Purely scalar perturbations are characterized by the fact that vectors and tensors are derived from scalar potentials, i. Scalar perturbative quantities are formed from the potentials via the 3D Laplacian, e. Purely vector perturbations are characterized by. The KK momentum density carries scalar and vector modes, and the KK anisotropic stress carries scalar, vector, and tensor modes:. Linearizing the conservation equations for a single adiabatic fluid, and the nonlocal conservation equations, we obtain. Equations , , and do not provide gauge-invariant equations for perturbed quantities, but their spatial gradients do.

The equations contain scalar, vector, and tensor modes, which can be separated out if desired. They are not a closed system of equations until is determined by a 5D analysis of the bulk perturbations. The metric-based approach does not have this drawback. An alternative approach to brane-world cosmological perturbations is an extension of the 4D metric-based gauge-invariant theory [ , ]. A review of this approach is given in [ 40 , ]. In an arbitrary gauge, and for a flat FRW background, the perturbed metric has the form. In the Gaussian normal gauge, the brane coordinate-position remains fixed under perturbation,.

In this gauge, we have. In the case of scalar perturbations, we have for example. All of the metric perturbation quantities are determined once a solution is found for the wave equation. The junction conditions 62 relate the off-brane derivatives of metric perturbations to the matter perturbations:.

The evolution of the bulk metric perturbations is determined by the perturbed 5D field equations in the vacuum bulk,. In the covariant approach, we define matter density and expansion velocity perturbation scalars, as in 4D general relativity,. We define the total effective dimensionless entropy S tot via.

The density perturbation equations on the brane are derived by taking the spatial gradients of Equations , , and , and using Equations and This leads to [ ]. If we can neglect this term on large scales, then the system of density perturbation equations closes on super-Hubble scales [ ]. An equivalent statement applies to the large-scale curvature perturbations [ ]. KK effects then introduce two new isocurvature modes on large scales associated with U and Q , and they modify the evolution of the adiabatic modes as well [ , ].

Thus on large scales the system of brane equations is closed, and we can determine the density perturbations without solving for the bulk metric perturbations. We can simplify the system as follows. The 3-Ricci tensor defined in Equation leads to a scalar covariant curvature perturbation variable,. Using these new variables, we find the closed system for large-scale perturbations:.

At low energies, and for constant w , the non-decaying attractor is the general relativity solution,. At very high energies, for , we get. The early growing and late time constant attractor solutions are seen explicitly in the plots. The presence of dark radiation in the background introduces new features.

In the radiation era , the non-decaying low-energy attractor becomes [ ]. In the very high energy limit,. The curvature perturbation on uniform density surfaces is defined in Equation The associated gauge-invariant quantity. On large scales, the perturbed dark energy conservation equation is [ ]. This is independent of brane-world modifications to the field equations, since it depends on energy conservation only. For the total, effective fluid, the curvature perturbation is defined as follows [ ]: It follows that the curvature perturbations on large scales, like the density perturbations, can be found on the brane without solving for the bulk metric perturbations.

The KK effects on the brane contribute a non-adiabatic mode, although at low energies. In the 4D longitudinal gauge of the metric perturbation formalism, the gauge-invariant curvature and metric perturbations on large scales are related by. In 4D general relativity, the right hand side of Equation is zero. The non-integrated Sachs-Wolfe formula has the same form as in general relativity:. The brane-world corrections to the general relativistic Sachs-Wolfe effect are then given by [ ].

Equation has been generalized to a 2-brane model, in which the radion makes a contribution to the Sachs-Wolfe effect [ ]. This is discussed below. The vorticity propagation equation on the brane is the same as in general relativity,. In general relativity, vector perturbations vanish when the vorticity is zero. By contrast, in brane-world cosmology, bulk KK effects can source vector perturbations even in the absence of vorticity [ ].

We define covariant dimensionless vector perturbation quantities for the vorticity and the KK gravi-vector term:. On large scales, we can find a closed system for these vector perturbations on the brane [ ]:. Vorticity in the brane matter is a source for the KK vector perturbation on large scales.

Vorticity decays unless the matter is ultra-relativistic or stiffer , and this source term typically provides a decaying mode. There is another pure KK mode, independent of vorticity, but this mode decays like vorticity. Inflation will redshift away the vorticity and the KK mode. Indeed, the massive KK vector modes are not excited during slow-roll inflation [ 40 , ]. The covariant description of tensor modes on the brane is via the shear, which satisfies the wave equation [ ].

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Unlike the density and vector perturbations, there is no closed system on the brane for large scales. The KK anisotropic stress is an unavoidable source for tensor modes on the brane. Thus it is necessary to use the 5D metric-based formalism. This is the subject of the next Section 7. The tensor perturbations are given by Equation , i. The transverse traceless f ij satisfies Equation , which implies, on splitting f ij into Fourier modes with amplitude f t,y ,.

By the transverse traceless part of Equation , the boundary condition is. The wave equation cannot be solved analytically except if the background metric functions are separable, and this only happens for maximally symmetric branes, i. We can calculate the spectrum of gravitational waves generated during brane inflation [ , , , ], if we approximate slow-roll inflation by a succession of de Sitter phases.

The linearized wave equation is separable. Note that localization fails for an AdS 4 brane [ , ]. The non-zero value of the Hubble parameter implies the existence of a mass gap [ ],.

## Brane-World Gravity

This result has been generalized: For dS 4 brane s with bulk scalar field, a universal lower bound on the mass gap of the KK tower is [ ]. The massive modes decay during inflation, according to Equation , leaving only the zero mode, which is effectively a 4D gravitational wave. The zero mode, satisfying the boundary condition. At high energies, the amplitude is considerably enhanced:. The factor F determines the modification of the gravitational wave amplitude relative to the standard 4D result:.

The modifying factor F can also be interpreted as a change in the effective Planck mass [ ]. This enhanced zero mode produced by brane inflation remains frozen outside the Hubble radius, as in general relativity, but when it re-enters the Hubble radius during radiation or matter domination, it will no longer be separated from the massive modes, since H will not be constant. Instead, massive modes will be excited during re-entry. In other words, energy will be lost from the zero mode as 5D gravitons are emitted into the bulk, i.

A phenomenological model of the damping of the zero mode due to 5D graviton emission is given in [ ]. Self-consistent low-energy approximations to compute this effect are developed in [ , 91 ]. At zero order, the low-energy approximation is based on the following [ , , 22 ]. In the radiation era, at low energy, the background metric functions obey. To lowest order, the wave equation therefore separates, and the mode functions can be found analytically [ , , 22 ]. The massive modes in the bulk, f m y , are the same as for a Minkowski brane. On large scales, or at late times, the mode functions on the brane are given in conformal time by.

The massive modes decay on super-Hubble scales, unlike the zero-mode. The massive modes on sub-Hubble scales are sourced by the initial zero mode that is re-entering the Hubble radius [ 91 ]:. Damping of brane-world gravity waves on horizon re-entry due to massive mode generation.

The solid curve is the numerical solution, the short-dashed curve the low-energy approximation, and the long-dashed curve the standard general relativity solution. For the CMB anisotropies, one needs to consider a multi-component source. Linearizing the general nonlinear expressions for the total effective energy-momentum tensor, we obtain.

The perturbation equations in the previous Section 7 form the basis for an analysis of scalar and tensor CMB anisotropies in the brane-world. The full system of equations on the brane, including the Boltzmann equation for photons, has been given for scalar [ ] and tensor [ ] perturbations. But the systems are not closed, as discussed above, because of the presence of the KK anisotropic stress , which acts a source term.

At high energies, earlier in the radiation era, the decoupled equation is fourth order [ ]:. The formalism and machinery are ready to compute the temperature and polarization anisotropies in brane-world cosmology, once a solution, or at least an approximation, is given for. The resulting power spectra will reveal the nature of the brane-world imprint on CMB anisotropies, and would in principle provide a means of constraining or possibly falsifying the brane-world models.

Once this is achieved, the implications for the fundamental underlying theory, i. However, the first step required is the solution for.

## Brane-World Gravity

This solution will be of the form given in Equation Once and F k are determined or estimated, the numerical integration in Equation can in principle be incorporated into a modified version of a CMB numerical code. The full solution in this form represents a formidable problem, and one is led to look for approximations.

However, two boundary conditions are needed to determine all functions of integration. This is achieved by introducing a second brane, as in the RS 2-brane scenario. This brane is to be thought of either as a regulator brane, whose backreaction on the observable brane is neglected which will only be true for a limited time , or as a shadow brane with physical fields, which have a gravitational effect on the observable brane.

The physical distance between the branes is , and. The dark energy obeys , where C is a constant. The approximation for the KK Weyl energy-momentum tensor on the observable brane is. The perturbation equations can then be derived as generalizations of the standard equations. For example, the equation is [ ]. The radion perturbation itself satisfies the wave equation. The regulator brane is assumed to be far enough away that its effects on the physical brane can be neglected over the timescales of interest.

By Equation it follows that. With these assumptions, and further assuming adiabatic perturbations for the matter, there is only one independent brane-world parameter, i. This assumption has a remarkable consequence on large scales: On small scales, beyond the first acoustic peak, the brane-world corrections are negligible. On scales up to the first acoustic peak, brane-world effects can be significant, changing the height and the location of the first peak.

However, it is not clear to what extent these features are general brane-world features within the low-energy approximation , and to what extent they are consequences of the simple assumptions imposed on the background. Further work remains to be done. A related low-energy approximation, using the moduli space approximation, has been developed for certain 2-brane models with bulk scalar field [ , 37 ]. The effective gravitational action on the physical brane, in the Einstein frame, is.

Figure taken from [ , 37 ]. Simple brane-world models of RS type provide a rich phenomenology for exploring some of the ideas that are emerging from M theory. The higher-dimensional degrees of freedom for the gravitational field, and the confinement of standard model fields to the visible brane, lead to a complex but fascinating interplay between gravity, particle physics, and geometry, that enlarges and enriches general relativity in the direction of a quantum gravity theory. This review has attempted to show some of the key features of brane-world gravity from the perspective of astrophysics and cosmology, emphasizing a geometric approach to dynamics and perturbations.

It has focused on 1-brane RS-type brane-worlds which have some attractive features:. The review has highlighted both the successes and the remaining open problems of the RS models and their generalizations. The open problems stem from a common basic difficulty, i. The key open problems of relevance to astrophysics and cosmology are. The RS-type models are the simplest brane-worlds with curved extra dimension that allow for a meaningful approach to astrophysics and cosmology. One also needs to consider generalizations that attempt to make these models more realistic, or that explore other aspects of higher-dimensional gravity which are not probed by these simple models.

Two important types of generalization are the following:. In summary, brane-world gravity opens up exciting prospects for subjecting M theory ideas to the increasingly stringent tests provided by high-precision astronomical observations. At the same time, brane-world models provide a rich arena for probing the geometry and dynamics of the gravitational field and its interaction with matter.

I thank my many collaborators and friends for discussions and sharing of ideas. National Center for Biotechnology Information , U. Living Reviews in Relativity. Published online Jun Accepted Apr Heuristics of higher-dimensional gravity One of the fundamental aspects of string theory is the need for extra spatial dimensions. Brane-worlds and M theory String theory thus incorporates the possibility that the fundamental scale is much less than the Planck scale felt in 4 dimensions. Open in a separate window.

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 4-dimensional gravity. In this model [ ], there is only one, positive tension, brane. Then the energy scales are related via. Covariant Approach to Brane-World Geometry and Dynamics The RS models and the subsequent generalization from a Minkowski brane to a Friedmann-Robertson-Walker FRW brane [ 27 , , , , , , , 99 , ] were derived as solutions in particular coordinates of the 5D Einstein equations, together with the junction conditions at the Z 2 -symmetric brane.

The 5D field equations determine the 5D curvature tensor; in the bulk, they are. Field equations on the brane Using Equations 44 and 48 , it follows that. These nonlocal corrections cannot be determined purely from data on the brane. Note that the covariant formalism applies also to the two-brane case. This splits into the gravito-electric and gravito-magnetic fields on the brane: Generalized Raychaudhuri equation expansion propagation: Gravito-electric propagation Maxwell-Weyl E-dot equation: Gravito-magnetic propagation Maxwell-Weyl H-dot equation: Gravito-electric divergence Maxwell-Weyl div-E equation: Gravito-magnetic divergence Maxwell-Weyl div-H equation: Gravitational Collapse and Black Holes on the Brane The physics of brane-world compact objects and gravitational collapse is complicated by a number of factors, especially the confinement of matter to the brane, while the gravitational field can access the extra dimension, and the nonlocal from the brane viewpoint gravitational interaction between the brane and the bulk.

The black string The projected Weyl term vanishes in the simplest candidate for a black hole solution. Taylor expansion into the bulk One can use a Taylor expansion, as in Equation 82 , in order to probe properties of a static black hole on the brane [ 72 ]. Realistic black holes Thus a simple brane-based approach, while giving useful insights, does not lead to a realistic black hole solution. There are contradictory indications about the nature of the realistic black hole solution on the brane: Numerical simulations of highly relativistic static stars on the brane [ ] indicate that general relativity remains a good approximation.

Oppenheimer-Snyder collapse gives a non-static black hole The simplest scenario in which to analyze gravitational collapse is the Oppenheimer-Snyder model, i. The collapse region has the metric. Quantum backreaction due to Hawking radiation in the 4D picture is described as classical dynamics in the 5D picture. The black hole evaporates as a classical process in the 5D picture, and there is thus no stationary black hole solution in RS 1-brane. Primordial black holes in 1-brane RS-type cosmology have been investigated in [ , 61 , , , 62 , ]. High-energy effects in the early universe see the next Section 5 can significantly modify the evaporation and accretion processes, leading to a prolonged survival of these black holes.

In natural static coordinates, the bulk metric is. Brane-world inflation In 1-brane RS-type brane-worlds, where the bulk has only a vacuum energy, inflation on the brane must be driven by a 4D scalar field trapped on the brane. The field satisfies the Klein-Gordon equation. High-energy inflation on the brane also generates a zero-mode 4D graviton mode of tensor perturbations, and stretches it to super-Hubble scales, as will be discussed below.

This zero-mode has the same qualitative features as in general relativity, remaining frozen at constant amplitude while beyond the Hubble horizon. Its amplitude is enhanced at high energies, although the enhancement is much less than for scalar perturbations [ ]: Vector perturbations in the bulk metric can support vector metric perturbations on the brane, even in the absence of matter perturbations see the next Section 6. However, there is no normalizable zero mode, and the massive KK modes stay in the vacuum state during brane-world inflation [ 39 ].

Therefore, as in general relativity, we can neglect vector perturbations in inflationary cosmology. Brane-world instanton The creation of an inflating brane-world can be modelled as a de Sitter instanton in a way that closely follows the 4D instanton, as shown in [ ]. Models with non-empty bulk The single-brane cosmological model can be generalized to include stresses other than A 5 in the bulk: This has the form of Equation , with. The dark stiff matter does not arise from massive KK modes of the graviton.

In the simplest case, when there is no coupling between the bulk field and brane matter, this gives. Perturbations The background dynamics of brane-world cosmology are simple because the FRW symmetries simplify the bulk and rule out nonlocal effects. Metric-based perturbations An alternative approach to brane-world cosmological perturbations is an extension of the 4D metric-based gauge-invariant theory [ , ]. Density perturbations on large scales In the covariant approach, we define matter density and expansion velocity perturbation scalars, as in 4D general relativity,.

Curvature perturbations and the Sachs-Wolfe effect The curvature perturbation on uniform density surfaces is defined in Equation Vector perturbations The vorticity propagation equation on the brane is the same as in general relativity,. Tensor perturbations The covariant description of tensor modes on the brane is via the shear, which satisfies the wave equation [ ]. The simplest model The simplest model is the one in which. Conclusion Simple brane-world models of RS type provide a rich phenomenology for exploring some of the ideas that are emerging from M theory.

It has focused on 1-brane RS-type brane-worlds which have some attractive features: They provide a simple 5D phenomenological realization of the Horava-Witten supergravity solutions in the limit where the hidden brane is removed to infinity, and the moduli effects from the 6 further compact extra dimensions may be neglected. They develop a new geometrical form of dimensional reduction based on a strongly curved rather than flat extra dimension.

They lead to cosmological models whose background dynamics are completely understood and reproduce general relativity results with suitable restrictions on parameters. The inclusion of dynamical interaction between the brane s and a bulk scalar field , so that the action is. An extra-dimensional mechanism for initiating inflation or the hot radiation era with super-Hubble correlations via brane interaction building on the initial work in [ 90 , , , , , , , , , 21 , 69 , , 29 , , ]. An extra-dimensional explanation for the dark energy and possibly also dark matter puzzles: Could dark energy or late-time acceleration of the universe be a result of gravitational effects on the visible brane of the shadow brane, mediated by the bulk scalar field?

The addition of stringy and quantum corrections to the Einstein-Hilbert action , including the following: The Gauss-Bonnet combination in particular has unique properties in 5D, giving field equations which are second-order in the bulk metric and linear in the second derivatives , and being ghost-free. In this sense, the Gauss-Bonnet correction removes an unstable and singular solution.

In the early universe, the Gauss-Bonnet corrections to the Friedmann equation have the dominant form. Quantum field theory corrections arising from the coupling between brane matter and bulk gravitons, leading to an induced 4D Ricci term in the brane action. The original induced gravity brane-world [ 89 , 66 , , ] was put forward as an alternative to the RS mechanism: The bulk is flat Minkowski 5D spacetime and as a consequence there is no normalizable zero-mode of the bulk graviton , and there is no brane tension.

Another viewpoint is to see the induced-gravity term in the action as a correction to the RS action: The late-universe 5D behaviour of gravity can naturally produce a late-time acceleration, even without dark energy , although the fine-tuning problem is not evaded. In the late universe at low energies, instead of recovering general relativity, there may be strong deviations from general relativity, and late-time acceleration from 5D gravity effects rather than negative pressure energy is typical.

Thus we have a striking result that both forms of correction to the gravitational action, i. Cosmologies with both induced-gravity and Gauss-Bonnet corrections to the RS action are considered in [ ]. Acknowledgments I thank my many collaborators and friends for discussions and sharing of ideas. Anisotropy and inflation in Bianchi I braneworlds. For a related online version see: Akama, [Online Los Alamos Preprint]: Natural quintessence and large extra dimensions. Final reheating temperature on a single brane. Embeddings in non-vacuum spacetimes. A possible new dimension at a few TeV.

New dimensions at a millimeter to a fermi and superstrings at a TeV. The hierarchy problem and new dimensions at a millimeter. Boundary inflation in the moduli space approximation. Suppression of entropy perturbations in multifield inflation on the brane. Thin-shell limit of branes in the presence of Gauss-Bonnet interactions. Stacking a 4D geometry into an Einstein-Gauss-Bonnet bulk.

Living on the edge: Kaluza-Klein anisotropy in the CMB. Baryogenesis by brane collision. Cosmological tensor perturbations in the Randall-Sundrum model: Evolution in the near-brane limit. Supergravity Inflation on the Brane. Tachyonic inflation in the braneworld scenario. Avoidance of naked singularities in dilatonic brane world scenarios with a Gauss-Bonnet term. Brane cosmological evolution in a bulk with cosmological constant. The radion in brane cosmology. Cosmological perturbations generated in the colliding bubble braneworld universe.

Perturbations on a moving D3-brane and mirage cosmology. On new gravitational instantons describing creation of brane-worlds. General brane cosmologies and their global spacetime structure. Initial conditions for brane inflation. Big Bang nucleosynthesis constraints on brane cosmologies. Fluctuating brane in a dilatonic bulk. Cosmology and brane worlds: Cosmic vorticity on the brane. Cosmological perturbations in the bulk and on the brane. Singularities on the brane are not isotropic.

M-theory moduli space and cosmology. Calcagni, [Online Los Alamos Preprint]: Shortcuts in the fifth dimension. Braneworld cosmological models with anisotropy. Bulk effects in the cosmological dynamics of brane-world scenarios. Evolution of cosmological models in the brane-world scenario. The subsequent normalized solutions have diminishing contributions. Moreover, we find out that the phenomenology of the hybrid brane is not different from the usual thick domain wall.

The use of numerical techniques for solving the equations of the massive modes is useful for matching possible phenomenological measurements in the gravitational law as a probe to warped extra dimensions. We use cookies to help provide and enhance our service and tailor content and ads. By continuing you agree to the use of cookies.

Author links open overlay panel D. If there are extra dimensions, the messengers that potentially herald their existence are particles known as Kaluza-Klein modes. These KK particles have the same charges as the particles we know, but they have momentum in the extra dimensions. They would thus appear to us as heavy particles with a characteristic mass spectrum determined by the extra dimensions' size and shape. Each particle we know of would have these KK partners, and we would expect to find them if the extra dimensions were large.

The fact that we have not yet seen KK particles in the energy regimes we have explored experimentally puts a bound on the extra dimensions' size. As I mentioned, the TeV energy scale of cm has been explored experimentally. Since we haven't yet seen KK modes and cm would yield KK particles of about a TeV in mass, that means all sizes up to are permissible for the possible extra dimensions. That's significantly larger than 10 33 cm, but it's still too small to be significant. This is how things stood in the world of extra dimensions until very recently.

It was thought that extra dimensions might be present but that they would be extremely small. Branes are essentially membranes—lower-dimensional objects in a higher-dimensional space. To picture this, think of a shower curtain, virtually a two-dimensional object in a three-dimensional space.

### Introduction

Branes are special, particularly in the context of string theory, because there's a natural mechanism to confine particles to the brane; thus not everything need travel in the extra dimensions even if those dimensions exist. Particles confined to the brane would have momentum and motion only along the brane, like water spots on the surface of your shower curtain. Branes allow for an entirely new set of possibilities in the physics of extra dimensions, because particles confined to the brane would look more or less as they would in a three-plus-one-dimension world; they never venture beyond it.

Protons, electrons, quarks, all sorts of fundamental particles could be stuck on the brane. In that case, you may wonder why we should care about extra dimensions at all, since despite their existence the particles that make up our world do not traverse them. However, although all known standard-model particles stick to the brane, this is not true of gravity. The mechanisms for confining particles and forces mediated by the photon or electrogauge proton to the brane do not apply to gravity. Gravity, according to the theory of general relativity, must necessarily exist in the full geometry of space.

Furthermore, a consistent gravitational theory requires that thegraviton, the particle that mediates gravity, has to couple to any source of energy, whether that source is confined to the brane or not. Finally, there is a string-theory explanation of why the graviton is not stuck to any brane: The graviton is associated with the closed string, and only open strings can be anchored to a brane. A scenario in which particles are confined to a brane and only gravity is sensitive to the additional dimensions permits extra dimensions that are considerably larger than previously thought.

The reason is that gravity is not nearly as well tested as other forces, and if it is only gravity that experiences extra dimensions, the constraints are much more permissive. We haven't studied gravity as well as we've studied most other particles, because it's an extremely weak force and therefore more difficult to precisely test. Physicists have showed that even dimensions almost as big as a millimeter would be permitted, if it were only gravity out in the higher-dimensional bulk. This size is huge compared with the scales we've been talking about.

It is a macroscopic, visible size! But because photons which we see with are stuck to the brane, too, the dimensions would not be visible to us, at least in the conventional ways. Once branes are included in the picture, you can start talking about crazily large extra dimensions.

If the extra dimensions are very large, that might explain why gravity is so weak. Gravity might not seem weak to you, but it's the entire earth that's pulling you down; the result of coupling an individual graviton to an individual particle is quite small. From the point of view of particle physics, which looks at the interactions of individual particles, gravity is an extremely weak force.

This weakness of gravity is a reformulation of the so-called hierarchy problem—that is, why the huge Planck mass suppressing gravitational interactions is sixteen orders of magnitude bigger than the mass associated with particles we see. But if gravity is spread out over large extra dimensions, its force would indeed be diluted.

The gravitational field would spread out in the extra dimensions and consequently be very weak on the brane—an idea recently proposed by theorists Nima Arkani Hamed, Savas Dimopoulos, and Gia Dvali. The problem with this scenario is the difficulty of explaining why the dimensions should be so large.

The problem of the large ratio of masses is transmuted into the problem of the large size of curled-up dimensions. Raman Sundrum, currently at Johns Hopkins University, and I recognized that a more natural explanation for the weakness of gravity could be the direct result of the gravitational attraction associated with the brane itself. In addition to trapping particles, branes carry energy. We showed that from the perspective of general relativity this means that the brane curves the space around it, changing gravity in its vicinity.

When the energy in space is correlated with the energy on the brane so that a large flat three-dimensional brane sits in the higher-dimensional space, the graviton the particle communicating the gravitational force is highly attracted to the brane. Rather than spreading uniformly in an extra dimension, gravity stays localized, very close to the brane.

For the particular geometry that solves Einstein's equations, when you go out some distance in an extra dimension, you see an exponentially suppressed gravitational force. This is remarkable because it means that a huge separation of mass scales—sixteen orders of magnitude—can result from a relatively modest separation of branes. If we are living on the second brane not the Planck brane , we would find that gravity was very weak.

Such a moderate distance between branes is not difficult to achieve and is many orders of magnitude smaller than that necessary for the large-extra-dimensions scenario just discussed. A localized graviton plus a second brane separated from the brane on which the standard model of particle physics is housed provides a natural solution to the hierarchy problem—the problem of why gravity is so incredibly weak.

The strength of gravity depends on location, and away from the Planck brane it is exponentially suppressed. This theory has exciting experimental implications, since it applies to a particle physics scale—namely, the TeV scale. In this theory's highly curved geometry, Kaluza-Klein particles—those particles with momentum in the extra dimensions—would have mass of about a TeV; thus there is a real possibility of producing them at colliders in the near future. They would be created like any other particle and they would decay in much the same way.

Experiments could then look at their decay products and reconstruct the mass and spin that is their distinguishing property. The graviton is the only particle we know about that has spin 2. The many Kaluza-Klein particles associated with the graviton would also have spin 2 and could therefore be readily identified. Observation of these particles would be strong evidence of the existence of additional dimensions and would suggest that this theory is correct.

As exciting as this explanation of the existence of very different mass scales is, Raman and I discovered something perhaps even more surprising. Conventionally, it was thought that extra dimensions must be curled up or bounded between two branes, or else we would observe higher-dimensional gravity.

The aforementioned second brane appeared to serve two purposes: It explained the hierarchy problem because of the small probability for the graviton to be there, and it was also responsible for bounding the extra dimension so that at long distances bigger than the dimension's size only three dimensions are seen. The concentration of the graviton near the Planck brane can, however, have an entirely different implication.

If we forget the hierarchy problem for the moment, the second brane is unnecessary! That is, even if there is an infinite extra dimension and we live on the Planck brane in this infinite dimension, we wouldn't know about it. In this "warped geometry," as the space with exponentially decreasing graviton amplitude is known, we would see things as if this dimension did not exist and the world were only three-dimensional. Because the graviton has such a small probability of being located away from the Planck brane, anything going on far away from the Planck brane should be irrelevant to physics on or near it.

The physics far away is in fact so entirely irrelevant that the extra dimension can be infinite, with absolutely no problem from a three-dimensional vantage point. Because the graviton makes only infrequent excursions into the bulk, a second brane or a curled-up dimension isn't necessary to get a theory that describes our three-dimensional world, as had previously been thought.

We might live on the Planck brane and address the hierarchy problem in some other manner—or we might live on a second brane out in the bulk, but this brane would not be the boundary of the now infinite space. It doesn't matter that the graviton occasionally leaks away from the Planck brane; it's so highly localized there that the Planck brane essentially mimics a world of three dimensions, as though an extra dimension didn't exist at all.

A four-spatial-dimensions world, say, would look almost identical to one with three spatial dimensions. Thus all the evidence we have for three spatial dimensions could equally well be evidence for a theory in which there are four spatial dimensions of infinite extent. It's an exciting but frustrating game. We used to think the easiest thing to rule out would be large extra dimensions, because large extra dimensions would be associated with low energies, which are more readily accessible.

Now, however, because of the curvature of space, there is a theory permitting an infinite fourth dimension of space in a configuration that so closely mimics three dimensions that the two worlds are virtually indistinguishable.