3 corresponds to an exponential
warp Q = which is different from Ors and
does not satisfy the RS warp equations (28) and
3/1 ' Q
: —KjA
(30)
The stability 'of the extrema depends on the sign of the second variation of the radion potential. This variation defines the radion mass. Consequently, stable background solutions must be associated with a positive sign. Consider the first variation of the 4-dimensional integral of V given in Eq. (27). if the equation of state (15) of the conformal bulk fields is independent of the radion perturbation and Q = f).RS we find
= -8x + D. (31)
For x-Xi = l we conclude that
V is negative. This implies
negative mass lachyen and that our braneworld solutions are unstable.
5 Conclusions
In this paper we have considered the set of exact 5-dimensional dynamical solutions with gravity localized near the brane which are associated with conformal bulk fields of weight -4 and which describe the
(27) dynamics of inhomogeneous dust, ■ generalized dark radiation and homogeneous polytropic matter on the brane. We have studied their behviour under radion field perturbations. We have shown that these solutions are extrema of the radion potential. We have also shown that if the metric function responsible for the localization of gravity is the exponential RS warp and the equation of state characterizing the conformal bulk fluid is independent of the radion field then the
(28) braneworld solutions are unstable. We have also found that the radion potential has another extremum which is not a solution of the complete set of Einstein equations. This point is connected with a different warp factor and its existence suggests that stable braneworlds associated with the same state of the conformal bulk matter should correspond to- warp functions other than
(29) the standard exponential RS warp. This is indeed true. Stable solutions are defined by a new set of warp functions given by
Q( y) = e
„-¡y\il
1 +
Ps J\y\n
4Aa
(32)
y
where pi <0 is a negative constant fraction of the 5-dimensional pressure of the conformal bulk fields of weight-4. On the brane these solutions also describe the dynamics of inhomogeneous dust, generalized dark radiation and homogeneous polytropic matter. More details will be presented in a forthcoming publication [30].
Acknowledgements
We are grateful for financial support from Fundagao para a Ciencia e a Tecnologia (FCT) and Fundo Social Europeu (FSE) under the contracts SFRH/BPD/7182/2001 and POCT1/32694/F1S/2QOQ (III Quadro Cornunidrio de Apoio) as well as from Centro Multidisciplinar de Astrofisica (CENTRA). We would also like to thank Cenalo Vaz for enlighting discussions and to the Physics Department of the University of Cincinnati for kind hospitality during a visit where part of this work was completed.
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