The visible universe could lie on a membrane floating within a higher-dimensional space. The extra dimensions would help unify the forces of nature and could contain parallel universes by Nima Arkani-Hamed, Savas Dimopoulos and Georgi Dvali

The classic 1884 story Flatland: A Romance of Many Dimensions, by Edwin A. Abbott, describes the adventures of "A. Square," a character who lives in a two-dimensional world populated by animated geometric figures—triangles, squares, pentagons, and so on. Toward the end of the story, on the first day of 2000, a spherical creature from three-dimensional "Spaceland" passes through Flatland and carries A. Square up off his planar domain to show him the true three-dimensional nature of the larger world. As he comes to grasp what the sphere is showing him, A. Square speculates that Spaceland may itself exist as a small subspace of a still larger four-dimensional universe.

Amazingly, in the past two years physicists have begun seriously examining a very similar idea: that everything we can see in our universe is confined to a three-dimensional "membrane" that lies within a higher-dimensional realm. But unlike A. Square, who had to rely on divine intervention from Spaceland for his insights, physicists may soon be able to detect and verify the existence of reality's extra dimensions, which could extend over distances as large as a millimeter (V25 of an inch). Experiments are already looking for the extra dimensions' effect on the force of gravity. If the theory is correct, upcoming high-energy particle experiments in Europe could see unusual processes involving quantum gravity, such as the creation of transitory micro black holes. More than just an idle romance of many dimensions, the theory is based on some of the most recent developments in string theory and would solve some long-standing puzzles of particle physics and cosmology.

The exotic concepts of string theory and multidimensions actually arise from attempts to understand the most familiar of forces: gravity. More than three centuries after Isaac Newton proposed his law of gravitation, physics still does not ex plain why gravity is so much weaker than all the other forces. The feebleness of gravity is dramatic. A small magnet readily overcomes the gravitational pull of the entire mass of the earth when it lifts a nail off the ground. The gravitational attraction between two electrons is 1043 times weaker than the repulsive electric force between them. Gravity seems important to us—keeping our feet on the ground and the earth orbiting the sun—only because these large aggregates of matter are electrically neutral, making the electrical forces vanishingly small and leaving gravity, weak as it is, as the only noticeable force left over.

Electrons would have to be 1022 times more massive for the electric and gravitational forces between two of them to be equal. To produce such a heavy particle would take 1019 gigaelectron volts (GeV) of energy, a quantity known as the Planck energy. A related quantity is the Planck length, a tiny 10-35 meter. By comparison, the nucleus of a hydrogen atom, a proton, is about 1019 times as large and has a mass of about 1 GeV. The Planck scale of energy and length is far out of reach of the most powerful accelerators. Even the Large Hadron Collider at CERN will probe distances only down to about 10-19 meter when it commences operations five years from now [see "The Large Hadron Collider," by Chris Llewellyn Smith; Scientific American, July]. Because gravity becomes comparable in strength to electro-magnetism and the other forces at the Planck scale, physicists have traditionally assumed that the theory unifying gravity with the other interactions would reveal itself only at these energies. The nature of the ultimate unified theory would then be hopelessly out of reach of direct experimental

investigation in the foreseeable future [see "A Unified Physics by 2050?" by Steven Weinberg; Scientific American, December 1999].

Today's most powerful accelerators probe the energy realm between 100 and 1,000 GeV (one teraelectron volt, or TeV). In this range, experimenters have seen the electromagnetic force and the weak interaction (a force between subatomic particles responsible for certain types of radioactive decay) become unified. We would understand gravity's extraordinary weakness if we understood the factor of 1016 that separates the electroweak scale from the Planck scale.

Alas, physicists' extremely successful theory of particle physics, called the Standard Model, cannot explain the size of this huge gap, because the theory is carefully adjusted to fit the observed electroweak scale. The good news is that this adjustment (along with about 16 others) serves once and for all to fit myriad observations. The bad news is that we must fine-tune the underlying theory to an accuracy of about one part in 1032; otherwise, quantum effects—instabilities—would drag the electroweak scale all the way back up to the Planck scale. The presence of such delicate balancing in the theory is like walking into a room and finding a pencil standing perfectly on its tip in the middle of a table. Though not impossible, the situation is highly unstable, and we are left wondering how it came about.

For 20 years, theorists have attacked this conundrum, called the hierarchy problem, by altering the nature of particle physics near 10-19 meter (or 1 TeV) to stabilize the electro-

weak scale. The most popular modification of the Standard Model that achieves this goal involves a new symmetry called supersymmetry. Going back to our pencil analogy, supersymmetry acts like an invisible thread holding up the pencil and preventing it from falling over. Although accelerators have not yet turned up any direct evidence for supersymmetry, some suggestive indirect evidence supports the supersymmetric extension of the Standard Model. For example, when the measured strengths of the strong, weak and electromagnetic forces are theoretically extrapolated to shorter distances, they meet very accurately at a common value only if supersymmetric rules govern the extrapolation. This result hints at a supersymmet-ric unification of these three forces at about 10-32 meter, about 1,000 times larger than the Planck length but still far beyond the range of particle colliders.

For two decades, the only viable framework for tackling the hierarchy problem has been to change particle physics near 10-19 meter by introducing new processes such as supersymmetry. But in the past two years theorists have proposed a radically different approach, modifying space-time, gravity and the Planck scale itself. The key insight is that the extraordinary size of the Planck scale, accepted for a century since Planck first introduced it, is based on an untested assumption about how gravity behaves over short distances.

Newton's inverse square law of gravity—which says the force between two masses falls as the square of the distance between them—works extremely well over macroscopic distances, explaining the earth's orbit around the sun, the moon's around the earth, and so on. But because gravity is so weak, the law has been experimentally tested down to distances of only about a millimeter, and we must extrapolate across 32 orders of magnitude to conclude that gravity only becomes strong at a Planck scale of 10-35 meter.

The inverse square law is natural in three-dimensional space [see upper illustration on opposite page]. Consider lines of gravitational force emanating uniformly from the earth. Farther from the earth, the lines are spread over a spherical shell of greater area. The surface area increases as the square of the distance, and so the force is diluted at that rate. Suppose there were one more dimension, making space four-dimensional. Then the field lines emanating from a point would get spread over a four-dimensional shell whose surface would increase as the cube of the distance, and gravity would follow an inverse cube law.

The inverse cube law certainly doesn't describe our uni verse, but now imagine that the extra dimension is curled up into a small circle of radius R and that we're looking at field lines coming from a tiny point mass [see lower illustration on opposite page]. When the field lines are much closer to the mass than the distance R, they can spread uniformly in all four dimensions, and so the force of gravity falls as the inverse cube of distance. Once the lines have spread fully around the circle, however, only three dimensions remain for them to continue spreading through, and so for distances much greater than R the force varies as the inverse square of the distance.

The same effect occurs if there are many extra dimensions, all curled up into circles of radius R. For n extra spatial dimensions at distances smaller than R, the force of gravity will follow an inverse 2 + n power law. Because we have measured gravity only down to a millimeter, we would be oblivious to changes in gravity caused by extra dimensions whose size R is smaller than a millimeter. Furthermore, the 2 + n power law would cause gravity to reach "Planck-scale strength" well above 10-35 meter. That is, the Planck length (defined by where gravity becomes strong) would not be that small, and the hierarchy problem would be reduced.

One can solve the hierarchy problem completely by postulating enough extra dimensions to move the Planck scale very

Dimensions. Our universe seems to have four dimensions: three of space (up-down, left-right, forward-backward) and one of time. Although we can barely imagine additional di-mensions,mathematicians and physicists have long analyzed the properties of theoretical spaces that have any number of dimensions.

Size of dimensions. The four known space-time dimensions of our universe are vast.The dimension of time extends back at least 13 billion years into the past and may extend infinitely into the future. The three spatial dimensions may be infinite; our telescopes have detected objects more than 12 billion light-years away.Dimensions can also be finite.For exam ple, the two dimensions of the surface of the earth extend only about 40,000 kilometers—the length of a great circle.

Small extra dimensions. Some modern physics theories postulate additional real dimensions that are wrapped up in circles so small (perhaps 10-35-meter radius) that we have not detected them.Think of a thread of cotton: to a good approximation, it is one-dimensional. A single number can specify where an ant stands on the thread. But using a microscope, we see dust mites crawling on the thread's two-dimensional surface: along the large length dimension and around the short circumference dimension.

Large extra dimensions. Recently physicists realized that extra dimensions as "big"as a millimeter could exist and remain invisible to us. Surprisingly, no known experimental data rule out the theory, and it could explain several mysteries of particle physics and cosmolo-gy.We and all the contents of our known three-dimensional universe (except for gravity) would be stuck on a "membrane," like pool balls moving on the two-dimensional green baize of a pool table.

Dimensions and gravity. The behavior of gravity—particularly its strength—is intimately related to how many dimensions it pervades. Studies of gravity acting over distances smaller than a millimeter could thus reveal large extra dimensions to us. Such experiments are under way. These dimensions would also enhance the production of bizarre quantum gravity objects such as micro black holes,graviton particles and super-strings, all of which could be detected sometime this decade at high-energy particle accelerators.

—Graham P. Collins, staff writer

BALLS ON A POOL TABLE ■', are analogous to fundamental particles on the membrane that is our known universe.

Billiard-ball collisions radiate energy into three dimensions as sound waves (red), analogous to gravitons. Precise studies of the balls' motions could detect the "missing" energy and thus the higher dimensions.

BALLS ON A POOL TABLE ■', are analogous to fundamental particles on the membrane that is our known universe.

Billiard-ball collisions radiate energy into three dimensions as sound waves (red), analogous to gravitons. Precise studies of the balls' motions could detect the "missing" energy and thus the higher dimensions.

GRAVITATIONAL LINES OF FORCE spread out from the earth in three dimensions. As distance from the earth increases, the force becomes diluted by being spread across a larger surface area (spheres). The surface area of each sphere increases as the square of its radius, so gravity falls as the inverse square of distance in three dimensions.

close to the electroweak scale. The ultimate unification of gravity with the other forces would then take place near 10-19 meter rather than 10-35 meter as traditionally assumed. How many dimensions are needed depends on how large they are. Conversely, for a given number of extra dimensions we can compute how large they must be to make gravity strong near 10-19 meter. If there is only one extra dimension, its radius R must be roughly the distance between the earth and the sun. Therefore, this case is already excluded by observation. Two extra dimensions, however, can solve the hierarchy problem if they are about a millimeter in size—precisely where our direct knowledge of gravity ends. The dimensions are smaller still if we add more of them, and for seven extra dimensions we need them to be around 10-14 meter big, about the size of a uranium nucleus. This is tiny by everyday standards but huge by the yardstick of particle physics.

Postulating extra dimensions may seem bizarre and ad hoc, but to physicists it is an old, familiar idea that dates back to the 1920s, when Polish mathematician Theodor Kaluza and Swedish physicist Oskar Klein developed a remarkable unified theory of gravity and electromagnetism that required one extra dimension. The idea has been revived in modern string theories, which require a total of 10 spatial dimensions for internal mathematical consistency. In the past, physicists have assumed that the extra dimensions are curled up into tiny circles with a size near the traditional Planck length of 10-35 meter, making them undetectable but also leaving the conundrum of the hierarchy problem. In contrast, in the new theory that we are discussing, the extra dimensions are wrapped into big circles of at least 10-14 meter radius and perhaps as enormous as a millimeter.

If these dimensions are that large, why haven't we seen them yet? Extra dimensions a millimeter big would be discernible to the naked eye and obvious through a microscope. And although we have not measured gravity below about a millimeter, we have a wealth of experimental knowledge concerning all the other forces at far shorter distances approaching 10-19 meter, all of it consistent only with three-dimensional space. How could there possibly be large extra dimensions?

The answer is at once simple and peculiar: all the matter and forces we know of—with the sole exception of gravity— are stuck to a "wall" in the space of the extra dimensions [see illustration on next page]. Electrons and protons and photons and all the other particles in the Standard Model cannot move in the extra dimensions; electric and magnetic field lines cannot spread into the higher-dimensional space. The wall has only three dimensions, and as far as these particles are concerned, the universe might as well be three-dimensional. Only gravitational field lines can extend into the higher-dimensional space, and only the particle that transmits gravity, the gravi-ton, can travel freely into the extra dimensions. The presence of the extra dimensions can be felt only through gravity.

To make an analogy, imagine that all the particles in the Standard Model, like electrons and protons, are billiard balls moving on the surface of a vast pool table. As far as they are concerned, the universe is two-dimensional. Nevertheless, pool-table inhabitants made out of "billiard balls" could still detect the higher-dimensional world: when two balls hit each other sufficiently hard, they produce sound waves, which travel in all three dimensions, carrying some energy away from the table surface [see illustration on opposite page]. The sound waves are analogous to gravitons, which can travel in

SMALL EXTRA DIMENSION wrapped in a circle (circumference of tube) modifies how gravity (red lines) spreads in space. At distances smaller than the circle radius (blue patches), the lines of force spread apart rapidly through all the dimensions. At much larger distances (yellow circle), the lines have filled the extra dimension, and it has no further effect on them.

OUR UNIVERSE MAY EXIST ON A WALL, or membrane, in the extra dimensions. The line along the cylinder (below right) and the flat plane represent our three-dimensional universe, to which all the known particles and forces except gravity are stuck. Gravity (red lines) propagates through all the dimensions. The extra dimensions may be as large as one millimeter without violating any existing observations.

OUR UNIVERSE MAY EXIST ON A WALL, or membrane, in the extra dimensions. The line along the cylinder (below right) and the flat plane represent our three-dimensional universe, to which all the known particles and forces except gravity are stuck. Gravity (red lines) propagates through all the dimensions. The extra dimensions may be as large as one millimeter without violating any existing observations.

the full higher-dimensional space. In high-energy particle collisions, we expect to observe missing energy, the result of gravitons escaping into the extra dimensions.

Although it may seem strange that some particles should be confined to a wall, similar phenomena are quite familiar. For instance, electrons in a copper wire can move only along the one-dimensional space of the wire and do not travel into the surrounding three-dimensional space. Likewise, water waves travel primarily on the surface of the ocean, not throughout its depth. The specific scenario we are describing, in which all particles except gravity are stuck to a wall, can arise naturally in string theory. In fact, one of the major insights triggering recent breakthroughs in string theory has been the recognition that the theory contains such "walls," known as D-branes, where "brane" comes from the word "membrane" and "D" stands for "Dirichlet," which indicates a mathematical property of the branes. D-branes have precisely the required features: particles such as electrons and photons are represented by tiny lengths of string that each have two endpoints that must be stuck to a D-brane. Gravitons, on the other hand, are tiny closed loops of string that can wander into all the dimensions because they have no endpoints anchoring them to a D-brane.

Is It Alive?

One of the first things good theorists do when they have a new theory is to try to kill it by finding an inconsistency with known experimental results. The theory of large extra dimensions changes gravity at macroscopic distances and alters other physics at high energies, so surely it is easy to kill. Remarkably, however, despite its radical departure from our usual picture of the universe, this theory does not contradict any known experimental results. A few examples of the sorts of tests that are passed shows how surprising this conclusion is.

One might initially worry that changing gravity would affect objects held together by gravity, such as stars and galaxies. But they are not affected. Gravity changes only at distances shorter than a millimeter, whereas in a star, for example, gravity acts across thousands of kilometers to hold distant parts of the star together. More generally, even though the ex

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