9
Anisotropic extra dimensions Eleftherios Papantonopoulos, Antonios Papazoglou, and Minas Tsoukalas Department of Physics, National Technical University of Athens, Zografou Campus GR 157 73, Athens, Greece (Received 22 May 2011; published 19 July 2011) We consider the scenario where, in a five-dimensional theory, the extra spatial dimension has different scaling than the other four dimensions. We find background maximally symmetric solutions, when the bulk is filled with a cosmological constant and at the same time it has a three-brane embedded in it. These background solutions are reminiscent of Randall-Sundrum warped metrics, with bulk curvature depending on the parameters of the breaking of diffeomorphism invariance. Subsequently, we consider the scalar perturbation sector of the theory and show that it has certain pathologies and the striking feature that in the limit where the diffeomorphism invariance is restored, there remain ghost scalar mode(s) in the spectrum. DOI: 10.1103/PhysRevD.84.025016 PACS numbers: 11.10.Kk I. INTRODUCTION Theories with extra spacetime dimensions were intro- duced in the 1920s by the work of Kaluza and Klein (KK) and gained further attention with the advance of string theory, where they were a crucial ingredient for the self- consistency of the theory. During the last decade, the interest in model building and phenomenology of extra dimensional theories has been renewed with the discovery that they can play an important role in physics of energies/ distances that are probed in current experiments and obser- vations [1]. There have been a wealth of models where extra dimensions open up at the electroweak scale and provide new insights of standard high energy problems [2], or at macroscopic scales and modify gravity in the infrared [3]. A common factor of these models is that the starting point in model building is a higher-dimensional diffeomorphism invariant theory. The background solutions of the metric, i.e. the gravitational vacuum of the theory, spontaneously breaks the higher-dimensional diffeomorphism group down to its four-dimensional subgroup. This constitutes a gravi- tational Higgs mechanism and provides masses for the towers of KK gravitons, vectors, and scalars [46]. Recently, Hor ˇava proposed a theory of gravity which also breaks diffeomorphism invariance [7]. The new theory is supposed to be an adequate ultraviolet (UV) completion of gneral relativity above the Planck scale. Its basic as- sumption is the existence of a preferred foliation by three- dimensional constant time hypersurfaces, which splits spacetime into space and time. This allows the addition ofhigher order spatial derivatives of the metric to the action, without introducing higher order time derivatives. This is supposed to improve the UV behavior of the graviton propagator and render the theory power-counting renormalizable without introducing ghost modes, which are common when adding higher order curvature invariants to the action in a covariant manner [8]. Such a theory cannot be invariant under the full set of diffeomorphisms, but it can still be invariant under the more limiting foliation-preserving diffeomorphisms. This breaking of the full diffeomorphisms group down to its foliation-preserving subgroup is, however, explicit and not spontaneous. This has been the source of problems, related with a scalar degree of freedom, whose behavior creates a serious viability issue for all versions of the theory. In the version with ‘‘projectability’’ and for maximally symmetric backgrounds, the scalar is either (classically) unstable or it becomes a ghost (quantum- mechanically unstable) [912]. 1 Also the nonprojectable version of the theory has strong coupling problems and instabilities [1416]. In the present paper, we will revisit the extra dimen- sional theories and break explicitly, i.e. at the action level, the higher-dimensional diffeomorphism invariance to its foliation-preserving subgroup. The foliation that we will choose is adapted to an extra space dimension, as opposed to the Hor ˇava-type of theories where it was adapted to the time dimension. This will leave intact the four-dimensional spacetime diffeomorphisms contrary to Hor ˇava theory. It is evident that in the resulting theory we cannot demand the theory to be renormalisable, as it was the motivation in the Hor ˇava theory, since that would indicate the inclusion of higher order four-dimensional curvature invariants, thus introducing ghosts. On the other hand, depending on the difference in scaling of the extra dimension compared with the scaling of the four ordinary spacetime dimensions, higher powers of the extrinsic curvature of the four- dimensional hypersurfaces may play a part in the effective theory, before reaching energies where inevitably ghost- bearing higher order dimensional operators appear. The aim of this paper is to study such a theory where diffeomorphism invariance in an extra dimensional model is explicitly broken. We will pursue this aim in a theory, which in the limit of diffeomorphism invariance restoration tends to the Randall-Sundrum warped model with one brane and infinite extra dimension. We will formulate the theory, find background solutions, and study the scalar sector perturbations around this background. This study 1 See, however, [13] and references therein for conditions under which the classical instability does not show up. PHYSICAL REVIEW D 84, 025016 (2011) 1550-7998= 2011=84(2)=025016(9) 025016-1 Ó 2011 American Physical Society

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Page 1: Anisotropic extra dimensions

Anisotropic extra dimensions

Eleftherios Papantonopoulos, Antonios Papazoglou, and Minas Tsoukalas

Department of Physics, National Technical University of Athens, Zografou Campus GR 157 73, Athens, Greece(Received 22 May 2011; published 19 July 2011)

We consider the scenario where, in a five-dimensional theory, the extra spatial dimension has different

scaling than the other four dimensions. We find background maximally symmetric solutions, when the

bulk is filled with a cosmological constant and at the same time it has a three-brane embedded in it. These

background solutions are reminiscent of Randall-Sundrum warped metrics, with bulk curvature depending

on the parameters of the breaking of diffeomorphism invariance. Subsequently, we consider the scalar

perturbation sector of the theory and show that it has certain pathologies and the striking feature that in the

limit where the diffeomorphism invariance is restored, there remain ghost scalar mode(s) in the spectrum.

DOI: 10.1103/PhysRevD.84.025016 PACS numbers: 11.10.Kk

I. INTRODUCTION

Theories with extra spacetime dimensions were intro-duced in the 1920s by the work of Kaluza and Klein (KK)and gained further attention with the advance of stringtheory, where they were a crucial ingredient for the self-consistency of the theory. During the last decade, theinterest in model building and phenomenology of extradimensional theories has been renewed with the discoverythat they can play an important role in physics of energies/distances that are probed in current experiments and obser-vations [1]. There have been awealth ofmodels where extradimensions open up at the electroweak scale and providenew insights of standard high energy problems [2], or atmacroscopic scales and modify gravity in the infrared [3].

A common factor of thesemodels is that the starting pointin model building is a higher-dimensional diffeomorphisminvariant theory. The background solutions of the metric,i.e. the gravitational vacuum of the theory, spontaneouslybreaks the higher-dimensional diffeomorphism group downto its four-dimensional subgroup. This constitutes a gravi-tational Higgs mechanism and provides masses for thetowers of KK gravitons, vectors, and scalars [4–6].

Recently, Horava proposed a theory of gravity whichalso breaks diffeomorphism invariance [7]. The new theoryis supposed to be an adequate ultraviolet (UV) completionof gneral relativity above the Planck scale. Its basic as-sumption is the existence of a preferred foliation by three-dimensional constant time hypersurfaces, which splitsspacetime into space and time. This allows the additionofhigher order spatial derivatives of the metric to theaction, without introducing higher order time derivatives.This is supposed to improve the UV behavior of thegraviton propagator and render the theory power-countingrenormalizable without introducing ghost modes, whichare common when adding higher order curvature invariantsto the action in a covariant manner [8].

Such a theory cannot be invariant under the full set ofdiffeomorphisms, but it can still be invariant under themore limiting foliation-preserving diffeomorphisms. Thisbreaking of the full diffeomorphisms group down to its

foliation-preserving subgroup is, however, explicit andnot spontaneous. This has been the source of problems,related with a scalar degree of freedom, whose behaviorcreates a serious viability issue for all versions of thetheory. In the version with ‘‘projectability’’ and formaximally symmetric backgrounds, the scalar is either(classically) unstable or it becomes a ghost (quantum-mechanically unstable) [9–12].1 Also the nonprojectableversion of the theory has strong coupling problems andinstabilities [14–16].In the present paper, we will revisit the extra dimen-

sional theories and break explicitly, i.e. at the action level,the higher-dimensional diffeomorphism invariance to itsfoliation-preserving subgroup. The foliation that we willchoose is adapted to an extra space dimension, as opposedto the Horava-type of theories where it was adapted to thetime dimension. This will leave intact the four-dimensionalspacetime diffeomorphisms contrary to Horava theory. It isevident that in the resulting theory we cannot demand thetheory to be renormalisable, as it was the motivation in theHorava theory, since that would indicate the inclusion ofhigher order four-dimensional curvature invariants, thusintroducing ghosts. On the other hand, depending on thedifference in scaling of the extra dimension compared withthe scaling of the four ordinary spacetime dimensions,higher powers of the extrinsic curvature of the four-dimensional hypersurfaces may play a part in the effectivetheory, before reaching energies where inevitably ghost-bearing higher order dimensional operators appear.The aim of this paper is to study such a theory where

diffeomorphism invariance in an extra dimensional modelis explicitly broken. We will pursue this aim in a theory,which in the limit of diffeomorphism invariance restorationtends to the Randall-Sundrum warped model with onebrane and infinite extra dimension. We will formulate thetheory, find background solutions, and study the scalarsector perturbations around this background. This study

1See, however, [13] and references therein for conditionsunder which the classical instability does not show up.

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will reveal that such an explicit breaking of the five-dimensional diffeomorphism invariance is dangerous andintroduces ghost scalar mode(s).

The pathological behavior of gravity at small distancesis a problem that plunges many gravity theories. In thedirection of proposing a UV-completion theory of generalrelativity, higher order curvature terms were introduced.The so-called modified gravity theory includes fðRÞ grav-ity models [17] and the Gauss-Bonnet (GB) models [18].The fðRÞmodels have a better behavior in the UV but theyhave serious cosmological problems [19] (see, however,[20]). For GB models it was shown that tensor perturba-tions are typically plagued by instabilities in the UV.

Similar problems arise by considering modifications ofgravity at large distances. The Dvali-Gabadadze-Porratibraneworld model [3] and its extension [21] with a GBterm in the bulk was proposed as an alternative to theacceleration of theUniversewithout the need of dark energy[22,23]. However, it was soon realized that the Dvali-Gabadadze-Porrati model contains a ghost mode [24],which casts doubts on the viability of the self-acceleratingsolution. Other theories that modify gravity at large dis-tances were proposed by introducing a mass term to thegraviton via a potential term. This was explored many yearsago by Fierz and Pauli [25]. Later it was realized thatgravitational theories with a mass term are behaving quiteunlike other theories and are typically accompanied byseveral problems, most notably ghost/tachyon instabilitiesand strong coupling issues. Also massive gravity theorieshave a very characteristic feature: the Van Dam-Veltman-Zakharov (vDVZ) discontinuity [26,27], i.e. the fact that asthe mass of the graviton goes to zero the scalar gravitonmode fails to decouple (see, however, a way out in curvedspace [28,29]). To cure this problem one has to advocatenonlinear dynamics, employing the Vainshtein mechanism[30], but then a ghost appears in the spectrum first noticedby Boulware and Deser [31].

A similar behavior like the vDVZ discontinuity in mas-sive gravity is observed in our theory: although the back-ground solutions tend to the Randall-Sundrum solutions asthe diffeomorphism invariance is restored, the problematicscalar modes are not removed from the spectrum in thesame limit. We attribute this behavior to the explicit break-ing of the diffeomorphism invariance in the theory.

II. BREAKING THE DIFFEOMORPHISMINVARIANCE

Our starting point is five-dimensional Einstein gravity inthe presence of a cosmological constant and a brane em-bedded at some point y ¼ 0 of the extra dimension. Weassume Z2 symmetry along the y direction with y ¼ 0 asthe fixed point. The corresponding action is given by

S ¼Z

d5xðRð5Þ � 2�5Þffiffiffiffiffiffiffiffi�G

p �Zbrane

d4xffiffiffiffiffiffiffi�g

p�; (2.1)

where Rð5Þ is the five-dimensional Einstein-Hilbert termand �5, �, and GMN is the five-dimensional cosmologicalconstant, the brane tension ,and the metric, respectively. Inthe following we wish to modify the above theory in a waythat diffeomorphism invariance is broken along the extradimension.First, let us make the following Arnowitt-Deser-Misner

(ADM) splitting along the extra dimension y,

ds2¼dy2N2c2þg��ðN�dyþdx�ÞðN�dyþdx�Þ; (2.2)

with g�� the four-dimensional metric. In the modification

that we will discuss, we shall adopt an anisotropic scalingof the different dimensions with ½x�� ¼ �1 and ½y� ¼�w, where the extra dimension singles out. For such atheory, the ADM metric components scale as ½g��� ¼ 0,

½N� ¼ 0 and ½N�� ¼ w� 1. The five-dimensional action(2.1) can be generalized as

S¼Zd4xdyN

ffiffiffiffiffiffiffi�gp �

2ðRð4Þ�2�5Þ� 2

�2ðK��K

����K2�

�Zbrane

d4xffiffiffiffiffiffiffi�g

p�; (2.3)

where the extrinsic curvature K�� is given by ½K��� ¼ w,

with

K�� ¼ 1

2Nð@yg�� �r�N� �r�N�Þ; (2.4)

and the scaling of the various quantities are ½�� ¼ wþ 2,

½�� ¼ w�42 , ½Rð4Þ� ¼ ½�5� ¼ 2. The above action receives

radiative corrections and can be extended by includinghigher-dimensional operators. Depending on the value ofw, there can be higher powers of the extrinsic curvatureimportant at energies lower than the one that unitarity ofthe theory is lost (typically when R2 terms dominate). Wewill discuss that possibility in a later section. For themoment, we will restrict ourselves to the classical gener-alized action (2.3).The parameter � is crucial for the following discussion.

It represents the breaking of the five-dimensional diffeo-morphism invariance to its four-dimensional subgroup.The restricted symmetry allows for a different weightingof the K2

�� and K2 terms in the action, contrary to the fully

covariant theory.The field equations coming from variation of (2.3) with

respect to the fields N, N�, and g�� are given by

0 ¼ 1ffiffiffiffiffiffiffi�gp �S

�N¼ �

2ðRð4Þ � 2�5Þ þ 2

�2ðK��K

�� � �K2Þ;

(2.5)

0 ¼ 1ffiffiffiffiffiffiffi�gp �S

�N�

¼ � 4

�2r��

��; (2.6)

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0 ¼ 1ffiffiffiffiffiffiffi�gp �S

�g��

¼ ��

2ðG��

ð4Þ þ g���5ÞN þ 2

�2½@y��� þ NK���

þ 2r�ð��ð�N�ÞÞ � N�r���� þ 2NK������

� 1

2g�� 2

�2ðK��K

�� � �K2Þ � 1

2��ðyÞg��; (2.7)

where the y-canonical momentum of the four-dimensionalmetric is

��� ¼ K�� � �Kg��: (2.8)

As in the original case of Randall-Sundrum, it is impor-tant to consider the junction conditions of the system.These junction conditions involvng the brane tension arefound by identifying the distributional terms in the equa-tions of motion. It is fairly easy to see from (2.7) that theonly distributional term is @y�

��. Integrating along the

extra dimension and taking the limit close to the branegives the junction conditions, which takes the followingfamiliar form:

2

�2½����þ� ¼ 1

2g���: (2.9)

Under the Z2 symmetry the above relation can be rewrittenas

4

�2��� ¼ 1

2g���: (2.10)

In the following section, we will present backgroundsolutions for this theory that possess maximal symmetry infour dimensions. We will see that these solutions bear greatsimilarity with the Randall-Sundrum solutions [2] and theircurved version [32], with �-dependent bulk curvatures.

III. SOLUTIONS WITH MAXIMALLYSYMMETRIC BACKGROUNDS

In this section we will consider solutions of the fieldequations where the four-dimensional metric g�� is maxi-

mally symmetric. This metric evaluated at the brane posi-tion is also the induced metric on the brane at y ¼ 0, sincethe brane is static. Therefore, the solutions for the bulkmetric, satisfying the appropriate boundary conditions,define families of maximally symmetric brane solutions.

A. Flat brane

Let us first examine the case where g�� is flat. In this

case we will consider the following ansatz for the metriccomponents:

N� ¼ 0; g�� ¼ e2fðyÞ��; N ¼ 1; (3.1)

allowing for a warp factor fðyÞ along the extra dimension.Then, Eq. (2.5) becomes

��5�þ 8ð�1þ 4�Þðf0ðyÞÞ2�

¼ 0; (3.2)

and Eq. (2.6) is automatically satisfied while Eq. (2.7)yields

�2�5�þ �2��ðyÞ þ 8ð�1þ 4�Þðf0ðyÞÞ2þ 4ð�1þ 4�Þf00ðyÞ ¼ 0: (3.3)

Depending on the value of the parameter � we distin-guish the three following cases of solutions that we need toexamine.

1. Case 1 ð� < 14Þ

In this case, (3.2) gives

fðyÞ ¼ � jyj�2

ffiffiffi2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�5�

1� 4�

s: (3.4)

Clearly, ��5 must be positive. Substituting the abovesolution to (3.3) we see that it is trivially satisfied.

2. Case 2 ð� > 14Þ

In this case, we have the same form of solution, but now��5 is negative. Again, Eq. (3.3) is satisfied. For � ¼ 1 weget a Randall-Sundrum-like solution [2].

3. Case 3 ð� ¼ 14Þ

For this case, we get that �5 ¼ 0 and fðyÞ is arbitrary.This is a critical point of the theory, where the bulk theoryis conformally invariant.Substituting g�� in (2.10) we get

�2�þ 8ð�1þ 4�Þf0ðyÞ ¼ 0: (3.5)

This equation can also be reproduced by integrating thedistributional parts of (3.3). Substituting fðyÞ gives thefollowing expression for the tension

� ¼ �2ffiffiffi2

p ð1� 4�Þ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�5�

1� 4�

s: (3.6)

Note that in the case, where the bulk is conformallyinvariant, the brane is tensionless. Furthermore, in the limitwhere � ! 1, we get the familiar result of a positivetension brane.

B. Curved brane

Instead of using a flat g��, we can introduce a metric of

constant nonzero curvature. Namely, we will have

ds2ð5Þ ¼ ds2ð4Þ þ dy2; (3.7)

where

ds2ð4Þ ¼ ðyÞ2ð�dt2 þ e2Ht�ijdxidxjÞ (3.8)

is a conformally flat, Einstein space of constant nonzerocurvature [33]. Again, Eq. (2.6) is automatically satisfied.

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For the other two Eqs. (2.5) and (2.7), we have, respec-tively,

��5�þ 1

�2ðyÞ2 ð6H2�2�þð8�32�Þð0ðyÞÞ2Þ¼0; (3.9)

and

3H2�

2ðyÞ4 ��5�

2ðyÞ2 ��

2ðyÞ2 �ðyÞ

þ 2ð1� 4�Þ�2ðyÞ4 ðð0ðyÞÞ2 þ ðyÞ00ðyÞÞ ¼ 0: (3.10)

Solving (3.9) for H2 and substituting in (3.10) we obtain

�2ðyÞð�5�þ2��ðyÞÞþ8ð�1þ4�Þ00ðyÞ¼0: (3.11)

This is an equation for the warp factor aðyÞ and its solutionwill depend on the diffeomorphism breaking parameter �.It is useful to see that Eqs. (3.9) and (3.11) in the limitwhere � ! 1 match with equations (6), (7) of [33] with�2 ! 1

M3 , � ! 4M3, and �5 ! �5=4M3.

Taking into account (2.10), we obtain as before threecases:

1. Case 1 (� < 14 )

In the case where � < 14 , the solution is

ðyÞ ¼ cosðmyÞ � �m

��5

sinðmjyjÞ; (3.12)

with

m2 ¼ � �2�

8j1� 4�j�5; (3.13)

�5 is negative and

H2 ¼ 1

48

�8�5 � �2�2

�ð1� 4�Þ�¼ 1

6�5�2ð�2m2 þ�2

5�2Þ:

(3.14)

2. Case 2 (� ¼ 14 )

In the conformal point�5 ¼ 0,H2 ¼ 0, and the brane istensionless (� ¼ 0).

3. Case 1 (� > 14 )

Finally, for the case where � > 14 , we have the following

solution:

ðyÞ ¼ coshðmyÞ þ �m

��5

sinhðmjyjÞ; (3.15)

with

m2 ¼ � �2�

8j1� 4�j�5: (3.16)

In the above relation, �5 is negative and furthermore

H2 ¼ 4

3

j1� 4�j�3�2

m2

�25

ð�2m2 ��25�

2Þ: (3.17)

Summarizing, the jH2j factor for the three cases is given by

jH2j

¼

8>>>>>>><>>>>>>>:

43j1�4�j�3�2

m2

�25

ð�2m2��25�

2Þ; j�5jm <�

� for dS4 branes;

0; j�5jm ¼ �

� for flat branes;

43j1�4�j�3�2

m2

�25

ð�25�

2��2m2Þ; j�5jm >�

� for AdS4 branes:

(3.18)

For positive values of �5 we have the same solutions butfor the opposite domains of � and with positive sign form2

in (3.16). The solutions we found are similar to the solu-tions discussed in [32] for curved backgrounds.

IV. HIGHER EXTRINSICCURVATURE OPERATORS

The action (2.3) receives naturally radiative correctionsin the form of higher-dimensional operators. The dimen-sion of these operators depends onw if they involveK��, or

arew-independent if they involve only Rð4Þ. It is reasonableto consider the theory only at energies that ðRð4ÞÞ2 terms aresubdominant, since these are bound to introduce ghosts. Forsuch energies, depending on the scaling w of the extradimension, one could add higher powers of the extrinsic

curvature in the action. Since ½ðRð4ÞÞ2� ¼ 4 and ½Kij� ¼ w,

we need to consider these powers of K that fulfill nw < 4.As an example, for n ¼ 3 we obtain that a value of w< 4

3

allows for cubic powers of the extrinsic curvature to beimportant at the energy region where the theory is stillunitary.Under these assumptions we are led to expand (2.3) by

introducing the following terms:

�S ¼Z

dydx4ffiffiffiffiffiffiffi�g

pN

�1

�4

1

�ðK3 þ �K��K

��K

þ K��K��K

�� �; (4.1)

where as before ½�� ¼ w�42 . The coupling � scales as ½�� ¼

4 and ;�; are dimensionless constants. The aboveaction is split into three pieces �S ¼ �Sþ�S� þ �S , which we are going to vary separately.

The variations of the above terms with respect to g��

give

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1ffiffiffiffiffiffiffi�gp ��S

�g��

¼ N1

�4

1

�½Nð�K3g�� � 3K2K��Þ � 3

2@yðK2g��Þ � 3rðK2gð�N�ÞÞ þ 3

2NrðK2g��Þ�; (4.2)

1ffiffiffiffiffiffiffi�gp ��S�

�g��

¼ 1

�4

1

��

��Nð2KK��K�

� þ K�K�K�� þ K2K��Þ � ð@yðKK��Þ þ 1

2@yðK�K�g

��Þ þ 2rðKð�N�ÞKÞ

þ rðgað�N�ÞK��K��Þ � NrðKK��Þ � 1

2NrðK�lK�lg

��ÞÞ�; (4.3)

1ffiffiffiffiffiffiffi�gp ��S

�g��

¼ 1

�4

1

�N

�1

2g��K��K

��K�� � 3K�

�K��K�� � 3

2KK�

�K��

��@y

�3

2K�

�K��

�þ 3rðK

�Kð��N�ÞÞ � Nr

�3

2K�

�K��

���: (4.4)

The variations with respect to N� give

1ffiffiffiffiffiffiffi�gp ��S

�N�

¼ 1

�4

1

�ð3r�ðK2g��ÞÞ; (4.5)

1ffiffiffiffiffiffiffi�gp ��S�

�N�

¼ 1

�4

1

��ðr�ð2K��K þ K��K��g

��ÞÞ; (4.6)

1ffiffiffiffiffiffiffi�gp ��S

�N�

¼ 1

�4

1

� ðr�ð3K�

�K��ÞÞ: (4.7)

Finally, the variation with respect to N is

1ffiffiffiffiffiffiffi�gp ��S

�N¼ 1

�4

1

�ð�2ÞðK3 þ �K��K

��K

þ K��K��K�

�Þ: (4.8)

As before we need to examine the junction conditionsacross the brane. Integrating (4.11) and focusing on thedistributional part, we see that the junction condition reads�2

�2���þ 1

�4

1

��

3

2K2g����

�KK��þ1

2K��K��g

��

� 3

2K�

�K��

��þ�¼1

2�g��: (4.9)

Taking into account the above variations and comparingwith the equations of motion of the quadratic terms of theextrinsic curvature, we see that in the case where ¼�4�� 16 we get exactly the same results both for flatand curved branes. This is because in this limit (4.1)vanishes. In the case of flat branes, even deviating fromthis limit, we obtain solutions of the same form withredefined constants.

A. Flat branes

For the case of flat branes we consider the ansatz (3.1).Then (2.5) along side with (4.8) becomes

���5þ8ð1�4�Þðf0ðyÞÞ2�2

�8ð16þ4�þ Þ��4

ðf0ðyÞÞ3¼0:

(4.10)

Equation (2.6) together with Eqs. (4.5), (4.6), and (4.7) aresatisfied identically. Equation (3.3) alongside withEqs. (4.2), (4.3), and (4.4) give the following expression:

��

2g���5Nþ 2

�2½@y���þNK���þ2NK����

��

�1

2g��

�2

�2ðK��K

����K2��1

2�g���ðyÞ

þ 1

�4

1

��K3g���3K2K���3

2@yðK2g��Þ

þ�

��2KK��K�

��K��K��K���@yðKK��Þ

�1

2@yðK��K��g

��Þ�þ

�1

2g��K��K

��K��

�3K��K��K�

��3

2KK�

�K���3

2@yðK�

�K��Þ

��¼0;

(4.11)

from which we get

��4�5�þ8ðf0ðyÞÞ2ð��2ð�1þ4�Þþð16þ4�þ Þf0ðyÞÞþ��4��ðyÞþ2ð2��2ð�1þ4�Þþ3ð16þ4�þ Þf0ðyÞÞf00ðyÞ¼0: (4.12)

Depending on the value of the expression ð16þ 4�þ Þwe can distinguish the following cases:

1. Case 1 ð16þ 4�þ Þ ¼ 0

It is clear that in the case where the dimensionlessconstants satisfy ð16þ 4�þ Þ ¼ 0, we end with thesame results as in the case of the K2 terms.

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2. Case 2 ð16þ 4�þ Þ � 0

When the above constraint is relaxed, solving (4.10) forðf0ðyÞÞ3 and substituting it to (4.12) we get

2ð2��2ð�1þ 4�Þ þ 3ð16þ 4�þ Þf0ðyÞÞf00ðyÞ ¼ 0:

(4.13)

Now if we have that

2��2ð�1þ 4�Þ þ 3ð16þ 4�þ Þf0ðyÞ ¼ 0; (4.14)

then

fðyÞ ¼ 2jyj��2ð1� 4�Þ3ð16þ 4�þ Þ ; (4.15)

and

�5 ¼ � 32�2�2ð�1þ 4�Þ327ð16þ 4�þ Þ2� : (4.16)

We note here that, contrary to the K2 terms, the behaviorof the warp factor fðyÞ, if it is growing or decaying,depends on the value of ð1� 4�Þ=ð16þ 4�þ Þ.Furthermore, in this case of the K3 terms the sign of �5,depends on the value of �.

If on the other hand f00ðyÞ ¼ 0, i.e.,

fðyÞ ¼ Ayþ B; (4.17)

we have that

�5¼�8A2ð��2ð�1þ4�Þþð16þ4�þ ÞAÞ��4�

: (4.18)

At the conformal point where � ! 1=4, we have thatf0ðyÞf00ðyÞ ¼ 0, so either fðyÞ ¼ const and �5 ¼ 0, orfðyÞ ¼ Ayþ B and �5 ¼ �8ð16þ 4�þ Þ=��4�.Again we see that the behavior of the warp factor and thecosmological constant depend on the choice of parameters.

Applying in the above the junction conditions (4.9), weget

��4��ðyÞ þ 4��2ð�1þ 4�Þ½f0ðyÞ�þ�þ 6ð16þ 4�þ Þ½ðf0ðyÞÞ2�þ� ¼ 0: (4.19)

Because of the Z2 symmetry, the quadratic term vanishes,resulting in the same junction equations as in the case ofK2

terms. Substituting the solution for fðyÞ [(4.15)] we get

� ¼ 16

3�

ð1� 4�Þ2ð16þ 4�þ Þ ; (4.20)

and for the solution (4.17)

� ¼ � 8

�2ð�1þ 4�ÞA: (4.21)

Again in the conformal limit the brane is tensionless. Inorder to get significant changes to the junction conditionswe have to move to K4 terms, since these terms willproduce terms ðf0ðyÞÞ3 which are even.

V. SCALAR PERTURBATIONS

In this section we study the scalar sector of perturbationsof the theory that has up to quadratic terms of the extrinsiccurvature. For that purpose, wewill use the flat vacua of thetheory, analyzed inSec. III A. We consider the followingmetric ansatz:2

N¼eðx�;yÞ; N�¼@��ðx�;yÞ; g��¼e2ðfðyÞþ�ðx�;yÞÞ��;

(5.1)

which differs from the most general scalar perturbationpossibly by a perturbation of g�� of the form 2@�@�E,

which, however, can be gauged away (see [15]).Using the above ansatz, we compute in the Appendix the

various invariants appearing in the action. Inserting themback in the action and keeping terms up to quadratic orderin perturbations, we obtain the following quadratic bulkaction:

S ¼Z

dx4dye2f��ð3

�ð@�Þ2 � hð4Þ�Þ � e2f

�1þ 4� þ þ 4� þ 2

2þ 8�2

��5

� 2

�2ð4e2fð1� 4�Þðð@yfÞ2 þ 2@yf@y� þ ð@y�Þ2 � ð@yfÞ2 � 2@yf@y� þ 2

2ð@yfÞ2

þ 4�ð@yfÞ2 þ 8�@yf@y� � 4�ð@yfÞ2 þ 8�2ð@yfÞ2Þ þ 2ð1� 4�Þ@yfhð4Þ�� 4ð1� 4�Þ@yf�hð4Þ�

� 2ð1� 4�Þ@y�hð4Þ�� 4ð1� 4�Þ@yf@��@��þ e�2fð1� �Þðhð4Þ�Þ2Þ�: (5.2)

This action has one nondynamical degree of freedom,. Varying the action with respect to produces a constraint, to beimposed to the system, which reads

3�hð4Þ�þ 4

�2ð1�4�Þ@yfhð4Þ�¼ 8

�2e2fð1�4�Þðð@yfÞ2þ2@yf@y��ð@yfÞ2þ4�ð@yfÞ2Þ�e2f��5ð1þ4�þÞ: (5.3)

2Note that the lapse function N depends also on the four-dimensional spacetime coordinates. Therefore, in general, terms dependingon derivatives of N are allowed in the action [14]. This would significantly increase the number of terms in the action, and to keep theanalysis tractable we assume their couplings to be small so they can be ignored.

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Using Eq. (5.3) we can eliminate in favor of the � and � in the action (5.2) and then we obtain

S ¼Z

dx4dye2f�24@yfð@y�Þhð4Þ�

ð�1þ 4�Þ�5�

2þ 16@yf@y�ð�1þ 4�Þe2f ð1þ 4�Þ

�2þ 3�ð@�Þ2

þ 9�

4�5

e�2fðhð4Þ�Þ2 � 3

2�2e�2fðhð4Þ�Þ2 � 6ð@yfÞe�2f ð�1þ 4�Þ

�2�5

hð4Þ�hð4Þ� � 2e2fð1þ 4� þ 8�2Þ�5�

�Z

dx4dye2fð1þ 4� þ 8�2Þe2f��ðyÞ: (5.4)

The last term in the above action is the brane boundaryterm appearing in (2.3). The above action, as it is explainedin the Appentix, after appropriate partial integrations(while assuming appropriate boundary conditions), canbe brought to the final form

S ¼Z

dx4dye2f�9�

4�5

e�2fðhð4Þ�Þ2 � 3

2�2e�2fðhð4Þ�Þ2

� 6ð@yfÞe�2f ð�1þ 4�Þ�2�5

hð4Þ�hð4Þ�

þ 6ffiffiffi2

p�

ð1� 4�Þ�5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�5�

1� 4�

sð@�Þ2�ðyÞ

�: (5.5)

This is the main result of our work. The terms in the firstline are higher derivative terms for the two dynamicaldegrees of freedom � and �, which have kinetic mixingbetween them. As a check of the correctness of the aboveaction, one can vary it with respect to �. Then we obtainthe same result with the variation of (5.2) that reads

ð1� �Þe�2fhð4Þ� ¼ ð1� 4�Þð@y� � @yfÞ: (5.6)

The equivalence can be seen by substituting (5.3) in (5.6).Similarly, the variation with respect to � gives the sameresult once the constraint is taken into account.

The action (5.5) has certain characteristics. First, all thebulk terms involve four derivatives of brane coordinates,and certain terms have four time derivatives. Second, thebrane term is a ghostlike kinetic term for � . Once the wavefunctions for the two dynamical modes have been substi-tuted, the four-dimensional action will consisted of theghost kinetic term for � , plus higher derivative terms.These terms will appear in the action multiplied withdifferent scales. However, whatever the hierarchy of thesescales may be, there will always be some ghost problem inthe spectrum: either from the quadratic kinetic term, orfrom the higher derivative terms.

These modes will be present even after the restoration ofthe five-dimensional diffeomorphism symmetry, i.e. when� ! 1. This last characteristic is very similar to whathappens in massive gravity, namely, the vDVZ discontinu-ity [26,27]. In that case, similarly the longitudinal mode ofthe massive graviton does not decouple in the limit of avanishing Pauli-Fierz mass. In our example, we have aneven worse result, since the remaining modes have ghostbehavior. These problems are probably due to the explicit

breaking of the diffeomorphism invariance in the theory.The theory may not appear problematic at the level ofbackground solutions, but nevertheless these problemsreveal themselves once the theory is perturbed.

VI. CONCLUSIONS

We investigated the consequences of an assumption thatthe fifth extra dimension scales differently than the otherfour dimensions. To achieve this we considered a five-dimensional theory with a cosmological constant and athree-brane embedded in it. In this theory, the full five-dimensional diffeomorphism group is explicitly brokendown to its foliation-preserving four-dimensional sub-group. The foliation we used involves an extra spacedimension and therefore the four-dimensional Lorentz in-variance is intact. Because of the different scaling of theextra spacial dimension, higher powers of the extrinsiccurvature of the four-dimensional hypersurfaces are al-lowed in the theory up to the energies that ghost-bearinghigher order dimensional operators appear.We made a systematic study of the local solutions of this

theory. For maximally symmetric backgrounds and up tosecond order in extrinsic curvature, we found all solutionsfor flat and curved branes. These solutions are similar tothe previously obtained solutions for flat and curvedbranes, they are, however, characterized by a parameter �that expresses the breaking of the five-dimensional diffeo-morphism invariance. We also obtained solutions in the flatbrane limit by including higher order in extrinsic curvatureterms. These solutions, in addition to their dependence on�, also depend on the coefficients by which the higherorder extrinsic curvature terms enter in the action.Having explicitly broken the Lorentz invariance in five

dimensions, we looked for possible effects on the four-dimensional spacetime. We performed a scalar perturba-tions analysis of the theory for up to quadratic terms in theextrinsic curvature. We found that, as a result of the break-ing of the diffeomorphism invariance, two dynamical sca-lar degrees of freedom appear. Both of them arecharacterized by the fact that they come as higher deriva-tive corrections to the action. Moreover, one of thesemodes appears with a ghostlike brane kinetic term. Thesepathologies can be attributed to the explicit breaking ofdiffeomrphism invariance along the extra space dimension.

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ACKNOWLEDGMENTS

M.T. would like to thank the Institute of Cosmology andGravitation of Portsmouth for kind hospitality during thefirst stages of the work. The authors would like to thankAntonio Padilla for useful discussions.

APPENDIX: TECHNICAL DETAILS FORSCALAR PERTURBATIONS

In this Appendix we give the technical details for deriv-ing the action (5.5) in Sec. V. We start with the generalizedaction (2.3) ignoring for the moment the boundary braneterm

S¼ZN

ffiffiffiffiffiffiffi�gp

dydx4��

2ðRð4Þ �2�5Þ� 2

�2ðK��K

����K2�:

(A1)

We perturb the above action, in the flat brane limit dis-cussed in Sec. III A, using the perturbation ansatz (5.1) forthe scalar perturbations. The extrinsic curvature and itstrace is given by

K�� ¼ e�4ðfþ�þ4Þ½e2ðfþ�Þ@yðfþ �Þ�� � @�@��

þ @��@��þ @��@��� ��@��@���; (A2)

K ¼ e�2ðfþ�þ2Þ½4e2ðfþ�Þ@yðfþ �Þ �hð4Þ�� 2@��@���;

(A3)

where hð4Þ ¼ ��@�@�, while the Ricci scalar is

R ¼ �6e�2ðfþ�Þðhð4Þ� þ ð@�Þ2Þ: (A4)

Collecting the above terms, the perturbed action, up toquadratic order, reads

S¼Zdx4dye2f

��

�3ðð@�Þ2�hð4Þ�Þ�e2f

�1þ4�þþ4�þ2

2þ8�2

��5

�� 2

�2

�4e2fð1�4�Þðð@yfÞ2þ2@yf@y�

þð@y�Þ2�ð@yfÞ2�2@yf@y�þ2

2ð@yfÞ2þ4�ð@yfÞ2þ8�@yf@y��4�ð@yfÞ2þ8�2ð@yfÞ2

þ2ð1�4�Þ@yfhð4Þ��4ð1�4�Þ@yf�hð4Þ��2ð1�4�Þ@y�hð4Þ��4ð1�4�Þ@yf@��@��þe�2fð1��Þðhð4Þ�Þ2��

:

(A5)

The above action contains one nondynamical field, namely. Varying the action with respect to, we get a constraint,

3�hð4Þ� þ 4

�2ð1� 4�Þ@yfhð4Þ�

¼ 8

�2e2fð1� 4�Þðð@yfÞ2 þ 2@yf@y� � ð@yfÞ2

þ 4�ð@yfÞ2Þ � e2f��5ð1þ 4� þ Þ; (A6)

while the equation for � is the following:

ð1� �Þe�2fhð4Þ� ¼ ð1� 4�Þð@y� � @yfÞ: (A7)

Using (A6) we can eliminate from the action (A5) andget the following expression for the action where we havetwo dynamical fields appearing in the action, � and �:

S ¼Z

dx4dye2f�24@yfð@y�Þhð4Þ�

ð�1þ 4�Þ�5�

2

þ 16@yf@y�ð�1þ 4�Þe2f ð1þ 4�Þ�2

þ 3�ð@�Þ2

þ 9�

4�5

e�2fðh�Þ2 � 3

2�2e�2fðh�Þ2 � 6ð@yfÞe�2f

� ð�1þ 4�Þ�2�5

h�h� � 2e2fð1þ 4� þ 8�2Þ�5�

�Z

dx4dye2fð1þ 4� þ 8�2Þe2f��ðyÞ: (A8)

Note that we have restated the boundary brane term of theoriginal action which we had neglected so far. This actioncan be simplified as follows. Consider the following terms:

Zdx4dye2f

�16@yf@y�ð�1þ 4�Þe2f 1þ 4�

�2

� 2e2fð1þ 4� þ 8�2Þ�5�

�; (A9)

and perform a partial integration with respect to the extradimension using (3.4). Then these terms are equal to

�Z

dx4dye4f2ffiffiffi2

p ð1� 4�Þ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�5�

1� 4�

sð1þ 4� þ 8�2Þ�ðyÞ;

which is exactly the same as the contribution of the bound-ary term appearing in (A8) with an opposite sign, afterhaving used the value of the brane tension (3.6). From theremaining terms of the action (A8) consider the followingterms:

Zdx4dye2f

�24@yfð@y�Þhð4Þ�

ð�1þ 4�Þ�5�

2þ 3�ð@�Þ2

�:

(A10)

Observing that

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Zdx4dye2f@y½@��@��� ¼

Zdx4dye2fð�2ð@y�Þhð4Þ�Þ; (A11)

where we have performed a partial integration with respect to the brane coordinates and assumed appropriate boundaryconditions, the above terms become

Zdx4dy

�3�ð@�Þ2e2f þ 12ð�2ð@y�Þhð4Þ�Þ@yfe2f ð1� 4�Þ

�5�2

�¼

Zdx4dy

�3�ð@�Þ2e2f þ 12@yðð@�Þ2Þ@yfe2f ð1� 4�Þ

�5�2

¼Z

dx4dye2f6

ffiffiffi2

p�

ð1� 4�Þ�5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�5�

1� 4�

sð@�Þ2�ðyÞ: (A12)

Using (A12) the final result of the perturbed action (2.3) under the scalar perturbations of the form (5.1) is

S¼Zdx4dye2f

24 9�

4�5

e�2fðh�Þ2� 3

2�2e�2fðh�Þ2� 6ð@yfÞe�2f ð�1þ 4�Þ

�2�5

h�h�þ 6ffiffiffi2

p�

ð1� 4�Þ�5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�5�

1� 4�

sð@�Þ2�ðyÞ

35:

(A13)

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