BraneWorld Gravity
Abstract
The observable universe could be a 1+3surface (the “brane”) embedded in a 1+3+dimensional spacetime (the “bulk”), with Standard Model particles and fields trapped on the brane while gravity is free to access the bulk. At least one of the extra spatial dimensions could be very large relative to the Planck scale, which lowers the fundamental gravity scale, possibly even down to the electroweak (TeV) level. This revolutionary picture arises in the framework of recent developments in M theory. The 1+10dimensional M theory encompasses the known 1+9dimensional superstring theories, and is widely considered to be a promising potential route to quantum gravity. General relativity cannot describe gravity at high enough energies and must be replaced by a quantum gravity theory, picking up significant corrections as the fundamental energy scale is approached. At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity “leaks” into the bulk, behaving in a truly higherdimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting and potentially testable implications for highenergy astrophysics, black holes and cosmology. Braneworld models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. This review discusses the geometry, dynamics and perturbations of simple braneworld models for cosmology and astrophysics, mainly focusing on warped 5dimensional braneworlds based on the RandallSundrum models.
I Introduction
At high enough energies, Einstein’s theory of general relativity breaks down, and will be superceded by a quantum gravity theory. The classical singularities predicted by general relativity in gravitational collapse and in the hot big bang will be removed by quantum gravity. But even below the fundamental energy scale that marks the transition to quantum gravity, significant corrections to general relativity will arise. These corrections could have a major impact on the behaviour of gravitational collapse, black holes and the early universe, and they could leave a trace–a “smoking gun”–in various observations and experiments. Thus it is important to estimate these corrections and develop tests for detecting them or ruling them out. In this way, quantum gravity can begin to be subject to testing by astrophysical and cosmological observations.
Developing a quantum theory of gravity and a unified theory of all the forces and particles of nature are the two main goals of current work in fundamental physics. There is as yet no generally accepted (pre)quantum gravity theory. Two of the main contenders are M theory mtheory and quantum geometry (loop quantum gravity) loop . It is important to explore the astrophysical and cosmological predictions of both these approaches. This review considers only models that arise within the framework of M theory, and mainly the 5dimensional warped braneworlds.
i.1 Heuristics of higherdimensional gravity
One of the fundamental aspects of string theory is the need for extra spatial dimensions. This revives the original higherdimensional ideas of Kaluza and Klein in the 1920’s, but in a new context of quantum gravity. An important consequence of extra dimensions is that the 4dimensional Planck scale is no longer the fundamental scale, which is , where is the number of extra dimensions. This can be seen from the modification of the gravitational potential. For an EinsteinHilbert gravitational action we have,
(1)  
(2) 
where and is the gravitational coupling constant,
(3) 
The static weak field limit of the field equations leads to the dimensional Poisson equation, whose solution is the gravitational potential,
(4) 
If the length scale of the extra dimensions is , then on scales , the potential is dimensional, . By contrast, on scales large relative to , where the extra dimensions do not contribute to variations in the potential, behaves like a 4dimensional potential, i.e., in the extra dimensions, and . This means that the usual Planck scale becomes an effective coupling constant, describing gravity on scales much larger than the extra dimensions, and related to the fundamental scale via the volume of the extra dimensions:
(5) 
If the extradimensional volume is Planck scale, i.e. , then . But if the extradimensional volume is significantly above Planckscale, then the true fundamental scale can be much less than the effective scale . In this case, we understand the weakness of gravity as due to the fact that it “spreads” into extra dimensions and only a part of it is felt in 4 dimensions.
A lower limit on is given by null results in tabletop experiments to test for deviations from Newton’s law in 4 dimensions, . These experiments currently exp probe submillimetre scales, so that
(6) 
Stronger bounds for braneworlds with compact flat extra dimensions can be derived from null results in particle accelerators and in highenergy astrophysics cav ; cheung ; hanraf .
i.2 Braneworlds and M theory
String theory thus incorporates the possibility that the fundamental scale is much less than the Planck scale felt in 4 dimensions. There are five distinct 1+9dimensional superstring theories, all giving quantum theories of gravity. Discoveries in the mid90 s of duality transformations that relate these superstring theories and the 1+10dimensional supergravity theory, led to the conjecture that all of these theories arise as different limits of a single theory, which has come to be known as M theory. The 11th dimension in M theory is related to the string coupling strength; the size of this dimension grows as the coupling becomes strong. At low energies, M theory can be approximated by 1+10dimensional supergravity.
It was also discovered that pbranes, which are extended objects of higher dimension than strings (1branes), play a fundamental role in the theory. In the weak coupling limit, pbranes () become infinitely heavy, so that they do not appear in the perturbative theory. Of particular importance among pbranes are the Dbranes, on which open strings can end. Roughly speaking, open strings, which describe the nongravitational sector, are attached at their endpoints to branes, while the closed strings of the gravitational sector can move freely in the bulk. Classically, this is realised via the localization of matter and radiation fields on the brane, with gravity propagating in the bulk (see Fig. 1).
In the HoravaWitten solution hv , gauge fields of the standard model are confined on two 1+9branes located at the end points of an orbifold, i.e., a circle folded on itself across a diameter. The 6 extra dimensions on the branes are compactified on a very small scale, close to the fundamental scale, and their effect on the dynamics is felt through “moduli” fields, i.e. 5D scalar fields. A 5D realization of the HoravaWitten theory and the corresponding braneworld cosmology is given in low .
These solutions can be thought of as effectively 5dimensional, with an extra dimension that can be large relative to the fundamental scale. They provide the basis for the RandallSundrum 2brane models of 5dimensional gravity rs1 (see Fig. 2). The singlebrane RandallSundrum models rs2 with infinite extra dimension arise when the orbifold radius tends to infinity. The RS models are not the only phenomenological realizations of M theory ideas. They were preceded by the ArkaniHamedDimopoulosDvali (ADD) add braneworld models, which put forward the idea that a large volume for the compact extra dimensions would lower the fundamental Planck scale,
(7) 
where is the electroweak scale. If is close to the lower limit in Eq. (7), then this would address the longstanding “hierarchy” problem, i.e. why there is such a large gap between and .
In the ADD models, more than one extra dimension is required for agreement with experiments, and there is “democracy” amongst the equivalent extra dimensions, which are typically flat. By contrast, the RS models have a “preferred” extra dimension, with other extra dimensions treated as ignorable (i.e., stabilized except at energies near the fundamental scale). Furthermore, this extra dimension is curved or “warped” rather than flat: the bulk is a portion of anti de Sitter (AdS) spacetime. As in the HoravaWitten solutions, the RS branes are symmetric (mirror symmetry), and have a tension, which serves to counter the influence of the negative bulk cosmological constant on the brane. This also means that the selfgravity of the branes is incorporated in the RS models. The novel feature of the RS models compared to previous higherdimensional models is that the observable 3 dimensions are protected from the large extra dimension (at low energies) by curvature rather than straightforward compactification.
The RS braneworlds and their generalizations (to include matter on the brane, scalar fields in the bulk, etc.) provide phenomenological models that reflect at least some of the features of M theory, and that bring exciting new geometric and particle physics ideas into play. The RS2 models also provide a framework for exploring holographic ideas that have emerged in M theory. Roughly speaking, holography suggests that higherdimensional gravitational dynamics may be determined from knowledge of the fields on a lowerdimensional boundary. The AdS/CFT correspondence is an example, in which the classical dynamics of the higherdimensional gravitational field are equivalent to the quantum dynamics of a conformal field theory (CFT) on the boundary. The RS2 model with its AdS metric satisfies this correspondence to lowest perturbative order acft (see also acftcosmo for the AdS/CFT correspondence in a cosmological context).
In this review, I focus on RS braneworlds (mainly RS 1brane) and their generalizations, with the emphasis on geometry and gravitational dynamics (see m2 ; rev ; lan for previous reviews with a broadly similar approach). Other recent reviews focus on stringtheory aspects, e.g. que , or on particle physics aspects, e.g. r ; cav . Before turning to a more detailed analysis of RS braneworlds, I discuss the notion of KaluzaKlein (KK) modes of the graviton.
i.3 Heuristics of KK modes
The dilution of gravity via extra dimensions not only weakens gravity on the brane, it also extends the range of graviton modes felt on the brane beyond the massless mode of 4dimensional gravity. For simplicity, consider a flat brane with one flat extra dimension, compactified through the identification , where . The perturbative 5D graviton amplitude can be Fourier expanded as
(8) 
where are the amplitudes of the KK modes, i.e. the effective 4D modes of the the 5D graviton. To see that these KK modes are massive from the brane viewpoint, we start from the 5D wave equation that the massless 5D field satisfies (in a suitable gauge):
(9) 
It follows that the KK modes satisfy a 4D KleinGordon equation with an effective 4D mass, ,
(10) 
The massless mode, , is the usual 4D graviton mode. But there is a tower of massive modes, , which imprint the effect of the 5D gravitational field on the 4D brane. Compactness of the extra dimension leads to discreteness of the spectrum. For an infinite extra dimension, , the separation between the modes disappears and the tower forms a continuous spectrum. In this case, the coupling of the KK modes to matter must be very weak in order to avoid exciting the lightest massive modes with .
From a geometric viewpoint, the KK modes can also be understood via the fact that the projection of the null graviton 5momentum onto the brane is timelike. If the unit normal to the brane is , then the induced metric on the brane is
(11) 
and the 5momentum may be decomposed as
(12) 
where is the projection along the brane, depending on the orientation of the 5momentum relative to the brane. The effective 4momentum of the 5D graviton is thus . Expanding
(13) 
It follows that the 5D graviton has an effective mass on the brane. The usual 4D graviton corresponds to the zero mode, , when is tangent to the brane.
The extra dimensions lead to new scalar and vector degrees of freedom on the brane. In 5D, the spin2 graviton is represented by a metric perturbation that is transverse traceless:
(14) 
In a suitable gauge, contains a 3D transverse traceless perturbation , a 3D transverse vector perturbation and a scalar perturbation , which each satisfy the 5D wave equation durkoc :
(15)  
(16)  
(17) 
The other components of are determined via constraints once these wave equations are solved. The 5 degrees of freedom (polarizations) in the 5D graviton are thus split into 2 () + 2 () +1 () degrees of freedom in 4D. On the brane, the 5D graviton field is felt as

a 4D spin2 graviton (2 polarizations)

a 4D spin1 gravivector (graviphoton) (2 polarizations)

a 4D spin0 graviscalar .
The massive modes of the 5D graviton are represented via massive modes in all 3 of these fields on the brane. The standard 4D graviton corresponds to the massless zeromode of .
In the general case of extra dimensions, the number of degrees of freedom in the graviton follows from the irreducible tensor representations of the isometry group as .
Ii RandallSundrum braneworlds
RS braneworlds do not rely on compactification to localize gravity at the brane, but on the curvature of the bulk (sometimes called “warped compactification”). What prevents gravity from ‘leaking’ into the extra dimension at low energies is a negative bulk cosmological constant,
(18) 
where is the curvature radius of AdS and is the corresponding energy scale. The curvature radius determines the magnitude of the Riemann tensor:
(19) 
The bulk cosmological constant acts to “squeeze” the gravitational field closer to the brane. We can see this clearly in Gaussian normal coordinates based on the brane at , for which the AdS metric takes the form
(20) 
with the Minkowski metric. The exponential warp factor reflects the confining role of the bulk cosmological constant. The symmetry about the brane at is incorporated via the term. In the bulk, this metric is a solution of the 5D Einstein equations,
(21) 
i.e., in Eq. (2). The brane is a flat Minkowski spacetime, , with selfgravity in the form of brane tension. One can also use Poincare coordinates, which bring the metric into manifestly conformally flat form:
(22) 
where .
The two RS models are distinguished as follows:

RS 2BRANE:
There are two branes in this model rs1 , at and , with symmetry identifications
(23) The branes have equal and opposite tensions, , where
(24) The positivetension brane has fundamental scale and is “hidden”. Standard model fields are confined on the negative tension (or “visible”) brane. Because of the exponential warping factor, the effective scale on the visible brane at is , where
(25) So the RS 2brane model gives a new approach to the hierarchy problem. Because of the finite separation between the branes, the KK spectrum is discrete. Furthermore, at low energies gravity on the branes becomes BransDickelike, with the sign of the BransDicke parameter equal to the sign of the brane tension gt . In order to recover 4D general relativity at low energies, a mechanism is required to stabilize the interbrane distance, which corresponds to a scalar field degree of freedom known as the radion goldwise ; 2b .

RS 1BRANE:
In this model rs2 , there is only one, positive tension, brane. It may be thought of as arising from sending the negative tension brane off to infinity, . Then the energy scales are related via
(26) The infinite extra dimension makes a finite contribution to the 5D volume because of the warp factor:
(27) Thus the effective size of the extra dimension probed by the 5D graviton is .
I will concentrate mainly on RS 1brane from now on, referring to RS 2brane occasionally. The RS 1brane models are in some sense the most simple and geometrically appealing form of braneworld model, while at the same time providing a framework for the AdS/CFT correspondence acft ; acftcosmo . The RS 2brane models introduce the added complication of radion stabilization, as well as possible complications arising from negative tension. However, they remain important and will occasionally be discussed.
In RS 1brane, the negative is offset by the positive brane tension . The finetuning in Eq. (24) ensures that there is zero effective cosmological constant on the brane, so that the brane has the induced geometry of Minkowski spacetime. To see how gravity is localized at low energies, we consider the 5D graviton perturbations of the metric rs2 ; gt ; morepert :
(28) 
(See Fig. 3.) This is the RS gauge, which is different from the gauge used in Eq. (15), but which also has no remaining gauge freedom. The 5 polarizations of the 5D graviton are contained in the 5 independent components of in the RS gauge.
We split the amplitude of into 3D Fourier modes, and the linearized 5D Einstein equations lead to the wave equation ()
(29) 
Separability means we can write
(30) 
and the wave equation reduces to
(31)  
(32) 
The zero mode solution is
(33)  
(34) 
and the solutions are
(35)  
(36) 
The boundary condition for the perturbations arises from the junction conditions, Eq. (61), discussed below, and leads to , since the transverse traceless part of the perturbed energymomentum tensor on the brane vanishes. This implies
(37) 
The zero mode is normalizable, since
(38) 
Its contribution to the gravitational potential . The contribution of the massive KK modes sums to a correction of the 4D potential. For , one obtains gives the 4D result,
(39) 
which simply reflects the fact that the potential becomes truly 5D on small scales. For ,
(40) 
which gives the small correction to 4D gravity at low energies from extradimensional effects. These effects serve to slightly strengthen the gravitational field, as expected.
Tabletop tests of Newton’s laws currently find no deviations down to , so that mm in Eq. (40). Then by Eqs. (24) and (26), this leads to lower limits on the brane tension and the fundamental scale of the RS 1brane model:
(41) 
These limits do not apply to the 2brane case.
For the 1brane model, the boundary condition, Eq. (37), admits a continuous spectrum of KK modes. In the 2brane model, must hold in addition to Eq. (37). This leads to conditions on , so that the KK spectrum is discrete:
(42) 
The limit Eq. (41) indicates that there are no observable collider, i.e. , signatures for the RS 1brane model. The 2brane model by contrast, for suitable choice of and so that , does predict collider signatures that are distinct from those of the ADD models hanraf .
Iii Covariant approach to braneworld geometry and dynamics
The RS models and the subsequent generalization from a Minkowski brane to a FriedmannRobertsonWalker (FRW) brane bdel ; morers2 ; gs , were derived as solutions in particular coordinates of the 5D Einstein equations, together with the junction conditions at the symmetric brane. A broader perspective, with useful insights into the interplay between 4D and 5D effects, can be obtained via the covariant ShiromizuMaedaSasaki approach sms , in which the brane and bulk metrics remain general. The basic idea is to use the GaussCodazzi equations to project the 5D curvature along the brane. (The general formalism for relating the geometries of a spacetime and of hypersurfaces within that spacetime is given in wald .)
The 5D field equations determine the 5D curvature tensor; in the bulk, they are
(43) 
where represents any 5D energymomentum of the gravitational sector (e.g., dilaton and moduli scalar fields, form fields).
Let be a Gaussian normal coordinate orthogonal to the brane (which is at without loss of generality), so that , with the unit normal. The 5D metric in terms of the induced metric on surfaces is locally given by
(44) 
The extrinsic curvature of surfaces describes the embedding of these surfaces. It can be defined via the Lie derivative or via the covariant derivative:
(45) 
so that
(46) 
where square brackets denote antisymmetrization. The Gauss equation gives the 4D curvature tensor in terms of the projection of the 5D curvature, with extrinsic curvature corrections:
(47) 
and the Codazzi equation determines the change of along via
(48) 
where .
Some other useful projections of the 5D curvature are:
(49)  
(50)  
(51) 
The 5D curvature tensor has Weyl (tracefree) and Ricci parts:
(52) 
iii.1 Field equations on the brane
Using Eqs. (43) and (47), it follows that
(53)  
where
(54) 
is the projection of the bulk Weyl tensor orthogonal to . This tensor satisfies
(55) 
by virtue of the Weyl tensor symmetries. Evaluating Eq. (53) on the brane (strictly, as , since is not defined on the brane sms ) will give the field equations on the brane.
First, we need to determine at the brane from the junction conditions. The total energymomentum tensor on the brane is
(56) 
where is the energymomentum tensor of particles and fields confined to the brane (so that ). The 5D field equations, including explicitly the contribution of the brane, are then
(57) 
Here the delta function enforces in the classical theory the string theory idea that Standard Model fields are confined to the brane. This is not a gravitational confinement, since there is in general a nonzero acceleration of particles normal to the brane m1 .
Integrating Eq. (57) along the extra dimension from to , and taking the limit , leads to the IsraelDarmois junction conditions at the brane,
(58)  
(59) 
where . The symmetry means that when you approach the brane from one side and go through it, you emerge into a bulk that looks the same, but with the normal reversed, . Then Eq. (45) implies that
(60) 
so that we can use the junction condition Eq. (59) to determine the extrinsic curvature on the brane:
(61) 
where , we have dropped the and we evaluate quantities on the brane by taking the limit .
Finally we arrive at the induced field equations on the brane, by substituting Eq. (61) into Eq. (53):
(62) 
The 4D gravitational constant is an effective coupling constant inherited from the fundamental coupling constant, and the 4D cosmological constant is nonzero when the RS balance between the bulk cosmological constant and the brane tension is broken:
(63)  
(64) 
The first correction term relative to Einstein’s theory is quadratic in the energymomentum tensor, arising from the extrinsic curvature terms in the projected Einstein tensor:
(65) 
The second correction term is the projected Weyl term. The last correction term on the right of Eq. (62), which generalizes the field equations in sms , is
(66) 
where describes any stresses in the bulk apart from the cosmological constant (see mw for the case of a scalar field).
What about the conservation equations? Using Eqs. (43), (48) and (61), one obtains
(67) 
Thus in general there is exchange of energymomentum between the bulk and the brane. From now on, we will assume that
(68) 
so that
(69)  
(70) 
and one then recovers from Eq. (67) the standard 4D conservation equations,
(71) 
This means that there is no exchange of energymomentum between the bulk and the brane; their interaction is purely gravitational. Then the 4D contracted Bianchi identities (), applied to Eq. (62), lead to
(72) 
which shows qualitatively how 1+3 spacetime variations in the matterradiation on the brane can source KK modes.
The induced field equations (70) show two key modifications to the standard 4D Einstein field equations arising from extradimensional effects:

is the highenergy correction term, which is negligible for , but dominant for :
(73) 
, the projection of the bulk Weyl tensor on the brane, encodes corrections from 5D graviton effects (the KK modes in the linearized case). From the braneobserver viewpoint, the energymomentum corrections in are local, whereas the KK corrections in are nonlocal, since they incorporate 5D gravity wave modes. These nonlocal corrections cannot be determined purely from data on the brane. In the perturbative analysis of RS 1brane which leads to the corrections in the gravitational potential, Eq. (40), the KK modes that generate this correction are responsible for a nonzero ; this term is what carries the modification to the weakfield field equations. The 9 independent components in the tracefree are reduced to 5 degrees of freedom by Eq. (72); these arise from the 5 polarizations of the 5D graviton.
Note that the covariant formalism applies also to the twobrane case. In that case, the gravitational influence of the second brane is felt via its contribution to .
iii.2 5dimensional equations and the initialvalue problem
The effective field equations are not a closed system. One needs to supplement them by 5D equations governing , which are obtained from the 5D Einstein and Bianchi equations. This leads to the coupled system ssm :
(74)  
(75)  
(76)  
(77) 
where the “magnetic” part of the bulk Weyl tensor, counterpart to the “electric” part , is
(78) 
These equations are to be solved subject to the boundary conditions at the brane,
(79)  
(80) 
where denotes .
The above equations have been used to develop a covariant analysis of the weak field ssm . They can also be used to develop a Taylor expansion of the metric about the brane. In Gaussian normal coordinates, Eq. (44), we have . Then we find
(81) 
In a noncovariant approach based on a specific form of the bulk metric in particular coordinates, the 5D Bianchi identities would be avoided and the equivalent problem would be one of solving the 5D field equations, subject to suitable 5D initial conditions and to the boundary conditions Eq. (61) on the metric. The advantage of the covariant splitting of the field equations and Bianchi identities along and normal to the brane is the clear insight that it gives into the interplay between the 4D and 5D gravitational fields. The disadvantage is that the splitting is not well suited to dynamical evolution of the equations. Evolution off the timelike brane in the spacelike normal direction does not in general constitute a welldefined initial value problem antav . One needs to specify initial data on a 4D spacelike (or null) surface, with boundary conditions at the brane(s) ensuring a consistent evolution ichnak . Clearly the evolution of the observed universe is dependent upon initial conditions which are inaccessible to branebound observers; this is simply another aspect of the fact that the brane dynamics is not determined by 4D but by 5D equations. The initial conditions on a 4D surface could arise from models for creation of the 5D universe gs ; ksb , from dynamical attractor behaviour mukcol or from suitable conditions (such as no incoming gravitational radiation) at the past Cauchy horizon if the bulk is asymptotically AdS.
iii.3 The brane viewpoint: a 1+3covariant analysis
A systematic analysis can be developed from the viewpoint of a branebound observer, following m1 . The effects of bulk gravity are conveyed, from a brane observer viewpoint, via the local () and nonlocal () corrections to Einstein’s equations. (In the more general case, bulk effects on the brane are also carried by , which describes any 5D fields.) The term cannot in general be determined from data on the brane, and the 5D equations above (or their equivalent) need to be solved in order to find .
The general form of the brane energymomentum tensor for any matter fields (scalar fields, perfect fluids, kinetic gases, dissipative fluids, etc.), including a combination of different fields, can be covariantly given in terms of a chosen 4velocity as
(82) 
Here and are the energy density and isotropic pressure, and
(83) 
projects into the comoving rest space orthogonal to on the brane. The momentum density and anisotropic stress obey
(84) 
where angled brackets denote the spatially projected, symmetric and tracefree part:
(85) 
In an inertial frame at any point on the brane, we have
(86) 
where .
The tensor , which carries local bulk effects onto the brane, may then be irreducibly decomposed as
(87)  
This simplifies for a perfect fluid or minimallycoupled scalar field:
(88) 
The trace free carries nonlocal bulk effects onto the brane, and contributes an effective “dark” radiative energymomentum on the brane, with energy density , pressure , momentum density and anisotropic stress :
(89) 
We can think of this as a KK or Weyl “fluid”. The brane “feels” the bulk gravitational field through this effective fluid. More specifically:

The KK (or Weyl) anisotropic stress incorporates the scalar or spin0 (“Coulomb”), the vector (transverse) or spin1 (gravimagnetic) and the tensor (transverse traceless) or spin2 (gravitational wave) 4D modes of the spin2 5D graviton.

The KK momentum density incorporates spin0 and spin1 modes, and defines a velocity of the Weyl fluid relative to via .

The KK energy density , often called the “dark radiation”, incorporates the spin0 mode.
In special cases, symmetry will impose simplifications on this tensor. For example, it must vanish for a conformally flat bulk, including AdS,
(90) 
The RS models have a Minkowski brane in an AdS bulk. This bulk is also compatible with an FRW brane. However, the most general vacuum bulk with a Friedmann brane is Schwarzschildanti de Sitter spacetime birk . Then it follows from the FRW symmetries that
(91) 
where only if the mass of the black hole in the bulk is zero. The presence of the bulk black hole generates via Coulomb effects the dark radiation on the brane.
For a static spherically symmetric brane (e.g. the exterior of a static star or black hole) dmpr ,
(92) 
This condition also holds for a Bianchi I brane mss . In these cases, is not determined by the symmetries, but by the 5D field equations. By contrast, the symmetries of a Gödel brane fix tsab .
The braneworld corrections can conveniently be consolidated into an effective total energy density, pressure, momentum density and anisotropic stress.
(93)  
(94)  
(95)  
(96) 
These general expressions simplify in the case of a perfect fluid (or minimally coupled scalar field, or isotropic oneparticle distribution function), i.e., for :
(97)  
(98)  
(99)  
(100) 
Note that nonlocal bulk effects can contribute to effective imperfect fluid terms even when the matter on the brane has perfect fluid form: there is in general an effective momentum density and anisotropic stress induced on the brane by massive KK modes of the 5D graviton.