Strong Brane Gravity and the Radion at Low Energies

24 Pages·0.29 MB·English

by  T. Wiseman

Strong Brane Gravity and the Radion at Low
Energies
Toby Wiseman∗
2
0
Department of Applied Mathematics and Theoretical Physics,
0
2 Center for Mathematical Sciences,

n Wilberforce Road,
a
Cambridge CB3 0WA, UK
J

6
January 16, 2002
1

1
v
7 Abstract
2
1 Forthe2-braneRandall-Sundrummodel,wecalculatethebulkgeometryforstronggravity,
1 in the low matter density regime, for slowly varying matter sources. This is relevant for
0 astrophysical or cosmological applications. The warped compactification means the radion
2 can not be written as a homogeneous mode in the orbifold coordinate, and we introduce
0 it by extending the coordinate patch approach of the linear theory to the non-linear case.
/
h The negative tension brane is taken to be in vacuum. For conformally invariant matter on
t the positive tension brane, we solve the bulk geometry as a derivative expansion, formally
-
p summing the ‘Kaluza-Klein’ contributions to all orders. For general matter we compute the
e Einstein equations to leading order, finding a scalar-tensor theory with ω(Ψ) ∝ Ψ/(1−Ψ),
h and geometrically interpret the radion. We comment that this radion scalar may become
: large in the context of strong gravity with low density matter. Equations of state allowing
v
i (ρ−3P)tobenegative,canexhibitbehaviorwherethematterdecreasesthedistancebetween
X the2branes,whichweillustratenumerically forstaticstarsolutions usinganincompressible
r fluid. For increasing stellar density, the branes become close before the upper mass limit,
a
but after violation of the dominant energy condition. This raises the interesting question of
whether astrophysically reasonable matter, and initial data, could cause branes to collide at
low energy, such as in dynamical collapse.
DAMTP-2002-4
hep-th/0201127
∗e-mail: [email protected]
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Strong Brane Gravity and the Radion at Low Energies Toby Wiseman
1 Introduction
Much progress has been made in understanding the long range gravitational response of branes
at orbifold fixed planes, to localized matter. The Randall-Sundrum models, with one or two
branes[1–3]simplyhavegravityandacosmologicalconstantinthebulk,makingthelinearproblem
verytractable. Linearcalculations[4–7]showthatintheonebranecase,longrange4-dimensional
gravity is recoveredon the brane. Little is known of the general non-linear behavior [8,9]. There
is no mass gap, and thus the non-linear problem is essentially a 5-dimensional one, even for long
wavelengthsources. Secondorderperturbationtheory[10,11],andfully non-linearstudies[12,13]
are again consistent with recovering usual 4-dimensional strong gravity. In particular [12] shows
that non-perturbative phenomena, such as the upper mass limit for static stars, extend smoothly
from large to small objects, whose characteristic sizes are taken relative to the AdS length.
For the two brane case, linear theory [4] found that the effective gravity is Brans-Dicke, for
an observer on the positive tension brane, and a vacuum negative tension brane. For reasonable
brane separations phenomenologically acceptable gravity is recovered. For observers on the neg-
ative tension brane it was found the response was incompatible with observation for any brane
separation, although stabilizing the distance between the branes does allow one to recover stan-
dard 4-dimensional gravity [14–20]. In this paper we consider an observer on the positive tension
brane, the negative tension brane to be in vacuum, and the orbifold radius to be unstabilized.
This allowsus to study the dynamicsofthe radioninstronggravity. Ofcourse,our methods may
be extended to include the stabilized case too. Although moduli have previously been assumed
to be fixed at late times, a crucial feature of the recent cyclic Ekpyrotic scenario [21,22], is that
the radius of the orbifold is not stabilized, the cyclic nature of this model being such that the
separation always remains finite.
Thetwobranestronggravitycaseappearsmoretractablethanforonebrane,asthelinearthe-
oryallowsaKaluza-Kleinstylereductionofthepropagator[4]. InKaluza-Kleincompactifications,
only the homogeneous zero modes are excited on long wavelengths, and the matter is thought to
comprisefieldsoverthe wholeinternalspace. However,formattertobe supportedonthe orbifold
branes there must be modes excited which are not homogeneous in the extra dimension. Thus
one cannotsimply write downa non-linearansatzfor the metric, as is familiar fromKaluza-Klein
compactifications.
Inpreviousworkonthe Horava-Wittencompactification[23],the low energy,long rangeeffec-
tive theory was constructed using consistent orbifold reductions [24–28]. The methods developed
treated the homogeneous component of the metric as a background, and inhomogeneous pertur-
bations about it as the contribution of the massive ‘Kaluza-Klein’ modes. The metric was then
solvedto leading orderin a derivative expansion. Strong gravitywas notdiscussedin these works
except when considering inflation [28]. In order to use these constructions, all the zero modes of
the orbifold must be homogeneous. One important result of the warped compactification, is that
whilst the graviton modes are homogeneous, the radion zero mode is not [29]. This appears to
be a general feature of warped models [30–32], and one cannot directly use these Horava-Witten
methods.
TheaimofthispaperistoapplytheorbifoldreductiontothewarpedRandall-Sundrummodel,
and then examine the non-linear behavior of the radion. We begin in section 2 by illustrating the
reduction proposed in [24–28] with a field theory example and discuss its application to strong
gravity. In order to apply the method we must find a way to include the inhomogeneous radion
zeromode. Insteadof explicitly using anansatz including the radion,we include it by ‘deflecting’
the brane relative to a non-linear extension of the ‘Randall-Sundrum gauge’ [4,5]. In sections 4
and 5 we solve the bulk metric for conformally invariant, low energy, long wavelength matter on
the positive tension brane. For conformally invariant matter, this coordinate system is Gaussian
normal to the brane. The bulk geometry is constructed using a derivative expansion, which is
formally summed to all orders.
In section 6, we extend the linear analysis to the non-linear, for general low density, long
wavelengthmatter,allowingthebranetobecomedeflectedrelativetothiscoordinatesystem. This
scalardeflectionbecomestheradion,andistreatednon-linearly. Thisiscrucial,asthis fieldtakes
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Strong Brane Gravity and the Radion at Low Energies Toby Wiseman
values roughly of order the Newtonian potential, for static systems. Therefore, the case of strong
gravityisexactlywheretheradionmustbetreatednon-linearly. Wecalculatealocalactionforthe
effective induced Einstein equations, to leading order in the derivative expansion. The resulting
theory is a scalar-tensor model, which reduces to Brans-Dicke in the linear approximation. This
is the firstnon-linear,covariantderivationofthe effectiveaction. Consideringthe zeromodes [33]
does not tell one the conformalmetric that matter couples to. Whilst cosmologicalsolutionssuch
as [15,17,34,35] may derive radion dynamics, this only applies for a homogeneous radion, and
thus is not a covariant derivation. Indeed the examples [15,17] illustrate this point well. The
form of the metric chosen, whilst coincidentally reproducing the correcteffective action [33],does
not solve the correct linearized equations [29], and thus only applies to the case of cosmological
symmetry.
We discuss in detail the validity of the approximation for considering low density, long wave-
length strong brane gravity. Terms that are quadratic in the energy density are neglected in
the orbifold reduction method, consistent with the low energy approximation. The characteristic
length scale of the matter must be largecomparedto the compactificationscale for the derivative
expansiontobevalid. Forstronggravity,theapproximationthenholdsprovidedthe4-dimensional
induced brane curvature invariants are small, compared to the compactification scales. Thus this
method will not allow a global solution to a black hole geometry, or any other spacetimes with
curvature singularities. However, it will apply to all other cases of strong gravity, such as static
relativisticstars,anexamplebeingneutronstars,ordynamicalnon-linearsystems,suchasbinary
neutron star systems, or collapse of matter up to the point where curvatures become singular.
Havingpointedoutthatthe radionmaytakelargevaluesfor stronggravityconfigurations,we
illustratethisusingtheexampleofrelativisticstars. Previouswork[36,37]hasconsideredstarson
branesusingtheprojectionformalismof[38],allowingthequadraticstresstensorcorrectionstobe
calculatedbymakingansatzesonthebulkgeometry. However,thesecorrectionsareassumedtobe
negligibleinourlowenergydensityassumption,anditisthebulkgeometriccorrectionswhichare
relevant, and cannotbe calculatedin the projectionapproach. In [12]the full bulk solutions were
numericallyconstructedfortheonebraneRandall-Sundrumcase. Usingthemethodshere,weare
able to analytically construct the geometry for large stars in the 2-brane case. In section 9, using
the leading order action, we numerically consider the static relativistic star, for incompressible
fluid matter. The linear theory shows that for perturbative stars with positive density the branes
aredeflectedapart. However,itindicatesthatifρ 3P becomesnegativetheoppositecouldoccur.
−
Now understanding the radion and bulk geometry non-linearly, we consider this, finding that as
non-lineareffectsbecomeimportantthe branesdoindeedbecomecloser. Furthermore,thebranes
appear to meet before the upper mass limit is reached. This will occur for any brane separation.
However, for phenomenologically acceptable separations [4], we find it does not occur before the
dominant energy condition is violated. Neutron stars are believed to have polytropic equations
of state that do not support negative ρ 3P. It then remains a very interesting, and tractable
−
1+1 problemto understandwhether dynamicalsystems may cause branes to collide, for realistic
matter and initial data. We then have the possibility that physical matter at low energies and
curvatures, compared to compactification scales, might cause brane collisions requiring a Planck
energy physics description.
2 Orbifold Reduction: A Field Theory Example
We illustrate the method of consistent orbifold reduction using a simple scalar field theory. The
methodwasoriginallyusedinthecontextofHorava-Wittenreductions[24–28]. Wetakethescalar
equation,
(cid:3) Ψ+(∂ Ψ)2+∂2Ψ+(∂ Ψ)2 =0 (1)
d µ y y
in a Minkowski (d+1)-dimensional bulk with signature ,+,+,... and coordinates xµ,y, with
−
µ = 0,1,...(d 1). We consider a finite range for y, choosing units such that 0 y 1. The
− ≤ ≤
operator (cid:3) is the Laplacian on d-dimensional Minkowski space formed from xµ. The analogy of
d
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Strong Brane Gravity and the Radion at Low Energies Toby Wiseman
a Kaluza-Kleincompactifications is to use periodic boundary conditions in y, identifying the field
at y =0 and y =1. For orbifold brane compactifications the boundary conditions to consider are
Neumann, with the field gradient ∂ Ψ = ρ(x) at y = 0 and being zero gradient at y = 1. This
y
will later correspond to the case of matter on the positive tension brane, and a vacuum negative
tension brane. The zero modes of the system, with ρ(x)=0 are simply,
Ψ(x,y)=Ψ(x), where, (cid:3) Ψ+(∂ Ψ)2 =0 (2)
d µ
Having no y gradient this induces no ρ(x) on the boundary, and in brane language, is analogous
to the zero mode ansatz discussed in [29]. Of course this solution also solves the Kaluza-Klein
periodic boundary conditions, as these are identical to the Neumann ones if ρ(x) = 0. Then the
full non-linear equations are solved in terms of a lower dimensional system. However, for non-
trivial ρ(x) the solution cannot be independent of y, and the question is whether one can reduce
the problem to a d-dimensional one.
If ρ << 1 we can linearize giving (cid:3) Ψ+∂2Ψ = 0, whose solutions can obviously be found
| | d y
exactly. For slowly varying matter, we may use a derivative expansion in (cid:3) as,
d
1 1 1
Ψ(x,y)= + +y y2 +O((cid:3) ) ρ(x) (3)
(cid:18)(cid:3) (cid:18)−3 − 2 (cid:19) d (cid:19)
d
whereweimplicitlyassumethatthed-dimensionalboundaryconditions,relevantfortheparticular
problem, are taken into account when evaluating the inverse Laplacian. It is these boundary
conditions that specify the zero mode component in the solution Ψ. Howeverwe now see that for
large enough sources, of characteristic spacetime scale L and density ρ, the leading term will be
large for ρL2 1. Then the non-linear terms in the original equation cannot be neglected. For
∼
low density matter, ρ << 1, we require L >> 1 for such strong gravitational effects, and then
| |
we expect higher terms in the derivative expansion to become smaller. Note that if L . 1, the
problem can be tackled simply using the linear theory, as Ψ <<1.
| |
Forsourceswith L 1orsmaller,nothingcanbe done inthe non-linearregime. The problem
≃
is essentially a (d+1)-dimensional one. However, for L>>1, in the above derivative expansion,
we see the leading term has no y dependence. One expects the remaining terms to be small,
and thus we see might hope the non-linearity is not truly (d+1)-dimensional, but rather simply
d-dimensional. We now explicitly see how to realize this.
2.1 Orbifold Reduction
We now illustrate the orbifold reduction technique employed in [27,28], the issue being whether
onecandobetterthanperturbationtheoryforslowlyvaryingsources. Toproceedweobservethat
the leading term in the expansion (3), whilst being large, is independent of y. Therefore instead
of linearizing the equation about Ψ(x,y)=0 as above, we try separating a homogeneous piece of
the solution from the inhomogeneous part, as,
Ψ(x,y)=H(x)+φ(x,y) (4)
where we choose that φ(x,0) = 0 to define the splitting. The aim is to absorb the leading term
of (3) into H(x). Then φ(x,y) will consist of the remaining terms in the expansion(3), but these
areallsmall. Note thatH(x)is azeromode solutiononlywhenρ(x)=0. The equationbecomes,
(cid:3) H(x)+(∂ H)2+(cid:3) φ+2(∂ H)(∂µφ)+(∂ φ)2+∂2φ+(∂ φ)2 =0 (5)
d µ d µ µ y y
and assuming φ <<1, can be linearized to eliminate quadratic terms in φ,
| |
(cid:3) H(x)+(∂ H)2+(cid:3) φ+2(∂ H)(∂µφ)+∂2φ=0 (6)
d µ d µ y
Removingsuchquadratictermsisthecrucialstepwhichwillreducetheproblemtoad-dimensional
one.
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Strong Brane Gravity and the Radion at Low Energies Toby Wiseman
2.2 ‘Strong Gravity’
In order to characterize the magnitude of the terms we define,
ρ
∂pρ(x) (7)
| µ |∼ Lp
for any integer p 0. We characterize,
≥
1
ρ(x) L2ρ=Φ (8)
| (cid:3) |∼
d
whichforlargeLallowsΦ O(1)evenforsmallρ. Wenowassumethatwecanperformthe split
∼
so that,
h
H(x)= 0 +h +h (cid:3) +... ρ(x) O(Φ)=O(L2ρ)
(cid:20)(cid:3) 1 2 d (cid:21) ∼
d
φ(x,y)=[f (y)+f (y)(cid:3) +...]ρ(x) O(ρ) (9)
1 2 d
∼
where h are constants and f (y) are only functions of y. Then we assess the terms above as,
i i
(cid:3) H (∂ H)2 (∂ H)(∂µφ) ∂2φ (cid:3) φ
d µ µ y d
1 (L2ρ)=O(ρ) (1(L2ρ))2 =O(Φρ) (1(L2ρ))(1(ρ))=O(ρ2) O(ρ) O( 1 ρ)
L2 L L L L2
and now we see that in fact the cross term, (∂ H)(∂µφ) is of order O(ρ2), and can again be
µ
neglected. We are left with the equation,
(cid:3) H +(∂ H)2+∂2φ= (cid:3) φ (10)
d µ y − d
where terms on the left hand side are O(ρ),O(ρΦ) and on the right are order O(ρ/L2). We
∼ ∼
also see that H and φ have decoupled. The boundary condition for the problem is that, φ=0 at
y =0 and ∂ φ=0 at y =1. We may solve the system by taking some H(x) and then solving the
y
linear problem,
(cid:3) φ(x,y)+∂2φ(x,y)= D [H(x)] (11)
d y − d
where D [H(x)] = (cid:3) H(x)+(∂ H)2 and provides a homogeneous source term for φ, of order
d d µ
O(ρ), even when H 1. This equation can be solved exactly using a Greens function, and then
∼
derivative expanded as L>>1 as,
1 1 1 1
φ(x,y)= y y2 + y y3+ y4 (cid:3) +O((cid:3)2) D [H(x)] (12)
(cid:20)(cid:18) − 2 (cid:19) (cid:18)3 − 6 24 (cid:19) d d (cid:21) d
Note thatwhilst wecansolvethis equationexactly,it is onlyuseful ifL>>1,asotherwiseterms
in the expansion will not decrease in magnitude. However for L . 1, Φ << 1 in any case, and
linear theory can be used.
Now we have solved the (d+1)-dimensional problem as Ψ(x,y) = H(x)+φ(x,y), with the
source ρ(x)=∂ Ψ ,
y y=0
|
1
ρ(x)= 1+ (cid:3) +O((cid:3)2) D [H(x)] (13)
(cid:20) 3 d d (cid:21) d
and the non-linear (d+1)-dimensional problem is reduced to a d-dimensional one, which was the
purpose of the exercise. Note also that we have not inverted Laplacians, and thus all issues of
5
Strong Brane Gravity and the Radion at Low Energies Toby Wiseman
d-dimensional boundary conditions are implicit in the reduced equation. Specifying these, one
may then invert the problem,
1
D [H(x)]= 1+ 1(cid:3) +O((cid:3)2) − ρ(x)
d (cid:20) 3 d d (cid:21)
=ρ(x)+KK corrections (14)
where the zero modes are solutions for ρ(x)=0.
2.3 Regimes of Interest
Therearethentworegimesofinteresttous. FirstlytheregimewhereΦ 1. Inthiscasethesub-
∼
leading terminthe derivativeexpansion(cid:3) ρ(x) O(ρ/L2) is oforderO(ρ2). Thus in this strong
d
∼
gravity regime, only the leading term in this derivative expansion is of relevance as we have not
calculatedthe otherO(ρ2)correctionsdueto linearizinginφ. Howeverthe homogeneoussplithas
allowed us to solve the non-linear problem up to O(ρ2) corrections. This is the case analogous to
stronggravityonthe brane forlargeobjects, whereeffective correctionsto the Einsteinequations
will be of order O(l2κ4ρ2), with l the AdS length, κ2 the d-dimensional gravitational constant,
d 4
and ρ the characteristic matter energy density. These are extremely small, and not of relevance
in the strong field regime.
ThesecondcaseiswhereΦ<<1,theweakgravityanalogy. Inthiscasewemustcontrastnon-
lineartermsoforderO(Φρ)withtermsinthederivativeexpansion,oforderO(ρ/L2p). Theformer
isapurelyd-dimensionalcorrection,whereasthelatterdependsonthesizeoftheobjectcompared
to the fundamental length scale of the compactification. Note, in this example we chose units so
thatthisfundamentalscalewasone. Inthebranecase,weareconcernedwiththeratioL/l,withl
theAdSlength. ForlargeL,thenon-lineartermsinH(x), O(Φρ),willbelargecomparedtothe
∼
derivative corrections. Simple linear theory would eliminate these non-linear terms in H(x), and
therefore this method allows usual d-dimensional non-linearity to be automatically included. Of
course the linear Greens function solutions tell one about all sizes of object and therefore contain
vastly more information than the above method can yield. However,in cases where one is simply
using the linear theory to derive the bulk metric in the long wavelength regime, it is far more
powerful to use the techniques here.
Note that for large enough p, one expects the derivative expansion term O(ρ/L2p) to be of
order O(ρ2), and then further terms no longer give meaningful corrections without calculating
non-derivative O(ρ2) corrections too. The number of terms in the derivative expansion that are
relevantbeforeO(ρ2)isreacheddependsontheexactvaluesofρ,L. Certainlythefirstsub-leading
term, p=1, is always important if Φ<<1, having magnitude O(ρ2/Φ).
3 Non-Linear Metric Decomposition
In following sections we use the orbifold reduction to solve the non-linear field equations for an
orbifold with matter on the positive tension brane, and a vacuum negative tension brane. The
solutionwillrelatethe(d+1)-dimensionalgeometrytotheusuald-dimensionalnon-linearEinstein
equations. We take the (d+1)-dimensional bulk metric,
l2
ds2 = (g (x,z)dxµdxν +dz2) (15)
z2 µν
withGreekindicestakingvaluesoverthebranespacetimed-dimensions. Ifg (x)istheMinkowski
µν
metricthenthisissimplyAdSinPoincarecoordinates,supportedbyabulkcosmologicalconstant
κ2 Λ= d(d 1)/(2l2), solving the bulk Einstein equations,
d+1 − −
G = κ2 Λg(d+1) (16)
AB − d+1 AB
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Strong Brane Gravity and the Radion at Low Energies Toby Wiseman
where A,B are (d + 1)-dimensional spacetime indices and g(d+1) is the metric as in (15). In
AB
vacuum, positive andnegative tensionorbifoldbranescanbe supportedat z =z ,z respectively,
1 2
with z <z . Observers are taken to reside on the positive tension Z orbifold brane in addition
1 2 2
to localized stress energy. We leave the negative tension brane in vacuum simply for convenience.
However, all the methods outlined in this paper can be applied relaxing this condition. Note
that the linear theory [4] already shows that an observer on the negative tension brane sees a
phenomenologicallyunacceptablegravitytheory,unlesstheorbifoldisstabilized[18]. Theorbifold
Z planesandlocalizedmatter IT aretreatedinthe thin wallapproximation,the planeshaving
2 µν
tensions κ2 σ = 2(d 1)/l.
d+1 ± −
Following the methods of [24–28], illustrated previously for the scalar field example in section
2,wedecomposeg intoahomogeneousandinhomogeneouspiecewithrespecttothecoordinate
µν
z, as
g (x,z)=0g (x)+h (x,z) (17)
µν µν µν
There is obviously freedom in the above decomposition. To uniquely define it, we require that
g (x,z )=0g (x) for some constantz , which will be the positionof the positive tension brane
µν 1 µν 1
if the matter has vanishing stress energy trace. Then the intrinsic metric on this surface z will
1
just be 0g (x)l2/z2. If the metric 0g is Ricci flat, the (d+1)-dimensional Einstein equations
µν 1 µν
are solved for vanishing h [39]. These geometric deformations are then the gravitational zero
µν
modes of the orbifold.
As discussed, we are not able to include the radion zero mode in the background,and have it
remain homogeneous, due to the warped geometry [29]. Thus we only include the homogeneous
gravitons, 0g (x). Instead we will include the radion by extending the brane deflection ideas
µν
of the linear theory. Thus we do not perturb the zz metric component as we will include any
perturbative back-reactionof the radion in the non-linear deflection of the brane.
The procedure we outline below is to find a suitable 0g (x) suchthat h (x,z) remainssmall
µν µν
for a low density, long wavelength matter perturbations on the orbifold brane, even when the
intrinsic geometry is non-linear. Thus we aim to absorb all the non-linearity into 0g (x), which
µν
effectively shapes the induced geometry on the brane. One can think of h as the contribution
µν
of the massive KK modes, which provide only small corrections to the induced geometry on the
brane, but play the essential role of supporting the localized matter.
Asinthescalarfieldexampleinsection2,wedefinetwodimensionlessquantities. Thefirst,ρ,
characterizesthe matter density or curvature scalecomparedto the AdS energydensity. The sec-
ond,L,comparesthe lengthscaleassociatedwiththe matter,to the AdSlength. Forconvenience
we considerthe parameterǫ=1/L2 whichis smallinthe relevantlargeobjectlimit. Formallywe
take,
ρ=l2 IRµν
αβ
k k
ρǫ=l4 I(cid:3)IRµν (18)
αβ
k k
and then expect,
ρǫp O( (l2I(cid:3))pl2IRµν ) (19)
αβ
∼ k k
where p 0 and indicates the maximumabsolute value of the tensor overallspace,with l, the
≥ kk
AdS length, introduced to make ρ,ǫ dimensionless. The curvature tensors are formed from the
induced brane metric, as is the Laplacian I(cid:3), and have indices arrangedas they would appear in
curvatureinvariants. InordertolinearizetheEinsteinequationsintheKaluza-Kleincontribution
h , we will require that the matter be low density comparedto AdS scales. In addition, in order
µν
to use a derivative expansion, we will require the object to have a large size or dynamical time
compared to the AdS length. Thus we make the requirements that,
ρ, ǫ<<1 (20)
7
Strong Brane Gravity and the Radion at Low Energies Toby Wiseman
As mentioned previously, for ρ<<1 and ǫ&1, the problem can be solved simply using standard
linear theory. An important point is that this restricts attention to non-singular geometries, as
the curvatures must remain bounded, and our approximation will work only when they are well
below the AdS curvature scale 1/l2. For consistency we take the induced metric, g˜ , to be of
µν
order,
ρ
(l∂ )2pg˜ O( ǫp) for p 1 (21)
µ µν
k k∼ ǫ ≥
Of course, when considering g˜ , the Ricci flat zero modes must be taken into account. Thus
µν
k k
we characterize the quantity,
ρ
Φ= ∆g˜ O( ) (22)
µν
k k∼ ǫ
where we understand ∆g˜ to be the difference of the metric from some Ricci flat zero mode
µν
background due to the presence of matter. We note that Φ will be of order the Newtonian
potential for static field configurations.
As a technical note, we will also consider the magnitudes in equation (19) to hold, with the
tensor indices arrangeddifferently. This implicitly assumes that the metric is non-degenerateand
non-singular.
4 The Bulk Metric
In this section, we solve the bulk geometry using a coordinate system which extends the Randall-
Sundrum gauge to the non-linear case [4,5]. As in the linear theory, this coordinate system is
such that constantz hyper-surfaceshave scalarinduced extrinsic curvature equalto d/l. We take
a positive tension brane with localized matter at z = z , and a vacuum negative tension brane
1
at z = z , subject to the restrictions (20), and the further condition that z ,z O(l). The
2 1 2
∼
geometry will be consistent only for positive tension brane matter with a vanishing stress energy
trace. In the subsequent sections we remove this restriction, bending the brane relative to the
surface z = z , again in analogy with the linear theory, although now these ‘radion’ deflections
1
may be large, but slowly varying.
Using the metric (15) and zero mode decomposition (17), the Einstein equations can be lin-
earized in the Kaluza-Klein perturbation h . As indicated in the scalar field example of section
µν
2, we will find that h l∂ h O(ρ) and therefore in linearizing we neglect terms of
µν z µν
k k ∼ k k ∼
order O(h2/l2) O(ρ2/l2), consistent with the low curvature condition. Away from the branes,
∼
the linearized Einstein equations, with critical bulk cosmologicalconstant, are,
d 1
G κ2 Λg(d+1) = Rˆ − ∂ h+O(h2/l2)
zz − d+1 zz − − 2z z
1
G = (0 ∂ hα 0 ∂ h)+O(h2/l2)
zµ α z µ µ z
2 ∇ − ∇
1 d 1
G κ2 Λg(d+1) =Gˆ (∂2h 0g ∂2h)+ − (∂ h 0g ∂ h)+O(h2/l2) (23)
µν − d+1 µν µν − 2 z µν − µν z 2z z µν − µν z
where Rˆ =0gµνRˆ , andRˆ ,Gˆ arethe RicciandEinstein curvatureofthe metricg (x,z)on
µν µν µν µν
constant z hyper-surfaces. Indices are raised and lowered with respect to 0g (x), h = hα and
µν α
0 is the covariant derivative of this homogeneous mode metric. Thus, we are using 0g in the
µν
∇
samewayoneusesabackgroundsolutioninusuallinearperturbationtheory. However,here,0g
µν
is not a solution to these equations in the presence of brane sources.
8
Strong Brane Gravity and the Radion at Low Energies Toby Wiseman
As h O(ρ), we may decompose the d-dimensionalcurvature terms. The Ricci curvature
µν
k k∼
term in G becomes
zz
Rˆ =0gµνRˆ =0gαβ0R 0(cid:3)h+0 α0 βh 0Rα h β +O(h2/l2)
µν αβ αβ β α
− ∇ ∇ −
=0R 0(cid:3)h+0 α0 βh +O(ρ2/l2) (24)
αβ
− ∇ ∇
Note that the term 0Rα h β is of order O(ρ2/l2), as both l20R and h are of order O(ρ). We
β α αβ αβ
now see that the linearized constraint equations, G κ2 Λg(d+1) =O(ρ2/l2), G =O(ρ2/l2)
zz − d+1 zz zµ
are satisfied if,
0R(x)=0
hα (x,z)=0 hα (x,z)=0 (25)
α α µ
∇
This can be recognized as the usual Randall-Sundrum gauge condition. We have simply imposed
thisaboutahomogeneousbackground0g ,whichsatisfiesthecondition0Gα =0. Theremaining
µν α
Einstein equation becomes,
1 ∂ h 1
0G zd 1∂ z µν = 0(cid:3)h +O(ρ2/l2) (26)
µν − 2 − z(cid:16) zd−1 (cid:17) 2 µν
We emphasize that 0G and 0(cid:3) are independent of z. Note that 0G , ∂2h O(ρ/l2),
µν k µνk k z µνk ∼
and 0(cid:3)h O(ρǫ/l2). For slowly varying matter, ǫ << 1, and we construct the solution as a
µν
∼
derivative expansion in 0(cid:3). For our brane configuration,with matter only on the positive tension
brane, we solve this differential equation using the ansatz,
hµν(x,z)=zd2f(z,0(cid:3))0Gµν(x) (27)
which indeed satisfies the constraints (25) if 0R = 0. The transverse condition is simply the
contracted Bianchi identity, and the traceless condition holds as 0R is zero. Then (26) reduces to
the formal ‘operator’ equation,
1 d2
∂z2f + z∂zf +(cid:18)0(cid:3)− 4z2(cid:19)f =2z−d2 (28)
which can be solved exactly to give,
f =fPI+A(0(cid:3))2(d20(cid:3)(cid:0))d2d4(cid:1)!Jd2(√0(cid:3)z)−B(0(cid:3))2d2 dπ 1 !(0(cid:3))d4Nd2(√0(cid:3)z) (29)
2 −
(cid:0) (cid:1)
for arbitrary functions A(0(cid:3)),B(0(cid:3)), which can be expanded as,
A
A(0(cid:3))=A +A 0(cid:3)+ 2(0(cid:3))2+... (30)
0 1
2!
and similarly for B, ensuring that the whole expression only contains positive powers of 0(cid:3). The
particular integral is taken as,
fPI = 02(cid:3)(cid:20)z−d2 + 2d2(d2π−1)!(0(cid:3))d4Nd2(√0(cid:3)z)(cid:21) d=odd (31)
 02(cid:3)(cid:20)z−d2 + 2d2(d2π−1)!(0(cid:3))d4 (cid:16)Nd2(√0(cid:3)z)− π1 log(0(cid:3))Jd2(√0(cid:3)z)(cid:17)(cid:21) d=even
which againcanbe expressedas a derivativeexpansionin 0(cid:3) of Taylorseriesform. Then for long
wavelength matter, defined through the condition (20), we expect leading terms to dominate the
9
Strong Brane Gravity and the Radion at Low Energies Toby Wiseman
series. By expanding the Bessel functions for √0(cid:3)z >>1 in the particular integral solution (31),
we see that the d-dimensional zero mode no longer well approximates the geometry, exactly as
one predicts for small objects which should behave in a manifestly (d+1)-dimensional manner.
If √0(cid:3) >> 1, ie. for small objects, ǫ & 1 and Φ = ρ/ǫ << 1 so linear theory can be used.
However, the case where it is z that is large, means one cannot study large non-linear objects in
the one brane Randall-Sundrum case, as one must remove the second brane to distances greater
than z & 1/√0(cid:3). This essentially shows why the one brane case is really a (d+1)-dimensional
problem.
We must now fix the two functions of integration, A(0(cid:3)),B(0(cid:3)), using the boundary condi-
tions of the problem. At z = z , the position of the negative tension vacuum brane, we require
2
∂ h (x,z ) = 0. The second boundary condition is simply the requirement that h (x,z ) = 0
z µν 2 µν 1
at z =z by definition of the metric splitting (17). Direct evaluation of the expressions (29) and
1
(31) for these conditions then determine A,B.
Using this formalsolution, representedby the operator (z,0(cid:3)) below,we mayexpand h in
µν
L
powers of 0(cid:3) acting on 0G , the term (l20(cid:3))p0G being of order O(ρǫp/l2), so that,
µν µν
h = (z,0(cid:3))0G = h (z)1+h (z)0(cid:3)+O(0(cid:3)2) 0G
µν µν 0 1 µν
L
(cid:2) (cid:3)
1
with, h (z)=B z2+A zd
0 0 0
− d 2
−
1 B z2 z4 + 1 A zd+2+B 1 z2+A zd d=3,d 5
2(d 2) 0 − 2(d 2)(d 4) 2(d+2) 0 1− d 2 1 ≥
h (z)= − − − −
1  41B0z2+ 112A0z6− 34+lo8g2−γz4+ 18(logz)z4+B1− 21z2+A1z4 d=4

and, A = 2 1 , B = z12 1 2Ω , Ω= z1 d−2 (32)
0 d(d−2)z2d−2 0 d−2(cid:18) − d (cid:19) (cid:18)z2(cid:19)
where A ,B are trivially calculated from the boundary conditions but are not shown explicitly
1 1
hereforclarity. Whilstwehaveexpandedtosub-leadingorder,O(l4(0(cid:3))0G ),asdiscussedinthe
µν
scalarfieldexample,insection2,itisonlytheleadingtermthatisrelevantforstronglynon-linear
configurations with Φ O(1). Then the sub-leading correction to h is of order O(ρ2), and one
µν
∼
must also calculate the corrections from the linearization of the Einstein equations in ∂ h which
z
enter at the same order. The sub-leading corrections are important in the case when Φ< 1, and
thefieldisweak,butonewishestotaked-dimensionalnon-lineartermsintoaccount,suchaswhen
calculating post-Newtonian corrections.
5 Conformally Invariant Matter
For a positive tension orbifold brane at constant z =z , the induced metric is,
1
l2
Ig = 0g (33)
µν z2 µν
1
as h = 0 at z = z by construction. For an orbifold brane with tension σ, the Israel matching
µν 1
conditions [40] yield the localized stress energy IT to be,
µν
κ2 σIg +κ2 IT =2[K Ig K]
− d+1 µν d+1 µν µν − µν z=z1
2(d 1) l
= − Ig ∂ h +O(ρ2/l) (34)
µν z µν
− l − z
1
10