S PACETIME REVIEW Extra Dimensions and Warped Geometries Lisa Randall

The field of extra dimensions, as well as the hypothesized sizes of extra area of a sphere drawn at the distance r dimensions, have grown by leaps and bounds over the past few years. I (because all force lines penetrate the sphere’s summarize the new results and the reasons for the recent activity in this surface). Because the gravitational force be- field. These include the observations that extra dimensions can be mac- tween two masses is proportional to the prod- roscopic or even infinite in size. Another new development is the appli- uct of their masses, the 1/r 2 form of the force cation of extra dimensions to the determination of particle physics pa- law has important consequences for heavy rameters and properties. macroscopic objects, such as planets. The force law is also measured on very small We generally take it for granted that we live balls (compactification) or the more recent scales with much smaller objects. Here, the in a world where there are three infinite spa- proposal by Sundrum and myself of focusing weakness of gravity is in evidence, and other, tial dimensions. In fact, we rarely give this of the gravitational potential in a lower di- stronger forces can interfere with the mea- fact much thought; we readily refer to left- mensional subspace (localization) (3, 4). surement. The best measurement to date right, forward-backward, and up-down. These ideas are important in and of them- comes from an impressively accurate exper- Yet the most exciting developments in selves; one or the other would be the reason iment by Adelberger’s group at the Universi- particle physics in the past few years have we so far haven’t observed evidence for extra ty of Washington (6), where it has been involved the recognition that additional di- dimensions, should they exist. I then go on to determined that the 1/r 2 force law persists mensions might exist and furthermore might explore another exciting development in the down to distances on the order of a 10th of a play a role in determining our observable field of extra dimensions. This is the fact that millimeter. Deviations from this form on world. New theoretical discoveries are evolv- even though the dimensions have not yet shorter distance scales are not excluded. ing at a very rapid rate. As we will see, the been seen directly, their existence might ex- This means that according to detailed ex- potential implications range from experimen- plain important features of the observed stan- perimental observations, physics appears to on October 15, 2010 tal signatures of extra dimensions, to under- dard model and will be observed in the near reflect three spatial dimensions on distance standing fundamental questions about the na- future should these conjectures prove correct. scales ranging from a 10th of a millimeter to ture of gravity, to new insights into the evo- astrophysical and probably cosmological dis- lution of our universe. Hidden Dimensions tances. Were there more than three spatial One of the chief motivations for consid- Before proceeding, it is useful to describe dimensions, the gravitational force should ering additional dimensions came from string several of the ways in which we determine spread out in all these dimensions, and the theory, which in turn is motivated by the that there are three dimensions of space. Cer- force law would fall off faster, 1/r3 say, for failure of classical gravity to work at very tainly, that is all that we see at a casual one additional spatial dimension. Somehow, short distance scales or, equivalently, at very glance, but science often consists of probing in order to agree with what we observe, this high energies, where quantum mechanical ef- beyond what is manifestly “evident.” The key better not be the case, and was one of the first www.sciencemag.org fects cannot be neglected. The only known to understanding the experimental and astro- issues that needed to be addressed when ad- way to consistently reconcile quantum me- physical determination of the dimensions of ditional dimensions were suggested. chanics with Einstein’s theory of gravity is space is to consider the gravitational force Kaluza first proposed an additional di- string theory, in which the fundamental ob- law that says that the force between two mension in his attempt to reconcile electro- 2 jects that constitute our universe are not par- masses is GNm1m2/r , where r is the separa- magnetism and gravity (1). His additional ticles but (very tiny) extended objects: tion between the two masses. I use units dimension had finite size; there are three h ϭ ϭ strings. In what follows, I use very little of where ( /2␲) c 1(h, Planck’s constant; c, infinite spatial dimensions, but a fourth one is Downloaded from the full formalism that has been developed to the speed of light). Newton’s constant, GN,is trapped in a circle of size rc in the extra describe string theory. However, I am moti- (10Ϫ33 cm)2, which is very small. It is in- dimension. Einstein, the referee of the paper, 2 vated by one very important fact. It appears versely proportional to MP , where MP, the objected that the size of this additional di- that one can only have a consistent string Planck mass, is about 1019 GeV. This mass mension was not determined. Publication was theory that can describe the known particles scale appears because it is associated with the delayed until 1926, when Klein observed that if there are many additional spatial dimen- graviton coupling, where the graviton is the a circle of size determined by the Planck sions: six or seven, depending on how one spin-2 particle that mediates the gravitational scale, 10Ϫ33 cm, is completely unobservable looks at it. The question is, then, why don’t force (5). Such energies can only be probed at (2). This is readily understood intuitively; we see these additional directions? What has the very tiny Planck length, 10Ϫ33 cm. The dimensions whose size we cannot resolve are become of them? Can they play any role in position dependence of the force law is readi- observationally indistinguishable from no ex- the physics we see? And is there any chance ly understood as a consequence of the isotro- tra dimension at all. One common example to we will observe them soon? py of space—the fact that the laws of physics illustrate this is a garden hose; as viewed at a Here I describe the two known ways to do not distinguish any particular direction. distance, it appears to be one dimensional. incorporate additional dimensions of space Imagine that one can draw the gravitational However, up close, one readily perceives that that are consistent with what we see, or rather force as a set of lines emanating in all direc- there are three dimensions. From the vantage what we don’t see; namely Kaluza’s (1, 2) tions from a massive source, so that the den- point of the force law argument given above, original idea of curling them up into little sity of lines determines the strength of the force law can be consistent with what we gravity. observe because isotropy is violated; the di- It is clear that as we measure the force at mensions of finite size are readily distin- Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA. E-mail: randall@physics. increasing distance r, the strength is propor- guished from the “compactified” circular ex- harvard.edu tional to 1/r 2, or the inverse of the surface tra dimension. The force lines can only

1422 24 MAY 2002 VOL 296 SCIENCE www.sciencemag.org S PACETIME spread a distance rc in the extra dimension. Planck mass are relevant. So it is too limiting to additional dimensions, because electromag- At larger distances, the force appears to be restrict ourselves to Planck-scale compactifica- netism, which is extremely well studied, be- just that of a four-dimensional (three space tion. If other possibilities exist, they should be haves just as it would in four dimensions, plus one time) universe. This argument can explored. In fact, experimental constraints tell regardless of the size of the additional dimen- also be used to see how the Planck scale of us only that the extra dimensions must be small- sions. However, the mechanism that confines a four-dimensional world is related to that er than about 10Ϫ16 mm, corresponding to the electromagnetism to the brane cannot be ap- of a higher dimensional world. In the n- mass scale 103 GeV probed by current particle plied to gravity; gravity must exist in the dimensional world, the force law dies off experiments. This means that from an experi- “bulk”; that is, the full spacetime. One way of with r as 1/rnϪ2 and is inversely propor- mental point of view, we really have no idea understanding this is that the graviton is con- tional to the higher dimensional “Planck” whether the dimensions are as small as the nected to the full spacetime geometry and is mass that determines the strength of the Planck length. They can be 16 orders of mag- coupled to energy anywhere. higher dimensional gravity, raised to the nitude larger [see (7) for example]. So the picture of ADD is that space is power n Ϫ 2. Because we average over the In fact, that is not the last word on the bounded by branes, where the standard model extra dimensions to see an effective lower possible size of extra dimensions. One of particles (but not gravity) are confined. This dimensional theory, this scale, M, is related the new observations of the past few years form of space had also been considered pre- to the measured four-dimensional Planck was that of Arkani-Hamed, Dimopoulos, viously by Horava and Witten (10, 11), with 2 ϭ nϪ2 mass scale by MP M V, where V is and Dvali (8) (ADD), who pointed out that the extra dimension being larger than the the volume of the additional n Ϫ 4 dimen- radii 1016 times larger still (about a milli- Planck scale but not nearly so large as the sions (in the simple Kaluza example, the meter) are also consistent with all known size proposed by ADD. The reason why the circumference of the circle. This means that observations if gravity only, and not other ADD bound is precisely that of Adelberger’s although physics at short distances appears particles and forces, existed in the addition- experiment (12) is that having thrown out the to be higher dimensional, as reflected in a al dimensions. Dimensions of this size constraints from ordinary particle experi- gravitational force that goes as 1/(Mr)nϪ2, could, however, be relevant to particle ments, it is the direct test of the form of gravity at short distances that limits the size of the additional dimensions.

An Infinite (but Hidden) Extra Dimension on October 15, 2010 In fact, extra dimensions can be larger still; they can be infinite in size. This even more revolutionary idea also follows from the ex- istence of branes, as Sundrum and I proposed. Gravity can be “trapped” and extra dimen- sions can have infinite spatial extent (4). This idea follows from a second property of branes: the fact that they carry energy. Be- www.sciencemag.org Ͼ Ͻ cause branes are not isotropic, the strength of Fig. 1. For r rc, force lines only spread in the infinite-sized extra dimensions. For r rc, force is that of higher dimensional space (solid line is equipotential). the gravitational force varies according to distance from the brane. It turns out that this at distances longer than rc, one would ob- physics, as I explain in the next section. dependence on position is very strong if there 2 serve 1/(MPr) (Fig. 1). This observation was a consequence of an- is one additional dimension; the strength de- We conclude that additional dimensions other ingredient from string theory that has creases exponentially with distance from the are acceptable so long as they are of suffi- gained prominence over the past 10 years: brane. A more precise formulation is the fol-

ciently small size. It was taken as convention- the existence of branes. lowing: If there is an energetic four-dimen- Downloaded from al wisdom by string theorists that the length In the string theoretical context, branes sional flat brane in a five-dimensional space- scale associated with the additional dimen- are precisely formulated extended objects time, the five-dimensional space does not sions is the Planck length, 10Ϫ33 cm. That is, that are necessary for the consistency of consist of flat, uniform, extra dimensions. the additional six (or seven) dimensions are string theory, the importance of which was To accomodate a flat brane requires that curled up in a manifold of extremely small demonstrated by Polchinski (9). The term in addition to the tension of the brane itself, (and therefore unmeasurable) size. The rea- “branes” derived from “membranes”; it is there is a bulk vacuum energy, closely son why the Planck length was the proposed perhaps simplest to envision these objects as aligned to the brane tension. The solution to scale is because there appears to be a single membranes floating in a higher dimensional Einstein’s equations is then described locally mass scale in string theory, and one would space. The properties of branes that are rele- as anti–de Sitter (AdS) space, a space with a expect other mass scales to be of similar vant to the study of extra dimensions are that negative vacuum energy. And in this space, magnitude. Because string theory’s major ac- (i) matter and forces can be confined to although it is fundamentally five dimensional complishment is to permit a consistent for- branes and (ii) branes carry energy, or ten- and there is therefore a five-dimensional mulation of quantum gravity, the Planck sion. The first property means that we can graviton, there exists a bound-state mode of scale associated with the gravitational force is have, for example, a brane with only three the graviton that is highly concentrated on the the natural scale. spatial dimensions, even though the full bulk brane and acts as if it were a four-dimension- However, we don’t yet understand the phys- space might have many extra dimensions. If a al graviton. In this geometry, the length of a ics leading to our four-dimensional world. photon is stuck on this brane, it would not yardstick depends on position. The space- From low-energy particle physics, we know explore the extra dimensions. This implies time is “warped”; it appears that the strength there is much physics other than gravitational that as far as electromagnetism is concerned, of the apparent four-dimensional gravity de- physics that is relevant to our world, and fur- the extra dimensions do not exist. Clearly, creases exponentially with distance away thermore, mass scales much lower than the this relaxes the constraints on the size of from the “Planck brane” that traps the gravi-

www.sciencemag.org SCIENCE VOL 296 24 MAY 2002 1423 S PACETIME ton. Another way of stating this is that al- Gabadadze, and Porrati (14, 15). Clearly, model particle masses (quarks and leptons) though the strength of the gravitational cou- there remain many possibilities to explore. are generated. The Planck scale that we have pling is the same everywhere, physical mass referred to several times is the mass scale that scales decrease exponentially with distance Extra Dimensions and the Hierarchy determines Newton’s constant. It is also the from the brane, so that gravity far from the Problem scale where quantum gravity effects should brane is weak. The fifth dimension does not So far, I have focused on gravity itself and become important and new physics such as have to be finite in size, because unlike the how one can accommodate a richer spacetime string theory should be relevant. In fact, we case with flat extra dimensions, the gravita- manifold than we think about intuitively, have already seen that Newton’s constant is tional force is spreading very little in the leading to new possibilities for what can be proportional to the inverse of the Planck mass direction perpendicular to the brane. our true geometry. In this section, I discuss squared. The hierarchy problem can be restat- To derive this form for the gravitational one way in which additional dimensions ed as the problem of why gravity is so ex- force, one solves Einstein’s equations of gen- might be relevant to determining four-dimen- ceptionally weak. The problem is that with- eral relativity in the presence of the brane. sional physics. The possibilities presented out any additional structure, there is no rea- General relativity tells us that not only do here generally involve a further layer of spec- son for two mass scales in the same theory to gravitational forces affect matter, but matter be so different. The problem is not only that determines the surrounding gravitational po- we don’t understand this mass scale, but it is tential. In this case, the presence of the mas- difficult to consistently maintain this separa- sive brane leads to a gravitational force that is tion of scales in the context of the field theory highly concentrated near the brane. So al- through which we study the electroweak forc- though the extra dimension can be very large es. Any interaction would tend to align these (or even infinite), the gravitational force is two highly disparate mass scales. Before the highly concentrated near the brane (Fig. 2). study of extra dimensions, it was thought that One is only sensitive to a short distance there was only one consistent resolution to scale (the scale over which the gravitational this problem that has not already been ex- force decreases), rather than the size of the cluded experimentally: namely, supersymme- extra dimension. This is why the hidden di- try. Supersymmetry postulates the existence mension can even be infinite in size (4). of additional particles that have the same on October 15, 2010 Gravity is localized near the brane. mass as and related interactions with the The model of (4) (known as RS2) is a known particles. These new particles cancel concrete example in which space has five effects that tend to align masses, so that the dimensions, but the world looks four dimen- electroweak scale can be as low, as we ob- sional. If one sits on the Planck brane, the serve. However, we have no firm evidence world looks four dimensional up to very high yet that weak-scale supersymmetry resolves energies, on the order of the Planck scale. As the hierarchy problem; this will only be one ventures out into the fifth dimension, one known with the exploration of higher mass would still measure a four-dimensional force scales at the Tevatron, currently operating in law, but it would only apply for successively Batavia, Illinois, and in the future with the www.sciencemag.org lower energies. Large Hadron Collider, located at CERN, More recently, with Karch, I studied a near Geneva. One potential piece of evidence theory that is in some ways even more sur- for supersymmetry is that the three couplings prising (13). In that theory, space looks four associated with the strengths of the strong, dimensional on the brane and at some dis- electromagnetic, and weak couplings unify at tance away from the brane. This is because a common value at a high energy scale close

there is again a mode that looks like a four- to the Planck scale if the further assumption Downloaded from Fig. 2. Force lines and equipotential surfaces for dimensional graviton. However, the majority the warped geometry. Force lines are denser of no additional charged particles between of the space is not sensitive to the force near the brane. the TeV and Planck scales is made (16–20). mediated by this four-dimensional graviton, However, we will see that extra-dimensional because it only couples in a small portion of theories can also be consistent with unifica- the space. The part of the space where the ulation in order to incorporate the physics tion. Furthermore, there are potential prob- trapped mode does not couple sees itself as that ties extra dimensions to observable low- lems with supersymmetric models. From ex- five dimensional. This leads one to consider energy scales. Such theories, in which the periment, we know that supersymmetry is not the possibility of gravitational “islands”; the extra dimensions are tied to relatively low- exact, so the masses of supersymmetric part- dimensionality of space you think you see energy scales, have the enticing possibility ners are not precisely degenerate with their depends on where you are in the bulk. The that they can be observed in the next gener- known standard-model counterparts. There is brane can be considered to be a four-dimen- ation of colliders. So whether or not we be- no simple credible theory of supersymmetry sional sinkhole. This is truly a possibility of lieve these scenarios, experiments in the near breaking that satisfactorily explains all nature; we only ever see a finite region of future should determine their validity. known observations. Until experiment nar- space, even with cosmological observations. One of the major goals of particle physi- rows the field, it is certainly worthwhile to Our observation that the world we see looks cists in the past 20 years has been to solve a consider alternative explanations for the four dimensional can be merely an acci- problem known as the hierarchy problem. hierarchy. dent of our location. The rest of the universe The hierarchy problem is the fact that the The first proposal to address the hierarchy can be five-, or even 10 dimensional, and we electroweak mass scale of about TeV ϭ 103 problem in the context of extra dimensions wouldn’t necessarily know it. Another in- GeV is 16 orders of magnitude smaller than was also by ADD (8). Their fundamental teresting possibility with an infinite flat the Planck mass scale of 1019 GeV. The insight was that the gravity and electroweak extra dimension was proposed by Dvali, electroweak scale is the scale where standard- scales can be so different because of addition-

1424 24 MAY 2002 VOL 296 SCIENCE www.sciencemag.org S PACETIME al structure in the theory of gravity, as op- traps gravity, there is an additional brane into known particles such as an electron and posed to all previous ideas that tried to intro- separated from the first. Quarks, leptons, positron that we can observe. This is a very duce new structure in the particle physics photons, and other ingredients of the standard distinctive signature; these particles would sector associated with the electroweak scale. model are stuck on this brane. Then the elec- have spin 2, like the graviton, and would ADD observed that with sufficiently large troweak force sees only the second (TeV) come with definite mass relations. There are dimensions, one can equate the fundamental brane, while gravity probes the entire space. other possibilities as well. In a variant of the gravitational and weak interaction mass Because the electroweak mass scale decreas- original proposal (26), in which the second scales. This follows from the relation be- es exponentially with distance from the brane brane does not end space but resides in an tween Planck scales given above; a large that traps gravity, a hierarchy in masses on infinite extra dimension (essentially combin- 16 volume permits MP to be large, whereas M, the order of 10 only requires a distance ing RS1 and RS2), one would have missing the gravitational scale in the higher dimen- scale of order log1016 Ϸ 35. If one can energy signatures identical to those one sional theory, is far lower, on the order of 103 naturally stabilize the length at this value, would obtain with six large ADD-type extra GeV. This does not resolve the hierarchy but there is a natural solution to the hierarchy dimensions. Other ranges of parameters for transforms it into a different problem, that of problem. The large number that separates the which low-energy tests, such as tests of grav- explaining the very large size of the extra TeV and Planck scales arises from the fact ity over short distances, might be relevant dimensions. This proposal has many interest- that the gravitational coupling changes so were considered (27). ing experimental consequences. It turns out rapidly (exponentially) over this relatively Another remarkable feature of the warped that with two extra dimensions, their size modest distance. Unlike the previous scenar- metric solution to the hierarchy problem would be on the order of a millimeter, which io, this is not a very large extra dimension but (RS1) is that the unification of couplings at a is precisely the size that is explored in current one of a relatively natural size. In this picture, high energy scale can be readily incorporated precision tests of gravity. This was one of the there are separate physical theories confined (28, 29). This is possible because, unlike the chief reasons for the excitement associated to the two different branes. The TeV brane on large extra dimension scenario, the TeV scale with these theories and motivated the work of which we live would house all the ingredients is not the highest energy scale accessible to Adelberger (6), which ruled out deviations of the standard model. The Planck brane the full higher-dimensional theory. Incorpo- from Newton’s law on scales of a millimeter. could be host to all sorts of other interactions rating this feature means that RS1 can be Furthermore, large extra dimensions that ad- we don’t see. The only reason why the Planck considered as a theory with all forces unified, dress the hierarchy problem would lead to brane is important to us is that it traps gravity, thereby achieving a major goal of particle on October 15, 2010 observable consequences at the same mass thereby explaining the hierarchy (Fig. 3). physics. scale we mentioned above in association with However, because this scenario relied cru- Another interesting feature of this scenar- supersymmetry. The same experiments that cially on the separation of branes, it was essen- io is that because of the inclusion of high- search for supersymmetry can also search for tial to have a mechanism that could stabilize energy scales, conventional inflation (30) can large extra dimensions. For the ADD scenar- this distance. Goldberger and Wise (23) showed readily be incorporated. Moreover, it has also io, the signature would be missing energy; that this stabilization could be achieved in the been shown to reproduce the known low- particles can collide to produce gravitational presence of an additional scalar field, which is a energy cosmology (24). This makes this the- particles that escape into the extra dimension particle whose energy is minimized for a par- ory a realistic candidate for the solution to the and are therefore not observed. Phenomeno- ticular value of the size of the fifth dimension. hierarchy. logical and astrophysical constraints and im- Subsequently, much work was done on this www.sciencemag.org plications of this scenario were considered in scenario. Recently, Giddings et al.(24) showed Other Implications for Particle Physics (21, 22). an example of a stabilized hierarchy derived Extra dimensions can have other important Certainly one unsatisfying feature of the explicitly from string theory based on an idea of ramifications for particle physics in our ob- large-dimension proposal is the difficulty in Verlinde (25). servable world. We have already discussed stabilizing large extra dimensions. But if one As with the large extra dimension sce- two ways in which they might address ques- has uniform isotropic extra dimensions, the narios, the experimental consequences of tions about the relative size of mass scales.

large volume is essential to explain the hierar- this warped geometry scenario (RS1) are There is another big difference between phys- Downloaded from chy. The weakness of gravity that we see as quite dramatic. Al- four-dimensional observers is due precisely to though in the sim- the fact that the gravitational force is spread out plest scenario no new over a large volume. Sundrum and I, in a theory physics effects will referred to as RS1 (3), realized that the very occur in gravity exper- different geometry we had found, given a brane iments at a millimeter, in a single extra dimension, can also address the there will be signifi- hierarchy but with a rather modestly sized extra cant effects in high- ψ(r) dimension if there is a second brane some energy particle physics distance away from the first. The geometry is experiments, should very similar to RS2 but with space ending on this scenario be cor- the second brane. rect. In the version This is due to the form of gravity; the of our theory present- strength of gravity decreases exponentially ed in (3), there would with distance from the brane because of the be particles asso- exponential rescaling of masses. The strength ciated with the gra- of gravity is not uniform; the gravitational viton (those that car- force is weak away from the brane even rymomentum in the Planck brane Tev brane without diluting the force over a large vol- fifth dimension) that ume. The proposal is the following. Suppose would be observed to Fig. 3. ⌿(r) is the graviton wavefunction. Gravity is weak because of the that in addition to the Planck brane, which decay in the detector exponential suppression of ⌿(r) on the TeV brane.

www.sciencemag.org SCIENCE VOL 296 24 MAY 2002 1425 S PACETIME ics with additional dimensions and that with esting recent work has sought to apply les- existence of a four-dimensional domain in a four: Particles and forces can reside some- sons we learned from extra dimensions to higher dimensional space and, amazingly, a where else in an additional dimension. We four-dimensional particle physics directly. massive graviton. These features have been briefly explain why this “sequestering” can There are essentially two ways in which this understood from a four-dimensional vantage help address many questions about why cer- has been done. point (49). It is very likely that AdS/CFT tain symmetries are more accurate than we The first, known as deconstruction, (41, 42), correspondence between a four-dimensional would expect in a universe with only four originates with the observation that if a non- conformal field theory (CFT) and a five- dimensions. gravitational higher dimensional theory is put dimensional AdS space will enlighten us fur- In four-dimensional field theory, once a on a lattice, with only the extra dimensions ther about gravity and string theory. symmetry is violated in one interaction, it being discretized, one has a theory that is fun- will filter into all possible symmetry-violat- damentally four dimensional but has an energy Conclusions ing terms. And basically, anything that is not regime cutoff in the ultraviolet by the lattice The study of extra dimensions is a rapidly prevented by symmetry will be present. Even spacing and in the infrared by the dimension of evolving field. New theoretical properties are though supersymmetry is very constraining, the lattice volume (these parameters are deter- being discovered at a very fast pace. Some of once supersymmetry is broken, one has a mined by the number of lattice sites and the these teach us fundamental features of grav- plethora of interactions that are permitted. gauge coupling), in which the theory appears to itational theories. Some have shed light on Particularly dangerous is the fact that in a be higher dimensional. This is revealed by the possible ways to address problems in particle broken supersymmetric theory, it is difficult spectrum, which agrees with that for an extra physics. And some have revealed new ideas to prevent “flavor violation”: interactions dimensional theory over this finite energy re- in the context of purely four-dimensional terms that change one type of quark into gime. It is clear that the theory is four dimen- physics. Finally, it is possible that one or another. Because there are strong experimen- sional at high energies, where all you see are several of these ideas will be relevant to the tal constraints on such processes, they should local four-dimensional forces corresponding to question of how string theory evolves from a be forbidden or suppressed by the theory. lattice sites in addition to particles transforming higher dimensional theory to one that repro- Sequestering (31) supersymmetry break- under these forces. At low energies, the theory duces observed four-dimensional physics. ing can suppress the unwanted flavor-violat- is four dimensional, because one cannot resolve Possible ideas along these lines are discussed ing interactions. The idea is that the super- distance scales necessary to see an additional (50, 51). Although it is clear that not all the symmetry-breaking sector resides on a differ- dimension (equivalently, there are no Kaluza- ideas I have presented will be realized in on October 15, 2010 ent brane than the observed standard-model Klein modes). However, at intermediate energy nature, it is not at all out of the question that particles. Isolating the physics in this way scales, the theory appears to be five (or higher) some of them are. The ideas that gravity can prevents the interactions that couple the su- dimensional. The idea of sequestering readily be localized and that large parameters can be persymmetry-breaking and standard-model carries over to this scenario. Other lessons from generated in a large or highly curved bulk sectors that can violate flavor. There is an- extra dimensions can in principle be repro- might well have important implications for other interesting feature of these theories. duced, and theories with no higher dimensional our world. Because direct interactions between the su- realization can be implemented. And it could be that the universe has a very persymmetry-breaking and standard-model The second means for constructing new rich structure, with many different branes, on sectors are forbidden, supersymmetry break- and interesting four-dimensional theories re- which there exist very different physics, living ing might be communicated only by particles lies on a fact that is now well studied by in an as yet unknown geometry. www.sciencemag.org associated with gravity. When this is true, string theorists: the fact that a five-dimen- there is a predictive spectrum, because the sional AdS space is equivalent to a four- References and Notes supersymmetry-breaking masses are deter- dimensional scale-invariant field theory, in 1. T. Kaluza, Preuss. Akad. Wiss. 966 (1921). 2. O. Klein, Z. Phys. 37, 895 (1926). mined by the known gravitational coupling the sense that all properties of the four-di- 3. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 and the known standard-model couplings. mensional theory can be computed from the (1999).࿜࿜࿜࿜ The simplest models have phenomenological five-dimensional gravitational theory, and in 4. , Phys. Rev. Lett. 83, 4690 (1999). 5. It is conventional for particle physicists to measure

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REVIEW Measuring Spacetime: From the Big Bang to Black Holes Max Tegmark

Space is not a boring static stage on which events unfold over time, but a ed, like say a hypersphere or doughnut, so on October 15, 2010 dynamic entity with curvature, fluctuations, and a rich life of its own. that traveling in a straight line could in prin- Spectacular measurements of the cosmic microwave background, gravita- ciple bring you back home—from the other tional lensing, type Ia supernovae, large-scale structure, spectra of the direction? The metric determines the local Lyman ␣ forest, stellar dynamics, and x-ray binaries are probing the shape of spacetime (i.e., the distances and properties of spacetime over 22 orders of magnitude in scale. Current time intervals we measure) and is mathemat- measurements are consistent with an infinite flat everlasting universe ically specified by a 4 ϫ 4 matrix at each containing about 30% cold dark matter, 65% dark energy, and at least two point in spacetime. distinct populations of black holes. GR consists of two parts, each providing a tool for measuring the metric. The first part of Traditionally, space was merely a three- spacetime and evidence for black holes. In GR states that, in the absence of nongravita- www.sciencemag.org dimensional (3D) static stage where the cos- the process, I will combine constraints from tional forces, test particles (objects not heavy mic drama played out over time. Einstein’s the cosmic microwave background (CMB) enough to have a noticeable effect on the theory of general relativity (1) replaced this (2), gravitational lensing, supernovae Ia, metric) move along geodesics in spacetime, concept with 4D spacetime, a dynamic geo- large-scale structure (LSS), the hydrogen Ly- in generalized straight lines, so the observed metric entity with a life of its own, capable of man ␣ forest (Ly␣F) (3), stellar dynamics, motions of photons and astronomical objects expanding, fluctuating, and even curving into and x-ray binaries. Although it is fashionable allow the metric to be reconstructed. I will

black holes. Now, the focus of research is to use cosmological data to measure a small refer to this as geometric measurements of Downloaded from increasingly shifting from the cosmic actors number of free “cosmological parameters,” I the metric. The second part of GR states that to the stage itself. Triggered by progress in will argue that improved data allow raising the curvature of spacetime (expressions in- detector, space, and computer technology, an the ambition level beyond this, testing rather volving the metric and its first two deriva- avalanche of astronomical data is revolution- than assuming the underlying physics. I will tives) is related to its matter content—in most izing our ability to measure the spacetime we discuss how, with a minimum of assump- cosmological situations simply the density inhabit on scales ranging from the cosmic tions, one can measure key properties of and pressure, but sometimes also bulk mo- horizon down to the event horizons of sus- spacetime itself in terms of a few cosmolog- tions and stress energy. I will refer to such pected black holes, using photons and astro- ical functions—the expansion history of the measurements of the metric as indirect, be- nomical objects as test particles. The goal of universe, the spacetime fluctuation spectrum, cause they assume the validity of the Einstein this article is to review these measurements and its growth. field equations (EFE) of GR. and future prospects, focusing on four key Before embarking on a survey of space- The current consensus in the cosmological issues: (i) the global topology and curvature time, a brief review is in order of what it is we community is that spacetime is extremely of space, (ii) the expansion history of space- want to measure, the basic tools at our dis- smooth, homogeneous, and isotropic (trans- time and evidence for dark energy, (iii) the posal (4, 5), and the general picture of how lationally and rotationally invariant) on large fluctuation history of spacetime and evidence spacetime relates to the structure of the uni- (ϳ1023 to 1026 m) scales, with small fluctu- for dark matter, and (iv) strongly curved verse. According to general relativity theory ations that have grown over time to form (GR), spacetime is what mathematicians call objects like galaxies and stars on smaller a manifold, characterized by a topology and a scales. CMB observations (2) have shown metric. The topology gives the global struc- that space is almost isotropic on the scale of Department of Physics, University of Pennsylvania, ϳ 26 Philadelphia, PA 19104, USA. E-mail: max@ ture (Fig. 1, top), and we can ask: Is space our cosmic horizon ( 10 m), with the met- physics.upenn.edu infinite in all directions or multiply connect- ric fluctuating by only about one part in 105

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